diff --git a/books/Makefile b/books/Makefile new file mode 100644 index 0000000..d91e0ea --- /dev/null +++ b/books/Makefile @@ -0,0 +1,205 @@ +AXIOM=/usr/local/axiom/mnt/fedora5 +TANGLE=${AXIOM}/bin/lib/notangle +WEAVE=${AXIOM}/bin/lib/noweave +LATEX=latex +MAKEINDEX=makeindex +DVIPS=dvips -Ppdf +PS2PDF=ps2pdf +RM=rm -f + +all: + +remake: bookvol0 bookvol1 bookvol2 bookvol3 bookvol4 bookvol5 bookvol6 \ + bookvol7 bookvol8 bookvol9 bookvol10 bookvol11 bookvol12 toc + +toc: + @echo "\\documentclass{book}" >toc.tex + @echo "\\begin{document}" >>toc.tex + @echo "{\\bf Volume ~ 0}: Axiom {\\sl Jenks and Sutor}\\\\" >toc.toc + @echo "{\\bf Volume ~ 1}: Axiom {\\sl Tutorial}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 2}: Axiom {\\sl Users Guide}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 3}: Axiom {\\sl Programmers Guide}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 4}: Axiom {\\sl Developers Guide}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 5}: Axiom {\\sl Interpreter}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 6}: Axiom {\\sl Command}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 7}: Axiom {\\sl Hyperdoc}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 8}: Axiom {\\sl Graphics}\\\\" >>toc.toc + @echo "{\\bf Volume ~ 9}: Axiom {\\sl Compiler}\\\\" >>toc.toc + @echo "{\\bf Volume 10}: Axiom {\\sl Algebra}\\\\" >>toc.toc + @echo "{\\bf Volume 11}: Axiom {\\sl Browser}\\\\" >>toc.toc + @echo "{\\bf Volume 12}: Axiom {\\sl Crystal}\\\\" >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\tableofcontents" >>toc.tex + @echo "\\end{document}" >>toc.tex + @echo "\\section*{Volume 0: Axiom Jenks and Sutor}" >>toc.toc + @cat bookvol0.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 1: Axiom Tutorial}" >>toc.toc + @cat bookvol1.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 2: Axiom Users Guide}" >>toc.toc + @cat bookvol2.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 3: Axiom Programmers Guide}" >>toc.toc + @cat bookvol3.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 4: Axiom Developers Guide}" >>toc.toc + @cat bookvol4.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 5: Axiom Interpreter}" >>toc.toc + @cat bookvol5.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 6: Axiom Command}" >>toc.toc + @cat bookvol6.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 7: Axiom Hyperdoc}" >>toc.toc + @cat bookvol7.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 8: Axiom Graphics}" >>toc.toc + @cat bookvol8.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 9: Axiom Compiler}" >>toc.toc + @cat bookvol9.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 10: Axiom Algebra}" >>toc.toc + @cat bookvol10.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 11: Axiom Browser}" >>toc.toc + @cat bookvol11.toc >>toc.toc + @echo "\\newpage" >>toc.toc + @echo "\\section*{Volume 12: Axiom Crystal}" >>toc.toc + @cat bookvol12.toc >>toc.toc + @${LATEX} toc.tex + @${DVIPS} toc.dvi 2>/dev/null + @${PS2PDF} toc.ps >/dev/null + @${RM} -f *.toc toc.aux toc.dvi toc.log toc.ps toc.tex toc.toc + +bookvol0: + @${LATEX} bookvol0.pamphlet + @${LATEX} bookvol0.pamphlet >/dev/null + @${DVIPS} bookvol0.dvi 2>/dev/null + @${PS2PDF} bookvol0.ps >/dev/null + @${RM} bookvol0.aux bookvol0.dvi bookvol0.log bookvol0.ps \ + bookvol0.idx bookvol0.tex + +bookvol1: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol1.pamphlet >bookvol1.tex) + @${LATEX} bookvol1.tex + @${LATEX} bookvol1.tex >/dev/null + @${DVIPS} bookvol1.dvi 2>/dev/null + @${PS2PDF} bookvol1.ps >/dev/null + @${RM} bookvol1.aux bookvol1.dvi bookvol1.log bookvol1.ps \ + bookvol1.idx bookvol1.tex + +bookvol2: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol2.pamphlet >bookvol2.tex) + @${LATEX} bookvol2.tex + @${LATEX} bookvol2.tex >/dev/null + @${DVIPS} bookvol2.dvi 2>/dev/null + @${PS2PDF} bookvol2.ps >/dev/null + @${RM} bookvol2.aux bookvol2.dvi bookvol2.log bookvol2.ps \ + bookvol2.idx bookvol2.tex + +bookvol3: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol3.pamphlet >bookvol3.tex) + @${LATEX} bookvol3.tex + @${LATEX} bookvol3.tex >/dev/null + @${DVIPS} bookvol3.dvi 2>/dev/null + @${PS2PDF} bookvol3.ps >/dev/null + @${RM} bookvol3.aux bookvol3.dvi bookvol3.log bookvol3.ps \ + bookvol3.idx bookvol3.tex + +bookvol4: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol4.pamphlet >bookvol4.tex) + @${LATEX} bookvol4.tex + @${LATEX} bookvol4.tex >/dev/null + @${DVIPS} bookvol4.dvi 2>/dev/null + @${PS2PDF} bookvol4.ps >/dev/null + @${RM} bookvol4.aux bookvol4.dvi bookvol4.log bookvol4.ps \ + bookvol4.idx bookvol4.tex + +bookvol5: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol5.pamphlet >bookvol5.tex) + @${LATEX} bookvol5.tex + @${LATEX} bookvol5.tex >/dev/null + @${DVIPS} bookvol5.dvi 2>/dev/null + @${PS2PDF} bookvol5.ps >/dev/null + @${RM} bookvol5.aux bookvol5.dvi bookvol5.log bookvol5.ps \ + bookvol5.idx bookvol5.tex + +bookvol6: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol6.pamphlet >bookvol6.tex) + @${LATEX} bookvol6.tex + @${LATEX} bookvol6.tex >/dev/null + @${DVIPS} bookvol6.dvi 2>/dev/null + @${PS2PDF} bookvol6.ps >/dev/null + @${RM} bookvol6.aux bookvol6.dvi bookvol6.log bookvol6.ps \ + bookvol6.idx bookvol6.tex + +bookvol7: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol7.pamphlet >bookvol7.tex) + @${LATEX} bookvol7.tex + @${LATEX} bookvol7.tex >/dev/null + @${DVIPS} bookvol7.dvi 2>/dev/null + @${PS2PDF} bookvol7.ps >/dev/null + @${RM} bookvol7.aux bookvol7.dvi bookvol7.log bookvol7.ps \ + bookvol7.idx bookvol7.tex + +bookvol8: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol8.pamphlet >bookvol8.tex) + @${LATEX} bookvol8.tex + @${LATEX} bookvol8.tex >/dev/null + @${DVIPS} bookvol8.dvi 2>/dev/null + @${PS2PDF} bookvol8.ps >/dev/null + @${RM} bookvol8.aux bookvol8.dvi bookvol8.log bookvol8.ps \ + bookvol8.idx bookvol8.tex + +bookvol9: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol9.pamphlet >bookvol9.tex) + @${LATEX} bookvol9.tex + @${LATEX} bookvol9.tex >/dev/null + @${DVIPS} bookvol9.dvi 2>/dev/null + @${PS2PDF} bookvol9.ps >/dev/null + @${RM} bookvol9.aux bookvol9.dvi bookvol9.log bookvol9.ps \ + bookvol9.idx bookvol9.tex + +bookvol10: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol10.pamphlet >bookvol10.tex) + @${LATEX} bookvol10.tex + @${LATEX} bookvol10.tex >/dev/null + @${DVIPS} bookvol10.dvi 2>/dev/null + @${PS2PDF} bookvol10.ps >/dev/null + @${RM} bookvol10.aux bookvol10.dvi bookvol10.log bookvol10.ps \ + bookvol10.idx bookvol10.tex + +bookvol11: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol11.pamphlet >bookvol11.tex) + @${LATEX} bookvol11.tex + @${LATEX} bookvol11.tex >/dev/null + @${DVIPS} bookvol11.dvi 2>/dev/null + @${PS2PDF} bookvol11.ps >/dev/null + @${RM} bookvol11.aux bookvol11.dvi bookvol11.log bookvol11.ps \ + bookvol11.idx bookvol11.tex + +bookvol12: + @(export AXIOM=${AXIOM} && \ + ${WEAVE} -delay bookvol12.pamphlet >bookvol12.tex) + @${LATEX} bookvol12.tex + @${LATEX} bookvol12.tex >/dev/null + @${DVIPS} bookvol12.dvi 2>/dev/null + @${PS2PDF} bookvol12.ps >/dev/null + @${RM} bookvol12.aux bookvol12.dvi bookvol12.log bookvol12.ps \ + bookvol12.idx bookvol12.tex + + diff --git a/books/axiom.sty b/books/axiom.sty new file mode 100755 index 0000000..3581b97 --- /dev/null +++ b/books/axiom.sty @@ -0,0 +1,1026 @@ +% axiom.sty -- LaTeX support for Axiom +% DON'T edit this file! Use src/doc/axiom.sty.pamphlet instead. + +\usepackage{alltt} + +\newenvironment{chunk}[1]{% we need the chunkname as an argument +\noindent% make sure we are in column 1 +{\small $<<${#1}$>>=$}% <>= +\begin{alltt}% use the verbatim +\small}% with a small font +{\end{alltt}% the end{chunk} ends the verbatim +\par{}% we add a newline +\noindent{}% start in column 1 +@% put an @ +\par% and a newline +\normalsize}% and return to the rest of the document + +{\obeyspaces\AtBeginDocument{\global\let =\ }} % from texbook, p 381 +\def\nwopt@nomargintag{\let\nwmargintag=\@gobble} +\def\nwopt@margintag{% + \def\nwmargintag##1{\leavevmode\llap{##1\kern\nwmarginglue\kern\codemargin}}} +\def\nwopt@margintag{% + \def\nwmargintag##1{\leavevmode\kern-\codemargin\nwthemargintag{##1}\kern\codemargin}} +\def\nwthemargintag#1{\llap{#1\kern\nwmarginglue}} +\nwopt@margintag +\newdimen\nwmarginglue +\nwmarginglue=0.3in +\def\nwtagstyle{\footnotesize\Rm} +% make \hsize in code sufficient for 88 columns +\setbox0=\hbox{\tt m} +\newdimen\codehsize +\codehsize=91\wd0 % 88 columns wasn't enough; I don't know why +\newdimen\codemargin +\codemargin=0pt +\newdimen\nwdefspace +\nwdefspace=\codehsize +% need to use \textwidth in {\LaTeX} to handle styles with +% non-standard margins (David Bruce). Don't know why we sometimes +% wanted \hsize. 27 August 1997. +%% \advance\nwdefspace by -\hsize\relax +\ifx\textwidth\undefined + \advance\nwdefspace by -\hsize\relax +\else + \advance\nwdefspace by -\textwidth\relax +\fi +\chardef\other=12 +\def\setupcode{% + \chardef\\=`\\ + \chardef\{=`\{ + \chardef\}=`\} + \catcode`\$=\other + \catcode`\&=\other + \catcode`\#=\other + \catcode`\%=\other + \catcode`\~=\other + \catcode`\_=\other + \catcode`\^=\other + \catcode`\"=\other % fixes problem with german.sty + \obeyspaces\Tt +} +\let\nwlbrace=\{ +\let\nwrbrace=\} +\def\nwendquote{\relax\ifhmode\spacefactor=1000 \fi} +{\catcode`\^^M=\active % make CR an active character + \gdef\newlines{\catcode`\^^M=\active % make CR an active character + \def^^M{\par\startline}}% + \gdef\eatline#1^^M{\relax}% +} +%%% DON'T \gdef^^M{\par\startline}}% in case ^^M appears in a \write +\def\startline{\noindent\hskip\parindent\ignorespaces} +\def\nwnewline{\ifvmode\else\hfil\break\leavevmode\hbox{}\fi} +\def\setupmodname{% + \catcode`\$=3 + \catcode`\&=4 + \catcode`\#=6 + \catcode`\%=14 + \catcode`\~=13 + \catcode`\_=8 + \catcode`\^=7 + \catcode`\ =10 + \catcode`\^^M=5 + \let\{\nwlbrace + \let\}\nwrbrace + % bad news --- don't know what catcode to give " + \Rm} +\def\LA{\begingroup\maybehbox\bgroup\setupmodname\It$\langle$} +\def\RA{\/$\rangle$\egroup\endgroup} +\def\code{\leavevmode\begingroup\setupcode\newlines} +\def\edoc{\endgroup} +\let\maybehbox\relax +\newbox\equivbox +\setbox\equivbox=\hbox{$\equiv$} +\newbox\plusequivbox +\setbox\plusequivbox=\hbox{$\mathord{+}\mathord{\equiv}$} +% \moddef can't have an argument because there might be \code...\edoc +\def\moddef{\leavevmode\kern-\codemargin\LA} +\def\endmoddef{\RA\ifmmode\equiv\else\unhcopy\equivbox\fi + \nobreak\hfill\nobreak} +\def\plusendmoddef{\RA\ifmmode\mathord{+}\mathord{\equiv}\else\unhcopy\plusequivbox\fi + \nobreak\hfill\nobreak} +\def\chunklist{% +\errhelp{I changed \chunklist to \nowebchunks. +I'll try to avoid such incompatible changes in the future.}% +\errmessage{Use \string\nowebchunks\space instead of \string\chunklist}} +\def\nowebchunks{\message{}} +\def\nowebindex{\message{}} +% here is support for the new-style (capitalized) font-changing commands +% thanks to Dave Love +\ifx\documentstyle\undefined + \let\Rm=\rm \let\It=\it \let\Tt=\tt % plain +\else\ifx\selectfont\undefined + \let\Rm=\rm \let\It=\it \let\Tt=\tt % LaTeX OFSS +\else % LaTeX NFSS + \def\Rm{\reset@font\rm} + \def\It{\reset@font\it} + \def\Tt{\reset@font\tt} + \def\Bf{\reset@font\bf} +\fi\fi +\ifx\reset@font\undefined \let\reset@font=\relax \fi +\def\noweboptions#1{% + \def\@nwoptionlist{#1}% + \@for\@nwoption:=\@nwoptionlist\do{% + \@ifundefined{nwopt@\@nwoption}{% + \@latexerr{There is no such noweb option as '\@nwoption'}\@eha}{% + \csname nwopt@\@nwoption\endcsname}}} +\codemargin=10pt +\advance\codehsize by \codemargin % make room for indentation of code +\advance\nwdefspace by \codemargin % and fix adjustment for def/use +\def\setcodemargin#1{% + \advance\codehsize by -\codemargin % make room for indentation of code + \advance\nwdefspace by -\codemargin % and fix adjustment for def/use + \codemargin=#1 + \advance\codehsize by \codemargin % make room for indentation of code + \advance\nwdefspace by \codemargin % and fix adjustment for + % def/use +} +\def\nwopt@shift{% + \dimen@=-0.8in + \if@twoside % Values for two-sided printing: + \advance\evensidemargin by \dimen@ + \else % Values for one-sided printing: + \advance\evensidemargin by \dimen@ + \advance\oddsidemargin by \dimen@ + \fi +% \advance \marginparwidth -\dimen@ +} +\let\nwopt@noshift\@empty +\def\nwbegincode#1{% + \begingroup + \topsep \nwcodetopsep + \@beginparpenalty \@highpenalty + \@endparpenalty -\@highpenalty + \@begincode } +\def\nwendcode{\endtrivlist \endgroup \filbreak} % keeps code on 1 page + +\newenvironment{webcode}{% + \@begincode +}{% + \endtrivlist} +\def\@begincode{% + \trivlist \item[]% + \leftskip\@totalleftmargin \advance\leftskip\codemargin + \rightskip\hsize \advance\rightskip -\codehsize + \parskip\z@ \parindent\z@ \parfillskip\@flushglue + \linewidth\codehsize + \@@par + \def\par{\leavevmode\null \@@par \penalty\nwcodepenalty}% + \obeylines + \@noligs \ifx\verbatim@nolig@list\undefined\else + \let\do=\nw@makeother \verbatim@nolig@list \do@noligs\` + \fi + \setupcode \frenchspacing \@vobeyspaces + \nowebsize \setupcode + \let\maybehbox\mbox } + \newskip\nwcodetopsep \nwcodetopsep = 3pt plus 1.2pt minus 1pt + \let\nowebsize=\normalsize + \def\nwopt@tinycode{\let\nowebsize=\tiny} + \def\nwopt@footnotesizecode{\let\nowebsize=\footnotesize} + \def\nwopt@scriptsizecode{\let\nowebsize=\scriptsize} + \def\nwopt@smallcode{\let\nowebsize=\small} + \def\nwopt@normalsizecode{\let\nowebsize=\normalsize} + \def\nwopt@largecode{\let\nowebsize=\large} + \def\nwopt@Largecode{\let\nowebsize=\Large} + \def\nwopt@LARGEcode{\let\nowebsize=\LARGE} + \def\nwopt@hugecode{\let\nowebsize=\huge} + \def\nwopt@Hugecode{\let\nowebsize=\Huge} +\newcount\nwcodepenalty \nwcodepenalty=\@highpenalty +\def\nw@makeother#1{\catcode`#1=12 } +\def\nwbegindocs#1{\ifvmode\noindent\fi} +\let\nwenddocs=\relax +\let\nwdocspar=\filbreak +\raggedbottom +\def\code{\leavevmode\begingroup\setupcode\@vobeyspaces\obeylines} +\let\edoc=\endgroup +\newdimen\@original@textwidth +\def\ps@noweb{% + \@original@textwidth=\textwidth + \let\@mkboth\@gobbletwo + \def\@oddfoot{}\def\@evenfoot{}% No feet. + \if@twoside % If two-sided printing. + \def\@evenhead{\hbox to \@original@textwidth{% + \Rm \thepage\qquad{\Tt\leftmark}\hfil\today}}% Left heading. + \def\@oddhead{\hbox to \@original@textwidth{% + \Rm \today\hfil{\Tt\leftmark}\qquad\thepage}}% Right heading. + \else % If one-sided printing. + \def\@oddhead{\hbox to \@original@textwidth{% + \Rm \today\hfil{\Tt\leftmark}\qquad\thepage}}% Right heading. + \let\@evenhead\@oddhead + \fi + \let\chaptermark\@gobble + \let\sectionmark\@gobble + \let\subsectionmark\@gobble + \let\subsubsectionmark\@gobble + \let\paragraphmark\@gobble + \let\subparagraphmark\@gobble + \def\nwfilename{\begingroup\let\do\@makeother\dospecials + \catcode`\{=1 \catcode`\}=2 \nw@filename} + \def\nw@filename##1{\endgroup\markboth{##1}{##1}\let\nw@filename=\nw@laterfilename}% +} +\def\nw@laterfilename#1{\endgroup\clearpage \markboth{#1}{#1}} +\let\nwfilename=\@gobble +\def\nwcodecomment#1{\@@par\penalty\nwcodepenalty + \if@firstnwcodecomment + \vskip\nwcodecommentsep\penalty\nwcodepenalty\@firstnwcodecommentfalse + \fi% + \hspace{-\codemargin}{% + \rightskip=0pt plus1in + \interlinepenalty\nwcodepenalty + \let\\\relax\footnotesize\Rm #1\@@par\penalty\nwcodepenalty}} +\def\@nwalsodefined#1{\nwcodecomment{\@nwlangdepdef\ \nwpageprep\ \@pagesl{#1}.}} +\def\@nwused#1{\nwcodecomment{\@nwlangdepcud\ \nwpageprep\ \@pagesl{#1}.}} +\def\@nwnotused#1{\nwcodecomment{\@nwlangdeprtc.}} +\def\nwoutput#1{\nwcodecomment{\@nwlangdepcwf\ {\Tt \@stripstar#1*\stripped}.}} +\def\@stripstar#1*#2\stripped{#1} +\providecommand{\nwprevdefptr}[1]{% + \mbox{$\mathord{\triangleleft}\,\mathord{\mbox{\subpageref{#1}}}$}} +\providecommand{\nwnextdefptr}[1]{% + \mbox{$\mathord{\mbox{\subpageref{#1}}}\,\mathord{\triangleright}$}} + +\providecommand{\@nwprevnextdefs}[2]{% + {\nwtagstyle + \ifx\relax#1\else ~~\nwprevdefptr{#1}\fi + \ifx\relax#2\else ~~\nwnextdefptr{#2}\fi}} +\providecommand{\@nwusesondefline}[1]{{\nwtagstyle~~(\@pagenumsl{#1})}} +\providecommand{\@nwstartdeflinemarkup}{\nobreak\hskip 1.5em plus 1fill\nobreak} +\providecommand{\@nwenddeflinemarkup}{\nobreak\hskip \nwdefspace minus\nwdefspace\nobreak} +\def\nwopt@longxref{% + \let\nwalsodefined\@nwalsodefined + \let\nwused\@nwused + \let\nwnotused\@nwnotused + \let\nwprevnextdefs\@gobbletwo + \let\nwusesondefline\@gobble + \let\nwstartdeflinemarkup\relax + \let\nwenddeflinemarkup\relax +} +\def\nwopt@shortxref{% + \let\nwalsodefined\@gobble + \let\nwused\@gobble + \let\nwnotused\@gobble + \let\nwprevnextdefs\@nwprevnextdefs + \let\nwusesondefline\@nwusesondefline + \let\nwstartdeflinemarkup\@nwstartdeflinemarkup + \let\nwenddeflinemarkup\@nwenddeflinemarkup +} +\def\nwopt@noxref{% + \let\nwalsodefined\@gobble + \let\nwused\@gobble + \let\nwnotused\@gobble + \let\nwprevnextdefs\@gobbletwo + \let\nwusesondefline\@gobble + \let\nwstartdeflinemarkup\relax + \let\nwenddeflinemarkup\relax +} +\nwopt@shortxref % to hell with backward compatibility! +\newskip\nwcodecommentsep \nwcodecommentsep=3pt plus 1pt minus 1pt +\newif\if@firstnwcodecomment\@firstnwcodecommenttrue +\newcount\@nwlopage\newcount\@nwhipage % range lo..hi-1 +\newcount\@nwlosub % subpage of lo +\newcount\@nwhisub % subpage of hi +\def\@nwfirstpage#1#2#3{% subpage page xref-tag + \@nwlopage=#2 \@nwlosub=#1 + \def\@nwloxreftag{#3}% + \advance\@nwpagecount by \@ne + \@nwhipage=\@nwlopage\advance\@nwhipage by \@ne } +\def\@nwnextpage#1#2#3{% subpage page xref-tag + \ifnum\@nwhipage=#2 + \advance\@nwhipage by \@ne + \advance\@nwpagecount by \@ne + \@nwhisub=#1 + \def\@nwhixreftag{#3}\else + \ifnum#2<\@nwlopage \advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa\@nwfirstpage{#1}{#2}{#3}\else + \ifnum#2>\@nwhipage \advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa\@nwfirstpage{#1}{#2}{#3}\else + \@nwlosub=0 \@nwhisub=0 + \fi\fi\fi + } +\newcount\@nwpagetemp +\newcount\@nwpagecount +\def\@nwfirstpagel#1{% label + \@ifundefined{r@#1}{\@warning{Reference `#1' on page \thepage \space undefined}% + \nwix@cons\nw@pages{\\{\bf ??}}}{% + \edef\@tempa{\noexpand\@nwfirstpage\subpagepair{#1}{#1}}\@tempa}} +\def\@nwnextpagel#1{% label + \@ifundefined{r@#1}{\@warning{Reference `#1' on page \thepage \space undefined}% + \nwix@cons\nw@pages{\\{\bf ??}}}{% + \edef\@tempa{\noexpand\@nwnextpage\subpagepair{#1}{#1}}\@tempa}} +\def\@pagesl#1{% list of labels + \gdef\nw@pages{}\@nwpagecount=0 + \def\\##1{\@nwfirstpagel{##1}\let\\=\@nwnextpagel}#1% + \advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa\def\\##1{\@nwhyperpagenum##1}% + \ifnum\@nwpagecount=1 \nwpageword \else \nwpagesword\fi~\commafy{\nw@pages}} +\def\@nwhyperpagenum#1#2{\nwhyperreference{#2}{#1}} + +\def\@pagenumsl#1{% list of labels -- doesn't include word `pages', commas, or `and' + \gdef\nw@pages{}\@nwpagecount=0 + \def\\##1{\@nwfirstpagel{##1}\let\\=\@nwnextpagel}#1% + \advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa% + \def\\##1{\@nwhyperpagenum##1\let\\=\@nwpagenumslrest}\nw@pages} +\def\@nwpagenumslrest#1{~\@nwhyperpagenum#1} +\def\subpages#1{% list of {{subpage}{page}} + \gdef\nw@pages{}\@nwpagecount=0 + \def\\##1{\edef\@tempa{\noexpand\@nwfirstpage##1{}}\@tempa + \def\\####1{\edef\@tempa{\noexpand\@nwnextpage####1}\@tempa}}#1% + \advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa\def\\##1{\@firstoftwo##1}% + \ifnum\@nwpagecount=1 \nwpageword \else \nwpagesword\fi~\commafy{\nw@pages}} +\def\@nwaddrange{\advance\@nwhipage by \m@ne + \ifnum\@nwhipage=\@nwlopage + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}}% + \else + \count@=\@nwhipage \advance\count@ by \m@ne + \ifnum\count@=\@nwlopage % consecutive pages + \edef\@tempa{\noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwlosub}{\number\@nwlopage}}% + {\@nwloxreftag}}% + \noexpand\noexpand\noexpand\\% + {{\nwthepagenum{\number\@nwhisub}{\number\@nwhipage}} + {\@nwhixreftag}}}% + \else \ifnum\@nwlopage<110 \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 \multiply\count@ by 100 + \ifnum\count@=\@nwlopage \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}\else + \count@=\@nwlopage \divide\count@ by 100 + \@nwpagetemp=\@nwhipage \divide\@nwpagetemp by 100 + \ifnum\count@=\@nwpagetemp % lo--least 2 digits of hi + \multiply\@nwpagetemp by 100 + \advance \@nwhipage by -\@nwpagetemp + \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \else \edef\@tempa{\noexpand\noexpand\noexpand\\{{\number\@nwlopage--\number\@nwhipage}{}}}% + \fi + \fi + \fi% + \fi + \fi% + \edef\@tempa{\noexpand\nwix@cons\noexpand\nw@pages{\@tempa}}\@tempa} +\def\nwpageword{\@nwlangdepchk} % chunk, was page +\def\nwpagesword{\@nwlangdepchks} % chunk, was page +\def\nwpageprep{\@nwlangdepin} % in, was on +\providecommand\nw@genericref[2]{% what to do, name of ref + \expandafter\nw@g@nericref\csname r@#2\endcsname#1{#2}} +\providecommand\nw@g@nericref[3]{% control sequence, what to do, name + \ifx#1\relax + \ref{#3}% trigger the standard `undefined ref' mechanisms + \else + \expandafter#2#1.\\% + \fi} +\def\nw@selectone#1#2#3\\{#1} +\def\nw@selecttwo#1#2#3\\{#2} +\def\nw@selectonetwo#1#2#3\\{{#1}{#2}} +\providecommand{\subpageref}[1]{% + \nwhyperreference{#1}{\nw@genericref\@subpageref{#1}}} +\def\@subpageref#1#2#3\\{% + \@ifundefined{2on#2}{#2}{\nwthepagenum{#1}{#2}}} +\providecommand{\subpagepair}[1]{% % produces {subpage}{page} + \@ifundefined{r@#1}% + {{0}{0}}% + {\nw@genericref\@subpagepair{#1}}} +\def\@subpagepair#1#2#3\\{% + \@ifundefined{2on#2}{{0}{#2}}{{#1}{#2}}} +\providecommand{\sublabel}[1]{% + \@bsphack + \nwblindhyperanchor{#1}% + \if@filesw {\let\thepage\relax + \def\protect{\noexpand\noexpand\noexpand}% + \edef\@tempa{\write\@auxout{\string + \newsublabel{#1}{{}{\thepage}}}}% + \expandafter}\@tempa + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +\providecommand{\nosublabel}[1]{% + \@bsphack\if@filesw {\let\thepage\relax + \def\protect{\noexpand\noexpand\noexpand}% + \edef\@tempa{\write\@auxout{\string + \newlabel{#1}{{0}{\thepage}}}}% + \expandafter}\@tempa + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +\providecommand\newsublabel{% + \nw@settrailers + \global\let\newsublabel\@newsublabel + \@newsublabel} +\providecommand{\@newsublabel}[2]{% + \edef\this@page{\@cdr#2\@nil}% + \ifx\this@page\last@page\else + \sub@page=\z@ + \fi + \edef\last@page{\this@page} + \advance\sub@page by \@ne + \ifnum\sub@page=\tw@ + \global\@namedef{2on\this@page}{}% + \fi + \pendingsublabel{#1}% + \edef\@tempa##1{\noexpand\newlabel{##1}% + {{\number\sub@page}{\this@page}\nw@labeltrailers}}% + \pending@sublabels + \def\pending@sublabels{}} +\providecommand\nw@settrailers{% -- won't work on first run + \@ifpackageloaded{nameref}% + {\gdef\nw@labeltrailers{{}{}{}}}% + {\gdef\nw@labeltrailers{}}} +\renewcommand\nw@settrailers{% + \@ifundefined{@secondoffive}% + {\gdef\nw@labeltrailers{}}% + {\gdef\nw@labeltrailers{{}{}{}}}} +\providecommand{\nextchunklabel}[1]{% + \nwblindhyperanchor{#1}% % looks slightly bogus --- nr + \@bsphack\if@filesw {\let\thepage\relax + \edef\@tempa{\write\@auxout{\string\pendingsublabel{#1}}}% + \expandafter}\@tempa + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +\providecommand\pendingsublabel[1]{% + \def\@tempa{\noexpand\@tempa}% + \edef\pending@sublabels{\noexpand\@tempa{#1}\pending@sublabels}} +\def\pending@sublabels{} +\def\last@page{\relax} +\newcount\sub@page +\def\@alphasubpagenum#1#2{#2\ifnum#1=0 \else\@alph{#1}\fi} +\def\@nosubpagenum#1#2{#2} +\def\@numsubpagenum#1#2{#2\ifnum#1=0 \else.\@arabic{#1}\fi} +\def\nwopt@nosubpage{\let\nwthepagenum=\@nosubpagenum\nwopt@nomargintag} +\def\nwopt@numsubpage{\let\nwthepagenum=\@numsubpagenum} +\def\nwopt@alphasubpage{\let\nwthepagenum=\@alphasubpagenum} +\nwopt@alphasubpage +\newcount\@nwalph@n +\let\@nwalph@d\@tempcnta +\let\@nwalph@bound\@tempcntb +\def\@nwlongalph#1{{% + \@nwalph@n=#1\advance\@nwalph@n by-1 + \@nwalph@bound=26 + \loop\ifnum\@nwalph@n<\@nwalph@bound\else + \advance\@nwalph@n by -\@nwalph@bound + \multiply\@nwalph@bound by 26 + \repeat + \loop\ifnum\@nwalph@bound>1 + \divide\@nwalph@bound by 26 + \@nwalph@d=\@nwalph@n\divide\@nwalph@d by \@nwalph@bound + % d := d * bound ; n -:= d; d := d / bound --- saves a temporary + \multiply\@nwalph@d by \@nwalph@bound + \advance\@nwalph@n by -\@nwalph@d + \divide\@nwalph@d by \@nwalph@bound + \advance\@nwalph@d by 1 \@alph{\@nwalph@d}% + \repeat +}} +\newcount\nw@chunkcount +\nw@chunkcount=\@ne +\providecommand{\weblabel}[1]{% + \@bsphack + \nwblindhyperanchor{#1}% + \if@filesw {\let\thepage\relax + \def\protect{\noexpand\noexpand\noexpand}% + \edef\@tempa{\write\@auxout{\string + \newsublabel{#1}{{}{\number\nw@chunkcount}}}}% + \expandafter}\@tempa + \global\advance\nw@chunkcount by \@ne + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +\def\nwopt@webnumbering{% + \let\sublabel=\weblabel + \def\nwpageword{chunk}\def\nwpagesword{chunks}% + \def\nwpageprep{in}} +% \nwindexdefn{printable name}{identifying label}{label of chunk} +% \nwindexuse{printable name}{identifying label}{label of chunk} + +\def\nwindexdefn#1#2#3{\@auxix{\protect\nwixd}{#2}{#3}} +\def\nwindexuse#1#2#3{\@auxix{\protect\nwixu}{#2}{#3}} + +\def\@auxix#1#2#3{% {marker}{id label}{subpage label} + \@bsphack\if@filesw {\let\nwixd\relax\let\nwixu\relax + \def\protect{\noexpand\noexpand\noexpand}% + \edef\@tempa{\write\@auxout{\string\nwixadd{#1}{#2}{#3}}}% + \expandafter}\@tempa + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +% \nwixadd{marker}{idlabel}{subpage label} +\def\nwixadd#1#2#3{% + \@ifundefined{nwixl@#2}% + {\global\@namedef{nwixl@#2}{#1{#3}}}% + {\expandafter\nwix@cons\csname nwixl@#2\endcsname{#1{#3}}}} +\def\@nwsubscriptident#1#2{\mbox{$\mbox{#1}_{\mathrm{\subpageref{#2}}}$}} +\def\@nwnosubscriptident#1#2{#1} +\def\@nwhyperident#1#2{\leavevmode\nwhyperreference{#2}{#1}} +\def\nwopt@subscriptidents{% + \let\nwlinkedidentq\@nwsubscriptident + \let\nwlinkedidentc\@nwsubscriptident +} +\def\nwopt@nosubscriptidents{% + \let\nwlinkedidentq\@nwnosubscriptident + \let\nwlinkedidentc\@nwnosubscriptident +} +\def\nwopt@hyperidents{% + \let\nwlinkedidentq\@nwhyperident + \let\nwlinkedidentc\@nwhyperident +} +\def\nwopt@nohyperidents{% + \let\nwlinkedidentq\@nwnosubscriptident + \let\nwlinkedidentc\@nwnosubscriptident +} +\def\nwopt@subscriptquotedidents{% + \let\nwlinkedidentq\@nwsubscriptident +} +\def\nwopt@nosubscriptquotedidents{% + \let\nwlinkedidentq\@nwnosubscriptident +} +\def\nwopt@hyperquotedidents{% + \let\nwlinkedidentq\@nwhyperident +} +\def\nwopt@nohyperquotedidents{% + \let\nwlinkedidentq\@nwnosubscriptident +} +\nwopt@hyperidents +\newcount\@commacount +\def\commafy#1{% + {\nwix@listcount{#1}\@commacount=\nwix@counter + \let\@comma@each=\\% + \ifcase\@commacount\let\\=\@comma@each\or\let\\=\@comma@each\or + \def\\{\def\\{ \@nwlangdepand\ \@comma@each}\@comma@each}\else + \def\\{\def\\{, % + \advance\@commacount by \m@ne + \ifnum\@commacount=1 \@nwlangdepand~\fi\@comma@each}\@comma@each}\fi + #1}} +\def\nwix@cons#1#2{% {list}{\marker{element}} + {\toks0=\expandafter{#1}\def\@tempa{#2}\toks2=\expandafter{\@tempa}% + \xdef#1{\the\toks0 \the\toks2 }}} +\def\nwix@uses#1{% {label} + \def\nwixu{\\}\let\nwixd\@gobble\@nameuse{nwixl@#1}} +\def\nwix@defs#1{% {label} + \def\nwixd{\\}\let\nwixu\@gobble\@nameuse{nwixl@#1}} +\newcount\nwix@counter +\def\nwix@listcount#1{% {list with \\} + {\count@=0 + \def\\##1{\advance\count@ by \@ne }% + #1\global\nwix@counter=\count@ }} +\def\nwix@usecount#1{\nwix@listcount{\nwix@uses{#1}}} +\def\nwix@defcount#1{\nwix@listcount{\nwix@defs{#1}}} +\def\nwix@id@defs#1{% index pair + {{\Tt \@car#1\@nil}% + \def\\##1{~\subpageref{##1}}\nwix@defs{\@cdr#1\@nil}}} +\def\nwidentuses#1{% list of index pairs + \nwcodecomment{\@nwlangdepuss\ \let\\=\nwix@id@defs\commafy{#1}.}} +\def\nwix@totaluses#1{% list of index pairs + {\count@=0 + \def\\##1{\nwix@usecount{\@cdr##1\@nil}\advance\count@ by\nwix@counter}% + #1\global\nwix@counter\count@ }} +\def\nwix@id@uses#1#2{% {ident}{label} + \nwix@usecount{#2}\ifnum\nwix@counter>0 + {\advance\leftskip by \codemargin + \nwcodecomment{{\Tt #1}, \@nwlangdepusd\ \nwpageprep\ \@pagesl{\nwix@uses{#2}}.}}% + \else + \ifnw@hideunuseddefs\else + {\advance\leftskip by \codemargin \nwcodecomment{{\Tt #1}, \@nwlangdepnvu.}}% + \fi + \fi} +\def\nwidentdefs#1{% list of index pairs + \ifnw@hideunuseddefs\nwix@totaluses{#1}\else\nwix@listcount{#1}\fi + \ifnum\nwix@counter>0 + \nwcodecomment{\@nwlangdepdfs:}% + {\def\\##1{\nwix@id@uses ##1}#1}% + \fi} +\newif\ifnw@hideunuseddefs\nw@hideunuseddefsfalse +\def\nwopt@hideunuseddefs{\nw@hideunuseddefstrue} +\def\nwopt@noidentxref{% + \let\nwidentdefs\@gobble + \let\nwidentuses\@gobble} +\def\nw@underlinedefs{% {list with \nwixd, \nwixu} + \let\\=\relax\def\nw@comma{, } + \def\nwixd##1{\\\underline{\subpageref{##1}}\let\\\nw@comma}% + \def\nwixu##1{\\\subpageref{##1}\let\\\nw@comma}} + +\def\nw@indexline#1#2{% + {\indent {\Tt #1}: \nw@underlinedefs\@nameuse{nwixl@#2}\par}} + +\newenvironment{thenowebindex}{\parindent=-10pt \parskip=\z@ + \advance\leftskip by 10pt + \advance\rightskip by 0pt plus1in\par\@afterindenttrue + \def\\##1{\nw@indexline##1}}{} +\def\nowebindex{% + \@ifundefined{nwixs@i}% + {\@warning{The \string\nowebindex\space is empty}}% + {\begin{thenowebindex}\@nameuse{nwixs@i}\end{thenowebindex}}} +\def\nowebindex@external{% + {\let\nwixadds@c=\@gobble + \def\nwixadds@i##1{\nw@indexline##1}% + \def\nwixaddsx##1##2{\@nameuse{nwixadds@##1}{##2}}% + \begin{thenowebindex}\@input{\jobname.nwi}\end{thenowebindex}}} +\def\nwixlogsorted#1#2{% list data + \@bsphack\if@filesw + \toks0={#2}\immediate\write\@auxout{\string\nwixadds{#1}{\the\toks0}} + \if@nobreak \ifvmode\nobreak\fi\fi\fi\@esphack} +\def\nwixadds#1#2{% + \@ifundefined{nwixs@#1}% + {\global\@namedef{nwixs@#1}{\\{#2}}}% + {\expandafter\nwix@cons\csname nwixs@#1\endcsname{\\{#2}}}} +\let\nwixaddsx=\@gobbletwo +\def\nwopt@externalindex{% + \ifx\nwixadds\@gobbletwo % already called + \else + \let\nwixaddsx=\nwixadds \let\nwixadds=\@gobbletwo + \let\nowebindex=\nowebindex@external + \let\nowebchunks=\nowebchunks@external + \fi} +\def\nowebchunks{% + \@ifundefined{nwixs@c}% + {\@warning{The are no \string\nowebchunks}}% + {\begin{thenowebchunks}\@nameuse{nwixs@c}\end{thenowebchunks}}} +\def\nowebchunks@external{% + {\let\nwixadds@i=\@gobble + \def\nwixadds@c##1{\nw@onechunk##1}% + \def\nwixaddsx##1##2{\@nameuse{nwixadds@##1}{##2}}% + \begin{thenowebchunks}\@input{\jobname.nwi}\end{thenowebchunks}}} + \@namedef{r@nw@notdef}{{0}{(\@nwlangdepnvd)}} +\def\nw@chunkunderlinedefs{% {list of labels with \nwixd, \nwixu} + \let\\=\relax\def\nw@comma{, } + \def\nwixd##1{\\\underline{\subpageref{##1}}\let\\\nw@comma}% + \def\nwixu##1{\\\subpageref{##1}\let\\\nw@comma}} +\def\nw@onechunk#1#2#3{% {name}{label of first definition}{list with \nwixd, \nwixu} + \@ifundefined{r@#2}{}{% + \indent\LA #1~{\nwtagstyle\subpageref{#2}}\RA + \if@nwlongchunks{~\nw@chunkunderlinedefs#3}\fi\par}} +\newenvironment{thenowebchunks}{\vskip3pt + \parskip=\z@\parindent=-10pt \advance\leftskip by 10pt + \advance\rightskip by 0pt plus10pt \@afterindenttrue + \def\\##1{\nw@onechunk##1}}{} +\newif\if@nwlongchunks +\@nwlongchunksfalse +\let\nwopt@longchunks\@nwlongchunkstrue +\providecommand\@nw@hyper@ref{\hyperreference} % naras +\providecommand\@nw@hyper@anc{\blindhyperanchor} % naras +\providecommand\@nw@hyperref@ref[2]{\hyperlink{noweb.#1}{#2}} % nr +\providecommand\@nw@hyperref@anc[1]{\hypertarget{noweb.#1}{\relax}} % nr +%%\renewcommand\@nw@hyperref@ref[2]{{#2}} % nr +%%\renewcommand\@nw@hyperref@anc[1]{} % nr +\providecommand\nwhyperreference{% + \@ifundefined{hyperlink} + {\@ifundefined{hyperreference} + {\global\let\nwhyperreference\@gobble} + {\global\let\nwhyperreference\@nw@hyper@ref}} + {\global\let\nwhyperreference\@nw@hyperref@ref}% + \nwhyperreference +} + +\providecommand\nwblindhyperanchor{% + \@ifundefined{hyperlink} + {\@ifundefined{hyperreference} + {\global\let\nwblindhyperanchor\@gobble} + {\global\let\nwblindhyperanchor\@nw@hyper@anc}} + {\global\let\nwblindhyperanchor\@nw@hyperref@anc}% + \nwblindhyperanchor +} +\providecommand\nwanchorto{% + \begingroup\let\do\@makeother\dospecials + \catcode`\{=1 \catcode`\}=2 \nw@anchorto} +\providecommand\nw@anchorto[1]{\endgroup\def\nw@next{#1}\nw@anchortofin} +\providecommand\nw@anchortofin[1]{#1\footnote{See URL \texttt{\nw@next}.}} +\let\nwanchorname\@gobble +\newif\ifhtml +\htmlfalse +\let\nwixident=\relax +\def\nwbackslash{\char92} +\def\nwlbrace{\char123} +\def\nwrbrace{\char125} +\def\nwopt@english{% + \def\@nwlangdepdef{This definition is continued}% + \def\@nwlangdepcud{This code is used}% + \def\@nwlangdeprtc{Root chunk (not used in this document)}% + \def\@nwlangdepcwf{This code is written to file}% + \def\@nwlangdepchk{chunk}% + \def\@nwlangdepchks{chunks}% + \def\@nwlangdepin{in}% + \def\@nwlangdepand{and}% + \def\@nwlangdepuss{Uses}% + \def\@nwlangdepusd{used}% + \def\@nwlangdepnvu{never used}% + \def\@nwlangdepdfs{Defines}% + \def\@nwlangdepnvd{never defined}% +} +\let\nwopt@american\nwopt@english +\def\nwopt@portuges{% + \def\@nwlangdepdef{Defini\c{c}\~ao continuada em}% + % This definition is continued + \def\@nwlangdepcud{C\'odigo usado em}% + % This code is used + \def\@nwlangdeprtc{Fragmento de topo (sem uso no documento)}% + % Root chunk (not used in this document) + \def\@nwlangdepcwf{Este c\'odigo foi escrito no ficheiro}% + % This code is written to file + \def\@nwlangdepchk{fragmento}% + % chunk + \def\@nwlangdepchks{fragmentos}% + % chunks + \def\@nwlangdepin{no(s)}% + % in + \def\@nwlangdepand{e}% + % and + \def\@nwlangdepuss{Usa}% + % Uses + \def\@nwlangdepusd{usado}% + % used + \def\@nwlangdepnvu{nunca usado}% + % never used + \def\@nwlangdepdfs{Define}% + % Defines + \def\@nwlangdepnvd{nunca definido}% + % never defined +} +\def\nwopt@frenchb{% + \def\@nwlangdepdef{Cette d\'efinition suit}% + % This definition is continued + \def\@nwlangdepcud{Ce code est employ\'e}% + % This code is used + \def\@nwlangdeprtc{Morceau racine (pas employ\'e dans ce document)}% + % Root chunk (not used in this document) + \def\@nwlangdepcwf{Ce code est \'ecrit aux fichier}% + % This code is written to file + \def\@nwlangdepchk{le morceau}% + % chunk + \def\@nwlangdepchks{les morceaux}% + % chunks + \def\@nwlangdepin{dans}% + % in + \def\@nwlangdepand{et}% + % and + \def\@nwlangdepuss{Il emploie}% + % Uses + \def\@nwlangdepusd{employ\'{e}}% + % used + \def\@nwlangdepnvu{jamais employ\'{e}}% + % never used + \def\@nwlangdepdfs{Il d\'{e}fine}% + % Defines + % Cannot use the accent here: \def\@nwlangdepnvd{jamais d\'{e}fini}% + \def\@nwlangdepnvd{jamais defini}% + % never defined +} +\let\nwopt@french\nwopt@frenchb +\def\nwopt@german{% + \def\@nwlangdepdef{Diese Definition wird fortgesetzt}% + % This definition is continued + \def\@nwlangdepcud{Dieser Code wird benutzt}% + % This code is used + \def\@nwlangdeprtc{Hauptteil (nicht in diesem Dokument benutzt)}% + % Root chunk (not used in this document) + \def\@nwlangdepcwf{Dieser Code schreibt man zum File}% + % This code is written to file + \def\@nwlangdepchk{Teil}% + % chunk + \def\@nwlangdepchks{Teils}% + % chunks + \def\@nwlangdepin{im}% + % in + \def\@nwlangdepand{und}% + % and + \def\@nwlangdepuss{Benutztt}% + % Uses + \def\@nwlangdepusd{benutzt}% + % used + \def\@nwlangdepnvu{nicht benutzt}% + % never used + \def\@nwlangdepdfs{Definiert}% + % Defines + \def\@nwlangdepnvd{nicht definiert}% + % never defined +} +\let\nwopt@ngerman\nwopt@german +\ifx\languagename\undefined % default is English + \noweboptions{english} +\else + \@ifundefined{nwopt@\languagename} + {\noweboptions{english}} + {\expandafter\noweboptions\expandafter{\languagename}} +\fi +% Final names for the system and its components were in doubt so +% Language is used throughout the book to specify what the system is named. +\providecommand{\Language}{AXIOM} + +% HyperName was used to specify the name of the browser +\providecommand{\HyperName}{HyperDoc} + +% This is nothing more than inline math mode in Tex but has additional +% meaning within the browser. +\providecommand{\spad}[1]{${#1}$} + +% This is a way to say 8th, 100th, etc. +% We use providecommand so it doesn't clash with amsmath package +\providecommand{\eth}[1]{{#1}-th} + +% add the binom function for combfunc +\providecommand{\binom}[2]{\left(\begin{array}{c}#1\\#2\end{array}\right)} + +% spadcommands are the actual text that you type at the axiom prompt +\providecommand{\spadcommand}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% returnType is the type signature returned by the axiom interpreter +\providecommand{\returnType}[1]% +{\begin{flushright}{\tt #1}\end{flushright}\vskip .1cm} + +%%% Axiom commands are set off in a special boxed area. +%%% This is used to draw a line around that boxed area. +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +% The book begins with some introductory material that is not really +% listed as a chapter. This creates a header similar to \chapter. +\providecommand{\pseudoChapter}[1]% +{\vskip .5in \noindent {\Huge{\bf #1}}\vskip .5in} + +% The book begins with some introductory material that is not really +% listed as a section. This creates a header similar to \section. +\providecommand{\pseudoSection}[1]% +{\vskip .25in \noindent {\large{\bf #1}}\vskip .25in} + +% spadofFrom records the operation in the index and the domain in the index +\providecommand{\spadopFrom}[2]{\index{library!operations!#1 @\begingroup \string\tt{} #1 \endgroup}\index{#2}``{\tt #1}''} + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +% These are special markers within the text for Hypertex keywords +% They have no particular meaning in the book form. +\providecommand{\spadfun}[1]{{\it #1}} +\providecommand{\spadgloss}[1]{{\it #1}} +\providecommand{\spadkey}[1]{\index{#1 @\begingroup \string\tt{} #1 \endgroup}} + +% spadtype records the domain in the index +\providecommand{\spadtype}[1]{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}} + +% This is used to create a caption on an included image +\providecommand{\simpleCaption}[1]{\def\thefigure{\@arabic\c@figure}\caption{#1}} + +% This is the name of the ``glossy'' pages in the physical book. +\providecommand{\Gallery}{\Language{} Images} + +% spadofFrom records the operation in the index and the domain in the index +\providecommand{\spadopFrom}[2]{\index{library!operations!#1 @\begingroup \string\tt{} #1 \endgroup}\index{#2}``{\tt #1}''} + +%% typeset e.g. and i.e. +\providecommand{\eg}{\emph{e.g.}} +\providecommand{\ie}{\emph{i.e.}} + +%% Typeset in-line code. +\providecommand{\Code}[1]{\texttt{#1}} + +%% Typeset a file name +\providecommand{\File}[1]{{\itshape{#1}}} + +%% Typeset a tool name, e.g. Axiom +\providecommand{\Tool}[1]{{\bfseries\sffamily{#1}}} +\providecommand\AXIOM{\vskip\parindent\indent{\bf AXIOM}\vskip\parindent\noindent\ignorespaces} +\providecommand\maxima{\vskip\parindent\noindent{\bf MAXIMA}\vskip\parindent\noindent\ignorespaces} +\providecommand\yacas{\vskip\parindent\noindent{\bf YACAS}\vskip\parindent\noindent\ignorespaces} diff --git a/books/bookvol0.pamphlet b/books/bookvol0.pamphlet new file mode 100644 index 0000000..8de7c52 --- /dev/null +++ b/books/bookvol0.pamphlet @@ -0,0 +1,67732 @@ +\documentclass{book} +%\usepackage{axiom} +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + +% spadcommands are the actual text that you type at the axiom prompt +\newcommand{\spadcommand}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadgraph are the actual text that you type at the axiom prompt for draw +\newcommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% returnType is the type signature returned by the axiom interpreter +\newcommand{\returnType}[1]% +{\begin{flushright}{\tt #1}\end{flushright}\vskip .1cm} + +% spadsig gives the standard -> notation for signatures +\newcommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% The book begins with some introductory material that is not really +% listed as a chapter. This creates a header similar to \chapter. +\newcommand{\pseudoChapter}[1]% +{\vskip .5in \noindent {\Large{\bf #1}}\vskip .5in} + +% The book begins with some introductory material that is not really +% listed as a section. This creates a header similar to \section. +\newcommand{\pseudoSection}[1]% +{\vskip .25in \noindent {\large{\bf #1}}\vskip .25in} + +% spadofFrom records the operation in the index and the domain in the index +\newcommand{\spadopFrom}[2]{\index{library!operations!#1 @\begingroup \string\tt{} #1 \endgroup}\index{#2}``{\tt #1}''} + +% spadfunFrom records the function name and domain in the index +\newcommand{\spadfunFrom}[2]{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +% special meanings for math characters +\newcommand{\N}{\mbox{\bbold N}} +\newcommand{\Natural}{\mbox{\bbold N}} +\newcommand{\Z}{\mbox{\bbold Z}} +\newcommand{\Integer}{\mbox{\bbold Z}} +\newcommand{\Rational}{\mbox{\bbold Q}} +\newcommand{\Q}{\mbox{\bbold Q}} +\newcommand{\Complex}{\mbox{\bbold C}} +\newcommand{\C}{{\mathcal C}} +\newcommand{\Real}{\mbox{\bbold R}} +\newcommand{\F}{{\mathcal F}} +\newcommand{\R}{{\mathcal R}} + +% draw a box around a text block +\newcommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\newcommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\newcommand{\argDef}[1]{{\tt ({#1})}} +\newcommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\newcommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} + +\newcommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth +\newcommand\ExitBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} +\newcommand\ReturnBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} +\newcommand\HelpBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} +\newcommand\UpBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} +\newcommand{\tpd}[5]{{\setlength{\unitlength}{0.01in}\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 0: Axiom Jenks and Sutor}} +\end{titlepage} +\pagenumbering{roman} +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +This document is a complete ``re-implementation'' of the original Axiom +book by Jenks and Sutor. Virtually every line has been reviewed and +rewritten into the new Axiom pamphlet format. Changes were made to +reflect the new Axiom system. Additional material was added and some +previous examples were rewritten. This is intended to be a ``living'' +document with material referenced or gathered automatically from other +parts of the system documentation. Future plans include adding active +examples (moving graphics, in-line command prompts) using Active-DVI. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +%\noindent +{\Large{\bf Foreword}} +\vskip .25in + +You are holding in your hands an unusual book. Winston Churchill once +said that the empires of the future will be empires of the mind. This +book might hold an electronic key to such an empire. + +When computers were young and slow, the emerging computer science +developed dreams of Artificial Intelligence and Automatic Theorem +Proving in which theorems can be proved by machines instead of +mathematicians. Now, when computer hardware has matured and become +cheaper and faster, there is not too much talk of putting the burden +of formulating and proving theorems on the computer's shoulders. +Moreover, even in those cases when computer programs do prove +theorems, or establish counter-examples (for example, the solution of +the four color problem, the non-existence of projective planes of +order 10, the disproof of the Mertens conjecture), humans carry most +of the burden in the form of programming and verification. + +It is the language of computer programming that has turned out to be +the crucial instrument of productivity in the evolution of scientific +computing. The original Artificial Intelligence efforts gave birth to +the first symbolic manipulation systems based on LISP. The first +complete symbolic manipulation or, as they are called now, computer +algebra packages tried to imbed the development programming and +execution of mathematical problems into a framework of familiar +symbolic notations, operations and conventions. In the third decade +of symbolic computations, a couple of these early systems---REDUCE and +MACSYMA---still hold their own among faithful users. + +Axiom was born in the mid-70's as a system called Scratchpad +developed by IBM researchers. Scratchpad/Axiom was born big---its +original platform was an IBM mainframe 3081, and later a 3090. The +system was growing and learning during the decade of the 80's, and its +development and progress influenced the field of computer algebra. +During this period, the first commercially available computer algebra +packages for mini and and microcomputers made their debut. By now, +our readers are aware of Mathematica, Maple, Derive, and Macsyma. +These systems (as well as a few special purpose computer algebra +packages in academia) emphasize ease of operation and standard +scientific conventions, and come with a prepared set of mathematical +solutions for typical tasks confronting an applied scientist or an +engineer. These features brought a recognition of the enormous +benefits of computer algebra to the widest circles of scientists and +engineers. + +The Scratchpad system took its time to blossom into the beautiful +Axiom product. There is no rival to this powerful environment in +its scope and, most importantly, in its structure and organization. +Axiom contains the basis for any comprehensive and elaborate +mathematical development. It gives the user all Foundation and +Algebra instruments necessary to develop a computer realization of +sophisticated mathematical objects in exactly the way a mathematician +would do it. Axiom is also the basis of a complete scientific +cyberspace---it provides an environment for mathematical objects used +in scientific computation, and the means of controlling and +communicating between these objects. Knowledge of only a few Axiom +language features and operating principles is all that is required to +make impressive progress in a given domain of interest. The system is +powerful. It is not an interactive interpretive environment operating +only in response to one line commands---it is a complete language with +rich syntax and a full compiler. Mathematics can be developed and +explored with ease by the user of Axiom. In fact, during +Axiom's growth cycle, many detailed mathematical domains were +constructed. Some of them are a part of Axiom's core and are +described in this book. For a bird's eye view of the algebra +hierarchy of Axiom, glance inside the book cover. + +The crucial strength of Axiom lies in its excellent structural +features and unlimited expandability---it is open, modular system +designed to support an ever growing number of facilities with minimal +increase in structural complexity. Its design also supports the +integration of other computation tools such as numerical software +libraries written in FORTRAN and C. While Axiom is already a +very powerful system, the prospect of scientists using the system to +develop their own fields of Science is truly exciting---the day is +still young for Axiom. + +Over the last several years Scratchpad/Axiom has scored many +successes in theoretical mathematics, mathematical physics, +combinatorics, digital signal processing, cryptography and parallel +processing. We have to confess that we enjoyed using +Scratchpad/Axiom. It provided us with an excellent environment for +our research, and allowed us to solve problems intractable on other +systems. We were able to prove new diophantine results for $\pi$; +establish the Grothendieck conjecture for certain classes of linear +differential equations; study the arithmetic properties of the +uniformization of hyperelliptic and other algebraic curves; construct +new factorization algorithms based on formal groups; within +Scratchpad/Axiom we were able to obtain new identities needed for +quantum field theory (elliptic genus formula and double scaling limit +for quantum gravity), and classify period relations for CM varieties +in terms of hypergeometric series. + +The Axiom system is now supported and distributed by NAG, the group +that is well known for its high quality software products for +numerical and statistical computations. The development of Axiom +in IBM was conducted at IBM T.J. Watson Research Center at Yorktown, +New York by a symbolic computation group headed by Richard D. Jenks. +Shmuel Winograd of IBM was instrumental in the progress of symbolic +research at IBM. + +This book opens the wonderful world of Axiom, guiding the reader +and user through Axiom's definitions, rules, applications and +interfaces. A variety of fully developed areas of mathematics are +presented as packages, and the user is well advised to take advantage +of the sophisticated realization of familiar mathematics. The +Axiom book is easy to read and the Axiom system is easy to use. +It possesses all the features required of a modern computer +environment (for example, windowing, integration of operating system +features, and interactive graphics). Axiom comes with a detailed +hypertext interface (HyperDoc), an elaborate browser, and complete +on-line documentation. The HyperDoc allows novices to solve their +problems in a straightforward way, by providing menus for step-by-step +interactive entry. + +The appearance of Axiom in the scientific market moves symbolic +computing into a higher plane, where scientists can formulate their +statements in their own language and receive computer assistance in +their proofs. Axiom's performance on workstations is truly +impressive, and users of Axiom will get more from them than we, the +early users, got from mainframes. Axiom provides a powerful +scientific environment for easy construction of mathematical tools and +algorithms; it is a symbolic manipulation system, and a high +performance numerical system, with full graphics capabilities. We +expect every (computer) power hungry scientist will want to take full +advantage of Axiom. + +\vskip .25in +%\noindent +David V. Chudnovsky \hfill Gregory V. Chudnovsky +\vfill +\eject +\pagenumbering{arabic} +\pseudoChapter{Introduction to Axiom} +\label{ugNewIntro} +\section{Introduction to Axiom} +Welcome to the world of Axiom. +We call Axiom a scientific computation system: +a self-contained toolbox designed to meet +your scientific programming needs, +from symbolics, to numerics, to graphics. + +This introduction is a quick overview of what Axiom offers. + +\subsection{Symbolic Computation} +Axiom provides a wide range of simple commands for symbolic +mathematical problem solving. Do you need to solve an equation, to +expand a series, or to obtain an integral? If so, just ask Axiom +to do it. + +Given $$\int\left({{1\over{(x^3 \ {(a+b x)}^{1/3})}}}\right)dx$$ +we would enter this into Axiom as: + +\spadcommand{integrate(1/(x**3 * (a+b*x)**(1/3)),x)} +which would give the result: +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +-{2 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{\root{3}\of{a}}\ {{\root{3}\of{{b +\ x}+ a}}^2}}+{{{\root{3}\of{a}}^2}\ {\root{3}\of{{b \ x}+ +a}}}+ a}\right)}}+ +\\ +\\ +\displaystyle +{4 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{{\root{3}\of{a}}^ +2}\ {\root{3}\of{{b \ x}+ a}}}- a}\right)}}+ +\\ +\\ +\displaystyle +{{12}\ {b^2}\ {x^2}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {{\root{3}\of{a}}^ +2}\ {\root{3}\of{{b \ x}+ a}}}+{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ b \ x}-{9 \ a}\right)}\ {\sqrt{3}}\ {\root{3}\of{a}}\ {{\root{3}\of{{b +\ x}+ a}}^2}} +\end{array} +\right)}\over{{18}\ {a^2}\ {x^2}\ {\sqrt{3}}\ {\root{3}\of{a}}} +$$ +\returnType{Type: Union(Expression Integer,...)} +Axiom provides state-of-the-art algebraic machinery to handle your +most advanced symbolic problems. For example, Axiom's integrator +gives you the answer when an answer exists. If one does not, it +provides a proof that there is no answer. Integration is just one of +a multitude of symbolic operations that Axiom provides. + +\subsection{Numeric Computation} +Axiom has a numerical library that includes operations for linear +algebra, solution of equations, and special functions. For many of +these operations, you can select any number of floating point digits +to be carried out in the computation. + +Solve $x^{49}-49x^4+9$ to 49 digits of accuracy. +First we need to change the default output length of numbers: + +\spadcommand{digits(49)} +and then we execute the command: + +\spadcommand{solve(x**49-49*x**4+9 = 0,1.e-49)} +$$ +\begin{array}{@{}l} +\displaystyle +\left[{x = -{0.6546536706904271136718122105095984761851224331 +556}}, \right. +\\ +\\ +\displaystyle +\left.{x ={1.086921395653859508493939035954893289009213388763}}, + \right. +\\ +\\ +\displaystyle +\left.{x ={0.654653670725527173969468606613676483536148760766 +1}}\right] +\end{array} +$$ + + +\returnType{Type: List Equation Polynomial Float} +The output of a computation can be converted to FORTRAN to be used +in a later numerical computation. +Besides floating point numbers, Axiom provides literally +dozens of kinds of numbers to compute with. +These range from various kinds of integers, to fractions, complex +numbers, quaternions, continued fractions, and to numbers represented +with an arbitrary base. + +What is $10$ to the $90$-th power in base $32$? + +\spadcommand{radix(10**90,32)} +returns: + +%\noindent +{\tt FMM3O955CSEIV0ILKH820CN3I7PICQU0OQMDOFV6TP000000000000000000 } +\returnType{Type: RadixExpansion 32} + +The AXIOM numerical library can be enhanced with a +substantial number of functions from the NAG library of numerical and +statistical algorithms. These functions will provide coverage of a wide +range of areas including roots of functions, Fourier transforms, quadrature, +differential equations, data approximation, non-linear optimization, linear +algebra, basic statistics, step-wise regression, analysis of variance, +time series analysis, mathematical programming, and special functions. +Contact the Numerical Algorithms Group Limited, Oxford, England. + +\subsection{Graphics} +You may often want to visualize a symbolic formula or draw +a graph from a set of numerical values. +To do this, you can call upon the Axiom +graphics capability. + +Draw $J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$. + +\spadcommand{draw(5*besselJ(0,sqrt(x**2+y**2)), x=-20..20, y=-20..20)} +\begin{figure}[htbp] +\includegraphics[bbllx=1, bblly=39, bburx=298, bbury=290]{ps/bessintr.ps} +\caption{$J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$} +\label{tpdhere} +\end{figure} + +Graphs in Axiom are interactive objects you can manipulate with +your mouse. Just click on the graph, and a control panel pops up. +Using this mouse and the control panel, you can translate, rotate, +zoom, change the coloring, lighting, shading, and perspective on the +picture. You can also generate a PostScript copy of your graph to +produce hard-copy output. + +\subsection{HyperDoc} + +\begin{figure}[htbp] +\includegraphics[bbllx=1, bblly=1, bburx=298, bbury=290]{ps/h-root.ps} +\caption{Hyperdoc opening menu} +\label{fig-intro-br} +\end{figure} + +HyperDoc presents you windows on the world of Axiom, +offering on-line help, examples, tutorials, a browser, and reference +material. HyperDoc gives you on-line access to this document in a +``hypertext'' format. Words that appear in a different font (for +example, {\tt Matrix}, {\bf factor}, and +{\it category}) are generally mouse-active; if you click on one +with your mouse, HyperDoc shows you a new window for that word. + +As another example of a HyperDoc facility, suppose that you want to +compute the roots of $x^{49} - 49x^4 + 9$ to 49 digits (as in our +previous example) and you don't know how to tell Axiom to do this. +The ``basic command'' facility of HyperDoc leads the way. Through the +series of HyperDoc windows shown in Figure \ref{fig-intro-br} on +page~\pageref{fig-intro-br} and the specified mouse clicks, you and +HyperDoc generate the correct command to issue to compute the answer. + +\subsection{Interactive Programming } +Axiom's interactive programming language lets you define your +own functions. A simple example of a user-defined function is one +that computes the successive Legendre polynomials. Axiom lets +you define these polynomials in a piece-wise way. + +The first Legendre polynomial. + +\spadcommand{p(0) == 1} +\returnType{Type: Void} +The second Legendre polynomial. + +\spadcommand{p(1) == x} +\returnType{Type: Void} +The $n$-th Legendre polynomial for $(n > 1)$. + +\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1) * p(n-2))/n} +\returnType{Type: Void} + +In addition to letting you define simple functions like this, the +interactive language can be used to create entire application +packages. All the graphs in the Axiom images section were created by +programs written in the interactive language. + +The above definitions for $p$ do no computation---they simply +tell Axiom how to compute $p(k)$ for some positive integer +$k$. + +To actually get a value of a Legendre polynomial, you ask for it. +\index{Legendre polynomials} + +What is the tenth Legendre polynomial? + +\spadcommand{p(10)} +\begin{verbatim} + Compiling function p with type Integer -> Polynomial Fraction + Integer + Compiling function p as a recurrence relation. +\end{verbatim} +$$ +{{{46189} \over {256}} \ {x \sp {10}}} -{{{109395} \over {256}} \ {x \sp +8}}+{{{45045} \over {128}} \ {x \sp 6}} -{{{15015} \over {128}} \ {x \sp +4}}+{{{3465} \over {256}} \ {x \sp 2}} -{{63} \over {256}} +$$ +\returnType{Type: Polynomial Fraction Integer} +Axiom applies the above pieces for $p$ to obtain the value +of $p(10)$. But it does more: it creates an optimized, compiled +function for $p$. The function is formed by putting the pieces +together into a single piece of code. By {\it compiled}, we mean that +the function is translated into basic machine-code. By {\it +optimized}, we mean that certain transformations are performed on that +code to make it run faster. For $p$, Axiom actually +translates the original definition that is recursive (one that calls +itself) to one that is iterative (one that consists of a simple loop). + +What is the coefficient of $x^{90}$ in $p(90)$? + +\spadcommand{coefficient(p(90),x,90)} +$$ +{5688265542052017822223458237426581853561497449095175} \over +{77371252455336267181195264} +$$ +\returnType{Type: Polynomial Fraction Integer} + +In general, a user function is type-analyzed and compiled on first use. +Later, if you use it with a different kind of object, the function +is recompiled if necessary. + +\subsection{Data Structures} + +A variety of data structures are available for interactive use. These +include strings, lists, vectors, sets, multisets, and hash tables. A +particularly useful structure for interactive use is the infinite +stream: + +Create the infinite stream of derivatives of Legendre polynomials. + +\spadcommand{[D(p(i),x) for i in 1..]} +$$ +\begin{array}{@{}l} +\displaystyle +\left[ 1, {3 \ x}, {{{{15}\over 2}\ {x^2}}-{3 \over 2}}, + {{{{35}\over 2}\ {x^3}}-{{{15}\over 2}\ x}}, {{{{315}\over +8}\ {x^4}}-{{{105}\over 4}\ {x^2}}+{{15}\over 8}}, \right. +\\ +\\ +\displaystyle +\left.{{{{693}\over 8}\ {x^5}}-{{{315}\over 4}\ {x^3}}+{{{105}\over +8}\ x}}, {{{{3003}\over{16}}\ {x^6}}-{{{3465}\over{16}}\ {x^ +4}}+{{{945}\over{16}}\ {x^2}}-{{35}\over{16}}}, \right. +\\ +\\ +\displaystyle +\left.{{{{6435}\over{16}}\ {x^7}}-{{{9009}\over{16}}\ {x^5}}+ +{{{3465}\over{16}}\ {x^3}}-{{{315}\over{16}}\ x}}, \right. +\\ +\\ +\displaystyle +\left.{{{{109395}\over{128}}\ {x^8}}-{{{45045}\over{32}}\ {x^ +6}}+{{{45045}\over{64}}\ {x^4}}-{{{3465}\over{32}}\ {x^2}}+{{3 +15}\over{128}}}, \right. +\\ +\\ +\displaystyle +\left.{{{{230945}\over{128}}\ {x^9}}-{{{109395}\over{32}}\ {x^ +7}}+{{{135135}\over{64}}\ {x^5}}-{{{15015}\over{32}}\ {x^3}}+ +{{{3465}\over{128}}\ x}}, \ldots \right] +\end{array} +$$ +\returnType{Type: Stream Polynomial Fraction Integer} + + +Streams display only a few of their initial elements. Otherwise, they +are ``lazy'': they only compute elements when you ask for them. + +Data structures are an important component for building application +software. Advanced users can represent data for applications in +optimal fashion. In all, Axiom offers over forty kinds of +aggregate data structures, ranging from mutable structures (such as +cyclic lists and flexible arrays) to storage efficient structures +(such as bit vectors). As an example, streams are used as the +internal data structure for power series. + +What is the series expansion +of $\log(\cot(x))$ +about $x=\pi/2$? +%NOTE: The book has a different answer (see p6) + +\spadcommand{series(log(cot(x)),x = \%pi/2)} +$$ +\begin{array}{@{}l} +\displaystyle +{\log \left({{-{2 \ x}+ \pi}\over 2}\right)}+ +{{1 \over 3}\ {{\left(x -{\pi \over 2}\right)}^2}}+ +{{7 \over{90}}\ {{\left(x -{\pi \over 2}\right)}^4}}+ +{{{62}\over{2835}}\ {{\left(x -{\pi \over 2}\right)}^6}}+ +\\ +\\ +\displaystyle +{{{127}\over{18900}}\ {{\left(x -{\pi \over 2}\right)}^8}}+ +{{{146}\over{66825}}\ {{\left(x -{\pi \over 2}\right)}^{10}}}+ +{O \left({{\left(x -{\pi \over 2}\right)}^{11}}\right)} +\end{array} +$$ +\returnType{Type: GeneralUnivariatePowerSeries(Expression Integer,x,pi/2)} + +Series and streams make no attempt to compute {\it all} their +elements! Rather, they stand ready to deliver elements on demand. + +What is the coefficient of the $50$-th +term of this series? + +\spadcommand{coefficient(\%,50)} +$$ +{44590788901016030052447242300856550965644} \over +{7131469286438669111584090881309360354581359130859375} +$$ +\returnType{Type: Expression Integer} + +\subsection{Mathematical Structures} +Axiom also has many kinds of mathematical structures. These +range from simple ones (like polynomials and matrices) to more +esoteric ones (like ideals and Clifford algebras). Most structures +allow the construction of arbitrarily complicated ``types.'' + +Even a simple input expression can +result in a type with several levels. + +\spadcommand{matrix [ [x + \%i,0], [1,-2] ]} +$$ +\left[ +\begin{array}{cc} +{x+i} & 0 \\ +1 & -2 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Complex Integer} + +The Axiom interpreter builds types in response to user input. +Often, the type of the result is changed in order to be applicable to +an operation. + +The inverse operation requires that elements of the above matrices +are fractions. + +\spadcommand{inverse(\%)} +$$ +\left[ +\begin{array}{cc} +{1 \over {x+i}} & 0 \\ +{1 \over {{2 \ x}+{2 \ i}}} & -{1 \over 2} +\end{array} +\right] +$$ +\returnType{Type: Union(Matrix Fraction Polynomial Complex Integer,...)} + +\subsection{Pattern Matching} + +A convenient facility for symbolic computation is ``pattern +matching.'' Suppose you have a trigonometric expression and you want +to transform it to some equivalent form. Use a $rule$ command to +describe the transformation rules you \index{rule} need. Then give +the rules a name and apply that name as a function to your +trigonometric expression. + +Introduce two rewrite rules. + +\spadcommand{sinCosExpandRules := rule\\ +\ \ sin(x+y) == sin(x)*cos(y) + sin(y)*cos(x)\\ +\ \ cos(x+y) == cos(x)*cos(y) - sin(x)*sin(y)\\ +\ \ sin(2*x) == 2*sin(x)*cos(x)\\ +\ \ cos(2*x) == cos(x)**2 - sin(x)**2 +} + +\begin{verbatim} + {sin(y + x) == cos(x)sin(y) + cos(y)sin(x), + cos(y + x) == - sin(x)sin(y) + cos(x)cos(y), + sin(2x) == 2cos(x)sin(x), + 2 2 + cos(2x) == - sin(x) + cos(x) } +\end{verbatim} +\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} + +Apply the rules to a simple trigonometric expression. + +\spadcommand{sinCosExpandRules(sin(a+2*b+c))} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left(-{{\cos \left({a}\right)}\ {{\sin \left({b}\right)}^2}}- +{2 \ {\cos \left({b}\right)}\ {\sin \left({a}\right)}\ {\sin +\left({b}\right)}}+{{\cos \left({a}\right)}\ {{\cos \left({b}\right)}^ +2}}\right)}\ {\sin \left({c}\right)}}- +\\ +\\ +\displaystyle +{{\cos \left({c}\right)}\ {\sin \left({a}\right)}\ {{\sin \left({b}\right)}^ +2}}+{2 \ {\cos \left({a}\right)}\ {\cos \left({b}\right)}\ {\cos +\left({c}\right)}\ {\sin \left({b}\right)}}+ +\\ +\\ +\displaystyle +{{{\cos \left({b}\right)}^2}\ {\cos \left({c}\right)}\ {\sin +\left({a}\right)}} +\end{array} +$$ +\returnType{Type: Expression Integer} + + +Using input files, you can create your own library of transformation +rules relevant to your applications, then selectively apply the rules +you need. + +\subsection{Polymorphic Algorithms} +All components of the Axiom algebra library are written in the +Axiom library language. This language is similar to the +interactive language except for protocols that authors are obliged to +follow. The library language permits you to write ``polymorphic +algorithms,'' algorithms defined to work in their most natural +settings and over a variety of types. + +Define a system of polynomial equations $S$. + +\spadcommand{S := [3*x**3 + y + 1 = 0,y**2 = 4]} +$$ +\left[ +{{y+{3 \ {x \sp 3}}+1}=0}, {{y \sp 2}=4} +\right] +$$ +\returnType{Type: List Equation Polynomial Integer} + +Solve the system $S$ using rational number arithmetic and +30 digits of accuracy. + +\spadcommand{solve(S,1/10**30)} +$$ +\left[ +{\left[ {y=-2}, {x={{1757879671211184245283070414507} \over +{2535301200456458802993406410752}}} +\right]}, + {\left[ {y=2}, {x=-1} +\right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Fraction Integer} + +Solve $S$ with the solutions expressed in radicals. + +\spadcommand{radicalSolve(S)} +$$ +\begin{array}{@{}l} +\displaystyle +\left[{\left[{y = 2}, {x = - 1}\right]}, {\left[{y = 2}, +{x ={{-{\sqrt{- 3}}+ 1}\over 2}}\right]}, \right. +\\ +\\ +\displaystyle +\left.{\left[{y = 2}, {x ={{{\sqrt{- 3}}+ 1}\over 2}}\right]}, + {\left[{y = - 2}, {x ={1 \over{\root{3}\of{3}}}}\right]}, + \right. +\\ +\\ +\displaystyle +\left.{\left[{y = - 2}, {x ={{{{\sqrt{- 1}}\ {\sqrt{3}}}- 1}\over{2 +\ {\root{3}\of{3}}}}}\right]}, {\left[{y = - 2}, {x ={{-{{\sqrt{- + 1}}\ {\sqrt{3}}}- 1}\over{2 \ {\root{3}\of{3}}}}}\right]}\right] +\end{array} +$$ +\returnType{Type: List List Equation Expression Integer} + +While these solutions look very different, the results were produced +by the same internal algorithm! The internal algorithm actually works +with equations over any ``field.'' Examples of fields are the +rational numbers, floating point numbers, rational functions, power +series, and general expressions involving radicals. + +\subsection{Extensibility} + +Users and system developers alike can augment the Axiom library, +all using one common language. Library code, like interpreter code, +is compiled into machine binary code for run-time efficiency. + +Using this language, you can create new computational types and new +algorithmic packages. All library code is polymorphic, described in +terms of a database of algebraic properties. By following the +language protocols, there is an automatic, guaranteed interaction +between your code and that of colleagues and system implementers. +\vfill\eject +\pseudoChapter{A Technical Introduction} +\label{ugTechIntro} +Axiom has both an {\it interactive language} for user +interactions and a {\it programming language} for building library +modules. Like Modula 2, \index{Modula 2} PASCAL, \index{PASCAL} +FORTRAN, \index{FORTRAN} and Ada, \index{Ada} the programming language +emphasizes strict type-checking. Unlike these languages, types in +Axiom are dynamic objects: they are created at run-time in +response to user commands. + +Here is the idea of the Axiom programming language in a +nutshell. Axiom types range from algebraic ones (like +polynomials, matrices, and power series) to data structures (like +lists, dictionaries, and input files). Types combine in any +meaningful way. You can build polynomials of matrices, matrices of +polynomials of power series, hash tables with symbolic keys and +rational function entries, and so on. + +{\it Categories} define algebraic properties to ensure mathematical +correctness. They ensure, for example, that matrices of polynomials +are OK, but matrices of input files are not. Through categories, +programs can discover that polynomials of continued fractions have a +commutative multiplication whereas polynomials of matrices do not. + +Categories allow algorithms to be defined in their most natural +setting. For example, an algorithm can be defined to solve polynomial +equations over {\it any} field. Likewise a greatest common divisor +can compute the ``gcd'' of two elements from {\it any} Euclidean +domain. Categories foil attempts to compute meaningless ``gcds'', for +example, of two hashtables. Categories also enable algorithms to be +compiled into machine code that can be run with arbitrary types. + +The Axiom interactive language is oriented towards ease-of-use. +The Axiom interpreter uses type-inferencing to deduce the type +of an object from user input. Type declarations can generally be +omitted for common types in the interactive language. + +So much for the nutshell. +Here are these basic ideas described by ten design principles: + +\subsection{Types are Defined by Abstract Datatype Programs} + +Basic types are called {\it domains of computation}, or, +simply, {\it domains.} +\index{domain} +Domains are defined by Axiom programs of the form: + +\begin{verbatim} +Name(...): Exports == Implementation +\end{verbatim} + +Each domain has a capitalized {\tt Name} that is used to refer to the +class of its members. For example, {\tt Integer} denotes ``the +class of integers,'' {\tt Float}, ``the class of floating point +numbers,'' and {\tt String}, ``the class of strings.'' + +The ``{\tt ...}'' part following {\tt Name} lists zero or more +parameters to the constructor. Some basic ones like {\tt Integer} take +no parameters. Others, like {\tt Matrix}, {\tt Polynomial} and +{\tt List}, take a single parameter that again must be a domain. For +example, {\tt Matrix(Integer)} denotes ``matrices over the integers,'' +{\tt Polynomial (Float)} denotes ``polynomial with floating point +coefficients,'' and {\tt List (Matrix (Polynomial (Integer)))} denotes +``lists of matrices of polynomials over the integers.'' There is no +restriction on the number or type of parameters of a domain +constructor. + +SquareMatrix(2,Integer) is an example of a domain constructor that accepts +both a particular data value as well as an integer. In this case the +number 2 specifies the number of rows and columns the square matrix +will contain. Elements of the matricies are integers. + +The {\tt Exports} part specifies operations for creating and +manipulating objects of the domain. For example, type +{\tt Integer} exports constants $0$ and $1$, and +operations \spadopFrom{+}{Integer}, \spadopFrom{-}{Integer}, and +\spadopFrom{*}{Integer}. While these operations are common, others +such as \spadfunFrom{odd?}{Integer} and \spadfunFrom{bit?}{Integer} +are not. In addition the Exports section can contain symbols that +represent properties that can be tested. For example, the Category +{\tt EntireRing} has the symbol {\tt noZeroDivisors} which asserts +that if a product is zero then one of the factors must be zero. + +The {\tt Implementation} part defines functions that implement the +exported operations of the domain. These functions are frequently +described in terms of another lower-level domain used to represent the +objects of the domain. Thus the operation of adding two vectors of +real numbers can be described and implemented using the addition +operation from {\tt Float}. + +\subsection{The Type of Basic Objects is a Domain or Subdomain} + +Every Axiom object belongs to a {\it unique} domain. The domain +of an object is also called its {\it type.} Thus the integer $7$ +has type {\tt Integer} and the string {\tt "daniel"} has type +{\tt String}. + +The type of an object, however, is not unique. The type of integer +$7$ is not only {\tt Integer} but {\tt NonNegativeInteger}, +{\tt PositiveInteger}, and possibly, in general, any other +``subdomain'' of the domain {\tt Integer}. A {\it subdomain} +\index{subdomain} is a domain with a ``membership predicate''. +{\tt PositiveInteger} is a subdomain of {\tt Integer} with the +predicate ``is the integer $> 0$?''. + +Subdomains with names are defined by abstract datatype programs +similar to those for domains. The {\it Export} part of a subdomain, +however, must list a subset of the exports of the domain. The {\tt +Implementation} part optionally gives special definitions for +subdomain objects. + +\subsection{Domains Have Types Called Categories} + +Domain and subdomains in Axiom are themselves objects that have +types. The type of a domain or subdomain is called a {\it category}. +\index{category} Categories are described by programs of the form: + +\begin{verbatim} +Name(...): Category == Exports +\end{verbatim} +The type of every category is the distinguished symbol {\tt Category.} +The category {\tt Name} is used to designate the class of domains of +that type. For example, category {\tt Ring} designates the class +of all rings. Like domains, categories can take zero or more +parameters as indicated by the ``{\tt ...}'' part following {\tt +Name.} Two examples are {\tt Module(R)} and +{\tt MatrixCategory(R,Row,Col)}. + +The {\tt Exports} part defines a set of operations. For example, +{\tt Ring} exports the operations \spadopFrom{0}{Ring}, +\spadopFrom{1}{Ring}, \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, and +\spadopFrom{*}{Ring}. Many algebraic domains such as +{\tt Integer} and {\tt Polynomial (Float)} are rings. +{\tt String} and {\tt List (R)} (for any domain $R$) +are not. + +Categories serve to ensure the type-correctness. The definition of +matrices states {\tt Matrix(R: Ring)} requiring its single parameter +$R$ to be a ring. Thus a ``matrix of polynomials'' is allowed, +but ``matrix of lists'' is not. + +Categories say nothing about representation. Domains, which are +instances of category types, specify representations. + +\subsection{Operations Can Refer To Abstract Types} + +All operations have prescribed source and target types. Types can be +denoted by symbols that stand for domains, called ``symbolic +domains.'' The following lines of Axiom code use a symbolic +domain $R$: + +\begin{verbatim} +R: Ring +power: (R, NonNegativeInteger): R -> R +power(x, n) == x ** n +\end{verbatim} + +Line 1 declares the symbol $R$ to be a ring. Line 2 declares the +type of $power$ in terms of $R$. From the definition on +line 3, $power(3,2)$ produces 9 for $x = 3$ and $R =$ +{\tt Integer}. Also, $power(3.0,2)$ produces $9.0$ for +$x = 3.0$ and $R =$ {\tt Float}. +$power("oxford",2)$ however fails since $"oxford"$ has type +{\tt String} which is not a ring. + +Using symbolic domains, algorithms can be defined in their most +natural or general setting. + +\subsection{Categories Form Hierarchies} + +Categories form hierarchies (technically, directed-acyclic graphs). A +simplified hierarchical world of algebraic categories is shown below. +At the top of this world is {\tt SetCategory}, the class of +algebraic sets. The notions of parents, ancestors, and descendants is +clear. Thus ordered sets (domains of category {\tt OrderedSet}) +and rings are also algebraic sets. Likewise, fields and integral +domains are rings and algebraic sets. However fields and integral +domains are not ordered sets. + +\begin{verbatim} +SetCategory +---- Ring ---- IntegralDomain ---- Field + | + +---- Finite ---+ + | \ + +---- OrderedSet -----+ OrderedFinite +\end{verbatim} +\begin{center} +Figure 1. A simplified category hierarchy. +\end{center} + +\subsection{Domains Belong to Categories by Assertion} + +A category designates a class of domains. Which domains? You might +think that {\tt Ring} designates the class of all domains that +export $0$, $1$, \spadopFrom{+}{Integer}, +\spadopFrom{-}{Integer}, and \spadopFrom{*}{Integer}. But this is not +so. Each domain must {\it assert} which categories it belongs to. + +The {\tt Export} part of the definition for {\tt Integer} reads, +for example: + +\begin{verbatim} +Join(OrderedSet, IntegralDomain, ...) with ... +\end{verbatim} + +This definition asserts that {\tt Integer} is both an ordered set +and an integral domain. In fact, {\tt Integer} does not +explicitly export constants $0$ and $1$ and operations +\spadopFrom{+}{Ring}, \spadopFrom{-}{Ring} and \spadopFrom{*}{Ring} at +all: it inherits them all from $Ring$! Since +{\tt IntegralDomain} is a descendant of $Ring$, +{\tt Integer} is therefore also a ring. + +Assertions can be conditional. For example, {\tt Complex(R)} +defines its exports by: + +\begin{verbatim} +Ring with ... if R has Field then Field ... +\end{verbatim} +Thus {\tt Complex(Float)} is a field but {\tt Complex(Integer)} +is not since {\tt Integer} is not a field. + +You may wonder: ``Why not simply let the set of operations determine +whether a domain belongs to a given category?''. Axiom allows +operation names (for example, {\bf norm}) to have very different +meanings in different contexts. The meaning of an operation in +Axiom is determined by context. By associating operations with +categories, operation names can be reused whenever appropriate or +convenient to do so. As a simple example, the operation {\tt <} +might be used to denote lexicographic-comparison in an algorithm. +However, it is wrong to use the same {\tt <} with this definition +of absolute-value: $$abs(x) == if\ x < 0\ then -x\ else\ x$$ Such a +definition for {\tt abs} in Axiom is protected by context: +argument $x$ is required to be a member of a domain of category +{\tt OrderedSet}. + +\subsection{Packages Are Clusters of Polymorphic Operations} + +In Axiom, facilities for symbolic integration, solution of +equations, and the like are placed in ``packages''. A {\it package} +\index{package} is a special kind of domain: one whose exported +operations depend solely on the parameters of the constructor and/or +explicit domains. Packages, unlike Domains, do not specify the +representation. + +If you want to use Axiom, for example, to define some algorithms +for solving equations of polynomials over an arbitrary field $F$, +you can do so with a package of the form: + +\begin{verbatim} +MySolve(F: Field): Exports == Implementation +\end{verbatim} +where {\tt Exports} specifies the {\bf solve} operations +you wish to export from the domain and the {\tt Implementation} +defines functions for implementing your algorithms. Once Axiom has +compiled your package, your algorithms can then be used for any {\tt F}: +floating-point numbers, rational numbers, complex rational functions, +and power series, to name a few. + +\subsection{The Interpreter Builds Domains Dynamically} + +The Axiom interpreter reads user input then builds whatever types +it needs to perform the indicated computations. +For example, to create the matrix +$$M = \pmatrix{x^2+1&0\cr0&x / 2\cr}$$ +using the command: + +\spadcommand{M = [ [x**2+1,0],[0,x / 2] ]::Matrix(POLY(FRAC(INT)))} +$$ +M={\left[ +\begin{array}{cc} +x^2+1 & 0 \\ +0 & x/2 +\end{array} +\right]} +$$ +\returnType{Type: Matrix Polynomial Fraction Integer} +the interpreter first loads the modules {\tt Matrix}, +{\tt Polynomial}, {\tt Fraction}, and {\tt Integer} +from the library, then builds the {\it domain tower} ``matrices of +polynomials of rational numbers (i.e. fractions of integers)''. + +You can watch the loading process by first typing + +\spadcommand{)set message autoload on} +In addition to the named +domains above many additional domains and categories are loaded. +Most systems are preloaded with such common types. For efficiency +reasons the most common domains are preloaded but most (there are +more than 1100 domains, categories, and packages) are not. Once these +domains are loaded they are immediately available to the interpreter. + +Once a domain tower is built, it contains all the operations specific +to the type. Computation proceeds by calling operations that exist in +the tower. For example, suppose that the user asks to square the +above matrix. To do this, the function \spadopFrom{*}{Matrix} from +{\tt Matrix} is passed the matrix $M$ to compute $M * M$. +The function is also passed an environment containing $R$ +that, in this case, is {\tt Polynomial (Fraction (Integer))}. +This results in the successive calling of the \spadopFrom{*}{Fraction} +operations from {\tt Polynomial}, then from {\tt Fraction}, +and then finally from {\tt Integer}. + +Categories play a policing role in the building of domains. Because +the argument of {\tt Matrix} is required to be a {\tt Ring}, +Axiom will not build nonsensical types such as ``matrices of +input files''. + +\subsection{Axiom Code is Compiled} + +Axiom programs are statically compiled to machine code, then +placed into library modules. Categories provide an important role in +obtaining efficient object code by enabling: +\begin{itemize} +\item static type-checking at compile time; +\item fast linkage to operations in domain-valued parameters; +\item optimization techniques to be used for partially specified types +(operations for ``vectors of $R$'', for instance, can be open-coded even +though {\tt R} is unknown). +\end{itemize} + +\subsection{Axiom is Extensible} + +Users and system implementers alike use the Axiom language to +add facilities to the Axiom library. The entire Axiom +library is in fact written in the Axiom source code and +available for user modification and/or extension. + +Axiom's use of abstract datatypes clearly separates the exports +of a domain (what operations are defined) from its implementation (how +the objects are represented and operations are defined). Users of a +domain can thus only create and manipulate objects through these +exported operations. This allows implementers to ``remove and +replace'' parts of the library safely by newly upgraded (and, we hope, +correct) implementations without consequence to its users. + +Categories protect names by context, making the same names available +for use in other contexts. Categories also provide for code-economy. +Algorithms can be parameterized categorically to characterize their +correct and most general context. Once compiled, the same machine +code is applicable in all such contexts. + +Finally, Axiom provides an automatic, guaranteed interaction +between new and old code. For example: +\begin{itemize} +\item if you write a new algorithm that requires a parameter to be a +field, then your algorithm will work automatically with every field +defined in the system; past, present, or future. +\item if you introduce a new domain constructor that produces a field, +then the objects of that domain can be used as parameters to any algorithm +using field objects defined in the system; past, present, or future. +\end{itemize} + +These are the key ideas. For further information, we particularly +recommend your reading chapters 11, 12, and 13, where these ideas are +explained in greater detail. + +\section{Using Axiom as a Pocket Calculator} +At the simplest level Axiom can be used as a pocket calculator +where expressions involving numbers and operators are entered +directly in infix notation. In this sense the more advanced +features of the calculator can be regarded as operators (e.g +{\bf sin}, {\bf cos}, etc). + +\subsection{Basic Arithmetic} +An example of this might be to calculate the cosine of 2.45 (in radians). +To do this one would type: + +\spadcommand{cos 2.45} +$$ +-{0.7702312540 473073417} +$$ +\returnType{Type: Float} + +Before proceeding any further it would be best to explain the previous +three lines. Firstly the text ``(1) {\tt ->} '' is part of the prompt that the +Axiom system provides when in interactive mode. The full prompt has other +text preceding this but it is not relevant here. The number in parenthesis +is the step number of the input which may be used to refer to the +{\sl results} of previous calculations. The step number appears at the start +of the second line to tell you which step the result belongs to. Since the +interpreter probably loaded numberous libraries to calculate the result given +above and listed each one in the prcess, there could easily be several pages +of text between your input and the answer. + +The last line contains the type of the result. The type {\tt Float} is used +to represent real numbers of arbitrary size and precision (where the user is +able to define how big arbitrary is -- the default is 20 digits but can be +as large as your computer system can handle). The type of the result can help +track down mistakes in your input if you don't get the answer you expected. + +Other arithmetic operations such as addition, subtraction, and multiplication +behave as expected: + +\spadcommand{6.93 * 4.1328} +$$ +28.640304 +$$ +\returnType{Type: Float} + +\spadcommand{6.93 / 4.1328} +$$ +1.6768292682 926829268 +$$ +\returnType{Type: Float} + +but integer division isn't quite so obvious. For example, if one types: + +\spadcommand{4/6} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Integer} + +a fractional result is obtained. The function used to display fractions +attempts to produce the most readable answer. In the example: + +\spadcommand{4/2} +$$ +2 +$$ +\returnType{Type: Fraction Integer} + +the result is stored as the fraction 2/1 but is displayed as the integer 2. +This fraction could be converted to type {\tt Integer} with no loss of +information but Axiom will not do so automatically. + +\subsection{Type Conversion} +To obtain the floating point value of a fraction one must convert +({\bf conversions} are applied by the user and +{\bf coercions} are applied automatically by the interpreter) the result +to type {\tt Float} using the ``::'' operator as follows: + +\spadcommand{(4.6)::Float} +$$ +4.6 +$$ +\returnType{Type: Float} + +Although Axiom can convert this back to a fraction it might not be the +same fraction you started with as due to rounding errors. For example, the +following conversion appears to be without error but others might not: + +\spadcommand{\%::Fraction Integer} +$$ +{23} \over 5 +$$ +\returnType{Type: Fraction Integer} + +where ``\%'' represents the previous {\it result} (not the calculation). + +Although Axiom has the ability to work with floating-point numbers to +a very high precision it must be remembered that calculations with these +numbers are {\bf not} exact. Since Axiom is a computer algebra package and +not a numerical solutions package this should not create too many problems. +The idea is that the user should use Axiom to do all the necessary symbolic +manipulation and only at the end should actual numerical results be extracted. + +If you bear in mind that Axiom appears to store expressions just as you have +typed them and does not perform any evalutation of them unless forced to then +programming in the system will be much easier. It means that anything you +ask Axiom to do (within reason) will be carried out with complete accuracy. + +In the previous examples the ``::'' operator was used to convert values from +one type to another. This type conversion is not possible for all values. +For instance, it is not possible to convert the number 3.4 to an integer +type since it can't be represented as an integer. The number 4.0 can be +converted to an integer type since it has no fractional part. + +Conversion from floating point values to integers is performed using the +functions {\bf round} and {\bf truncate}. The first of these rounds a +floating point number to the nearest integer while the other truncates +(i.e. removes the fractional part). Both functions return the result as a +{\bf floating point} number. To extract the fractional part of a floating +point number use the function {\bf fractionPart} but note that the sign +of the result depends on the sign of the argument. Axiom obtains the +fractional partof $x$ using $x - truncate(x)$: + +\spadcommand{round(3.77623)} +$$ +4.0 +$$ +\returnType{Type: Float} + +\spadcommand{round(-3.77623)} +$$ +-{4.0} +$$ +\returnType{Type: Float} + +\spadcommand{truncate(9.235)} +$$ +9.0 +$$ +\returnType{Type: Float} + +\spadcommand{truncate(-9.654)} +$$ +-{9.0} +$$ +\returnType{Type: Float} + +\spadcommand{fractionPart(-3.77623)} +$$ +-{0.77623} +$$ +\returnType{Type: Float} + +\subsection{Useful Functions} +To obtain the absolute value of a number the {\bf abs} function can be used. +This takes a single argument which is usually an integer or a floating point +value but doesn't necessarily have to be. The sign of a value can be obtained +via the {\bf sign} function which rturns $-1$, $0$, or $1$ depending on the +sign of the argument. + +\spadcommand{abs(4)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{abs(-3)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{abs(-34254.12314)} +$$ +34254.12314 +$$ +\returnType{Type: Float} + +\spadcommand{sign(-49543.2345346)} +$$ +-1 +$$ +\returnType{Type: Integer} + +\spadcommand{sign(0)} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\spadcommand{sign(234235.42354)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Tests on values can be done using various functions which are generally more +efficient than using relational operators such as $=$ particularly if the +value is a matrix. Examples of some of these functions are: + +\spadcommand{positive?(-234)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{negative?(-234)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{zero?(42)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{one?(1)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{odd?(23)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{odd?(9.435)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{even?(-42)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{prime?(37)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{prime?(-37)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Some other functions that are quite useful for manipulating numerical values +are: + +\begin{verbatim} +sin(x) Sine of x +cos(x) Cosine of x +tan(x) Tangent of x +asin(x) Arcsin of x +acos(x) Arccos of x +atan(x) Arctangent of x +gcd(x,y) Greatest common divisor of x and y +lcm(x,y) Lowest common multiple of x and y +max(x,y) Maximum of x and y +min(x,y) Minimum of x and y +factorial(x) Factorial of x +factor(x) Prime factors of x +divide(x,y) Quotient and remainder of x/y +\end{verbatim} + +Some simple infix and prefix operators: +\begin{verbatim} ++ Addition - Subtraction +- Numerical Negation ~ Logical Negation +/\ Conjunction (AND) \/ Disjunction (OR) +and Logical AND (/\) or Logical OR (\/) +not Logical Negation ** Exponentiation +* Multiplication / Division +quo Quotient rem Remainder +< less than > greater than +<= less than or equal >= greater than or equal +\end{verbatim} + +Some useful Axiom macros: +\begin{verbatim} +%i The square root of -1 +%e The base of the natural logarithm +%pi Pi +%infinity Infinity +%plusInfinity Positive Infinity +%minusInfinity Negative Infinity +\end{verbatim} + +\section{Using Axiom as a Symbolic Calculator} +In the previous section all the examples involved numbers and simple +functions. Also none of the expressions entered were assigned to anything. +In this section we will move on to simple algebra (i.e. expressions involving +symbols and other features available on more sophisticated calculators). + +\subsection{Expressions Involving Symbols} +Expressions involving symbols are entered just as they are written down, +for example: + +\spadcommand{xSquared := x**2} +$$ +x \sp 2 +$$ +\returnType{Type: Polynomial Integer} + +where the assignment operator ``:='' represents immediate assignment. Later +it will be seen that this form of assignment is not always desirable and +the use of the delayed assignment operator ``=='' will be introduced. The +type of the result is {\tt Polynomial Integer} which is used to represent +polynomials with integer coefficients. Some other examples along similar +lines are: + +\spadcommand{xDummy := 3.21*x**2} +$$ +{3.21} \ {x \sp 2} +$$ +\returnType{Type: Polynomial Float} + +\spadcommand{xDummy := x**2.5} +$$ +{x \sp 2} \ {\sqrt {x}} +$$ +\returnType{Type: Expression Float} + +\spadcommand{xDummy := x**3.3} +$$ +{x \sp 3} \ {{\root {{10}} \of {x}} \sp 3} +$$ +\returnType{Type: Expression Float} + +\spadcommand{xyDummy := x**2 - y**2} +$$ +-{y \sp 2}+{x \sp 2} +$$ +\returnType{Type: Polynomial Integer} + +Given that we can define expressions involving symbols, how do we actually +compute the result when the symbols are assigned values? The answer is to +use the {\bf eval} function which takes an expression as its first argument +followed by a list of assignments. For example, to evaluate the expressions +{\bf XDummy} and {xyDummy} resulting from their respective assignments above +we type: + +\spadcommand{eval(xDummy,x=3)} +$$ +37.5405075985 29552193 +$$ +\returnType{Type: Expression Float} + +\spadcommand{eval(xyDummy, [x=3, y=2.1])} +$$ +4.59 +$$ +\returnType{Type: Polynomial Float} + +\subsection{Complex Numbers} +For many scientific calculations real numbers aren't sufficient and support +for complex numbers is also required. Complex numbers are handled in an +intuitive manner and Axiom, which uses the {\bf \%i} macro to represent +the square root of $-1$. Thus expressions involving complex numbers are +entered just like other expressions. + +\spadcommand{(2/3 + \%i)**3} +$$ +-{{46} \over {27}}+{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +The real and imaginary parts of a complex number can be extracted using +the {\bf real} and {\bf imag} functions and the complex conjugate of a +number can be obtained using {\bf conjugate}: + +\spadcommand{real(3 + 2*\%i)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{imag(3+ 2*\%i)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{conjugate(3 + 2*\%i)} +$$ +3 -{2 \ i} +$$ +\returnType{Type: Complex Integer} + +The function {\bf factor} can also be applied to complex numbers but the +results aren't quite so obvious as for factoring integer: + +\spadcommand{144 + 24*\%i} +$$ +{144}+{{24} \ i} +$$ +\returnType{Type: Complex Integer} + +\subsection{Number Representations} +By default all numerical results are displayed in decimal with real numbers +shown to 20 significant figures. If the integer part of a number is longer +than 20 digits then nothing after the decimal point is shown and the integer +part is given in full. To alter the number of digits shown the function +{\bf digits} can be called. The result returned by this function is the +previous setting. For example, to find the value of $\pi$ to 40 digits +we type: + +\spadcommand{digits(40)} +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{\%pi::Float} +$$ +3.1415926535\ 8979323846\ 2643383279\ 502884197 +$$ +\returnType{Type: Float} + +As can be seen in the example above, there is a gap after every ten digits. +This can be changed using the {\bf outputSpacing} function where the argument +is the number of digits to be displayed before a space is inserted. If no +spaces are desired then use the value $0$. Two other functions controlling +the appearance of real numbers are {\bf outputFloating} and {\bf outputFixed}. +The former causes Axiom to display floating-point values in exponent notation +and the latter causes it to use fixed-point notation. For example: + +\spadcommand{outputFloating(); \%} +$$ +0.3141592653 5897932384 6264338327 9502884197 E 1 +$$ +\returnType{Type: Float} + +\spadcommand{outputFloating(3); 0.00345} +$$ +0.345 E -2 +$$ +\returnType{Type: Float} + +\spadcommand{outputFixed(); \%} +$$ +0.00345 +$$ +\returnType{Type: Float} + +\spadcommand{outputFixed(3); \%} +$$ +0.003 +$$ +\returnType{Type: Float} + +\spadcommand{outputGeneral(); \%} +$$ +0.00345 +$$ +\returnType{Type: Float} + +Note that the semicolon ``;'' in the examples above allows several +expressions to be entered on one line. The result of the last expression +is displayed. remember also that the percent symbol ``\%'' is used to +represent the result of a previous calculation. + +To display rational numbers in a base other than 10 the function {\bf radix} +is used. The first argument of this function is the expression to be +displayed and the second is the base to be used. + +\spadcommand{radix(10**10,32)} +$$ +{\rm 9A0NP00 } +$$ +\returnType{Type: RadixExpansion 32} + +\spadcommand{radix(3/21,5)} +$$ +0.{\overline {032412}} +$$ +\returnType{Type: RadixExpansion 5} + +Rational numbers can be represented as a repeated decimal expansion using +the {\bf decimal} function or as a continued fraction using +{\bf continuedFraction}. Any attempt to call these functions with irrational +values will fail. + +\spadcommand{decimal(22/7)} +$$ +3.{\overline {142857}} +$$ +\returnType{Type: DecimalExpansion} + +\spadcommand{continuedFraction(6543/210)} +$$ +{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} +$$ +\returnType{Type: ContinuedFraction Integer} + +Finally, partial fractions in compact and expanded form are available via the +functions {\bf partialFraction} and {\bf padicFraction} respectively. The +former takes two arguments, the first being the numerator of the fraction +and the second being the denominator. The latter function takes a fraction +and expands it further while the function {\bf compactFraction} does the +reverse: + +\spadcommand{partialFraction(234,40)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{padicFraction(\%)} +$$ +6 -{1 \over 2} -{1 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{compactFraction(\%)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{padicFraction(234/40)} +$$ +{117} \over {20} +$$ +\returnType{Type: PartialFraction Fraction Integer} + +To extract parts of a partial fraction the function {\bf nthFractionalTerm} +is available and returns a partial fraction of one term. To decompose this +further the numerator can be obtained using {\bf firstNumer} and the +denominator with {\bf firstDenom}. The whole part of a partial fraction can +be retrieved using {\bf wholePart} and the number of fractional parts can +be found using the function {\bf numberOf FractionalTerms}: + +\spadcommand{t := partialFraction(234,40)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{wholePart(t)} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{numberOfFractionalTerms(t)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{p := nthFractionalTerm(t,1)} +$$ +-{3 \over {2 \sp 2}} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{firstNumer(p)} +$$ +-3 +$$ +\returnType{Type: Integer} + +\spadcommand{firstDenom(p)} +$$ +2 \sp 2 +$$ +\returnType{Type: Factored Integer} + +\subsection{Modular Arithmetic} +By using the type constructor {\tt PrimeField} it is possible to do +arithmetic modulo some prime number. For example, arithmetic module $7$ +can be performed as follows: + +\spadcommand{x : PrimeField 7 := 5} +$$ +5 +$$ +\returnType{Type: PrimeField 7} + +\spadcommand{x**5 + 6} +$$ +2 +$$ +\returnType{Type: PrimeField 7} + +\spadcommand{1/x} +$$ +3 +$$ +\returnType{Type: PrimeField 7} + +The first example should be read as: +\begin{center} +{\tt Let $x$ be of type PrimeField(7) and assign to it the value $5$} +\end{center} + +Note that it is only possible to invert non-zero values if the arithmetic +is performed modulo a prime number. Thus arithmetic modulo a non-prime +integer is possible but the reciprocal operation is undefined and will +generate an error. Attempting to use the {\tt PrimeField} type constructor +with a non-prime argument will generate an error. An example of non-prime +modulo arithmetic is: + +\spadcommand{y : IntegerMod 8 := 11} +$$ +3 +$$ +\returnType{Type: IntegerMod 8} + +\spadcommand{y*4 + 27} +$$ +7 +$$ +\returnType{Type: IntegerMod 8} + +Note that polynomials can be constructed in a similar way: + +\spadcommand{(3*a**4 + 27*a - 36)::Polynomial PrimeField 7} +$$ +{3 \ {a \sp 4}}+{6 \ a}+6 +$$ +\returnType{Type: Polynomial PrimeField 7} + +\section{General Points about Axiom} +\subsection{Computation Without Output} +It is sometimes desirable to enter an expression and prevent Axiom from +displaying the result. To do this the expression should be terminated with +a semicolon ``;''. In a previous section it was mentioned that a set of +expressions separated by semicolons would be evaluated and the result +of the last one displayed. Thus if a single expression is followed by a +semicolon no output will be produced (except for its type): + +\spadcommand{2 + 4*5;} +\returnType{Type: PositiveInteger} + +\subsection{Accessing Earlier Results} +The ``\%'' macro represents the result of the previous computation. The +``\%\%'' macro is available which takes a single integer argument. If the +argument is positive then it refers to the step number of the calculation +where the numbering begins from one and can be seen at the end of each +prompt (the number in parentheses). If the argument is negative then it +refers to previous results counting backwards from the last result. That is, +``\%\%(-1)'' is the same as ``\%''. The value of ``\%\%(0)'' is not defined and +will generate an error if requested. + +\subsection{Splitting Expressions Over Several Lines} +Although Axiom will quite happily accept expressions that are longer than +the width of the screen (just keep typing without pressing the {\bf Return} +key) it is often preferable to split the expression being entered at a point +where it would result in more readable input. To do this the underscore +``\_'' symbol is placed before the break point and then the {\bf Return} +key is pressed. The rest of the expression is typed on the next line, +can be preceeded by any number of whitespace chars, for example: +\begin{verbatim} +2_ ++_ +3 +\end{verbatim} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +The underscore symbol is an escape character and its presence alters the +meaning of the characters that follow it. As mentions above whitespace +following an underscore is ignored (the {\bf Return} key generates a +whitespace character). Any other character following an underscore loses +whatever special meaning it may have had. Thus one can create the +identifier ``a+b'' by typing ``a\_+b'' although this might lead to confusions. +Also note the result of the following example: + +\spadcommand{ThisIsAVeryLong\_\\ +VariableName} +$$ +ThisIsAVeryLongVariableName +$$ +\returnType{Type: Variable ThisIsAVeryLongVariableName} + +\subsection{Comments and Descriptions} +Comments and descriptions are really only of use in files of Axiom code but +can be used when the output of an interactive session is being spooled to +a file (via the system command {\bf )spool}). A comment begins with two +dashes ``- -'' and continues until the end of the line. Multi-line +comments are only possible if each individual line begins with two dashes. + +Descriptions are the same as comments except that the Axiom compiler will +include them in the object files produced and make them availabe to the +end user for documentation purposes. + +A description is placed {\bf before} a calculation begins with three +``+++'' signs and a description placed after a calculation begins with +two plus symbols ``++''. The so-called ``plus plus'' comments are used +within the algebra files and are processed by the compiler to add +to the documentation. The so-called ``minus minus'' comments are ignored +everywhere. + +\subsection{Control of Result Types} +In earlier sections the type of an expression was converted to another +via the ``::'' operator. However, this is not the only method for +converting between types and two other operators need to be introduced +and explained. + +The first operator is ``\$'' and is used to specify the package to be +used to calculate the result. Thus: + +\spadcommand{(2/3)\$Float} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +tells Axiom to use the ``/'' operator from the {\tt Float} package to +evaluate the expression $2/3$. This does not necessarily mean that the +result will be of the same type as the domain from which the operator +was taken. In the following example the {\bf sign} operator is taken +from the {\tt Float} package but the result is of type {\tt Integer}. + +\spadcommand{sign(2.3)\$Float} +$$ +1 +$$ +\returnType{Type: Integer} + +The other operator is ``@'' which is used to tell Axiom what the desired +type of the result of the calculation is. In most situations all three +operators yield the same results but the example below should help +distinguish them. + +\spadcommand{(2 + 3)::String} +$$ +\mbox{\tt "5"} +$$ +\returnType{Type: String} + +\spadcommand{(2 + 3)@String} +\begin{verbatim} +An expression involving @ String actually evaluated to one of + type PositiveInteger . Perhaps you should use :: String . +\end{verbatim} + +\spadcommand{(2 + 3)\$String} +\begin{verbatim} + The function + is not implemented in String . +\end{verbatim} + +If an expression {\sl X} is converted using one of the three operators to +type {\sl T} the interpretations are: + +{\bf ::} means explicitly convert {\sl X} to type {\sl T} if possible. + +{\bf \$} means use the available operators for type {\sl T} to compute {\sl X}. + +{\bf @} means choose operators to compute {\sl X} so that the result is of +type {\sl T}. + +\section{Data Structures in Axiom} +This chapter is an overview of {\sl some} of the data structures provided +by Axiom. +\subsection{Lists} +The Axiom {\tt List} type constructor is used to create homogenous lists of +finite size. The notation for lists and the names of the functions that +operate over them are similar to those found in functional languages such +as ML. + +Lists can be created by placing a comma separated list of values inside +square brackets or if a list with just one element is desired then the +function {\bf list} is available: + +\spadcommand{[4]} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{list(4)} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{[1,2,3,5,7,11]} +$$ +\left[ +1, 2, 3, 5, 7, {11} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The function {\bf append} takes two lists as arguments and returns the list +consisting of the second argument appended to the first. A single element +can be added to the front of a list using {\bf cons}: + +\spadcommand{append([1,2,3,5],[7,11])} +$$ +\left[ +1, 2, 3, 5, 7, {11} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{cons(23,[65,42,19])} +$$ +\left[ +{23}, {65}, {42}, {19} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Lists are accessed sequentially so if Axiom is asked for the value of the +twentieth element in the list it will move from the start of the list over +nineteen elements before it reaches the desired element. Each element of a +list is stored as a node consisting of the value of the element and a pointer +to the rest of the list. As a result the two main operations on a list are +called {\bf first} and {\bf rest}. Both of these functions take a second +optional argument which specifies the length of the first part of the list: + +\spadcommand{first([1,5,6,2,3])} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{first([1,5,6,2,3],2)} +$$ +\left[ +1, 5 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{rest([1,5,6,2,3])} +$$ +\left[ +5, 6, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{rest([1,5,6,2,3],2)} +$$ +\left[ +6, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Other functions are {\bf empty?} which tests to see if a list contains no +elements, {\bf member?} which tests to see if the first argument is a member +of the second, {\bf reverse} which reverses the order of the list, {\bf sort} +which sorts a list, and {\bf removeDuplicates} which removes any duplicates. +The length of a list can be obtained using the ``\#'' operator. + +\spadcommand{empty?([7,2,-1,2])} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{member?(-1,[7,2,-1,2])} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{reverse([7,2,-1,2])} +$$ +\left[ +2, -1, 2, 7 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{sort([7,2,-1,2])} +$$ +\left[ +-1, 2, 2, 7 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{removeDuplicates([1,5,3,5,1,1,2])} +$$ +\left[ +1, 5, 3, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{\#[7,2,-1,2]} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Lists in Axiom are mutable and so their contents (the elements and the links) +can be modified in place. Functions that operator over lists in this way have +names ending in the symbol ``!''. For example, {\bf concat!} takes two lists +as arguments and appends the second argument to the first (except when the +first argument is an empty list) and {\bf setrest!} changes the link +emanating from the first argument to point to the second argument: + +\spadcommand{u := [9,2,4,7]} +$$ +\left[ +9, 2, 4, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{concat!(u,[1,5,42]); u} +$$ +\left[ +9, 2, 4, 7, 1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{endOfu := rest(u,4)} +$$ +\left[ +1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{partOfu := rest(u,2)} +$$ +\left[ +4, 7, 1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{setrest!(endOfu,partOfu); u} +$$ +\left[ +9, 2, {\overline {4, 7, 1}} +\right] +$$ +\returnType{Type: List PositiveInteger} + +From this it can be seen that the lists returned by {\bf first} and {\bf rest} +are pointers to the original list and {\sl not} a copy. Thus great care must +be taken when dealing with lists in Axiom. + +Although the {\sl n}th element of the list {\sl l} can be obtained by +applying the {\bf first} function to $n-1$ applications of {\bf rest} +to {\sl l}, Axiom provides a more useful access method in the form of +the ``.'' operator: + +\spadcommand{u.3} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.5} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.6} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{first rest rest u -- Same as u.3} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.first} +$$ +9 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u(3)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +The operation {\sl u.i} is referred to as {\sl indexing into u} or +{\sl elting into u}. The latter term comes from the {\bf elt} function +which is used to extract elements (the first element of the list is at +index $1$). + +\spadcommand{elt(u,4)} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +If a list has no cycles then any attempt to access an element beyond the +end of the list will generate an error. However, in the example above there +was a cycle starting at the third element so the access to the sixth +element wrapped around to give the third element. Since lists are mutable it +is possible to modify elements directly: + +\spadcommand{u.3 := 42; u} +$$ +\left[ +9, 2, {\overline {{42}, 7, 1}} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Other list operations are: +\spadcommand{L := [9,3,4,7]; \#L} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{last(L)} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{L.last} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{L.(\#L - 1)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Note that using the ``\#'' operator on a list with cycles causes Axiom to +enter an infinite loop. + +Note that any operation on a list {\sl L} that returns a list ${\sl L}L^{'}$ +will, in general, be such that any changes to ${\sl L}L^{'}$ will have the +side-effect of altering {\sl L}. For example: + +\spadcommand{m := rest(L,2)} +$$ +\left[ +4, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{m.1 := 20; L} +$$ +\left[ +9, 3, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n := L} +$$ +\left[ +9, 3, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n.2 := 99; L} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Thus the only safe way of copying lists is to copy each element from one to +another and not use the assignment operator: + +\spadcommand{p := [i for i in n] -- Same as `p := copy(n)'} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{p.2 := 5; p} +$$ +\left[ +9, 5, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +In the previous example a new way of constructing lists was given. This is +a powerful method which gives the reader more information about the contents +of the list than before and which is extremely flexible. The example + +\spadcommand{[i for i in 1..10]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +should be read as + +\begin{center} +``Using the expression {\sl i}, generate each element of the list by +iterating the symbol {\sl i} over the range of integers [1,10]'' +\end{center} + +To generate the list of the squares of the first ten elements we just use: + +\spadcommand{[i**2 for i in 1..10]} +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: List PositiveInteger} + +For more complex lists we can apply a condition to the elements that are to +be placed into the list to obtain a list of even numbers between 0 and 11: + +\spadcommand{[i for i in 1..10 | even?(i)]} +$$ +\left[ +2, 4, 6, 8, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +This example should be read as: +\begin{center} +``Using the expression {\sl i}, generate each element of the list +by iterating the symbol {\sl i} over the range of integers [1,10] such that +{\sl i} is even'' +\end{center} + +The following achieves the same result: + +\spadcommand{[i for i in 2..10 by 2]} +$$ +\left[ +2, 4, 6, 8, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\subsection{Segmented Lists} +A segmented list is one in which some of the elements are ranges of values. +The {\bf expand} function converts lists of this type into ordinary lists: + +\spadcommand{[1..10]} +$$ +\left[ +{1..{10}} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +\spadcommand{[1..3,5,6,8..10]} +$$ +\left[ +{1..3}, {5..5}, {6..6}, {8..{10}} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +\spadcommand{expand(\%)} +$$ +\left[ +1, 2, 3, 5, 6, 8, 9, {10} +\right] +$$ +\returnType{Type: List Integer} + +If the upper bound of a segment is omitted then a different type of +segmented list is obtained and expanding it will produce a stream (which +will be considered in the next section): + +\spadcommand{[1..]} +$$ +\left[ +{1..} +\right] +$$ +\returnType{Type: List UniversalSegment PositiveInteger} + +\spadcommand{expand(\%)} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\subsection{Streams} +Streams are infinite lists which have the ability to calculate the next +element should it be required. For example, a stream of positive integers +and a list of prime numbers can be generated by: + +\spadcommand{[i for i in 1..]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +\spadcommand{[i for i in 1.. | prime?(i)]} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +In each case the first few elements of the stream are calculated for display +purposes but the rest of the stream remains unevaluated. The value of items +in a stream are only calculated when they are needed which gives rise to +their alternative name of ``lazy lists''. + +Another method of creating streams is to use the {\bf generate(f,a)} function. +This applies its first argument repeatedly onto its second to produce the +stream $[a,f(a),f(f(a)),f(f(f(a)))\ldots]$. Given that the function +{\bf nextPrime} returns the lowest prime number greater than its argument we +can generate a stream of primes as follows: +\spadcommand{generate(nextPrime,2)\$Stream Integer} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +As a longer example a stream of Fibonacci numbers will be computed. The +Fibonacci numbers start at $1$ and each following number is the addition +of the two numbers that precede it so the Fibonacci sequence is: +$$1,1,2,3,5,8,\ldots$$. + +Since the generation of any Fibonacci number only relies on knowing the +previous two numbers we can look at the series through a window of two +elements. To create the series the window is placed at the start over +the values $[1,1]$ and their sum obtained. The window is now shifted to +the right by one position and the sum placed into the empty slot of the +window; the process is then repeated. To implement this we require a +function that takes a list of two elements (the current view of the window), +adds them, and outputs the new window. The result is the function +$[a,b]$~{\tt ->}~$[b,a+b]$: +\spadcommand{win : List Integer -> List Integer} +\returnType{Type: Void} + +\spadcommand{win(x) == [x.2, x.1 + x.2]} +\returnType{Type: Void} + +\spadcommand{win([1,1])} +$$ +\left[ +1, 2 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{win(\%)} +$$ +\left[ +2, 3 +\right] +$$ +\returnType{Type: List Integer} + +Thus it can be seen that by repeatedly applying {\bf win} to the {\sl results} +of the previous invocation each element of the series is obtained. Clearly +{\bf win} is an ideal function to construct streams using the {\bf generate} +function: +\spadcommand{fibs := [generate(win,[1,1])]} +$$ +\left[ +{\left[ 1, 1 +\right]}, + {\left[ 1, 2 +\right]}, + {\left[ 2, 3 +\right]}, + {\left[ 3, 5 +\right]}, + {\left[ 5, 8 +\right]}, + {\left[ 8, {13} +\right]}, + {\left[ {13}, {21} +\right]}, + {\left[ {21}, {34} +\right]}, + {\left[ {34}, {55} +\right]}, + {\left[ {55}, {89} +\right]}, + \ldots +\right] +$$ +\returnType{Type: Stream List Integer} + +This isn't quite what is wanted -- we need to extract the first element of +each list and place that in our series: +\spadcommand{fibs := [i.1 for i in [generate(win,[1,1])] ]} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Obtaining the 200th Fibonacci number is trivial: +\spadcommand{fibs.200} +$$ +280571172992510140037611932413038677189525 +$$ +\returnType{Type: PositiveInteger} + +One other function of interest is {\bf complete} which expands a finite +stream derived from an infinite one (and thus was still stored as an +infinite stream) to form a finite stream. + +\subsection{Arrays, Vectors, Strings, and Bits} +The simplest array data structure is the {\sl one-dimensional array} which +can be obtained by applying the {\bf oneDimensionalArray} function to a list: +\spadcommand{oneDimensionalArray([7,2,5,4,1,9])} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +One-dimensional array are homogenous (all elements must have the same type) +and mutable (elements can be changed) like lists but unlike lists they are +constant in size and have uniform access times (it is just as quick to read +the last element of a one-dimensional array as it is to read the first; this +is not true for lists). + +Since these arrays are mutable all the warnings that apply to lists apply to +arrays. That is, it is possible to modify an element in a copy of an array +and change the original: +\spadcommand{x := oneDimensionalArray([7,2,5,4,1,9])} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{y := x} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{y.3 := 20 ; x} +$$ +\left[ +7, 2, {20}, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +Note that because these arrays are of fixed size the {\bf concat!} function +cannot be applied to them without generating an error. If arrays of this +type are required use the {\bf FlexibleArray} constructor. + +One-dimensional arrays can be created using {\bf new} which specifies the size +of the array and the initial value for each of the elements. Other operations +that can be applied to one-dimensional arrays are {\bf map!} which applies +a mapping onto each element, {\bf swap!} which swaps two elements and +{\bf copyInto!(a,b,c)} which copies the array {\sl b} onto {\sl a} starting at +position {\sl c}. +\spadcommand{a : ARRAY1 PositiveInteger := new(10,3)} +$$ +\left[ +3, 3, 3, 3, 3, 3, 3, 3, 3, 3 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +(note that {\tt ARRAY1} is an abbreviation for the type +{\tt OneDimensionalArray}.) Other types based on one-dimensional arrays are +{\tt Vector}, {\tt String}, and {tt Bits}. + +\spadcommand{map!(i +-> i+1,a); a} +$$ +\left[ +4, 4, 4, 4, 4, 4, 4, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{b := oneDimensionalArray([2,3,4,5,6])} +$$ +\left[ +2, 3, 4, 5, 6 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{swap!(b,2,3); b} +$$ +\left[ +2, 4, 3, 5, 6 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{copyInto!(a,b,3)} +$$ +\left[ +4, 4, 2, 4, 3, 5, 6, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{a} +$$ +\left[ +4, 4, 2, 4, 3, 5, 6, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{vector([1/2,1/3,1/14])} +$$ +\left[ +{1 \over 2}, {1 \over 3}, {1 \over {14}} +\right] +$$ +\returnType{Type: Vector Fraction Integer} + +\spadcommand{"Hello, World"} +$$ +\mbox{\tt "Hello, World"} +$$ +\returnType{Type: String} + +\spadcommand{bits(8,true)} +$$ +\mbox{\tt "11111111"} +$$ +\returnType{Type: Bits} + +A vector is similar to a one-dimensional array except that if its +components belong to a ring then arithmetic operations are provided. + +\subsection{Flexible Arrays} +Flexible arrays are designed to provide the efficiency of one-dimensional +arrays while retaining the flexibility of lists. They are implemented by +allocating a fixed block of storage for the array. If the array needs to +be expanded then a larger block of storage is allocated and the contents +of the old block are copied into the new one. + +There are several operations that can be applied to this type, most of +which modify the array in place. As a result these functions all have +names ending in ``!''. The {\bf physicalLength} returns the actual length +of the array as stored in memory while the {\bf physicalLength!} allows this +value to be changed by the user. +\spadcommand{f : FARRAY INT := new(6,1)} +$$ +\left[ +1, 1, 1, 1, 1, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{f.1:=4; f.2:=3 ; f.3:=8 ; f.5:=2 ; f} +$$ +\left[ +4, 3, 8, 1, 2, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{insert!(42,f,3); f} +$$ +\left[ +4, 3, {42}, 8, 1, 2, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{insert!(28,f,8); f} +$$ +\left[ +4, 3, {42}, 8, 1, 2, 1, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{removeDuplicates!(f)} +$$ +\left[ +4, 3, {42}, 8, 1, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{delete!(f,5)} +$$ +\left[ +4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{g:=f(3..5)} +$$ +\left[ +{42}, 8, 2 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{g.2:=7; f} +$$ +\left[ +4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{insert!(g,f,1)} +$$ +\left[ +{42}, 7, 2, 4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{physicalLength(f)} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{physicalLength!(f,20)} +$$ +\left[ +{42}, 7, 2, 4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{merge!(sort!(f),sort!(g))} +$$ +\left[ +2, 2, 2, 3, 4, 7, 7, 8, {28}, {42}, {42}, +{42} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{shrinkable(false)\$FlexibleArray(Integer)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +There are several things to point out concerning these +examples. First, although flexible arrays are mutable, making copies +of these arrays creates separate entities. This can be seen by the +fact that the modification of element {\sl b.2} above did not alter +{\sl a}. Second, the {\bf merge!} function can take an extra argument +before the two arrays are merged. The argument is a comparison +function and defaults to ``{\tt <=}'' if omitted. Lastly, +{\bf shrinkable} tells the system whether or not to let flexible arrays +contract when elements are deleted from them. An explicit package +reference must be given as in the example above. + +\section{Functions, Choices, and Loops} +By now the reader should be able to construct simple one-line expressions +involving variables and different data structures. This section builds on +this knowledge and shows how to use iteration, make choices, and build +functions in Axiom. At the moment it is assumed that the reader has a rough +idea of how types are specified and constructed so that they can follow +the examples given. + +From this point on most examples will be taken from input files. + +\subsection{Reading Code from a File} +Input files contain code that will be fed to the command prompt. The +primary different between the command line and an input file is that +indentation matters. In an input file you can specify ``piles'' of code +by using indentation. + +The names of all input files in Axiom should end in ``.input'' otherwise +Axiom will refuse to read them. + +If an input file is named {\bf foo.input} you can feed the contents of +the file to the command prompt (as though you typed them) by writing: +{\bf )read foo.input}. + +It is good practice to start each input file with the {\bf )clear all} +command so that all functions and variables in the current environment +are erased. +\subsection{Blocks} +The Axiom constructs that provide looping, choices, and user-defined +functions all rely on the notion of blocks. A block is a sequence of +expressions which are evaluated in the order that they appear except +when it is modified by control expressions such as loops. To leave a +block prematurely use an expression of the form: +{\sl BoolExpr}~{\tt =>}~{\sl Expr} +where {\sl BoolExpr} is any Axiom expression that has type {\tt Boolean}. +The value and type of {\sl Expr} determines the value and type returned +by the block. + +If blocks are entered at the keyboard (as opposed to reading them from +a text file) then there is only one way of creating them. The syntax is: +$$( expression1 ; expression2; \ldots ; expressionN )$$ + +In an input file a block can be constructed as above or by placing all the +statements at the same indentation level. When indentation is used to +indicate program structure the block is called a {\sl pile}. As an example +of a simple block a list of three integers can be constructed using +parentheses: +\spadcommand{( a:=4; b:=1; c:=9; L:=[a,b,c])} +$$ +\left[ +4, 1, 9 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Doing the same thing using piles in an input file you could type: +\begin{verbatim} +L := + a:=4 + b:=1 + c:=9 + [a,b,c] +\end{verbatim} +$$ +\left[ +4, 1, 9 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Since blocks have a type and a value they can be used as arguments to +functions or as part of other expressions. It should be pointed out that +the following example is not recommended practice but helps to illustrate +the idea of blocks and their ability to return values: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} + +Note that indentation is {\bf extremely} important. If the example above +had the pile starting at ``a:='' moved left by two spaces so that the +``a'' was under the ``('' of the first line then the interpreter would +signal an error. Furthermore if the closing parenthesis ``)'' is moved +up to give +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c) +\end{verbatim} +\begin{verbatim} + Line 1: sqrt(4.0 + + ....A + Error A: Missing mate. + Line 2: a:=3.0 + Line 3: b:=1.0 + Line 4: c:=a + b + Line 5: c) + .........AB + Error A: (from A up to B) Ignored. + Error B: Improper syntax. + Error B: syntax error at top level + Error B: Possibly missing a ) + 5 error(s) parsing +\end{verbatim} +then the parser will generate errors. If the parenthesis is shifted right +by several spaces so that it is in line with the ``c'' thus: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +\begin{verbatim} + Line 1: sqrt(4.0 + + ....A + Error A: Missing mate. + Line 2: a:=3.0 + Line 3: b:=1.0 + Line 4: c:=a + b + Line 5: c + Line 6: ) + .........A + Error A: (from A up to A) Ignored. + Error A: Improper syntax. + Error A: syntax error at top level + Error A: Possibly missing a ) + 5 error(s) parsing +\end{verbatim} +a similar error will be raised. Finally, the ``)'' must be indented by +at least one space relative to the sqrt thus: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} +or an error will be generated. + +It can be seen that great care needs to be taken when constructing input +files consisting of piles of expressions. It would seem prudent to add +one pile at a time and check if it is acceptable before adding more, +particularly if piles are nested. However, it should be pointed out that +the use of piles as values for functions is not very readable and so +perhaps the delicate nature of their interpretation should deter programmers +from using them in these situations. Using piles should really be restricted +to constructing functions, etc. and a small amount of rewriting can remove +the need to use them as arguments. For example, the previous block could +easily be implemented as: +\begin{verbatim} +a:=3.0 +b:=1.0 +c:=a + b +sqrt(4.0 + c) +\end{verbatim} +\begin{verbatim} +a:=3.0 +\end{verbatim} +$$ +3.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +b:=1.0 +\end{verbatim} +$$ +1.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +c:=a + b +\end{verbatim} +$$ +4.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +sqrt(4.0 + c) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} + +which achieves the same result and is easier to understand. Note that this +is still a pile but it is not as fragile as the previous version. +\subsection{Functions} +Definitions of functions in Axiom are quite simple providing two things +are observed. First, the type of the function must either be completely +specified or completely unspecified. Second, the body of the function is +assigned to the function identifier using the delayed assignment operator +``==''. + +To specify the type of something the ``:'' operator is used. Thus to define +a variable {\sl x} to be of type {\tt Fraction Integer} we enter: +\spadcommand{x : Fraction Integer} +\returnType{Type: Void} + +For functions the method is the same except that the arguments are +placed in parentheses and the return type is placed after the symbol +``{\tt ->}''. Some examples of function definitions taking zero, one, +two, or three arguments and returning a list of integers are: + +\spadcommand{f : () -> List Integer} +\returnType{Type: Void} + +\spadcommand{g : (Integer) -> List Integer} +\returnType{Type: Void} + +\spadcommand{h : (Integer, Integer) -> List Integer} +\returnType{Type: Void} + +\spadcommand{k : (Integer, Integer, Integer) -> List Integer} +\returnType{Type: Void} + +Now the actual function definitions might be: +\spadcommand{f() == [\ ]} +\returnType{Type: Void} + +\spadcommand{g(a) == [a]} +\returnType{Type: Void} + +\spadcommand{h(a,b) == [a,b]} +\returnType{Type: Void} + +\spadcommand{k(a,b,c) == [a,b,c]} +\returnType{Type: Void} + +with some invocations of these functions: +\spadcommand{f()} +\begin{verbatim} + Compiling function f with type () -> List Integer +\end{verbatim} +$$ +\left[\ +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{g(4)} +\begin{verbatim} + Compiling function g with type Integer -> List Integer +\end{verbatim} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{h(2,9)} +\begin{verbatim} + Compiling function h with type (Integer,Integer) -> List Integer +\end{verbatim} +$$ +\left[ +2, 9 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{k(-3,42,100)} +\begin{verbatim} + Compiling function k with type (Integer,Integer,Integer) -> List + Integer +\end{verbatim} +$$ +\left[ +-3, {42}, {100} +\right] +$$ +\returnType{Type: List Integer} + +The value returned by a function is either the value of the last expression +evaluated or the result of a {\bf return} statement. For example, the +following are effectively the same: +\spadcommand{p : Integer -> Integer} +\returnType{Type: Void} + +\spadcommand{p x == (a:=1; b:=2; a+b+x)} +\returnType{Type: Void} + +\spadcommand{p x == (a:=1; b:=2; return(a+b+x))} +\returnType{Type: Void} + +Note that a block (pile) is assigned to the function identifier {\bf p} and +thus all the rules about blocks apply to function definitions. Also there was +only one argument so the parenthese are not needed. + +This is basically all that one needs to know about defining functions in +Axiom -- first specify the complete type and then assign a block to the +function name. The rest of this section is concerned with defining more +complex blocks than those in this section and as a result function definitions +will crop up continually particularly since they are a good way of testing +examples. Since the block structure is more complex we will use the {\bf pile} +notation and thus have to use input files to read the piles. + +\subsection{Choices} +Apart from the ``{\tt =>}'' operator that allows a block to exit before the end +Axiom provides the standard {\bf if-then-else} construct. The general +syntax is: +{\center{if {\sl BooleanExpr} then {\sl Expr1} else {\sl Expr2}}} + +where ``else {\sl Expr2}'' can be omitted. If the expression {\sl BooleanExpr} +evaluates to {\tt true} then {\sl Expr1} is executed otherwise {\sl Expr2} +(if present) will be executed. An example of piles and {\bf if-then-else} is: +(read from an input file) +\begin{verbatim} +h := 2.0 +if h > 3.1 then + 1.0 + else + z:= cos(h) + max(x,0.5) +\end{verbatim} +\begin{verbatim} +h := 2.0 +\end{verbatim} +$$ +2.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +if h > 3.1 then + 1.0 + else + z:= cos(h) + max(x,0.5) +\end{verbatim} +$$ +x +$$ +\returnType{Type: Polynomial Float} + +Note the indentation -- the ``else'' must be indented relative to the ``if'' +otherwise it will generate an error (Axiom will think there are two piles, +the second one beginning with ``else''). + +Any expression that has type {\tt Boolean} can be used as {\tt BooleanExpr} +and the most common will be those involving the relational operators ``$>$'', +``$<$'', and ``=''. Usually the type of an expression involving the equality +operator ``='' will be {\bf Boolean} but in those situations when it isn't +you may need to use the ``@'' operator to ensure that it is. + +\subsection{Loops} +Loops in Axiom are regarded as expressions containing another expression +called the {\sl loop body}. The loop body is executed zero or more times +depending on the kind of loop. Loops can be nested to any depth. + +\subsubsection{The {\tt repeat} loop} +The simplest kind of loop provided by Axiom is the {\bf repeat} loop. The +general syntax of this is: +{\center{{\bf repeat} {\sl loopBody}}} + +This will cause Axiom to execute {\sl loopBody} repeatedly until either a +{\bf break} or {\bf return} statement is encountered. If {\sl loopBody} +contains neither of these statements then it will loop forever. The +following piece of code will display the numbers from $1$ to $4$: +\begin{verbatim} +i:=1 +repeat + if i > 4 then break + output(i) + i:=i+1 +\end{verbatim} +\begin{verbatim} +i:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +repeat + if i > 4 then break + output(i) + i:=i+1 + + 1 + 2 + 3 + 4 +\end{verbatim} +\returnType{Type: Void} + +It was mentioned that loops will only be left when either a {\bf break} or +{\bf return} statement is encountered so why can't one use the ``{\tt =>}'' +operator? The reason is that the ``{\tt =>}'' operator tells Axiom to leave the +current block whereas {\bf break} leaves the current loop. The {\bf return} +statement leaves the current function. + +To skip the rest of a loop body and continue the next iteration of the loop +use the {\bf iterate} statement (the -- starts a comment in Axiom) +\begin{verbatim} +i := 0 +repeat + i := i + 1 + if i > 6 then break + -- Return to start if i is odd + if odd?(i) then iterate + output(i) +\end{verbatim} +\begin{verbatim} +i := 0 +\end{verbatim} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\begin{verbatim} +repeat + i := i + 1 + if i > 6 then break + -- Return to start if i is odd + if odd?(i) then iterate + output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +\subsubsection{The {\tt while} loop} +The while statement extends the basic {\bf repeat} loop to place the control +of leaving the loop at the start rather than have it buried in the middle. +Since the body of the loop is still part of a {\bf repeat} loop, {\bf break} +and ``{\tt =>}'' work in the same way as in the previous section. The general +syntax of a {\bf while} loop is: +{\center{while {\sl BoolExpr} repeat {\sl loopBody}}} + +As before, {\sl BoolExpr} must be an expression of type {\bf Boolean}. Before +the body of the loop is executed {\sl BoolExpr} is tested. If it evaluates to +{\tt true} then the loop body is entered otherwise the loop is terminated. +Multiple conditions can be applied using the logical operators such as +{\bf and} or by using several {\bf while} statements before the {\bf repeat}. +\begin{verbatim} +x:=1 +y:=1 +while x < 4 and y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +\begin{verbatim} +x:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 and y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] +\end{verbatim} +\returnType{Type: Void} + +\begin{verbatim} +x:=1 +y:=1 +while x < 4 while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +\begin{verbatim} +x:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] +\end{verbatim} +\returnType{Type: Void} + +Note that the last example using two {\bf while} statements is {\sl not} a +nested loop but the following one is: +\begin{verbatim} +x:=1 +y:=1 +while x < 4 repeat + while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +\begin{verbatim} +x:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 repeat + while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] + [4,7] + [5,9] +\end{verbatim} +\returnType{Type: Void} + +Suppose we that, given a matrix of arbitrary size, find the position and +value of the first negative element by examining the matrix in row-major +order: +\begin{verbatim} +m := matrix [ [ 21, 37, 53, 14 ],_ + [ 8, 22,-24, 16 ],_ + [ 2, 10, 15, 14 ],_ + [ 26, 33, 55,-13 ] ] + +lastrow := nrows(m) +lastcol := ncols(m) +r := 1 +while r <= lastrow repeat + c := 1 -- Index of first column + while c <= lastcol repeat + if elt(m,r,c) < 0 then + output [r,c,elt(m,r,c)] + r := lastrow + break -- Don't look any further + c := c + 1 + r := r + 1 +\end{verbatim} +\begin{verbatim} +m := matrix [ [ 21, 37, 53, 14 ],_ + [ 8, 22,-24, 16 ],_ + [ 2, 10, 15, 14 ],_ + [ 26, 33, 55,-13 ] ] +\end{verbatim} +$$ +\left[ +\begin{array}{cccc} +{21} & {37} & {53} & {14} \\ +8 & {22} & -{24} & {16} \\ +2 & {10} & {15} & {14} \\ +{26} & {33} & {55} & -{13} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\begin{verbatim} +lastrow := nrows(m) +\end{verbatim} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +lastcol := ncols(m) +\end{verbatim} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +r := 1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while r <= lastrow repeat + c := 1 -- Index of first column + while c <= lastcol repeat + if elt(m,r,c) < 0 then + output [r,c,elt(m,r,c)] + r := lastrow + break -- Don't look any further + c := c + 1 + r := r + 1 + + [2,3,- 24] +\end{verbatim} +\returnType{Type: Void} + +\subsubsection{The {\tt for} loop} +The last loop statement of interest is the {\bf for} loop. There are two +ways of creating a {\bf for} loop. The first way uses either a list or +a segment: +\begin{center} +for {\sl var} in {\sl seg} repeat {\sl loopBody}\\ +for {\sl var} in {\sl list} repeat {\sl loopBody} +\end{center} +where {\sl var} is an index variable which is iterated over the values in +{\sl seg} or {\sl list}. The value {\sl seg} is a segment such as $1\ldots10$ +or $1\ldots$ and {\sl list} is a list of some type. For example: +\begin{verbatim} +for i in 1..10 repeat + ~prime?(i) => iterate + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +\begin{verbatim} +for w in ["This", "is", "your", "life!"] repeat + output(w) + + This + is + your + life! +\end{verbatim} +\returnType{Type: Void} + +The second form of the {\bf for} loop syntax includes a ``{\bf such that}'' +clause which must be of type {\bf Boolean}: +\begin{center} +for {\sl var} | {\sl BoolExpr} in {\sl seg} repeat {\sl loopBody}\\ +for {\sl var} | {\sl BoolExpr} in {\sl list} repeat {\sl loopBody} +\end{center} +Some examples are: +\begin{verbatim} +for i in 1..10 | prime?(i) repeat + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +\begin{verbatim} +for i in [1,2,3,4,5,6,7,8,9,10] | prime?(i) repeat + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +You can also use a {\bf while} clause: +\begin{verbatim} +for i in 1.. while i < 7 repeat + if even?(i) then output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +Using the ``{\bf such that}'' clause makes this appear simpler: +\begin{verbatim} +for i in 1.. | even?(i) while i < 7 repeat + output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +You can use multiple {\bf for} clauses to iterate over several sequences +in parallel: +\begin{verbatim} +for a in 1..4 for b in 5..8 repeat + output [a,b] + + [1,5] + [2,6] + [3,7] + [4,8] +\end{verbatim} +\returnType{Type: Void} + +As a general point it should be noted that any symbols referred to in the +``{\bf such that}'' and {\bf while} clauses must be pre-defined. This +either means that the symbols must have been defined in an outer level +(e.g. in an enclosing loop) or in a {\bf for} clause appearing before the +``{\bf such that}'' or {\bf while}. For example: +\begin{verbatim} +for a in 1..4 repeat + for b in 7..9 | prime?(a+b) repeat + output [a,b,a+b] + + [2,9,11] + [3,8,11] + [4,7,11] + [4,9,13] +\end{verbatim} +\returnType{Type: Void} + +Finally, the {\bf for} statement has a {\bf by} clause to specify the +step size. This makes it possible to iterate over the segment in +reverse order: +\begin{verbatim} +for a in 1..4 for b in 8..5 by -1 repeat + output [a,b] + + [1,8] + [2,7] + [3,6] + [4,5] +\end{verbatim} +\returnType{Type: Void} + +Note that without the ``by -1'' the segment 8..5 is empty so there is +nothing to iterate over and the loop exits immediately. + +\setcounter{chapter}{0} % Chapter 1 + +\hyphenation{ +multi-set +Uni-var-iate-Poly-nomial +Mul-ti-var-iate-Poly-nomial +Distributed-Mul-ti-var-iate-Poly-nomial +Homo-gen-eous-Distributed-Mul-ti-var-iate-Poly-nomial +New-Distributed-Mul-ti-var-iate-Poly-nomial +General-Distributed-Mul-ti-var-iate-Poly-nomial +} + +\chapter{An Overview of Axiom} +\begin{quote} +When we start cataloging the gains in tools sitting on a computer, the +benefits of software are amazing. But, if the benefits of software are +so great, why do we worry about making it easier -- don't the ends pay +for the means? We worry becuase making such software is extraordinarily +hard and almost no one can do it -- the detail is exhausting, the +creativity required is extreme, the hours of failure upon failure +requiring patience and persistence would tax anyone claiming to be +sane. Yet we require people with such characteristics be found and +employed and employed cheaply. + +-- Christopher Alexander + +(from Patterns of Software by Richard Gabriel) + +\end{quote} +\label{ugIntro} + +Welcome to the Axiom environment for interactive computation and +problem solving. Consider this chapter a brief, whirlwind tour of the +Axiom world. We introduce you to Axiom's graphics and the +Axiom language. Then we give a sampling of the large variety of +facilities in the Axiom system, ranging from the various kinds +of numbers, to data types (like lists, arrays, and sets) and +mathematical objects (like matrices, integrals, and differential +equations). We conclude with the discussion of system commands and an +interactive ``undo.'' + +Before embarking on the tour, we need to brief those readers working +interactively with Axiom on some details. + +\section{Starting Up and Winding Down} +\label{ugIntroStart} +You need to know how to start the Axiom system and how to stop it. +We assume that Axiom has been correctly installed on your +machine (as described in another Axiom document). + +To begin using Axiom, issue the command {\bf axiom} to the +Axiom operating system shell. +\index{axiom @{\bf axiom}} There is a brief pause, some start-up +messages, and then one or more windows appear. + +If you are not running Axiom under the X Window System, there is +only one window (the console). At the lower left of the screen there +is a prompt that \index{prompt} looks like +\begin{verbatim} +(1) -> +\end{verbatim} + +When you want to enter input to Axiom, you do so on the same +line after the prompt. The ``1'' in ``(1)'', also called the equation +number, is the computation step number and is incremented +\index{step number} after you enter Axiom statements. +Note, however, that a system command such as {\tt )clear all} +may change the step number in other ways. We talk about step numbers +more when we discuss system commands and the workspace history facility. + +If you are running Axiom under the X Window System, there may be +two \index{X Window System} windows: the console window (as just +described) and the HyperDoc main menu. \index{Hyper@{HyperDoc}} +HyperDoc is a multiple-window hypertext system +that lets you \index{window} view Axiom documentation and +examples on-line, execute Axiom expressions, and generate +graphics. If you are in a graphical windowing environment, it is +usually started automatically when Axiom begins. If it is not +running, issue {\tt )hd} to start it. We discuss the basics of +HyperDoc in Chapter \ref{ugHyper} on page~\pageref{ugHyper}. + +To interrupt an Axiom computation, hold down the \index{interrupt} +{\bf Ctrl} (control) key and press {\bf c}. This brings you back to +the Axiom prompt. + +\boxed{4.6in}{ +\vskip 0.1cm +To exit from Axiom, move to the console window, \index{stopping +@{stopping Axiom}} type {\tt )quit} \index{exiting @{exiting +Axiom}} at the input prompt and press the {\bf Enter} key. +You will probably be prompted with the following +message: +\begin{center} +Please enter {\bf y} or {\bf yes} if you really want to leave the \\ +interactive environment and return to the operating system +\end{center} +You should respond {\bf yes}, for example, to exit Axiom.\\ +} + +We are purposely vague in describing exactly what your screen looks +like or what messages Axiom displays. Axiom runs on a number of +different machines, operating systems and window environments, and +these differences all affect the physical look of the system. You can +also change the way that Axiom behaves via {\it system commands} +described later in this chapter and in Appendix A. +System commands are special commands, like {\tt )set}, that begin with +a closing parenthesis and are used to change your environment. For +example, you can set a system variable so that you are not prompted +for confirmation when you want to leave Axiom. + +\subsection{Clef} +\label{ugAvailCLEF} +If you are using Axiom under the X Window System, the +\index{Clef} \index{command line editor} Clef command +line editor is probably available and installed. With this editor you +can recall previous lines with the up and down arrow keys. To move +forward and backward on a line, use the right and left arrows. You +can use the {\bf Insert} key to toggle insert mode on or off. When +you are in insert mode, the cursor appears as a large block and if you +type anything, the characters are inserted into the line without +deleting the previous ones. + +If you press the {\bf Home} key, the cursor moves to the beginning of +the line and if you press the {\bf End} key, the cursor moves to the +end of the line. Pressing {\bf Ctrl-End} deletes all the text from +the cursor to the end of the line. + +Clef also provides Axiom operation name completion for +\index{operation name completion} a limited set of operations. If you +enter a few letters and then press the {\bf Tab} key, Clef tries to +use those letters as the prefix of an Axiom operation name. If +a name appears and it is not what you want, press {\bf Tab} again to +see another name. + +You are ready to begin your journey into the world of Axiom. + +\section{Typographic Conventions} +\label{ugIntroTypo} +In this document we have followed these typographical conventions: +\begin{itemize} +% +\item Categories, domains and packages are displayed in this font: +{\tt Ring}, {\tt Integer}, {\tt DiophantineSolutionPackage}. +% +\item Prefix operators, infix operators, and punctuation symbols in +the Axiom language are displayed in the text like this: +{\tt +}, {\tt \$}, {\tt +->}. +% +\item Axiom expressions or expression fragments are displayed in this font:\\ +{\tt inc(x) == x + 1}. +% +\item For clarity of presentation, \TeX{} is often used to format expressions\\ +$g(x)=x^2+1$. +% +\item Function names and HyperDoc button names are displayed in the text in +this font: +{\bf factor}, {\bf integrate}, {\bf Lighting}. +% +\item Italics are used for emphasis and for words defined in the glossary: \\ +{\it category}. +\end{itemize} + +This document contains over 2500 examples of Axiom input and output. All +examples were run though Axiom and their output was created in \TeX{} +form by the Axiom {\tt TexFormat} package. We have deleted system +messages from the example output if those messages are not important +for the discussions in which the examples appear. + +\section{The Axiom Language} +\label{ugIntroExpressions} +The Axiom language is a rich language for performing interactive +computations and for building components of the Axiom library. +Here we present only some basic aspects of the language that you need +to know for the rest of this chapter. Our discussion here is +intentionally informal, with details unveiled on an ``as needed'' +basis. For more information on a particular construct, we suggest you +consult the index. + +\subsection{Arithmetic Expressions} +\label{ugIntroArithmetic} +For arithmetic expressions, use the ``{\tt +}'' and ``{\tt -}'' operator +as in mathematics. Use ``{\tt *}'' for multiplication, and ``{\tt **}'' +for exponentiation. To create a fraction, use ``{\tt /}''. When an +expression contains several operators, those of highest +{\it precedence} are evaluated first. For arithmetic operators, +``{\tt **}'' has highest precedence, ``{\tt *}'' and ``{\tt /}'' have the +next highest precedence, and ``{\tt +}'' and ``{\tt -}'' have the lowest +precedence. + +Axiom puts implicit parentheses around operations of higher +precedence, and groups those of equal precedence from left to right. +\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1} +$$ +-{{19} \over 4} +$$ +\returnType{Type: Fraction Integer} + +The above expression is equivalent to this. +\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1} +$$ +-{{19} \over 4} +$$ +\returnType{Type: Fraction Integer} + +If an expression contains subexpressions enclosed in parentheses, +the parenthesized subexpressions are evaluated first (from left to +right, from inside out). +\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))} +$$ +{11} \over 4 +$$ +\returnType{Type: Fraction Integer} + +\subsection{Previous Results} +\label{ugIntroPrevious} +Use the percent sign ``{\tt \%}'' to refer to the last result. +\index{result!previous} Also, use ``{\tt \%\%}' to refer to +previous results. \index{percentpercent@{\%\%}} ``{\tt \%\%(-1)}'' is +equivalent to ``{\tt \%}'', ``{\tt \%\%(-2)}'' returns the next to +the last result, and so on. ``{\tt \%\%(1)}'' returns the result from +step number 1, ``{\tt \%\%(2)}'' returns the result from step number 2, +and so on. ``{\tt \%\%(0)}'' is not defined. + +This is ten to the tenth power. +\spadcommand{10 ** 10} +$$ +10000000000 +$$ +\returnType{Type: PositiveInteger} + +This is the last result minus one. +\spadcommand{\% - 1} +$$ +9999999999 +$$ +\returnType{Type: PositiveInteger} + +This is the last result. +\spadcommand{\%\%(-1)} +$$ +9999999999 +$$ +\returnType{Type: PositiveInteger} + +This is the result from step number 1. +\spadcommand{\%\%(1)} +$$ +10000000000 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Some Types} +\label{ugIntroTypes} +Everything in Axiom has a type. The type determines what operations +you can perform on an object and how the object can be used. +Chapter~\ref{ugTypes} on page~\pageref{ugTypes} is dedicated to the +interactive use of types. Several of the final chapters discuss how +types are built and how they are organized in the Axiom library. + +Positive integers are given type {\bf PositiveInteger}. +\spadcommand{8} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +Negative ones are given type {\bf Integer}. This fine +distinction is helpful to the Axiom interpreter. + +\spadcommand{-8} +$$ +-8 +$$ +\returnType{Type: Integer} + +Here a positive integer exponent gives a polynomial result. +\spadcommand{x**8} +$$ +x \sp 8 +$$ +\returnType{Type: Polynomial Integer} + +Here a negative integer exponent produces a fraction. +\spadcommand{x**(-8)} +$$ +1 \over {x \sp 8} +$$ +\returnType{Type: Fraction Polynomial Integer} + +\subsection{Symbols, Variables, Assignments, and Declarations} +\label{ugIntroAssign} +A {\it symbol} is a literal used for the input of things like +the ``variables'' in polynomials and power series. + +We use the three symbols $x$, $y$, and $z$ in +entering this polynomial. +\spadcommand{(x - y*z)**2} +$$ +{{y \sp 2} \ {z \sp 2}} -{2 \ x \ y \ z}+{x \sp 2} +$$ +\returnType{Type: Polynomial Integer} + +A symbol has a name beginning with an uppercase or lowercase +alphabetic \index{symbol!naming} character, ``{\tt \%}'', or +``{\tt !}''. Successive characters (if any) can be any of the +above, digits, or ``{\tt ?}''. Case is distinguished: the symbol +{\tt points} is different from the symbol {\tt Points}. + +A symbol can also be used in Axiom as a {\it variable}. A variable +refers to a value. To {\sl assign} a value to a variable, +\index{variable!naming} the operator ``{\tt :=}'' \index{assignment} +is used.\footnote{Axiom actually has two forms of assignment: +{\it immediate} assignment, as discussed here, and {\it delayed +assignment}. See Section \ref{ugLangAssign} on page~\pageref{ugLangAssign} +for details.} A variable initially has no restrictions on the kinds +of \index{declaration} values to which it can refer. + +This assignment gives the value $4$ (an integer) to +a variable named $x$. +\spadcommand{x := 4} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +This gives the value $z + 3/5$ (a polynomial) to $x$. +\spadcommand{x := z + 3/5} +$$ +z+{3 \over 5} +$$ +\returnType{Type: Polynomial Fraction Integer} + +To restrict the types of objects that can be assigned to a variable, +use a {\it declaration} +\spadcommand{y : Integer} +\returnType{Type: Void} + +After a variable is declared to be of some type, only values +of that type can be assigned to that variable. +\spadcommand{y := 89} +$$ +89 +$$ +\returnType{Type: Integer} + +The declaration for $y$ forces values assigned to $y$ to +be converted to integer values. +\spadcommand{y := sin \%pi} +$$ +0 +$$ +\returnType{Type: Integer} + +If no such conversion is possible, +Axiom refuses to assign a value to $y$. +\spadcommand{y := 2/3} +\begin{verbatim} + Cannot convert right-hand side of assignment + 2 + - + 3 + + to an object of the type Integer of the left-hand side. +\end{verbatim} + +A type declaration can also be given together with an assignment. +The declaration can assist Axiom in choosing the correct +operations to apply. +\spadcommand{f : Float := 2/3} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +Any number of expressions can be given on input line. +Just separate them by semicolons. +Only the result of evaluating the last expression is displayed. + +These two expressions have the same effect as +the previous single expression. + +\spadcommand{f : Float; f := 2/3} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +The type of a symbol is either {\tt Symbol} +or {\tt Variable({\it name})} where {\it name} is the name +of the symbol. + +By default, the interpreter +gives this symbol the type {\tt Variable(q)}. + +\spadcommand{q} +$$ +q +$$ +\returnType{Type: Variable q} + +When multiple symbols are involved, {\tt Symbol} is used. +\spadcommand{[q, r]} +$$ +\left[ +q, r +\right] +$$ +\returnType{Type: List OrderedVariableList [q,r]} + +What happens when you try to use a symbol that is the name of a variable? +\spadcommand{f} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +Use a single quote ``{\tt '}'' before \index{quote} the name to get the symbol. + +\spadcommand{'f} +$$ +f +$$ +\returnType{Type: Variable f} + +Quoting a name creates a symbol by preventing evaluation of the name +as a variable. Experience will teach you when you are most likely +going to need to use a quote. We try to point out the location of +such trouble spots. + +\subsection{Conversion} +\label{ugIntroConversion} +Objects of one type can usually be ``converted'' to objects of several +other types. To {\sl convert} an object to a new type, use the ``{\tt ::}'' +infix operator.\footnote{Conversion is discussed in detail in +\ref{ugTypesConvert} on page~\pageref{ugTypesConvert}.} For example, +to display an object, it is necessary to convert the object to type +{\tt OutputForm}. + +This produces a polynomial with rational number coefficients. + +\spadcommand{p := r**2 + 2/3} +$$ +{r \sp 2}+{2 \over 3} +$$ +\returnType{Type: Polynomial Fraction Integer} + +Create a quotient of polynomials with integer coefficients +by using ``{\tt ::}''. + +\spadcommand{p :: Fraction Polynomial Integer } +$$ +{{3 \ {r \sp 2}}+2} \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +Some conversions can be performed automatically when Axiom tries +to evaluate your input. Others conversions must be explicitly +requested. + +\subsection{Calling Functions} +\label{ugIntroCallFun} +As we saw earlier, when you want to add or subtract two values, you +place the arithmetic operator ``{\tt +}'' or ``{\tt -}'' between the two +arguments denoting the values. To use most other Axiom +operations, however, you use another syntax: \index{function!calling} +write the name of the operation first, then an open parenthesis, then +each of the arguments separated by commas, and, finally, a closing +parenthesis. If the operation takes only one argument and the +argument is a number or a symbol, you can omit the parentheses. + +This calls the operation {\bf factor} with the single integer argument $120$. + +\spadcommand{factor(120)} +$$ +{2 \sp 3} \ 3 \ 5 +$$ +\returnType{Type: Factored Integer} + +This is a call to {\bf divide} with the two integer arguments +$125$ and $7$. +\spadcommand{divide(125,7)} +$$ +\left[ +{quotient={17}}, {remainder=6} +\right] +$$ +\returnType{Type: Record(quotient: Integer, remainder: Integer)} + +This calls {\bf quatern} with four floating-point arguments. +\spadcommand{quatern(3.4,5.6,2.9,0.1)} +$$ +{3.4}+{{5.6} \ i}+{{2.9} \ j}+{{0.1} \ k} +$$ +\returnType{Type: Quaternion Float} + +This is the same as {\bf factorial}(10). +\spadcommand{factorial 10} +$$ +3628800 +$$ +\returnType{Type: PositiveInteger} + +An operations that returns a {\tt Boolean} value (that is, +{\tt true} or {\tt false}) frequently has a name suffixed with +a question mark (``?''). For example, the {\bf even?} +operation returns {\tt true} if its integer argument is an even +number, {\tt false} otherwise. + +An operation that can be destructive on one or more arguments +usually has a name ending in a exclamation point (``!''). +This actually means that it is {\it allowed} to update its +arguments but it is not {\it required} to do so. For example, +the underlying representation of a collection type may not allow +the very last element to removed and so an empty object may be +returned instead. Therefore, it is important that you use the +object returned by the operation and not rely on a physical +change having occurred within the object. Usually, destructive +operations are provided for efficiency reasons. + +\subsection{Some Predefined Macros} +\label{ugIntroMacros} +Axiom provides several macros for your convenience.\footnote{See +\ref{ugUserMacros} on page~\pageref{ugUserMacros} for a discussion on +how to write your own macros.} Macros are names +\index{macro!predefined} (or forms) that expand to larger expressions +for commonly used values. + +\begin{center} +\begin{tabular}{ll} +{\it \%i} & The square root of -1. \\ +{\it \%e} & The base of the natural logarithm. \\ +{\it \%pi} & $\pi$. \\ +{\it \%infinity} & $\infty$. \\ +{\it \%plusInfinity} & $+\infty$. \\ +{\it \%minusInfinity} & $-\infty$. +\end{tabular} +\end{center} +\index{\%i} +\index{\%e} +\index{\%pi} +\index{pi@{$\pi$ (= \%pi)}} +\index{\%infinity} +\index{infinity@{$\infty$ (= \%infinity)}} +\index{\%plusInfinity} +\index{\%minusInfinity} + +To display all the macros (along with anything you have +defined in the workspace), issue the system command {\tt )display all}. + +\subsection{Long Lines} +\label{ugIntroLong} +When you enter Axiom expressions from your keyboard, there will +be times when they are too long to fit on one line. Axiom does +not care how long your lines are, so you can let them continue from +the right margin to the left side of the next line. + +Alternatively, you may want to enter several shorter lines and have +Axiom glue them together. To get this glue, put an underscore +(\_) at the end of each line you wish to continue. + +\begin{verbatim} +2_ ++_ +3 +\end{verbatim} +is the same as if you had entered +\begin{verbatim} +2+3 +\end{verbatim} + +Axiom statements in an input file +(see Section \ref{ugInOutIn} on page~\pageref{ugInOutIn}), +can use indentation to indicate the program structure . +(see Section \ref{ugLangBlocks} on page~\pageref{ugLangBlocks}). + +\subsection{Comments} +\label{ugIntroComments} +Comment statements begin with two consecutive hyphens or two +consecutive plus signs and continue until the end of the line. + +The comment beginning with ``{\tt --}'' is ignored by Axiom. +\spadcommand{2 + 3 -- this is rather simple, no?} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +There is no way to write long multi-line comments other than starting +each line with ``{\tt --}'' or ``{\tt ++}''. + +\section{Numbers} +\label{ugIntroNumbers} +Axiom distinguishes very carefully between different kinds of +numbers, how they are represented and what their properties are. Here +are a sampling of some of these kinds of numbers and some things you +can do with them. + +Integer arithmetic is always exact. +\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3} +$$ +25387751112538918594666224484237298 +$$ +\returnType{Type: PositiveInteger} + +Integers can be represented in factored form. +\spadcommand{factor 643238070748569023720594412551704344145570763243} +$$ +{{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} \sp 5} \ {{23} +\sp 3} \ {{29} \sp 2} +$$ +\returnType{Type: Factored Integer} + +Results stay factored when you do arithmetic. +Note that the $12$ is automatically factored for you. +\spadcommand{\% * 12} +\index{radix} +$$ +{2 \sp 2} \ 3 \ {{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} +\sp 5} \ {{23} \sp 3} \ {{29} \sp 2} +$$ +\returnType{Type: Factored Integer} + +Integers can also be displayed to bases other than 10. +This is an integer in base 11. +\spadcommand{radix(25937424601,11)} +$$ +10000000000 +$$ +\returnType{Type: RadixExpansion 11} + +Roman numerals are also available for those special occasions. +\index{Roman numerals} + +\spadcommand{roman(1992)} +$$ +{\rm MCMXCII } +$$ +\returnType{Type: RomanNumeral} + +Rational number arithmetic is also exact. + +\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9} +$$ +{55739} \over {2520} +$$ +\returnType{Type: Fraction Integer} + +To factor fractions, you have to pmap {\bf factor} onto the numerator +and denominator. + +\spadcommand{map(factor,r)} +$$ +{{139} \ {401}} \over {{2 \sp 3} \ {3 \sp 2} \ 5 \ 7} +$$ +\returnType{Type: Fraction Factored Integer} + +{\tt SingleInteger} refers to machine word-length integers. + +In English, this expression means ``$11$ as a small integer''. +\spadcommand{11@SingleInteger} +$$ +11 +$$ +\returnType{Type: SingleInteger} + +Machine double-precision floating-point numbers are also available for +numeric and graphical applications. +\spadcommand{123.21@DoubleFloat} +$$ +123.21000000000001 +$$ +\returnType{Type: DoubleFloat} + +The normal floating-point type in Axiom, {\tt Float}, is a +software implementation of floating-point numbers in which the +exponent and the mantissa may have any number of digits. +The types {\tt Complex(Float)} and +{\tt Complex(DoubleFloat)} are the corresponding software +implementations of complex floating-point numbers. + +This is a floating-point approximation to about twenty digits. +\index{floating point} The ``{\tt ::}'' is used here to change from +one kind of object (here, a rational number) to another (a +floating-point number). + +\spadcommand{r :: Float} +$$ +22.1186507936 50793651 +$$ +\returnType{Type: Float} + +Use \spadfunFrom{digits}{Float} to change the number of digits in +the representation. +This operation returns the previous value so you can reset it +later. +\spadcommand{digits(22)} +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +To $22$ digits of precision, the number +$e^{\pi {\sqrt {163.0}}}$ appears to be an integer. +\spadcommand{exp(\%pi * sqrt 163.0)} +$$ +26253741 2640768744.0 +$$ +\returnType{Type: Float} + +Increase the precision to forty digits and try again. +\spadcommand{digits(40); exp(\%pi * sqrt 163.0)} +$$ +26253741\ 2640768743.9999999999\ 9925007259\ 76 +$$ +\returnType{Type: Float} + +Here are complex numbers with rational numbers as real and +\index{complex numbers} imaginary parts. +\spadcommand{(2/3 + \%i)**3} +$$ +-{{46} \over {27}}+{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +The standard operations on complex numbers are available. +\spadcommand{conjugate \% } +$$ +-{{46} \over {27}} -{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +You can factor complex integers. +\spadcommand{factor(89 - 23 * \%i)} +$$ +-{{\left( 1+i +\right)} +\ {{\left( 2+i +\right)} +\sp 2} \ {{\left( 3+{2 \ i} +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +Complex numbers with floating point parts are also available. +\spadcommand{exp(\%pi/4.0 * \%i)} +$$ +{0.7071067811\ 8654752440\ 0844362104\ 8490392849} + +$$ +$$ +{{0.7071067811\ 8654752440\ 0844362104\ 8490392848} \ i} +$$ +\returnType{Type: Complex Float} + +The real and imaginary parts can be symbolic. +\spadcommand{complex(u,v)} +$$ +u+{v \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Of course, you can do complex arithmetic with these also. +\spadcommand{\% ** 2} +$$ +-{v \sp 2}+{u \sp 2}+{2 \ u \ v \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Every rational number has an exact representation as a +repeating decimal expansion +\spadcommand{decimal(1/352)} +$$ +0.{00284}{\overline {09}} +$$ +\returnType{Type: DecimalExpansion} + +A rational number can also be expressed as a continued fraction. + +\spadcommand{continuedFraction(6543/210)} +$$ +{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} +$$ +\returnType{Type: ContinuedFraction Integer} + +Also, partial fractions can be used and can be displayed in a +\index{partial fraction} +compact format +\index{fraction!partial} +\spadcommand{partialFraction(1,factorial(10))} +$$ +{{159} \over {2 \sp 8}} -{{23} \over {3 \sp 4}} -{{12} \over {5 \sp 2}}+{1 +\over 7} +$$ +\returnType{Type: PartialFraction Integer} + +or expanded format. +\spadcommand{padicFraction(\%)} +$$ +{1 \over 2}+{1 \over {2 \sp 4}}+{1 \over {2 \sp 5}}+{1 \over {2 \sp 6}}+{1 +\over {2 \sp 7}}+{1 \over {2 \sp 8}} -{2 \over {3 \sp 2}} -{1 \over {3 \sp +3}} -{2 \over {3 \sp 4}} -{2 \over 5} -{2 \over {5 \sp 2}}+{1 \over 7} +$$ +\returnType{Type: PartialFraction Integer} + +Like integers, bases (radices) other than ten can be used for rational +numbers. +Here we use base eight. +\spadcommand{radix(4/7, 8)} +$$ +0.{\overline 4} +$$ +\returnType{Type: RadixExpansion 8} + +Of course, there are complex versions of these as well. +Axiom decides to make the result a complex rational number. +\spadcommand{\% + 2/3*\%i} +$$ +{4 \over 7}+{{2 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +You can also use Axiom to manipulate fractional powers. +\index{radical} +\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)} +$$ +\root {3} \of {{{{14} \ {\sqrt {7}}}+5}} +$$ +\returnType{Type: AlgebraicNumber} + +You can also compute with integers modulo a prime. +\spadcommand{x : PrimeField 7 := 5} +$$ +5 +$$ +\returnType{Type: PrimeField 7} + +Arithmetic is then done modulo $7$. +\spadcommand{x**3} +$$ +6 +$$ +\returnType{Type: PrimeField 7} + +Since $7$ is prime, you can invert nonzero values. +\spadcommand{1/x} +$$ +3 +$$ +\returnType{Type: PrimeField 7} + +You can also compute modulo an integer that is not a prime. +\spadcommand{y : IntegerMod 6 := 5} +$$ +5 +$$ +\returnType{Type: IntegerMod 6} + +All of the usual arithmetic operations are available. +\spadcommand{y**3} +$$ +5 +$$ +\returnType{Type: IntegerMod 6} + +Inversion is not available if the modulus is not a prime number. +Modular arithmetic and prime fields are discussed in Section +\ref{ugxProblemFinitePrime} on page~\pageref{ugxProblemFinitePrime}. + +\spadcommand{1/y} +\begin{verbatim} + There are 12 exposed and 13 unexposed library operations named / + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op / + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named / + with argument type(s) + PositiveInteger + IntegerMod 6 + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +This defines $a$ to be an algebraic number, that is, +a root of a polynomial equation. +\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)} +$$ +a +$$ +\returnType{Type: Expression Integer} + +Computations with $a$ are reduced according to the polynomial equation. +\spadcommand{(a + 1)**10} +$$ +-{{85} \ {a \sp 4}} -{{264} \ {a \sp 3}} -{{378} \ {a \sp 2}} -{{458} \ +a} -{287} +$$ +\returnType{Type: Expression Integer} + +Define $b$ to be an algebraic number involving $a$. +\spadcommand{b := rootOf(b**4 + a,b)} +$$ +b +$$ +\returnType{Type: Expression Integer} + +Do some arithmetic. +\spadcommand{2/(b - 1)} +$$ +2 \over {b -1} +$$ +\returnType{Type: Expression Integer} + +To expand and simplify this, call {\it ratDenom} +to rationalize the denominator. +\spadcommand{ratDenom(\%)} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 1 +\end{array} +$$ + +\returnType{Type: Expression Integer} + +If we do this, we should get $b$. +\spadcommand{2/\%+1} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 3 +\end{array} +\right)}\over{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 1 +\end{array} +\right)} +$$ +\returnType{Type: Expression Integer} + +But we need to rationalize the denominator again. + +\spadcommand{ratDenom(\%)} +$$ +b +$$ +\returnType{Type: Expression Integer} + +Types {\tt Quaternion} and {\tt Octonion} are also available. +Multiplication of quaternions is non-commutative, as expected. + +\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} +$$ +-{8 \ i}+{{16} \ j} -{8 \ k} +$$ +\returnType{Type: Quaternion Integer} + +\section{Data Structures} +\label{ugIntroCollect} +Axiom has a large variety of data structures available. Many +data structures are particularly useful for interactive computation +and others are useful for building applications. The data structures +of Axiom are organized into {\sl category hierarchies}. + +A {\it list} \footnote{Lists are discussed in Section \ref{ListXmpPage} on +page~\pageref{ListXmpPage}} is the most commonly used data structure in +Axiom for holding objects all of the same type. The name {\it list} is +short for ``linked-list of nodes.'' Each node consists of a value +(\spadfunFrom{first}{List}) and a link (\spadfunFrom{rest}{List}) that +points to the next node, or to a distinguished value denoting the +empty list. To get to, say, the third element, Axiom starts at the +front of the list, then traverses across two links to the third node. + +Write a list of elements using square brackets with commas separating +the elements. +\spadcommand{u := [1,-7,11]} +$$ +\left[ +1, -7, {11} +\right] +$$ +\returnType{Type: List Integer} + +This is the value at the third node. Alternatively, you can say $u.3$. +\spadcommand{first rest rest u} +$$ +11 +$$ +\returnType{Type: PositiveInteger} + +Many operations are defined on lists, such as: {\bf empty?}, to test +that a list has no elements; {\bf cons}$(x,l)$, to create a new list +with {\bf first} element $x$ and {\bf rest} $l$; {\bf reverse}, to +create a new list with elements in reverse order; and {\bf sort}, to +arrange elements in order. + +An important point about lists is that they are ``mutable'': their +constituent elements and links can be changed ``in place.'' +To do this, use any of the operations whose names end with the +character ``{\tt !}''. + +The operation \spadfunFrom{concat!}{List}$(u,v)$ replaces the +last link of the list $u$ to point to some other list $v$. +Since $u$ refers to the original list, this change is seen by $u$. +\spadcommand{concat!(u,[9,1,3,-4]); u} +$$ +\left[ +1, -7, {11}, 9, 1, 3, -4 +\right] +$$ +\returnType{Type: List Integer} + +A {\it cyclic list} is a list with a ``cycle'': \index{list!cyclic} a +link pointing back to an earlier node of the list. \index{cyclic +list} To create a cycle, first get a node somewhere down the list. +\spadcommand{lastnode := rest(u,3)} +$$ +\left[ +9, 1, 3, -4 +\right] +$$ +\returnType{Type: List Integer} + +Use \spadfunFrom{setrest!}{List} to change the link emanating from +that node to point back to an earlier part of the list. + +\spadcommand{setrest!(lastnode,rest(u,2)); u} +$$ +\left[ +1, -7, {\overline {{11}, 9}} +\right] +$$ +\returnType{Type: List Integer} + +A {\it stream} is a structure that (potentially) has an infinite +number of distinct elements. Think of a stream as an +``infinite list'' where elements are computed successively. +\footnote{Streams are discussed in Section{StreamXmpPage} on +page~\pageref{StreamXmpPage}} + +Create an infinite stream of factored integers. Only a certain number +of initial elements are computed and displayed. + +\spadcommand{[factor(i) for i in 2.. by 2]} +$$ +\left[ +2, {2 \sp 2}, {2 \ 3}, {2 \sp 3}, {2 \ 5}, {{2 \sp 2} \ 3}, + {2 \ 7}, {2 \sp 4}, {2 \ {3 \sp 2}}, {{2 \sp 2} \ 5}, +\ldots +\right] +$$ +\returnType{Type: Stream Factored Integer} + +Axiom represents streams by a collection of already-computed +elements together with a function to compute the next element ``on +demand.'' Asking for the $n$-th element causes elements +$1$ through $n$ to be evaluated. +\spadcommand{\%.36} +$$ +{2 \sp 3} \ {3 \sp 2} +$$ +\returnType{Type: Factored Integer} + +Streams can also be finite or cyclic. +They are implemented by a linked list structure similar to lists +and have many of the same operations. +For example, {\bf first} and {\bf rest} are used to access +elements and successive nodes of a stream. + +A {\it one-dimensional array} is another data structure used to hold +objects of the same type \footnote{OnedimensionalArray is discussed in +Section \ref{OneDimensionalArrayXmpPage} on +page~\pageref{OneDimensionalArrayXmpPage}}. Unlike lists, +one-dimensional arrays are inflexible---they are +\index{array!one-dimensional} implemented using a fixed block of +storage. Their advantage is that they give quick and equal access +time to any element. + +A simple way to create a one-dimensional array is to apply the +operation {\bf oneDimensionalArray} to a list of elements. +\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]} +$$ +\left[ +1, -7, 3, {3 \over 2} +\right] +$$ +\returnType{Type: OneDimensionalArray Fraction Integer} + +One-dimensional arrays are also mutable: you can change their +constituent elements ``in place.'' +\spadcommand{a.3 := 11; a} +$$ +\left[ +1, -7, {11}, {3 \over 2} +\right] +$$ +\returnType{Type: OneDimensionalArray Fraction Integer} + +However, one-dimensional arrays are not flexible structures. +You cannot destructively {\bf concat!} them together. +\spadcommand{concat!(a,oneDimensionalArray [1,-2])} +\begin{verbatim} + There are 5 exposed and 0 unexposed library operations named concat! + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op concat! + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named + concat! with argument type(s) + OneDimensionalArray Fraction Integer + OneDimensionalArray Integer + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +Examples of datatypes similar to {\tt OneDimensionalArray} +are: {\tt Vector} (vectors are mathematical structures +implemented by one-dimensional arrays), {\tt String} (arrays +of ``characters,'' represented by byte vectors), and +{\tt Bits} (represented by ``bit vectors''). + +A vector of 32 bits, each representing the {\bf Boolean} value +${\tt true}$. +\spadcommand{bits(32,true)} +$$ +\mbox{\tt "11111111111111111111111111111111"} +$$ +\returnType{Type: Bits} + +A {\it flexible array} \footnote{FlexibleArray is discussed in Section +\ref{FlexibleArrayXmpPage} on page~\pageref{FlexibleArrayXmpPage}} is +a cross between a list \index{array!flexible} and a one-dimensional +array. Like a one-dimensional array, a flexible array occupies a fixed +block of storage. Its block of storage, however, has room to expand. +When it gets full, it grows (a new, larger block of storage is +allocated); when it has too much room, it contracts. + +Create a flexible array of three elements. +\spadcommand{f := flexibleArray [2, 7, -5]} +$$ +\left[ +2, 7, -5 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Insert some elements between the second and third elements. +\spadcommand{insert!(flexibleArray [11, -3],f,2)} +$$ +\left[ +2, {11}, -3, 7, -5 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Flexible arrays are used to implement ``heaps.'' A {\it heap} is an +example of a data structure called a {\it priority queue}, where +elements are ordered with respect to one another. A heap +\footnote{Heap is discussed in Section \ref{HeapXmpPage} on +page~\pageref{HeapXmpPage}} is organized so as to optimize insertion +and extraction of maximum elements. The {\bf extract!} operation +returns the maximum element of the heap, after destructively removing +that element and reorganizing the heap so that the next maximum +element is ready to be delivered. + +An easy way to create a heap is to apply the operation {\it heap} +to a list of values. +\spadcommand{h := heap [-4,7,11,3,4,-7]} +$$ +\left[ +{11}, 4, 7, -4, 3, -7 +\right] +$$ +\returnType{Type: Heap Integer} + +This loop extracts elements one-at-a-time from $h$ until the heap +is exhausted, returning the elements as a list in the order they were +extracted. +\spadcommand{[extract!(h) while not empty?(h)]} +$$ +\left[ +{11}, 7, 4, 3, -4, -7 +\right] +$$ +\returnType{Type: List Integer} + +A {\it binary tree} is a ``tree'' with at most two branches +\index{tree} per node: it is either empty, or else is a node +consisting of a value, and a left and right subtree (again, binary +trees). \footnote{BinarySearchTrees are discussed in Section +\ref{BinarySearchTreeXmpPage} on page~\pageref{BinarySearchTreeXmpPage}} +Examples of binary tree types are {\tt BinarySearchTree}, +{\tt PendantTree}, {\tt TournamentTree}, and {\tt BalancedBinaryTree}. + +A {\it binary search tree} is a binary tree such that, +\index{tree!binary search} for each node, the value of the node is +\index{binary search tree} greater than all values (if any) in the +left subtree, and less than or equal all values (if any) in the right +subtree. +\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} +$$ +\left[ +{\left[ 2, 3, 4 +\right]}, + 5, {\left[ 7, 9, {11} +\right]} +\right] +$$ +\returnType{Type: BinarySearchTree PositiveInteger} + +A {\it balanced binary tree} is useful for doing modular computations. +\index{balanced binary tree} Given a list $lm$ of moduli, +\index{tree!balanced binary} {\bf modTree}$(a,lm)$ produces +a balanced binary tree with the values $a \bmod m$ at its leaves. +\spadcommand{modTree(8,[2,3,5,7])} +$$ +\left[ +0, 2, 3, 1 +\right] +$$ +\returnType{Type: List Integer} + +A {\it set} is a collection of elements where duplication and order is +irrelevant. \footnote{Sets are discussed in Section \ref{SetXmpPage} +on page~\pageref{SetXmpPage}} Sets are always finite and have no +corresponding structure like streams for infinite collections. + +Create sets using braces ``\{`` and ``\}'' rather than brackets. + +\spadcommand{fs := set[1/3,4/5,-1/3,4/5]} +$$ +\left\{ +-{1 \over 3}, {1 \over 3}, {4 \over 5} +\right\} +$$ +\returnType{Type: Set Fraction Integer} + +A {\it multiset} is a set that keeps track of the number of duplicate +values. \footnote{Multisets are discussed in Section +\ref{MultisetXmpPage} on page~\pageref{MultisetXmpPage}} + +For all the primes $p$ between 2 and 1000, find the +distribution of $p \bmod 5$. +\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} +$$ +\left\{ +0, {{42} \mbox{\rm : } 3}, {{40} \mbox{\rm : } 1}, {{38} \mbox{\rm : +} 4}, {{47} \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset Integer} + +A {\it table} is conceptually a set of ``key--value'' pairs and is a +generalization of a multiset. For examples of tables, see +{\tt AssociationList}, {\tt HashTable}, {\tt KeyedAccessFile}, +{\tt Library}, {\tt SparseTable}, {\tt StringTable}, and {\tt Table}. The +domain {\tt Table(Key, Entry)} provides a general-purpose type for +tables with {\it values} of type $Entry$ indexed by {\it keys} of type +$Key$. + +Compute the above distribution of primes using tables. First, let +$t$ denote an empty table of keys and values, each of type {\tt Integer}. +\spadcommand{t : Table(Integer,Integer) := empty()} +$$ +{\rm table}() +$$ +\returnType{Type: Table(Integer,Integer)} + +We define a function {\bf howMany} to return the number of values +of a given modulus $k$ seen so far. It calls +{\bf search}$(k,t)$ which returns the number of values +stored under the key $k$ in table $t$, or {\tt ``failed''} +if no such value is yet stored in $t$ under $k$. + +In English, this says ``Define $howMany(k)$ as follows. +First, let $n$ be the value of {\it search}$(k,t)$. +Then, if $n$ has the value $"failed"$, return the value +$1$; otherwise return $n + 1$.'' +\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)} +\returnType{Type: Void} + +Run through the primes to create the table, then print the table. +The expression {\tt t.m := howMany(m)} updates the value in table $t$ +stored under key $m$. +\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t} +\begin{verbatim} + Compiling function howMany with type Integer -> Integer +\end{verbatim} +$$ +{\rm table } +\left( +{{2={47}}, {4={38}}, {1={40}}, {3={42}}, {0=1}} +\right) +$$ +\returnType{Type: Table(Integer,Integer)} + +A {\it record} is an example of an inhomogeneous collection of +objects.\footnote{See \ref{ugTypesRecords} on +page~\pageref{ugTypesRecords} for details.} A record consists of a +set of named {\it selectors} that can be used to access its +components. \index{Record@{\sf Record}} + +Declare that $daniel$ can only be +assigned a record with two prescribed fields. +\spadcommand{daniel : Record(age : Integer, salary : Float)} +\returnType{Type: Void} + +Give $daniel$ a value, using square brackets to enclose the values of +the fields. +\spadcommand{daniel := [28, 32005.12]} +$$ +\left[ +{age={28}}, {salary={32005.12}} +\right] +$$ +\returnType{Type: Record(age: Integer,salary: Float)} + +Give $daniel$ a raise. +\spadcommand{daniel.salary := 35000; daniel} +$$ +\left[ +{age={28}}, {salary={35000.0}} +\right] +$$ +\returnType{Type: Record(age: Integer,salary: Float)} + +A {\it union} is a data structure used when objects have multiple +types.\footnote{See \ref{ugTypesUnions} on +page~\pageref{ugTypesUnions} for details.} \index{Union@{\sf Union}} + +Let $dog$ be either an integer or a string value. +\spadcommand{dog: Union(licenseNumber: Integer, name: String)} +\returnType{Type: Void} + +Give $dog$ a name. +\spadcommand{dog := "Whisper"} +$$ +\mbox{\tt "Whisper"} +$$ +\returnType{Type: Union(name: String,...)} + +All told, there are over forty different data structures in Axiom. +Using the domain constructors described in Chapter \ref{ugDomains} on +page~\pageref{ugDomains}, you can add your own data structure or +extend an existing one. Choosing the right data structure for your +application may be the key to obtaining good performance. + +\section{Expanding to Higher Dimensions} +\label{ugIntroTwoDim} +To get higher dimensional aggregates, you can create one-dimensional +aggregates with elements that are themselves aggregates, for example, +lists of lists, one-dimensional arrays of lists of multisets, and so +on. For applications requiring two-dimensional homogeneous +aggregates, you will likely find {\it two-dimensional arrays} +\index{matrix} and {\it matrices} most useful. +\index{array!two-dimensional} + +The entries in {\tt TwoDimensionalArray} and {\tt Matrix} objects are +all the same type, except that those for {\tt Matrix} must belong to a +{\tt Ring}. You create and access elements in roughly the same way. +Since matrices have an understood algebraic structure, certain +algebraic operations are available for matrices but not for arrays. +Because of this, we limit our discussion here to {\tt Matrix}, that +can be regarded as an extension of {\tt TwoDimensionalArray}. See {\tt +TwoDimensionalArray} for more information about arrays. For more +information about Axiom's linear algebra facilities, see {\tt Matrix}, +{\tt Permanent}, {\tt SquareMatrix}, {\tt Vector}, see Section +\ref{ugProblemEigen} on page~\pageref{ugProblemEigen} (computation of +eigenvalues and eigenvectors), and Section \ref{ugProblemLinPolEqn} on +page~\pageref{ugProblemLinPolEqn} (solution of linear and polynomial +equations). + +You can create a matrix from a list of lists, \index{matrix!creating} +where each of the inner lists represents a row of the matrix. +\spadcommand{m := matrix([ [1,2], [3,4] ])} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +The ``collections'' construct (see \ref{ugLangIts} on +page~\pageref{ugLangIts}) is useful for creating matrices whose +entries are given by formulas. \index{matrix!Hilbert} +\spadcommand{matrix([ [1/(i + j - x) for i in 1..4] for j in 1..4])} +$$ +\left[ +\begin{array}{cccc} +-{1 \over {x -2}} & -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} \\ +-{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} \\ +-{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} \\ +-{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} & -{1 \over {x -8}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Polynomial Integer} + +Let $vm$ denote the three by three Vandermonde matrix. +\spadcommand{vm := matrix [ [1,1,1], [x,y,z], [x*x,y*y,z*z] ]} +$$ +\left[ +\begin{array}{ccc} +1 & 1 & 1 \\ +x & y & z \\ +{x \sp 2} & {y \sp 2} & {z \sp 2} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +Use this syntax to extract an entry in the matrix. + +\spadcommand{vm(3,3)} +$$ +z \sp 2 +$$ +\returnType{Type: Polynomial Integer} + +You can also pull out a {\bf row} or a {\bf column}. + +\spadcommand{column(vm,2)} +$$ +\left[ +1, y, {y \sp 2} +\right] +$$ +\returnType{Type: Vector Polynomial Integer} + +You can do arithmetic. + +\spadcommand{vm * vm} +$$ +\left[ +\begin{array}{ccc} +{{x \sp 2}+x+1} & {{y \sp 2}+y+1} & {{z \sp 2}+z+1} \\ +{{{x \sp 2} \ z}+{x \ y}+x} & {{{y \sp 2} \ z}+{y \sp 2}+x} & {{z \sp +3}+{y \ z}+x} \\ +{{{x \sp 2} \ {z \sp 2}}+{x \ {y \sp 2}}+{x \sp 2}} & {{{y \sp 2} \ {z \sp +2}}+{y \sp 3}+{x \sp 2}} & {{z \sp 4}+{{y \sp 2} \ z}+{x \sp 2}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +You can perform operations such as +{\bf transpose}, {\bf trace}, and {\bf determinant}. +\spadcommand{factor determinant vm} +$$ +{\left( y -x +\right)} +\ {\left( z -y +\right)} +\ {\left( z -x +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +\section{Writing Your Own Functions} +\label{ugIntroYou} +Axiom provides you with a very large library of predefined +operations and objects to compute with. You can use the Axiom +library of constructors to create new objects dynamically of quite +arbitrary complexity. For example, you can make lists of matrices of +fractions of polynomials with complex floating point numbers as +coefficients. Moreover, the library provides a wealth of operations +that allow you to create and manipulate these objects. + +For many applications, you need to interact with the interpreter and +write some Axiom programs to tackle your application. +Axiom allows you to write functions interactively, +\index{function} thereby effectively extending the system library. +Here we give a few simple examples, leaving the details to +Chapter \ref{ugUser} on page~\pageref{ugUser}. + +We begin by looking at several ways that you can define the +``factorial'' function in Axiom. The first way is to give a +\index{function!piece-wise definition} piece-wise definition of the +function. \index{piece-wise function definition} This method is best +for a general recurrence relation since the pieces are gathered +together and compiled into an efficient iterative function. +Furthermore, enough previously computed values are automatically saved +so that a subsequent call to the function can pick up from where it +left off. + +Define the value of {\bf fact} at $0$. +\spadcommand{fact(0) == 1} +\returnType{Type: Void} + +Define the value of {\bf fact}(n) for general $n$. +\spadcommand{fact(n) == n*fact(n-1)} +\returnType{Type: Void} + +Ask for the value at $50$. The resulting function created by +Axiom computes the value by iteration. + +\spadcommand{fact(50)} +\begin{verbatim} + Compiling function fact with type Integer -> Integer + Compiling function fact as a recurrence relation. +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A second definition uses an {\tt if-then-else} and recursion. +\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)} +\returnType{Type: Void} + +This function is less efficient than the previous version since +each iteration involves a recursive function call. +\spadcommand{fac(50)} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A third version directly uses iteration. +\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)} +\returnType{Type: Void} + +This is the least space-consumptive version. +\spadcommand{fa(50)} +\begin{verbatim} + Compiling function fac with type Integer -> Integer +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A final version appears to construct a large list and then reduces over +it with multiplication. +\spadcommand{f(n) == reduce(*,[i for i in 2..n])} +\returnType{Type: Void} + +In fact, the resulting computation is optimized into an efficient +iteration loop equivalent to that of the third version. +\spadcommand{f(50)} +\begin{verbatim} +Compiling function f with type + PositiveInteger -> PositiveInteger +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +The library version uses an algorithm that is different from the four +above because it highly optimizes the recurrence relation definition of +{\bf factorial}. + +\spadcommand{factorial(50)} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +You are not limited to one-line functions in Axiom. If you place your +function definitions in {\bf .input} files \index{file!input} (see +\ref{ugInOutIn} on page~\pageref{ugInOutIn}), you can have multi-line +functions that use indentation for grouping. + +Given $n$ elements, {\bf diagonalMatrix} creates an +$n$ by $n$ matrix with those elements down the diagonal. +This function uses a permutation matrix +that interchanges the $i$th and $j$th rows of a matrix +by which it is right-multiplied. + +This function definition shows a style of definition that can be used +in {\bf .input} files. Indentation is used to create {\sl blocks}: +sequences of expressions that are evaluated in sequence except as +modified by control statements such as {\tt if-then-else} and {\tt return}. + +\begin{verbatim} +permMat(n, i, j) == + m := diagonalMatrix + [(if i = k or j = k then 0 else 1) + for k in 1..n] + m(i,j) := 1 + m(j,i) := 1 + m +\end{verbatim} + +This creates a four by four matrix that interchanges the second and third +rows. +\spadcommand{p := permMat(4,2,3)} +\begin{verbatim} + Compiling function permMat with type (PositiveInteger, + PositiveInteger,PositiveInteger) -> Matrix Integer +\end{verbatim} +$$ +\left[ +\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Create an example matrix to permute. +\spadcommand{m := matrix [ [4*i + j for j in 1..4] for i in 0..3]} +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +5 & 6 & 7 & 8 \\ +9 & {10} & {11} & {12} \\ +{13} & {14} & {15} & {16} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Interchange the second and third rows of m. +\spadcommand{permMat(4,2,3) * m} +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +9 & {10} & {11} & {12} \\ +5 & 6 & 7 & 8 \\ +{13} & {14} & {15} & {16} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +A function can also be passed as an argument to another function, +which then applies the function or passes it off to some other +function that does. You often have to declare the type of a function +that has functional arguments. + +This declares {\bf t} to be a two-argument function that returns a +{\tt Float}. The first argument is a function that takes one +{\tt Float} argument and returns a {\tt Float}. + +\spadcommand{t : (Float -> Float, Float) -> Float} +\returnType{Type: Void} + +This is the definition of {\bf t}. + +\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2} +\returnType{Type: Void} + +We have not defined a {\bf cos} in the workspace. The one from the +Axiom library will do. + +\spadcommand{t(cos, 5.2058)} +$$ +1.0 +$$ +\returnType{Type: Float} + +Here we define our own (user-defined) function. +\spadcommand{cosinv(y) == cos(1/y)} +\returnType{Type: Void} + +Pass this function as an argument to {\bf t}. +\spadcommand{t(cosinv, 5.2058)} +$$ +1.7392237241\ 8005164925\ 4147684772\ 932520785 +$$ +\returnType{Type: Float} + +Axiom also has pattern matching capabilities for +\index{simplification} +simplification +\index{pattern matching} +of expressions and for defining new functions by rules. +For example, suppose that you want to apply regularly a transformation +that groups together products of radicals: +$$\sqrt{a}\sqrt{b} \mapsto \sqrt{ab}, \quad +(\forall a)(\forall b)$$ +Note that such a transformation is not generally correct. +Axiom never uses it automatically. + +Give this rule the name {\bf groupSqrt}. +\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))} +$$ +{ \%C \ {\sqrt {a}} \ {\sqrt {b}}} \mbox{\rm == } { \%C \ {\sqrt {{a \ +b}}}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Here is a test expression. +\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y}}}+ +{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x}}}\right)}\ {\sqrt{z}}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x}}\ {\sqrt{y}}}+ +{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ +y}+{x^2} +\end{array} +$$ +\returnType{Type: Expression Integer} + +The rule +{\bf groupSqrt} successfully simplifies the expression. +\spadcommand{groupSqrt a} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y \ z}}}+ +{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x \ z}}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x \ y}}}+ +{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ +y}+{x^2} +\end{array} +$$ +\returnType{Type: Expression Integer} + +\section{Polynomials} +\label{ugIntroVariables} +Polynomials are the commonly used algebraic types in symbolic +computation. \index{polynomial} Interactive users of Axiom +generally only see one type of polynomial: {\tt Polynomial(R)}. +This type represents polynomials in any number of unspecified +variables over a particular coefficient domain $R$. This type +represents its coefficients {\sl sparsely}: only terms with non-zero +coefficients are represented. + +In building applications, many other kinds of polynomial +representations are useful. Polynomials may have one variable or +multiple variables, the variables can be named or unnamed, the +coefficients can be stored sparsely or densely. So-called +``distributed multivariate polynomials'' store polynomials as +coefficients paired with vectors of exponents. This type is +particularly efficient for use in algorithms for solving systems of +non-linear polynomial equations. + +The polynomial constructor most familiar to the interactive user +is {\tt Polynomial}. +\spadcommand{(x**2 - x*y**3 +3*y)**2} +$$ +{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp +3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} +$$ +\returnType{Type: Polynomial Integer} + +If you wish to restrict the variables used, +{\tt UnivariatePolynomial} provides polynomials in one variable. + +\spadcommand{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)} +$$ +{{18} \ {x \sp 3}}+{{60} \ {x \sp 2}} -{{46} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The constructor {\tt MultivariatePolynomial} provides polynomials +in one or more specified variables. + +\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2} +$$ +{x \sp 4} -{2 \ {y \sp 3} \ {x \sp 3}}+{{\left( {y \sp 6}+{6 \ y} +\right)} +\ {x \sp 2}} -{6 \ {y \sp 4} \ x}+{9 \ {y \sp 2}} +$$ +\returnType{Type: MultivariatePolynomial([x,y],Integer)} + +You can change the way the polynomial appears by modifying the variable +ordering in the explicit list. +\spadcommand{m :: MPOLY([y,x],INT)} +$$ +{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp +3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} +$$ +\returnType{Type: MultivariatePolynomial([y,x],Integer)} + +The constructor {\tt DistributedMultivariatePolynomial} provides +polynomials in one or more specified variables with the monomials +ordered lexicographically. + +\spadcommand{m :: DMP([y,x],INT)} +$$ +{{y \sp 6} \ {x \sp 2}} -{6 \ {y \sp 4} \ x} -{2 \ {y \sp 3} \ {x \sp +3}}+{9 \ {y \sp 2}}+{6 \ y \ {x \sp 2}}+{x \sp 4} +$$ +\returnType{Type: DistributedMultivariatePolynomial([y,x],Integer)} + +The constructor +{\tt HomogeneousDistributedMultivariatePolynomial} is similar +except that the monomials are ordered by total order refined by +reverse lexicographic order. +\spadcommand{m :: HDMP([y,x],INT)} +$$ +{{y \sp 6} \ {x \sp 2}} -{2 \ {y \sp 3} \ {x \sp 3}} -{6 \ {y \sp 4} \ +x}+{x \sp 4}+{6 \ y \ {x \sp 2}}+{9 \ {y \sp 2}} +$$ +\returnType{Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)} + +More generally, the domain constructor +{\tt GeneralDistributedMultivariatePolynomial} allows the user to +provide an arbitrary predicate to define his own term ordering. These +last three constructors are typically used in Gr\"{o}bner basis +applications and +when a flat (that is, non-recursive) display is wanted and the term +ordering is critical for controlling the computation. + +\section{Limits} +\label{ugIntroCalcLimits} + +Axiom's {\bf limit} function is usually used to evaluate +limits of quotients where the numerator and denominator \index{limit} +both tend to zero or both tend to infinity. To find the limit of an +expression $f$ as a real variable $x$ tends to a limit +value $a$, enter {\tt limit(f, x=a)}. Use +{\bf complexLimit} if the variable is complex. Additional +information and examples of limits are in +Section \ref{ugProblemLimits} on page~\pageref{ugProblemLimits}. + +You can take limits of functions with parameters. +\index{limit!of function with parameters} +\spadcommand{g := csc(a*x) / csch(b*x)} +$$ +{\csc +\left( +{{a \ x}} +\right)} +\over {\csch +\left( +{{b \ x}} +\right)} +$$ +\returnType{Type: Expression Integer} + +As you can see, the limit is expressed in terms of the parameters. +\spadcommand{limit(g,x=0)} +$$ +b \over a +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +A variable may also approach plus or minus infinity: +\spadcommand{h := (1 + k/x)**x} +$$ +{{x+k} \over x} \sp x +$$ +\returnType{Type: Expression Integer} + +Use {\tt \%plusInfinity} and {\tt \%minusInfinity} to +denote $\infty$ and $-\infty$. +\spadcommand{limit(h,x=\%plusInfinity)} +$$ +e \sp k +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +A function can be defined on both sides of a particular value, but +may tend to different limits as its variable approaches that value from the +left and from the right. + +\spadcommand{limit(sqrt(y**2)/y,y = 0)} +$$ +\left[ +{leftHandLimit=-1}, {rightHandLimit=1} +\right] +$$ +\returnType{Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} + +As $x$ approaches $0$ along the real axis, {\tt exp(-1/x**2)} +tends to $0$. + +\spadcommand{limit(exp(-1/x**2),x = 0)} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +However, if $x$ is allowed to approach $0$ along any path in the +complex plane, the limiting value of {\tt exp(-1/x**2)} depends on the +path taken because the function has an essential singularity at $x=0$. +This is reflected in the error message returned by the function. +\spadcommand{complexLimit(exp(-1/x**2),x = 0)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\section{Series} +\label{ugIntroSeries} + +Axiom also provides power series. \index{series!power} By default, +Axiom tries to compute and display the first ten elements of a series. +Use {\tt )set streams calculate} to change the default value to +something else. For the purposes of this document, we have used this +system command to display fewer than ten terms. For more information +about working with series, see \ref{ugProblemSeries} on +page~\pageref{ugProblemSeries}. + +You can convert a functional expression to a power series by using the +operation {\bf series}. In this example, {\tt sin(a*x)} is +expanded in powers of $(x - 0)$, that is, in powers of $x$. +\spadcommand{series(sin(a*x),x = 0)} +$$ +{a \ x} -{{{a \sp 3} \over 6} \ {x \sp 3}}+{{{a \sp 5} \over {120}} \ {x +\sp 5}} -{{{a \sp 7} \over {5040}} \ {x \sp 7}}+{{{a \sp 9} \over {362880}} +\ {x \sp 9}} -{{{a \sp {11}} \over {39916800}} \ {x \sp {11}}}+{O +\left( +{{x \sp {12}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +This expression expands {\tt sin(a*x)} in powers of {\tt (x - \%pi/4)}. +\spadcommand{series(sin(a*x),x = \%pi/4)} +$$ +{\sin +\left({{{a \ \pi} \over 4}}\right)}+ +{a \ {\cos \left({{{a \ \pi} \over 4}} \right)} +\ {\left( x -{\pi \over 4} \right)}}- +\hbox{\hskip 2.0cm} +$$ +$$ +{{{{a \sp 2} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over 2} +\ {{\left( x -{\pi \over 4} \right)}\sp 2}} - +{{{{a \sp 3} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over 6} +\ {{\left( x -{\pi \over 4} \right)}\sp 3}} + +$$ +$$ +{{{{a \sp 4} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {24}} +\ {{\left( x -{\pi \over 4} \right)}\sp 4}} + +{{{{a \sp 5} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {120}} +\ {{\left( x -{\pi \over 4} \right)}\sp 5}} - +$$ +$$ +{{{{a \sp 6} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {720}} +\ {{\left( x -{\pi \over 4} \right)}\sp 6}} - +{{{{a \sp 7} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {5040}} +\ {{\left( x -{\pi \over 4} \right)}\sp 7}} + +$$ +$$ +{{{{a \sp 8} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {40320}} +\ {{\left( x -{\pi \over 4} \right)}\sp 8}} + +{{{{a \sp 9} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {362880}} +\ {{\left( x -{\pi \over 4} \right)}\sp 9}} - +$$ +$$ +{{{{a \sp {10}} \ {\sin \left({{{a \ \pi} \over 4}} \right)}} +\over {3628800}} +\ {{\left( x -{\pi \over 4} \right)}\sp {10}}} + +{O \left({{{\left( x -{\pi \over 4} \right)}\sp {11}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,pi/4)} + +Axiom provides \index{series!Puiseux} {\it Puiseux series:} +\index{Puiseux series} series with rational number exponents. The +first argument to {\bf series} is an in-place function that +computes the $n$-th coefficient. (Recall that the +``{\tt +->}'' is an infix operator meaning ``maps to.'') +\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x=0,4/3..,2)} +%%NOTE: the paper book shows O(x^4) but Axiom computes O(x^5) +$$ +{x \sp {4 \over 3}} -{{1 \over 6} \ {x \sp {{10} \over 3}}}+{O +\left( +{{x \sp 5}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +Once you have created a power series, you can perform arithmetic +operations on that series. We compute the Taylor expansion of $1/(1-x)$. +\index{series!Taylor} +\spadcommand{f := series(1/(1-x),x = 0)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +Compute the square of the series. +\spadcommand{f ** 2} +$$ +1+{2 \ x}+{3 \ {x \sp 2}}+{4 \ {x \sp 3}}+{5 \ {x \sp 4}}+{6 \ {x \sp +5}}+{7 \ {x \sp 6}}+{8 \ {x \sp 7}}+{9 \ {x \sp 8}}+{{10} \ {x \sp +9}}+{{11} \ {x \sp {10}}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +The usual elementary functions +({\bf log}, {\bf exp}, trigonometric functions, and so on) +are defined for power series. +\spadcommand{f := series(1/(1-x),x = 0)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{g := log(f)} +$$ +\begin{array}{@{}l} +x+ +{{1 \over 2} \ {x \sp 2}}+ +{{1 \over 3} \ {x \sp 3}}+ +{{1 \over 4} \ {x \sp 4}}+ +{{1 \over 5} \ {x \sp 5}}+ +{{1 \over 6} \ {x \sp 6}}+ +{{1 \over 7} \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{{1 \over 8} \ {x \sp 8}}+ +{{1 \over 9} \ {x \sp 9}}+ +{{1 \over {10}} \ {x \sp {10}}}+ +{{1 \over {11}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{exp(g)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +% Warning: currently there are (interpreter) problems with converting +% rational functions and polynomials to power series. + +Here is a way to obtain numerical approximations of +$e$ from the Taylor series expansion of {\bf exp}(x). +First create the desired Taylor expansion. +\spadcommand{f := taylor(exp(x))} +$$ +1+x+{{1 \over 2} \ {x \sp 2}}+{{1 \over 6} \ {x \sp 3}}+{{1 \over {24}} \ +{x \sp 4}}+{{1 \over {120}} \ {x \sp 5}}+{{1 \over {720}} \ {x \sp 6}} + +\hbox{\hskip 1.0cm} +$$ +$$ +{{1 +\over {5040}} \ {x \sp 7}} + +{{1 \over {40320}} \ {x \sp 8}}+{{1 \over +{362880}} \ {x \sp 9}}+{{1 \over {3628800}} \ {x \sp {10}}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + +Evaluate the series at the value $1.0$. +% Warning: syntax for evaluating power series may change. +As you see, you get a sequence of partial sums. +\spadcommand{eval(f,1.0)} +$$ +\left[ +{1.0}, {2.0}, {2.5}, {2.6666666666 666666667}, \hbox{\hskip 3.0cm} +\right. +$$ +$${2.7083333333 333333333}, {2.7166666666 666666667}, \hbox{\hskip 1.0cm} +$$ +$${2.7180555555 555555556}, {2.7182539682 53968254}, \hbox{\hskip 1.1cm} +$$ +$$\left. +{2.7182787698 412698413}, {2.7182815255 731922399}, \ldots +\hbox{\hskip 0.4cm} +\right] +$$ +\returnType{Type: Stream Expression Float} + +\section{Derivatives} +\label{ugIntroCalcDeriv} + +Use the Axiom function {\bf D} to differentiate an +\index{derivative} expression. \index{differentiation} + +To find the derivative of an expression $f$ with respect to a +variable $x$, enter {\bf D}(f, x). + +\spadcommand{f := exp exp x} +$$ +e \sp {e \sp x} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(f, x)} +$$ +{e \sp x} \ {e \sp {e \sp x}} +$$ +\returnType{Type: Expression Integer} + +An optional third argument $n$ in {\bf D} asks Axiom for the $n$-th +derivative of $f$. This finds the fourth derivative of $f$ with +respect to $x$. + +\spadcommand{D(f, x, 4)} +$$ +{\left( {{e \sp x} \sp 4}+{6 \ {{e \sp x} \sp 3}}+{7 \ {{e \sp x} \sp +2}}+{e \sp x} +\right)} +\ {e \sp {e \sp x}} +$$ +\returnType{Type: Expression Integer} + +You can also compute partial derivatives by specifying the order of +\index{differentiation!partial} +differentiation. +\spadcommand{g := sin(x**2 + y)} +$$ +\sin +\left( +{{y+{x \sp 2}}} +\right) +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(g, y)} +$$ +\cos +\left( +{{y+{x \sp 2}}} +\right) +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(g, [y, y, x, x])} +$$ +{4 \ {x \sp 2} \ {\sin +\left( +{{y+{x \sp 2}}} +\right)}} +-{2 \ {\cos +\left( +{{y+{x \sp 2}}} +\right)}} +$$ +\returnType{Type: Expression Integer} + +Axiom can manipulate the derivatives (partial and iterated) of +\index{differentiation!formal} expressions involving formal operators. +All the dependencies must be explicit. + +This returns $0$ since F (so far) does not explicitly depend on $x$. + +\spadcommand{D(F,x)} +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +Suppose that we have F a function of $x$, $y$, and $z$, +where $x$ and $y$ are themselves functions of $z$. + +Start by declaring that $F$, $x$, and $y$ are operators. +\index{operator} + +\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y} +$$ +y +$$ +\returnType{Type: BasicOperator} + +You can use F, $x$, and $y$ in expressions. + +\spadcommand{a := F(x z, y z, z**2) + x y(z+1)} +$$ +{x +\left( +{{y +\left( +{{z+1}} +\right)}} +\right)}+{F +\left( +{{x +\left( +{z} +\right)}, + {y +\left( +{z} +\right)}, + {z \sp 2}} +\right)} +$$ +\returnType{Type: Expression Integer} + +Differentiate formally with respect to $z$. +The formal derivatives appearing in $dadz$ are not just formal symbols, +but do represent the derivatives of $x$, $y$, and F. + +\spadcommand{dadz := D(a, z)} +$$ +\begin{array}{@{}l} +\displaystyle +{2 \ z \ {{F_{, 3}}\left({{x \left({z}\right)}, {y \left({z}\right)}, + {z^2}}\right)}}+{{{y_{\ }^{,}}\left({z}\right)}\ {{F_{, 2}}\left({{x +\left({z}\right)}, {y \left({z}\right)}, {z^2}}\right)}}+ + +\\ +\\ +\displaystyle +{{{x_{\ }^{,}}\left({z}\right)}\ {{F_{, 1}}\left({{x \left({z}\right)}, + {y \left({z}\right)}, {z^2}}\right)}}+{{{x_{\ }^{,}}\left({y +\left({z + 1}\right)}\right)}\ {{y_{\ }^{,}}\left({z + 1}\right)}} +\end{array} +$$ +\returnType{Type: Expression Integer} + +You can evaluate the above for particular functional values of +F, $x$, and $y$. If $x(z)$ is {\bf exp}(z) and $y(z)$ is {\bf log}(z+1), +then evaluates {\tt dadz}. + +\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^z}, + {\log \left({z + 1}\right)}, {z^2}}\right)}}+ +\\ +\\ +\displaystyle +{{F_{, 2}}\left({{e^ +z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ +\\ +\\ +\displaystyle +{{\left(z + 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log +\left({z + 1}\right)}, {z^2}}\right)}}+ z + 1 +\end{array} +\right)}\over{z + 1} +$$ +\returnType{Type: Expression Integer} + +You obtain the same result by first evaluating $a$ and +then differentiating. + +\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))} +$$ +{F +\left( +{{e \sp z}, {\log +\left( +{{z+1}} +\right)}, + {z \sp 2}} +\right)}+z+2 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(\%, z)} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^ +z}, {\log \left({z + 1}\right)}, {z^2}}\right)}}+ +\\ +\\ +\displaystyle +{{F_{, 2}}\left({{e^z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ +\\ +\\ +\displaystyle +{{\left(z ++ 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log \left({z ++ 1}\right)}, {z^2}}\right)}}+ z + 1 +\end{array} +\right)} +\over{z + 1} +$$ +\returnType{Type: Expression Integer} + +\section{Integration} +\label{ugIntroIntegrate} + +Axiom has extensive library facilities for integration. +\index{integration} + +The first example is the integration of a fraction with denominator +that factors into a quadratic and a quartic irreducible polynomial. +The usual partial fraction approach used by most other computer +algebra systems either fails or introduces expensive unneeded +algebraic numbers. + +We use a factorization-free algorithm. +\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} +$$ +{\arctan +\left( +{{{x \sp 3}+{3 \ {x \sp 2}}+{3 \ x}+1}} +\right)} +\over 3 +$$ +\returnType{Type: Union(Expression Integer,...)} + +When real parameters are present, the form of the integral can depend on +the signs of some expressions. + +Rather than query the user or make sign assumptions, Axiom returns +all possible answers. +\spadcommand{integrate(1/(x**2 + a),x)} +$$ +\left[ +{{\log +\left( +{{{{{\left( {x \sp 2} -a +\right)} +\ {\sqrt {-a}}}+{2 \ a \ x}} \over {{x \sp 2}+a}}} +\right)} +\over {2 \ {\sqrt {-a}}}}, {{\arctan +\left( +{{{x \ {\sqrt {a}}} \over a}} +\right)} +\over {\sqrt {a}}} +\right] +$$ +\returnType{Type: Union(List Expression Integer,...)} + +The {\bf integrate} operation generally assumes that all +parameters are real. The only exception is when the integrand has +complex valued quantities. + +If the parameter is complex instead of real, then the notion of sign +is undefined and there is a unique answer. You can request this +answer by ``prepending'' the word ``complex'' to the command name: + +\spadcommand{complexIntegrate(1/(x**2 + a),x)} +%%NOTE: the expression in the book is different but they differentiate +%%to exactly the same answer. +$$ +{{\log +\left( +{{{{x \ {\sqrt {-a}}}+a} \over {\sqrt {-a}}}} +\right)} +-{\log +\left( +{{{{x \ {\sqrt {-a}}} -a} \over {\sqrt {-a}}}} +\right)}} +\over {2 \ {\sqrt {-a}}} +$$ +\returnType{Type: Expression Integer} + +The following two examples illustrate the limitations of table-based +approaches. The two integrands are very similar, but the answer to +one of them requires the addition of two new algebraic numbers. + +This one is the easy one. +The next one looks very similar +but the answer is much more complicated. +\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} +$$ +{{\left( {{120} \ {b \sp 3} \ {x \sp 3}} -{{135} \ a \ {b \sp 2} \ {x +\sp 2}}+{{162} \ {a \sp 2} \ b \ x} -{{243} \ {a \sp 3}} +\right)} +\ {{\root {3} \of {{{b \ x}+a}}} \sp 2}} \over {{440} \ {b \sp 4}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Only an algorithmic approach is guaranteed to find what new constants +must be added in order to find a solution. + +\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} +$$ +\left( +\begin{array}{@{}l} +-{2 \ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log +\left( +{{{{\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp 2}}+{{{\root +{3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}}+a}} +\right)}}+ +\\ +\\ +\displaystyle +{4\ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log +\left( +{{{{{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}} -a}} +\right)}}+ +\\ +\\ +\displaystyle +{{12}\ {b \sp 2} \ {x \sp 2} \ {\arctan +\left( +{{{{2 \ {\sqrt {3}} \ {{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ +x}+a}}}}+{a \ {\sqrt {3}}}} \over {3 \ a}}} +\right)}}+ +\\ +\\ +\displaystyle +{{\left( +{{12} \ b \ x} -{9 \ a} +\right)} +\ {\sqrt {3}} \ {\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp +2}} +\end{array} +\right) +\over {{18} \ {a \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\root {3} \of +{a}}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Some computer algebra systems use heuristics or table-driven +approaches to integration. When these systems cannot determine the +answer to an integration problem, they reply ``I don't know.'' Axiom +uses an algorithm which is a {\sl decision procedure} for integration. +If Axiom returns the original integral that conclusively proves that +an integral cannot be expressed in terms of elementary functions. + +When Axiom returns an integral sign, it has proved that no answer +exists as an elementary function. + +\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} +$$ +\int \sp{\displaystyle x} {{{\log +\left( +{{{\sqrt {{b+{ \%Q \ a}}}}+1}} +\right)} +\over \%Q} \ {d \%Q}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Axiom can handle complicated mixed functions much beyond what you +can find in tables. + +Whenever possible, Axiom tries to express the answer using the +functions present in the integrand. + +\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} +%%NOTE: the book has the same answer with a trailing ``+4'' term. +%%This term is not generated by Axiom +$$ +{2 \ {\log +\left( +{{{-{2 \ {\cosh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)}} +-{2 \ x}} \over {{\sinh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)} +-{\cosh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)}}}} +\right)}} +-{2 \ {\sqrt {{x+b}}}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +A strong structure-checking algorithm in Axiom finds hidden algebraic +relationships between functions. + +\spadcommand{integrate(tan(atan(x)/3),x)} +%%NOTE: the book has a trailing ``+16'' term in the numerator +%%This is not generated by Axiom +$$ +\left( +\begin{array}{@{}l} +{8 \ {\log +\left( +{{{3 \ {{\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)} +\sp 2}} -1}} +\right)}} +-{3 \ {{\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)} +\sp 2}}+ +\\ +\\ +\displaystyle +{{18} \ x \ {\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)}} +\end{array} +\right) +\over {18} +$$ +\returnType{Type: Union(Expression Integer,...)} + +The discovery of this algebraic relationship is necessary for correct +integration of this function. +Here are the details: +\begin{enumerate} +\item +If $x=\tan t$ and $g=\tan (t/3)$ then the following +algebraic relation is true: $${g^3-3xg^2-3g+x=0}$$ +\item +Integrate $g$ using this algebraic relation; this produces: +$${{(24g^2 - 8)\log(3g^2 - 1) + (81x^2 + 24)g^2 + 72xg - 27x^2 - 16} +\over{54g^2 - 18}}$$ +\item +Rationalize the denominator, producing: +$${8\log(3g^2-1) - 3g^2 + 18xg + 16} \over {18}$$ +Replace $g$ by the initial definition +$g = \tan(\arctan(x)/3)$ +to produce the final result. +\end{enumerate} + +This is an example of a mixed function where +the algebraic layer is over the transcendental one. +\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)} +$$ +-{{2 \ {\sqrt {{{\log +\left( +{x} +\right)}+x}}}} +\over {{\log +\left( +{x} +\right)}+x}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +While incomplete for non-elementary functions, Axiom can +handle some of them. +\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} +$$ +{{{\left( {\erf +\left( +{x} +\right)} +-1 +\right)} +\ {\sqrt {\pi}} \ {\log +\left( +{{{{\erf +\left( +{x} +\right)} +-1} \over {{\erf +\left( +{x} +\right)}+1}}} +\right)}} +-{2 \ {\sqrt {\pi}}}} \over {{8 \ {\erf +\left( +{x} +\right)}} +-8} +$$ +\returnType{Type: Union(Expression Integer,...)} + +More examples of Axiom's integration capabilities are discussed in +Section \ref{ugProblemIntegration} on page~\pageref{ugProblemIntegration}. + +\section{Differential Equations} +\label{ugIntroDiffEqns} +The general approach used in integration also carries over to the +solution of linear differential equations. + +Let's solve some differential equations. +Let $y$ be the unknown function in terms of $x$. +\spadcommand{y := operator 'y} +$$ +y +$$ +\returnType{Type: BasicOperator} + +Here we solve a third order equation with polynomial coefficients. +\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4} +$$ +{{{x \sp 3} \ {{y \sb {{\ }} \sp {,,,}} +\left( +{x} +\right)}}+{{x +\sp 2} \ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}} +-{2 \ x \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}}+{2 +\ {y +\left( +{x} +\right)}}}={2 +\ {x \sp 4}} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x)} +%%NOTE: the book has a different solution and it appears to be +%%less complicated than this one. +$$ +\begin{array}{@{}l} +\left[ +{particular={{{x \sp 5} -{{10} \ {x \sp 3}}+{{20} \ {x \sp 2}}+4} \over +{{15} \ x}}}, +\right. +\\ +\\ +\displaystyle +\left. +{basis={\left[ {{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+1} +\over x}, {{{x \sp 3} -1} \over x}, {{{x \sp 3} -{3 \ {x \sp 2}} -1} +\over x} +\right]}} +\right] +\end{array} +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} + + +Here we find all the algebraic function solutions of the equation. +\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0} +$$ +{{{\left( {x \sp 2}+1 +\right)} +\ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}}+{3 +\ x \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}}+{y +\left( +{x} +\right)}}=0 +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x)} +$$ +\left[ +{particular=0}, +{basis={\left[ {1 \over {\sqrt {{{x \sp 2}+1}}}}, +{{\log +\left( +{{{\sqrt {{{x \sp 2}+1}}} -x}} +\right)} +\over {\sqrt {{{x \sp 2}+1}}}} +\right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} + +Coefficients of differential equations can come from arbitrary +constant fields. For example, coefficients can contain algebraic +numbers. + +This example has solutions whose logarithmic derivative is an +algebraic function of degree two. + +\spadcommand{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x} +$$ +{2 \ {x \sp 3} \ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}}+{3 +\ {x \sp 2} \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}} +-{2 \ {y +\left( +{x} +\right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{solve(eq,y,x).basis} +$$ +\left[ +{e \sp {\left( -{2 \over {\sqrt {x}}} +\right)}}, + {e \sp {2 \over {\sqrt {x}}}} +\right] +$$ +\returnType{Type: List Expression Integer} + +Here's another differential equation to solve. +\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)} +$$ +{{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}={{y +\left( +{x} +\right)} +\over {{{y +\left( +{x} +\right)} +\ {\log +\left( +{{y +\left( +{x} +\right)}} +\right)}}+x}} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x)} +$$ +{{{y +\left( +{x} +\right)} +\ {{\log +\left( +{{y +\left( +{x} +\right)}} +\right)} +\sp 2}} -{2 \ x}} \over {2 \ {y +\left( +{x} +\right)}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Rather than attempting to get a closed form solution of +a differential equation, you instead might want to find an +approximate solution in the form of a series. + +Let's solve a system of nonlinear first order equations and get a +solution in power series. Tell Axiom that $x$ is also an +operator. + +\spadcommand{x := operator 'x} +$$ +x +$$ +\returnType{Type: BasicOperator} + +Here are the two equations forming our system. +\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2} +$$ +{{x \sb {{\ }} \sp {,}} +\left( +{t} +\right)}={{{x +\left( +{t} +\right)} +\sp 2}+1} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)} +$$ +{{y \sb {{\ }} \sp {,}} +\left( +{t} +\right)}={{x +\left( +{t} +\right)} +\ {y +\left( +{t} +\right)}} +$$ +\returnType{Type: Equation Expression Integer} + +We can solve the system around $t = 0$ with the initial +conditions $x(0) = 0$ and $y(0) = 1$. Notice that since +we give the unknowns in the order $[x, y]$, the answer is a list +of two series in the order +$[{\rm series\ for\ }x(t), {\rm series\ for\ }y(t)]$. + +\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])} +$$ +\left[ +{\ t+ +{{1 \over 3} \ {t \sp 3}}+ +{{2 \over {15}} \ {t \sp 5}}+ +{{{17} \over {315}} \ {t \sp 7}}+ +{{{62} \over {2835}} \ {t \sp 9}}+ +{O \left({{t \sp {11}}} \right)}}, +\right. +\hbox{\hskip 2.0cm} +$$ +$$ +\hbox{\hskip 0.4cm} +\left. +{1+ +{{1 \over 2} \ {t \sp 2}}+ +{{5 \over {24}} \ {t \sp 4}}+ +{{{61} \over {720}} \ {t \sp 6}}+ +{{{277} \over {8064}} \ {t \sp 8}}+ +{{{50521} \over {3628800}} \ {t \sp {10}}}+ +{O \left({{t \sp {11}}}\right)}} +\right] +$$ +\returnType{Type: List UnivariateTaylorSeries(Expression Integer,t,0)} + +\section{Solution of Equations} +\label{ugIntroSolution} +Axiom also has state-of-the-art algorithms for the solution of +systems of polynomial equations. When the number of equations and +unknowns is the same, and you have no symbolic coefficients, you can +use {\bf solve} for real roots and {\bf complexSolve} for +complex roots. In each case, you tell Axiom how accurate you +want your result to be. All operations in the {\it solve} family +return answers in the form of a list of solution sets, where each +solution set is a list of equations. + +A system of two equations involving a symbolic parameter $t$. +\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]} +\returnType{Type: Void} + +Find the real roots of $S(19)$ with +rational arithmetic, correct to within $1/10^{20}$. +\spadcommand{solve(S(19),1/10**20)} +$$ +\left[ +{\left[ {y=5}, {x=-{{2451682632253093442511} \over +{295147905179352825856}}} +\right]}, +\right. +$$ +$$ +\left. +{\left[ {y=5}, {x={{2451682632253093442511} \over +{295147905179352825856}}} +\right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Fraction Integer} + +Find the complex roots of $S(19)$ with floating +point coefficients to $20$ digits accuracy in the mantissa. + +\spadcommand{complexSolve(S(19),10.e-20)} +$$ +\left[ +{\left[ {y={5.0}}, {x={8.3066238629 180748526}} \right]}, +\right. +$$ +$$ +{\left[ {y={5.0}}, {x=-{8.3066238629 180748526}} \right]}, +$$ +$$ +\left. +{\left[ {y=-{{3.0} \ i}}, {x={1.0}} \right]}, +{\left[ {y={{3.0} \ i}}, {x={1.0}} \right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Complex Float} + +If a system of equations has symbolic coefficients and you want +a solution in radicals, try {\bf radicalSolve}. +\spadcommand{radicalSolve(S(a),[x,y])} +$$ +\left[ +{\left[ {x=-{\sqrt {{a+{50}}}}}, {y=5} \right]}, +{\left[ {x={\sqrt {{a+{50}}}}}, {y=5} \right]}, +\right. +$$ +$$ +\hbox{\hskip 0.7cm} +\left. +{\left[ {x=1}, {y={\sqrt {{{-a+1} \over 2}}}} \right]}, +{\left[ {x=1}, {y=-{\sqrt {{{-a+1} \over 2}}}} \right]} +\right] +$$ +\returnType{Type: List List Equation Expression Integer} + +For systems of equations with symbolic coefficients, you can apply +{\bf solve}, listing the variables that you want Axiom to +solve for. For polynomial equations, a solution cannot usually be +expressed solely in terms of the other variables. Instead, the +solution is presented as a ``triangular'' system of equations, where +each polynomial has coefficients involving only the succeeding +variables. This is analogous to converting a linear system of +equations to ``triangular form''. + +A system of three equations in five variables. +\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]} +$$ +\left[ +{z -y+{x \sp 2}}, {{{x \sp 2} \ z} -{b \ y}+{x \sp 4}}, {{{y \sp 2} \ +z} -{b \ x} -a} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +Solve the system for unknowns $[x,y,z]$, +reducing the solution to triangular form. +\spadcommand{solve(eqns,[x,y,z])} +$$ +\left[ +{\left[ {x=-{a \over b}}, {y=0}, {z=-{{a \sp 2} \over {b \sp 2}}} +\right]}, +\right. +\hbox{\hskip 10.0cm} +$$ +$$ +\left. +\begin{array}{@{}l} +\left[ +{x={{{z \sp 3}+{2 \ b \ {z \sp 2}}+{{b \sp 2} \ z} -a} \over b}}, +{y={z+b}}, +\right. +\hbox{\hskip 10.0cm} +\\ +\\ +\displaystyle +{z \sp 6}+{4 \ b \ {z \sp 5}}+ +{6 \ {b \sp 2} \ {z \sp 4}}+ +{{\left( {4 \ {b \sp 3}} -{2 \ a} \right)}\ {z \sp 3}}+ +{{\left( {b \sp 4} -{4 \ a \ b} \right)}\ {z \sp 2}}- +\hbox{\hskip 4.0cm} +\\ +\\ +\displaystyle +\left. +{2 \ a \ {b \sp 2} \ z} -{b \sp 3}+{a \sp 2}=0 +\right] +\end{array} +\right] +\hbox{\hskip 7.0cm} +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\section{System Commands} +\label{ugIntroSysCmmands} +We conclude our tour of Axiom with a brief discussion of +{\it system commands}. System commands are special statements +that start with a closing parenthesis ({\tt )}). They are used +to control or display your Axiom environment, start the +HyperDoc system, issue operating system commands and leave +Axiom. For example, {\tt )system} is used to issue commands +to the operating system from Axiom. Here +is a brief description of some of these commands. For more +information on specific commands, see Appendix A +on page~\pageref{ugSysCmd}. + +Perhaps the most important user command is the {\tt )clear all} +command that initializes your environment. Every section and +subsection in this document has an invisible {\tt )clear all} that is +read prior to the examples given in the section. {\tt )clear all} +gives you a fresh, empty environment with no user variables defined +and the step number reset to $1$. The {\tt )clear} command +can also be used to selectively clear values and properties of system +variables. + +Another useful system command is {\tt )read}. A preferred way to +develop an application in Axiom is to put your interactive +commands into a file, say {\bf my.input} file. To get Axiom to +read this file, you use the system command {\tt )read my.input}. +If you need to make changes to your approach or definitions, go into +your favorite editor, change {\bf my.input}, then {\tt )read +my.input} again. + +Other system commands include: {\tt )history}, to display +previous input and/or output lines; {\tt )display}, to display +properties and values of workspace variables; and {\tt )what}. + +Issue {\tt )what} to get a list of Axiom objects that +contain a given substring in their name. +\spadcommand{)what operations integrate} +\begin{verbatim} + +Operations whose names satisfy the above pattern(s): + +HermiteIntegrate algintegrate complexIntegrate +expintegrate extendedIntegrate fintegrate +infieldIntegrate integrate internalIntegrate +internalIntegrate0 lazyGintegrate lazyIntegrate +lfintegrate limitedIntegrate monomialIntegrate +nagPolygonIntegrate palgintegrate pmComplexintegrate +pmintegrate primintegrate tanintegrate + +To get more information about an operation such as +limitedIntegrate , issue the command )display op limitedIntegrate + +\end{verbatim} + +\subsection{Undo} +\label{ugIntroUndo} +A useful system command is {\tt )undo}. Sometimes while computing +interactively with Axiom, you make a mistake and enter an +incorrect definition or assignment. Or perhaps you need to try one of +several alternative approaches, one after another, to find the best +way to approach an application. For this, you will find the +{\it undo} facility of Axiom helpful. + +System command {\tt )undo n} means ``undo back to step +$n$''; it restores the values of user variables to those that +existed immediately after input expression $n$ was evaluated. +Similarly, {\tt )undo -n} undoes changes caused by the last +$n$ input expressions. Once you have done an {\tt )undo}, +you can continue on from there, or make a change and {\bf redo} all +your input expressions from the point of the {\tt )undo} forward. +The {\tt )undo} is completely general: it changes the environment +like any user expression. Thus you can {\tt )undo} any previous +undo. + +Here is a sample dialogue between user and Axiom. + +``Let me define +two mutually dependent functions $f$ and $g$ piece-wise.'' +\spadcommand{f(0) == 1; g(0) == 1} +\returnType{Type: Void} + +``Here is the general term for $f$.'' +\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)} +\returnType{Type: Void} + +``And here is the general term for $g$.'' +\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)} +\returnType{Type: Void} + +``What is value of $f(3)$?'' +\spadcommand{f(3)} +$$ +-{x \sp 3}+{{\left( e+{{1 \over 3} \ d} +\right)} +\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} +-{{1 \over 9} \ {d \sp 2}} +\right)} +\ x}+{{1 \over 8} \ {e \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +``Hmm, I think I want to define $f$ differently. +Undo to the environment right after I defined $f$.'' +\spadcommand{)undo 2} + +``Here is how I think I want $f$ to be defined instead.'' +\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)} +\begin{verbatim} + 1 old definition(s) deleted for function or rule f +\end{verbatim} +\returnType{Type: Void} + +Redo the computation from expression $3$ forward. +\spadcommand{)undo )redo} +\begin{verbatim} +g(n) == -x*f(n-1) + d/3*g(n-1) + + Type: Void +f(3) + + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function f with type Integer -> Polynomial Fraction + Integer + Compiling function f as a recurrence relation. + ++++ |*1;f;1;G82322;AUX| redefined + ++++ |*1;f;1;G82322| redefined +\end{verbatim} +$$ +-{x \sp 3}+{d \ {x \sp 2}} -{{1 \over 3} \ {d \sp 2} \ x}+{{1 \over {27}} +\ {d \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +``I want my old definition of +$f$ after all. Undo the undo and restore +the environment to that immediately after $(4)$.'' +\spadcommand{)undo 4} + +``Check that the value of $f(3)$ is restored.'' +\spadcommand{f(3)} +\begin{verbatim} + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function f with type Integer -> Polynomial Fraction + Integer + Compiling function f as a recurrence relation. + ++++ |*1;f;1;G82322;AUX| redefined + ++++ |*1;f;1;G82322| redefined +\end{verbatim} +$$ +-{x \sp 3}+{{\left( e+{{1 \over 3} \ d} +\right)} +\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} +-{{1 \over 9} \ {d \sp 2}} +\right)} +\ x}+{{1 \over 8} \ {e \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +After you have gone off on several tangents, then backtracked to +previous points in your conversation using {\tt )undo}, you might +want to save all the ``correct'' input commands you issued, +disregarding those undone. The system command {\tt )history +)write mynew.input} writes a clean straight-line program onto the file +{\bf mynew.input} on your disk. + +\section{Graphics} +\label{ugIntroGraphics} +Axiom has a two- and three-dimensional drawing and rendering +\index{graphics} package that allows you to draw, shade, color, +rotate, translate, map, clip, scale and combine graphic output of +Axiom computations. The graphics interface is capable of +plotting functions of one or more variables and plotting parametric +surfaces. Once the graphics figure appears in a window, move your +mouse to the window and click. A control panel appears immediately +and allows you to interactively transform the object. + +This is an example of Axiom's two-dimensional plotting. +From the 2D Control Panel you can rescale the plot, turn axes and units +on and off and save the image, among other things. +This PostScript image was produced by clicking on the +{\bf PS} 2D Control Panel button. +\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} +% window was 256 x 256 +%\epsffile[72 72 300 300]{ps/rose-1.ps} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=176, bbury=186]{ps/p28a.eps} +\caption{$J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$} +\label{tpdhere} +\end{figure} + +This is an example of Axiom's three-dimensional plotting. +It is a monochrome graph of the complex arctangent +function. +The image displayed was rotated and had the ``shade'' and ``outline'' +display options set from the 3D Control Panel. +The PostScript output was produced by clicking on the +{\bf save} 3D Control Panel button and then +clicking on the {\bf PS} button. +See Section \ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric} +for more details and examples of Axiom's numeric and graphics capabilities. + +\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} +% window was 256 x 256 +%\epsffile[72 72 285 285]{ps/atan-1.ps} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=175, bbury=185]{ps/p28b.eps} +\caption{atan} +\label{tpdhere1} +\end{figure} + +An exhibit of Axiom images is given later. For a description of the +commands and programs that produced these figures, see +\ref{ugAppGraphics} on page~\pageref{ugAppGraphics}. PostScript +\index{PostScript} output is available so that Axiom images can be +printed.\footnote{PostScript is a trademark of Adobe Systems +Incorporated, registered in the United States.} See \ref{ugGraph} on +page~\pageref{ugGraph} for more examples and details about using +Axiom's graphics facilities. + +This concludes your tour of Axiom. +To disembark, issue the system command {\tt )quit} to leave Axiom +and return to the operating system. + +\setcounter{chapter}{1} + +\chapter{Using Types and Modes} +\begin{quote} +Only recently have I begun to realize that the problem is not merely +one of technical mastery or the competent application of the rules +\ldots +but that there is actually something else which is guiding these +rules. It actually involves a different level of mastery. It's quite +a different process to do it right; and every single act that you +do can be done in that sense well or badly. But even assuming that +you have got the technical part clear, the creation of this quality +is a much more complicated process of the most utterly absorbing and +fascinating dimensions. It is in fact a major creative or artistic +act -- every single little thing you do -- \ldots + +-- Christopher Alexander + +(from Patterns of Software by Richard Gabriel) + +\end{quote} +\label{ugTypes} + +In this chapter we look at the key notion of {\it type} and its +generalization {\it mode}. We show that every Axiom object has a type +that determines what you can do with the object. In particular, we +explain how to use types to call specific functions from particular +parts of the library and how types and modes can be used to create new +objects from old. We also look at {\tt Record} and {\tt Union} types +and the special type {\tt Any}. Finally, we give you an idea of how +Axiom manipulates types and modes internally to resolve ambiguities. + +\section{The Basic Idea} +\label{ugTypesBasic} + +The Axiom world deals with many kinds of objects. There are +mathematical objects such as numbers and polynomials, data structure +objects such as lists and arrays, and graphics objects such as points +and graphic images. Functions are objects too. + +Axiom organizes objects using the notion of domain of computation, or +simply {\it domain}. Each domain denotes a class of objects. The +class of objects it denotes is usually given by the name of the +domain: {\tt Integer} for the integers, {\tt Float} for floating-point +numbers, and so on. The convention is that the first letter of a +domain name is capitalized. Similarly, the domain +{\tt Polynomial(Integer)} denotes ``polynomials with integer +coefficients.'' Also, {\tt Matrix(Float)} denotes ``matrices with +floating-point entries.'' + +Every basic Axiom object belongs to a unique domain. The integer $3$ +belongs to the domain {\tt Integer} and the polynomial $x + 3$ belongs +to the domain {\tt Polynomial(Integer)}. The domain of an object is +also called its {\it type}. Thus we speak of ``the type +{\tt Integer}'' and ``the type {\tt Polynomial(Integer)}.'' + +After an Axiom computation, the type is displayed toward the +right-hand side of the page (or screen). +\spadcommand{-3} +$$ +-3 +$$ +\returnType{Type: Integer} + +Here we create a rational number but it looks like the last result. +The type however tells you it is different. You cannot identify the +type of an object by how Axiom displays the object. +\spadcommand{-3/1} +$$ +-3 +$$ +\returnType{Type: Fraction Integer} + +When a computation produces a result of a simpler type, Axiom leaves +the type unsimplified. Thus no information is lost. +\spadcommand{x + 3 - x} +$$ +3 +$$ +\returnType{Type: Polynomial Integer} + +This seldom matters since Axiom retracts the answer to the +simpler type if it is necessary. +\spadcommand{factorial(\%)} +$$ +6 +$$ +\returnType{Type: Expression Integer} + +When you issue a positive number, the type {\tt PositiveInteger} is +printed. Surely, $3$ also has type {\tt Integer}! The curious reader +may now have two questions. First, is the type of an object not +unique? Second, how is {\tt PositiveInteger} related to {\tt +Integer}? +\spadcommand{3} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +Any domain can be refined to a {\it subdomain} by a membership +{\tt predicate}. A {\tt predicate} is a function that, when applied to an +object of the domain, returns either {\tt true} or {\tt false}. For +example, the domain {\tt Integer} can be refined to the subdomain +{\tt PositiveInteger}, the set of integers $x$ such that $x > 0$, by giving +the Axiom predicate $x +-> x > 0$. Similarly, Axiom can define +subdomains such as ``the subdomain of diagonal matrices,'' ``the +subdomain of lists of length two,'' ``the subdomain of monic +irreducible polynomials in $x$,'' and so on. Trivially, any domain is +a subdomain of itself. + +While an object belongs to a unique domain, it can belong to any +number of subdomains. Any subdomain of the domain of an object can be +used as the {\it type} of that object. The type of $3$ is indeed both +{\tt Integer} and {\tt PositiveInteger} as well as any other subdomain +of integer whose predicate is satisfied, such as ``the prime +integers,'' ``the odd positive integers between 3 and 17,'' and so on. + +\subsection{Domain Constructors} +\label{ugTypesBasicDomainCons} + +In Axiom, domains are objects. You can create them, pass them to +functions, and, as we'll see later, test them for certain properties. + +In Axiom, you ask for a value of a function by applying its name +to a set of arguments. + +To ask for ``the factorial of $7$'' you enter this expression to +Axiom. This applies the function {\tt factorial} to the value $7$ to +compute the result. +\spadcommand{factorial(7)} +$$ +5040 +$$ +\returnType{Type: PositiveInteger} + +Enter the type {\tt Polynomial (Integer)} as an expression to Axiom. +This looks much like a function call as well. It is! The result is +appropriately stated to be of type {\tt Domain}, which according to +our usual convention, denotes the class of all domains. +\spadcommand{Polynomial(Integer)} +$$ +\mbox{\rm Polynomial Integer} +$$ +\returnType{Type: Domain} + +The most basic operation involving domains is that of building a new +domain from a given one. To create the domain of ``polynomials over +the integers,'' Axiom applies the function {\tt Polynomial} to the +domain {\tt Integer}. A function like {\tt Polynomial} is called a +{\it domain constructor} or, \index{constructor!domain} more simply, a +{\it constructor}. A domain constructor is a function that creates a +domain. An argument to a domain constructor can be another domain or, +in general, an arbitrary kind of object. {\tt Polynomial} takes a +single domain argument while {\tt SquareMatrix} takes a positive +integer as an argument to give its dimension and a domain argument to +give the type of its components. + +What kinds of domains can you use as the argument to {\tt Polynomial} +or {\tt SquareMatrix} or {\tt List}? Well, the first two are +mathematical in nature. You want to be able to perform algebraic +operations like ``{\tt +}'' and ``{\tt *}'' on polynomials and square +matrices, and operations such as {\bf determinant} on square +matrices. So you want to allow polynomials of integers {\it and} +polynomials of square matrices with complex number coefficients and, +in general, anything that ``makes sense.'' At the same time, you don't +want Axiom to be able to build nonsense domains such as ``polynomials +of strings!'' + +In contrast to algebraic structures, data structures can hold any kind +of object. Operations on lists such as \spadfunFrom{insert}{List}, +\spadfunFrom{delete}{List}, and \spadfunFrom{concat}{List} just +manipulate the list itself without changing or operating on its +elements. Thus you can build {\tt List} over almost any datatype, +including itself. + +Create a complicated algebraic domain. +\spadcommand{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))} +$$ +\mbox{\rm List List Matrix Polynomial Complex Fraction Integer} +$$ +\returnType{Type: Domain} + +Try to create a meaningless domain. +\spadcommand{Polynomial(String)} +\begin{verbatim} + Polynomial String is not a valid type. +\end{verbatim} + +Evidently from our last example, Axiom has some mechanism that tells +what a constructor can use as an argument. This brings us to the +notion of {\it category}. As domains are objects, they too have a +domain. The domain of a domain is a category. A category is simply a +type whose members are domains. + +A common algebraic category is {\tt Ring}, the class of all domains +that are ``rings.'' A ring is an algebraic structure with constants +$0$ and $1$ and operations \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, +and \spadopFrom{*}{Ring}. These operations are assumed ``closed'' +with respect to the domain, meaning that they take two objects of the +domain and produce a result object also in the domain. The operations +are understood to satisfy certain ``axioms,'' certain mathematical +principles providing the algebraic foundation for rings. For example, +the {\it additive inverse axiom} for rings states: \begin{center} +Every element $x$ has an additive inverse $y$ such that $x + y = 0$. +\end{center} The prototypical example of a domain that is a ring is +the integers. Keep them in mind whenever we mention {\tt Ring}. + +Many algebraic domain constructors such as {\tt Complex}, +{\tt Polynomial}, {\tt Fraction}, take rings as arguments and return rings +as values. You can use the infix operator ``$has$'' to ask a domain +if it belongs to a particular category. + +All numerical types are rings. Domain constructor {\tt Polynomial} +builds ``the ring of polynomials over any other ring.'' +\spadcommand{Polynomial(Integer) has Ring} +$$ +{\rm true} +$$ +\returnType{Type: Boolean} + +Constructor {\tt List} never produces a ring. +\spadcommand{List(Integer) has Ring} +$$ +{\rm false} +$$ +\returnType{Type: Boolean} + +The constructor {\tt Matrix(R)} builds ``the domain of all matrices +over the ring $R$.'' This domain is never a ring since the operations +``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' on matrices of arbitrary +shapes are undefined. +\spadcommand{Matrix(Integer) has Ring} +$$ +{\rm false} +$$ +\returnType{Type: Boolean} + +Thus you can never build polynomials over matrices. +\spadcommand{Polynomial(Matrix(Integer))} +\begin{verbatim} + Polynomial Matrix Integer is not a valid type. +\end{verbatim} + +Use {\tt SquareMatrix(n,R)} instead. For any positive integer $n$, it +builds ``the ring of $n$ by $n$ matrices over $R$.'' +\spadcommand{Polynomial(SquareMatrix(7,Complex(Integer)))} +$$ +\mbox{\rm Polynomial SquareMatrix(7,Complex Integer)} +$$ +\returnType{Type: Domain} + +Another common category is {\tt Field}, the class of all fields. +\index{field} A field is a ring with additional operations. For +example, a field has commutative multiplication and a closed operation +\spadopFrom{/}{Field} for the division of two elements. {\tt Integer} +is not a field since, for example, $3/2$ does not have an integer +result. The prototypical example of a field is the rational numbers, +that is, the domain {\tt Fraction(Integer)}. In general, the +constructor {\tt Fraction} takes an IntegralDomain, which is a ring +with additional properties, as an argument and returns a field. +\footnote{Actually, the argument domain must have some additional +so as to belong to the category {\tt IntegralDomain}} +Other domain constructors, such as {\tt Complex}, build fields only if their +argument domain is a field. + +The complex integers (often called the ``Gaussian integers'') do not form +a field. +\spadcommand{Complex(Integer) has Field} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +But fractions of complex integers do. +\spadcommand{Fraction(Complex(Integer)) has Field} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The algebraically equivalent domain of complex rational numbers is a field +since domain constructor {\tt Complex} produces a field whenever its +argument is a field. +\spadcommand{Complex(Fraction(Integer)) has Field} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The most basic category is {\tt Type}. \index{Type} It denotes the +class of all domains and subdomains. Note carefully that {\tt Type} +does not denote the class of all types. The type of all categories is +{\tt Category}. The type of {\tt Type} itself is undefined. Domain +constructor {\tt List} is able to build ``lists of elements from +domain $D$'' for arbitrary $D$ simply by requiring that $D$ belong to +category {\tt Type}. + +Now, you may ask, what exactly is a category? \index{category} Like +domains, categories can be defined in the Axiom language. A category +is defined by three components: +% +\begin{enumerate} +\item a name (for example, {\tt Ring}), +used to refer to the class of domains that the category represents; +\item a set of operations, used to refer to the operations that +the domains of this class support +(for example, ``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' for rings); and +\item an optional list of other categories that this category extends. +\end{enumerate} +% +This last component is a new idea. And it is key to the design of +Axiom! Because categories can extend one another, they form +hierarchies. Detailed charts showing the category hierarchies +in Axiom are displayed in Appendix (TPDHERE). There you see +that all categories are extensions of {\tt Type} and that {\tt Field} +is an extension of {\tt Ring}. + +The operations supported by the domains of a category are called the +{\sl exports} of that category because these are the operations made +available for system-wide use. The exports of a domain of a given +category are not only the ones explicitly mentioned by the category. +Since a category extends other categories, the operations of these +other categories---and all categories these other categories +extend---are also exported by the domains. + +For example, polynomial domains belong to {\tt PolynomialCategory}. +This category explicitly mentions some twenty-nine operations on +polynomials, but it extends eleven other categories (including +{\tt Ring}). As a result, the current system has over one hundred +operations on polynomials. + +If a domain belongs to a category that extends, say, {\tt Ring}, it is +convenient to say that the domain exports {\tt Ring}. The name of the +category thus provides a convenient shorthand for the list of +operations exported by the category. Rather than listing operations +such as \spadopFrom{+}{Ring} and \spadopFrom{*}{Ring} of {\tt Ring} +each time they are needed, the definition of a type simply asserts +that it exports category {\tt Ring}. + +The category name, however, is more than a shorthand. The name +{\tt Ring}, in fact, implies that the operations exported by rings are +required to satisfy a set of ``axioms'' associated with the name +{\tt Ring}. This subtle but important feature distinguishes Axiom from +other abstract datatype designs. + +Why is it not correct to assume that some type is a ring if it exports +all of the operations of {\tt Ring}? Here is why. Some languages +such as {\bf APL} \index{APL} denote the {\tt Boolean} constants +{\tt true} and {\tt false} by the integers $1$ and $0$ respectively, then use +``{\tt +}'' and ``{\tt *}'' to denote the logical operators {\bf or} and +{\bf and}. But with these definitions {\tt Boolean} is not a +ring since the additive inverse axiom is violated. That is, there is +no inverse element $a$ such that $1 + a = 0$, or, in the usual terms: +{\tt true or a = false}. This alternative definition of {\tt Boolean} +can be easily and correctly implemented in Axiom, since {\tt Boolean} +simply does not assert that it is of category {\tt Ring}. This +prevents the system from building meaningless domains such as +{\tt Polynomial(Boolean)} and then wrongfully applying algorithms that +presume that the ring axioms hold. + +Enough on categories. To learn more about them, see Chapter +\ref{ugCategories} on page~\pageref{ugCategories}. +We now return to our discussion of domains. + +Domains {\it export} a set of operations to make them available for +system-wide use. {\tt Integer}, for example, exports the operations +\spadopFrom{+}{Integer} and \spadopFrom{=}{Integer} given by the +signatures \spadopFrom{+}{Integer}: +\spadsig{(Integer,Integer)}{Integer} and \spadopFrom{=}{Integer}: +\spadsig{(Integer,Integer)}{Boolean}, respectively. Each of these +operations takes two {\tt Integer} arguments. The +\spadopFrom{+}{Integer} operation also returns an {\tt Integer} but +\spadopFrom{=}{Integer} returns a {\tt Boolean}: {\tt true} or {\tt false}. +The operations exported by a domain usually manipulate objects of the +domain---but not always. + +The operations of a domain may actually take as arguments, and return +as values, objects from any domain. For example, {\tt Fraction +(Integer)} exports the operations \spadopFrom{/}{Fraction}: +\spadsig{(Integer,Integer)}{Fraction(Integer)} and +\spadfunFrom{characteristic}{Fraction}: +\spadsig{}{NonNegativeInteger}. + +Suppose all operations of a domain take as arguments and return as +values, only objects from {\it other} domains. \index{package} This +kind of domain \index{constructor!package} is what Axiom calls a {\it +package}. + +A package does not designate a class of objects at all. Rather, a +package is just a collection of operations. Actually the bulk of the +Axiom library of algorithms consists of packages. The facilities for +factorization; integration; solution of linear, polynomial, and +differential equations; computation of limits; and so on, are all +defined in packages. Domains needed by algorithms can be passed to a +package as arguments or used by name if they are not ``variable.'' +Packages are useful for defining operations that convert objects of +one type to another, particularly when these types have different +parameterizations. As an example, the package {\tt PolynomialFunction2(R,S)} +defines operations that convert polynomials +over a domain $R$ to polynomials over $S$. To convert an object from +{\tt Polynomial(Integer)} to {\tt Polynomial(Float)}, Axiom builds the +package {\tt PolynomialFunctions2(Integer,Float)} in order to create +the required conversion function. (This happens ``behind the scenes'' +for you: see \ref{ugTypesConvert} on page~\pageref{ugTypesConvert} +for details on how to convert objects.) + +Axiom categories, domains and packages and all their contained +functions are written in the Axiom programming language and have been +compiled into machine code. This is what comprises the Axiom +{\it library}. We will show you how to use these +domains and their functions and how to write your own functions. + +\section{Writing Types and Modes} +\label{ugTypesWriting} + +We have already seen in the last section \ref{ugTypesBasic} on +page~\pageref{ugTypesBasic} several examples of types. Most of these +examples had either no arguments (for example, {\tt Integer}) or one +argument (for example, {\tt Polynomial (Integer)}). In this section +we give details about writing arbitrary types. We then define modes +and discuss how to write them. We conclude the section with a +discussion on constructor abbreviations. + +When might you need to write a type or mode? You need to do so when +you declare variables. +\spadcommand{a : PositiveInteger} +\returnType{Type: Void} + +You need to do so when you declare functions +(See Section \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}), +\spadcommand{f : Integer -> String} +\returnType{Type: Void} + +You need to do so when you convert an object from one type to another +(See Section \ref{ugTypesConvert} on page~\pageref{ugTypesConvert}). +\spadcommand{factor(2 :: Complex(Integer))} +$$ +-{i \ {{\left( 1+i +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +\spadcommand{(2 = 3)\$Integer} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +You need to do so when you give computation target type information +(See Section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}). +\spadcommand{(2 = 3)@Boolean} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\subsection{Types with No Arguments} +\label{ugTypesWritingZero} + +A constructor with no arguments can be written either +\index{type!using parentheses} with or without +\index{parentheses!using with types} trailing opening and closing +parentheses ``{\tt ()}''. + +\begin{center} +{\tt Boolean()} is the same as {\tt Boolean} \\ +{\tt Integer()} is the same as {\tt Integer} \\ +{\tt String()} is the same as {\tt String} \\ +{\tt Void()} is the same as {\tt Void} +\end{center} + +It is customary to omit the parentheses. + +\subsection{Types with One Argument} +\label{ugTypesWritingOne} + +A constructor with one argument can frequently be +\index{type!using parentheses} written with no +\index{parentheses!using with types} parentheses. Types nest from +right to left so that {\tt Complex Fraction Polynomial Integer} +is the same as {\tt Complex (Fraction (Polynomial (Integer)))}. +You need to use parentheses to force the application of a constructor +to the correct argument, but you need not use any more than is necessary +to remove ambiguities. + +Here are some guidelines for using parentheses (they are possibly slightly +more restrictive than they need to be). + +If the argument is an expression like $2 + 3$ +then you must enclose the argument in parentheses. +\spadcommand{e : PrimeField(2 + 3)} +\returnType{Type: Void} + +If the type is to be used with package calling +then you must enclose the argument in parentheses. +\spadcommand{content(2)\$Polynomial(Integer)} +$$ +2 +$$ +\returnType{Type: Integer} + +Alternatively, you can write the type without parentheses +then enclose the whole type expression with parentheses. +\spadcommand{content(2)\$(Polynomial Complex Fraction Integer)} +$$ +2 +$$ +\returnType{Type: Complex Fraction Integer} + +If you supply computation target type information +(See Section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}) +then you should enclose the argument in parentheses. +\spadcommand{(2/3)@Fraction(Polynomial(Integer))} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +If the type itself has parentheses around it and we are not in the +case of the first example above, then the parentheses can usually be +omitted. +\spadcommand{(2/3)@Fraction(Polynomial Integer)} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +If the type is used in a declaration and the argument is a single-word +type, integer or symbol, then the parentheses can usually be omitted. +\spadcommand{(d,f,g) : Complex Polynomial Integer} +\returnType{Type: Void} + +\subsection{Types with More Than One Argument} +\label{ugTypesWritingMore} + +If a constructor \index{type!using parentheses} has more than +\index{parentheses!using with types} one argument, you must use +parentheses. Some examples are \\ + +{\tt UnivariatePolynomial(x, Float)} \\ +{\tt MultivariatePolynomial([z,w,r], Complex Float)} \\ +{\tt SquareMatrix(3, Integer)} \\ +{\tt FactoredFunctions2(Integer,Fraction Integer)} + +\subsection{Modes} +\label{ugTypesWritingModes} + +A {\it mode} is a type that possibly is a question mark ({\tt ?}) or +contains one in an argument position. For example, the following are +all modes.\\ + +{\tt ?} \\ +{\tt Polynomial ?} \\ +{\tt Matrix Polynomial ?} \\ +{\tt SquareMatrix(3,?)} \\ +{\tt Integer} \\ +{\tt OneDimensionalArray(Float)} + +As is evident from these examples, a mode is a type with a part that +is not specified (indicated by a question mark). Only one ``{\tt ?}'' is +allowed per mode and it must appear in the most deeply nested argument +that is a type. Thus {\tt ?(Integer)}, {\tt Matrix(? (Polynomial))}, +{\tt SquareMatrix(?, Integer)} (it requires a numeric argument) +and {\tt SquareMatrix(?, ?)} are all +invalid. The question mark must take the place of a domain, not data. +This rules out, for example, the two {\tt SquareMatrix} expressions. + +Modes can be used for declarations (See Section \ref{ugTypesDeclare} +on page~\pageref{ugTypesDeclare}) and conversions (Section +\ref{ugTypesConvert} on page~\pageref{ugTypesConvert}). However, you +cannot use a mode for package calling or giving target type information. + +\subsection{Abbreviations} +\label{ugTypesWritingAbbr} + +Every constructor has an abbreviation that +\index{abbreviation!constructor} you can freely +\index{constructor!abbreviation} substitute for the constructor name. +In some cases, the abbreviation is nothing more than the capitalized +version of the constructor name. + +\boxed{4.6in}{ + +\vskip 0.1cm +Aside from allowing types to be written more concisely, abbreviations +are used by Axiom to name various system files for constructors (such +as library filenames, test input files and example files). Here are +some common abbreviations. + +\begin{center} +\begin{tabular}{ll} +\small{\tt COMPLEX} abbreviates {\tt Complex} & +\small{\tt DFLOAT} abbreviates {\tt DoubleFloat} \\ +\small{\tt EXPR} abbreviates {\tt Expression} & +\small{\tt FLOAT} abbreviates {\tt Float} \\ +\small{\tt FRAC} abbreviates {\tt Fraction} & +\small{\tt INT} abbreviates {\tt Integer} \\ +\small{\tt MATRIX} abbreviates {\tt Matrix} & +\small{\tt NNI} abbreviates {\tt NonNegativeInteger} \\ +\small{\tt PI} abbreviates {\tt PositiveInteger} & +\small{\tt POLY} abbreviates {\tt Polynomial} \\ +\small{\tt STRING} abbreviates {\tt String} & +\small{\tt UP} abbreviates {\tt UnivariatePolynomial}\\ +\end{tabular} +\end{center} +\vskip 0.1cm +} + +You can combine both full constructor names and abbreviations in a +type expression. Here are some types using abbreviations. + +\begin{center} +\begin{tabular}{rcl} +{\tt POLY INT} & is the same as & {\tt Polynomial(INT)} \\ +{\tt POLY(Integer)} & is the same as & {\tt Polynomial(Integer)} \\ +{\tt POLY(Integer)} & is the same as & {\tt Polynomial(INT)} \\ +{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt Fraction Complex Integer} \\ +{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt FRAC(Complex Integer)} +\end{tabular} +\end{center} + +There are several ways of finding the names of constructors and their +abbreviations. For a specific constructor, use {\tt )abbreviation +query}. \index{abbreviation} You can also use the {\tt )what} system +command to see the names and abbreviations of constructors. +\index{what} For more information about {\tt )what}, see +\ref{ugSysCmdwhat} on page~\pageref{ugSysCmdwhat}. + +{\tt )abbreviation query} can be abbreviated (no pun intended) to +{\tt )abb q}. +\spadcommand{)abb q Integer} +\begin{verbatim} + INT abbreviates domain Integer +\end{verbatim} + +The {\tt )abbreviation query} command lists the constructor name if +you give the abbreviation. Issue {\tt )abb q} if you want to see the +names and abbreviations of all Axiom constructors. +\spadcommand{)abb q DMP} +\begin{verbatim} + DMP abbreviates domain DistributedMultivariatePolynomial +\end{verbatim} + +Issue this to see all packages whose +names contain the string ``ode''. \index{what packages} +\spadcommand{)what packages ode} +\begin{verbatim} +---------------------- Packages ----------------------- + +Packages with names matching patterns: + ode + + EXPRODE ExpressionSpaceODESolver + FCPAK1 FortranCodePackage1 + GRAY GrayCode + LODEEF ElementaryFunctionLODESolver + NODE1 NonLinearFirstOrderODESolver + ODECONST ConstantLODE + ODEEF ElementaryFunctionODESolver + ODEINT ODEIntegration + ODEPAL PureAlgebraicLODE + ODERAT RationalLODE + ODERED ReduceLODE + ODESYS SystemODESolver + ODETOOLS ODETools + UTSODE UnivariateTaylorSeriesODESolver + UTSODETL UTSodetools +\end{verbatim} + +\section{Declarations} +\label{ugTypesDeclare} + +A {\it declaration} is an expression used to restrict the type of +values that can be assigned to variables. A colon ``{\tt :}'' is always +used after a variable or list of variables to be declared. + +\boxed{4.6in}{ +\vskip 0.1cm +For a single variable, the syntax for declaration is +\begin{center} +{\it variableName $:$ typeOrMode} +\end{center} + +For multiple variables, the syntax is +\begin{center} +{\tt ($\hbox{\it variableName}_{1}$, $\hbox{\it variableName}_{2}$, +\ldots $\hbox{\it variableName}_{N}$): {\it typeOrMode}} +\end{center} +\vskip 0.1cm +} + +You can always combine a declaration with an assignment. When you do, +it is equivalent to first giving a declaration statement, then giving +an assignment. For more information on assignment, see +Section \ref{ugIntroAssign} on page~\pageref{ugIntroAssign} and +Section \ref{ugLangAssign} on page~\pageref{ugLangAssign}. +To see how to declare your own functions, +see \ref{ugUserDeclare} on page~\pageref{ugUserDeclare}. + +This declares one variable to have a type. +\spadcommand{a : Integer} +\returnType{Type: Void} + +This declares several variables to have a type. +\spadcommand{(b,c) : Integer} +\returnType{Type: Void} + +$a$, $b$ and $c$ can only hold integer values. +\spadcommand{a := 45} +$$ +45 +$$ +\returnType{Type: Integer} + +If a value cannot be converted to a declared type, +an error message is displayed. +\spadcommand{b := 4/5} +\begin{verbatim} + + Cannot convert right-hand side of assignment + 4 + - + 5 + + to an object of the type Integer of the left-hand side. +\end{verbatim} + +This declares a variable with a mode. +\spadcommand{n : Complex ?} +\returnType{Type: Void} + +This declares several variables with a mode. +\spadcommand{(p,q,r) : Matrix Polynomial ?} +\returnType{Type: Void} + +This complex object has integer real and imaginary parts. +\spadcommand{n := -36 + 9 * \%i} +$$ +-{36}+{9 \ i} +$$ +\returnType{Type: Complex Integer} + +This complex object has fractional symbolic real and imaginary parts. +\spadcommand{n := complex(4/(x + y),y/x)} +$$ +{4 \over {y+x}}+{{y \over x} \ i} +$$ +\returnType{Type: Complex Fraction Polynomial Integer} + +This matrix has entries that are polynomials with integer +coefficients. +\spadcommand{p := [ [1,2],[3,4],[5,6] ]} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 \\ +5 & 6 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +This matrix has a single entry that is a polynomial with +rational number coefficients. +\spadcommand{q := [ [x - 2/3] ]} +$$ +\left[ +\begin{array}{c} +{x -{2 \over 3}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Fraction Integer} + +This matrix has entries that are polynomials with complex integer +coefficients. + +\spadcommand{r := [ [1-\%i*x,7*y+4*\%i] ]} +$$ +\left[ +\begin{array}{cc} +{-{i \ x}+1} & {{7 \ y}+{4 \ i}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Complex Integer} + +Note the difference between this and the next example. +This is a complex object with polynomial real and imaginary parts. + +\spadcommand{f : COMPLEX POLY ? := (x + y*\%i)**2} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +This is a polynomial with complex integer coefficients. The objects +are convertible from one to the other. See \ref{ugTypesConvert} on +page~\pageref{ugTypesConvert} for more information. + +\spadcommand{g : POLY COMPLEX ? := (x + y*\%i)**2} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +\section{Records} +\label{ugTypesRecords} + +A {\tt Record} is an object composed of one or more other objects, +\index{Record} each of which is referenced \index{selector!record} +with \index{record!selector} a {\it selector}. Components can all +belong to the same type or each can have a different type. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Record} type is \begin{center} {\tt +Record($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, +$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, +$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} \end{center} You must be +careful if a selector has the same name as a variable in the +workspace. If this occurs, precede the selector name by a single +\index{quote} quote.\\ +} + +Record components are implicitly ordered. All the components of a +record can be set at once by assigning the record a bracketed {\it +tuple} of values of the proper length. For example: +\spadcommand{r : Record(a:Integer, b: String) := [1, "two"]} +$$ +\left[ +{a=1}, {b= \mbox{\tt "two"} } +\right] +$$ +\returnType{Type: Record(a: Integer,b: String)} +To access a component of a record $r$, write the name $r$, followed by +a period, followed by a selector. + +The object returned by this computation is a record with two components: a +$quotient$ part and a $remainder$ part. +\spadcommand{u := divide(5,2)} +$$ +\left[ +{quotient=2}, {remainder=1} +\right] +$$ +\returnType{Type: Record(quotient: Integer,remainder: Integer)} + +This is the quotient part. +\spadcommand{u.quotient} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +This is the remainder part. +\spadcommand{u.remainder} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +You can use selector expressions on the left-hand side of an assignment +to change destructively the components of a record. +\spadcommand{u.quotient := 8978} +$$ +8978 +$$ +\returnType{Type: PositiveInteger} + +The selected component $quotient$ has the value $8978$, which is what +is returned by the assignment. Check that the value of $u$ was +modified. +\spadcommand{u} +$$ +\left[ +{quotient={8978}}, {remainder=1} +\right] +$$ +\returnType{Type: Record(quotient: Integer,remainder: Integer)} + +Selectors are evaluated. Thus you can use variables that evaluate to +selectors instead of the selectors themselves. +\spadcommand{s := 'quotient} +$$ +quotient +$$ +\returnType{Type: Variable quotient} + +Be careful! A selector could have the same name as a variable in the +workspace. If this occurs, precede the selector name by a single +quote, as in $u.'quotient$. \index{selector!quoting} +\spadcommand{divide(5,2).s} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Here we declare that the value of $bd$ has two components: a string, +to be accessed via {\tt name}, and an integer, to be accessed via +{\tt birthdayMonth}. +\spadcommand{bd : Record(name : String, birthdayMonth : Integer)} +\returnType{Type: Void} + +You must initially set the value of the entire {\tt Record} at once. +\spadcommand{bd := ["Judith", 3]} +$$ +\left[ +{name= \mbox{\tt "Judith"} }, {birthdayMonth=3} +\right] +$$ +\returnType{Type: Record(name: String,birthdayMonth: Integer)} + +Once set, you can change any of the individual components. +\spadcommand{bd.name := "Katie"} +$$ +\mbox{\tt "Katie"} +$$ +\returnType{Type: String} + +Records may be nested and the selector names can be shared at +different levels. +\spadcommand{r : Record(a : Record(b: Integer, c: Integer), b: Integer)} +\returnType{Type: Void} + +The record $r$ has a $b$ selector at two different levels. +Here is an initial value for $r$. +\spadcommand{r := [ [1,2], 3 ]} +$$ +\left[ +{a={\left[ {b=1}, {c=2} +\right]}}, + {b=3} +\right] +$$ +\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} + +This extracts the $b$ component from the $a$ component of $r$. +\spadcommand{r.a.b} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +This extracts the $b$ component from $r$. +\spadcommand{r.b} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +You can also use spaces or parentheses to refer to {\tt Record} +components. This is the same as $r.a$. +\spadcommand{r(a)} +$$ +\left[ +{b=1}, {c=2} +\right] +$$ +\returnType{Type: Record(b: Integer,c: Integer)} +This is the same as $r.b$. +\spadcommand{r b} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +This is the same as $r.b := 10$. +\spadcommand{r(b) := 10} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +Look at $r$ to make sure it was modified. +\spadcommand{r} +$$ +\left[ +{a={\left[ {b=1}, {c=2} +\right]}}, + {b={10}} +\right] +$$ +\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} + +\section{Unions} +\label{ugTypesUnions} + +Type {\tt Union} is used for objects that can be of any of a specific +finite set of types. \index{Union} Two versions of unions are +available, one with selectors (like records) and one without. +\index{union} + +\subsection{Unions Without Selectors} +\label{ugTypesUnionsWOSel} + +The declaration $x : Union(Integer, String, Float)$ states that $x$ +can have values that are integers, strings or ``big'' floats. If, for +example, the {\tt Union} object is an integer, the object is said to +belong to the {\tt Integer} {\it branch} of the {\tt Union}. Note +that we are being a bit careless with the language here. Technically, +the type of $x$ is always {\tt Union(Integer, String, Float)}. If it +belongs to the {\tt Integer} branch, $x$ may be converted to an object +of type {\tt Integer}. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Union} type without selectors is +\begin{center} +{\tt Union($\hbox{\it type}_{1}$, $\hbox{\it type}_{2}$, +\ldots, $\hbox{\it type}+{N}$)} +\end{center} +The types in a union without selectors must be distinct.\\ +} + +It is possible to create unions like {\tt Union(Integer, PositiveInteger)} +but they are difficult to work with because of the overlap in the branch +types. See below for the rules Axiom uses for converting something into +a union object. + +The {\tt case} infix \index{case} operator returns a {\tt Boolean} and can +be used to determine the branch in which an object lies. + +This function displays a message stating in which branch of the +{\tt Union} the object (defined as $x$ above) lies. + +\begin{verbatim} +sayBranch(x : Union(Integer,String,Float)) : Void == + output + x case Integer => "Integer branch" + x case String => "String branch" + "Float branch" +\end{verbatim} + +This tries {\bf sayBranch} with an integer. +\spadcommand{sayBranch 1} +\begin{verbatim} +Compiling function sayBranch with type Union(Integer,String,Float) + -> Void + Integer branch +\end{verbatim} +\returnType{Type: Void} + +This tries {\bf sayBranch} with a string. +\spadcommand{sayBranch "hello"} +\begin{verbatim} + String branch +\end{verbatim} +\returnType{Type: Void} + +This tries {\bf sayBranch} with a floating-point number. +\spadcommand{sayBranch 2.718281828} +\begin{verbatim} + Float branch +\end{verbatim} +\returnType{Type: Void} + +There are two things of interest about this particular +example to which we would like to draw your attention. +\begin{enumerate} +\item Axiom normally converts a result to the target value +before passing it to the function. +If we left the declaration information out of this function definition +then the {\bf sayBranch} call would have been attempted with an +{\tt Integer} rather than a {\tt Union}, and an error would have +resulted. +\item The types in a {\tt Union} are searched in the order given. +So if the type were given as + +%\noindent +{\tt sayBranch(x: Union(String,Integer,Float,Any)): Void} + +then the result would have been ``String branch'' because there +is a conversion from {\tt Integer} to {\tt String}. +\end{enumerate} + +Sometimes {\tt Union} types can have extremely long names. Axiom +therefore abbreviates the names of unions by printing the type of the +branch first within the {\tt Union} and then eliding the remaining +types with an ellipsis ({\tt ...}). + +Here the {\tt Integer} branch is displayed first. Use ``{\tt ::}'' to +create a {\tt Union} object from an object. +\spadcommand{78 :: Union(Integer,String)} +$$ +78 +$$ +\returnType{Type: Union(Integer,...)} + +Here the {\tt String} branch is displayed first. +\spadcommand{s := "string" :: Union(Integer,String)} +$$ +\mbox{\tt "string"} +$$ +\returnType{Type: Union(String,...)} + +Use {\tt typeOf} to see the full and actual {\tt Union} type. \index{typeOf} +\spadcommand{typeOf s} +$$ +Union(Integer,String) +$$ +\returnType{Type: Domain} + +A common operation that returns a union is \spadfunFrom{exquo}{Integer} +which returns the ``exact quotient'' if the quotient is exact, +\spadcommand{three := exquo(6,2)} +$$ +3 +$$ +\returnType{Type: Union(Integer,...)} + +and {\tt "failed"} if the quotient is not exact. +\spadcommand{exquo(5,2)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +A union with a {\tt "failed"} is frequently used to indicate the failure +or lack of applicability of an object. As another example, assign an +integer a variable $r$ declared to be a rational number. +\spadcommand{r: FRAC INT := 3} +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +The operation \spadfunFrom{retractIfCan}{Fraction} tries to retract +the fraction to the underlying domain {\tt Integer}. It produces a +union object. Here it succeeds. +\spadcommand{retractIfCan(r)} +$$ +3 +$$ +\returnType{Type: Union(Integer,...)} + +Assign it a rational number. +\spadcommand{r := 3/2} +$$ +3 \over 2 +$$ +\returnType{Type: Fraction Integer} + +Here the retraction fails. +\spadcommand{retractIfCan(r)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\subsection{Unions With Selectors} +\label{ugTypesUnionsWSel} + +Like records (\ref{ugTypesRecords} on page~\pageref{ugTypesRecords}), +you can write {\tt Union} types \index{selector!union} with selectors. +\index{union!selector} + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Union} type with selectors is +\begin{center} +{\tt Union($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, +$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, +$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} +\end{center} +You must be careful if a selector has the same name as a variable in +the workspace. If this occurs, precede the selector name by a single +\index{quote} quote. \index{selector!quoting} It is an error to use a +selector that does not correspond to the branch of the {\tt Union} in +which the element actually lies. \\ +} + +Be sure to understand the difference between records and unions with +selectors. \index{union!difference from record} Records can have more +than one component and the selectors are used to refer to the +components. \index{record!difference from union} Unions always have +one component but the type of that one component can vary. An object +of type {\tt Record(a: Integer, b: Float, c: String)} contains an +integer {\it and} a float {\it and} a string. An object of type +{\tt Union(a: Integer, b: Float, c: String)} contains an integer +{\it or} a float {\it or} a string. + +Here is a version of the {\bf sayBranch} function (cf. +\ref{ugTypesUnionsWOSel} on page~\pageref{ugTypesUnionsWOSel}) that +works with a union with selectors. It displays a message stating in +which branch of the {\tt Union} the object lies. + +\begin{verbatim} +sayBranch(x:Union(i:Integer,s:String,f:Float)):Void== + output + x case i => "Integer branch" + x case s => "String branch" + "Float branch" +\end{verbatim} + +Note that {\tt case} uses the selector name as its right-hand argument. +\index{case} If you accidentally use the branch type on the right-hand +side of {\tt case}, {\tt false} will be returned. + +Declare variable $u$ to have a union type with selectors. +\spadcommand{u : Union(i : Integer, s : String)} +\returnType{Type: Void} + +Give an initial value to $u$. +\spadcommand{u := "good morning"} +$$ +\mbox{\tt "good morning"} +$$ +\returnType{Type: Union(s: String,...)} + +Use $case$ to determine in which branch of a {\tt Union} an object lies. +\spadcommand{u case i} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{u case s} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +To access the element in a particular branch, use the selector. +\spadcommand{u.s} +$$ +\mbox{\tt "good morning"} +$$ +\returnType{Type: String} + +\section{The ``Any'' Domain} +\label{ugTypesAnyNone} + +With the exception of objects of type {\tt Record}, all Axiom data +structures are homogenous, that is, they hold objects all of the same +type. \index{Any} If you need to get around this, you can use type +{\tt Any}. Using {\tt Any}, for example, you can create lists whose +elements are integers, rational numbers, strings, and even other +lists. + +Declare $u$ to have type {\tt Any}. +\spadcommand{u: Any} +\returnType{Type: Void} + +Assign a list of mixed type values to $u$ +\spadcommand{u := [1, 7.2, 3/2, x**2, "wally"]} +$$ +\left[ +1, {7.2}, {3 \over 2}, {x \sp 2}, \mbox{\tt "wally"} +\right] +$$ +\returnType{Type: List Any} + +When we ask for the elements, Axiom displays these types. +\spadcommand{u.1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Actually, these objects belong to {\tt Any} but Axiom +automatically converts them to their natural types for you. +\spadcommand{u.3} +$$ +3 \over 2 +$$ +\returnType{Type: Fraction Integer} + +Since type {\tt Any} can be anything, it can only belong to type +{\tt Type}. Therefore it cannot be used in algebraic domains. +\spadcommand{v : Matrix(Any)} +\begin{verbatim} + Matrix Any is not a valid type. +\end{verbatim} + +Perhaps you are wondering how Axiom internally represents objects of +type {\tt Any}. An object of type {\tt Any} consists not only a data +part representing its normal value, but also a type part (a +{\it badge}) giving \index{badge} its type. For example, the value $1$ of +type {\tt PositiveInteger} as an object of type {\tt Any} internally +looks like $[1,{\tt PositiveInteger()}]$. + +When should you use {\tt Any} instead of a {\tt Union} type? For a +{\tt Union}, you must know in advance exactly which types you are +going to +allow. For {\tt Any}, anything that comes along can be accommodated. + +\section{Conversion} +\label{ugTypesConvert} + +\boxed{4.6in}{ +\vskip 0.1cm +Conversion is the process of changing an object of one type into an +object of another type. The syntax for conversion is: +$$ +{\it object} {\tt ::} {\it newType} +$$ +} + +By default, $3$ has the type {\tt PositiveInteger}. +\spadcommand{3} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +We can change this into an object of type {\tt Fraction Integer} +by using ``{\tt ::}''. +\spadcommand{3 :: Fraction Integer} +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +A {\it coercion} is a special kind of conversion that Axiom is allowed +to do automatically when you enter an expression. Coercions are +usually somewhat safer than more general conversions. The Axiom +library contains operations called {\bf coerce} and {\bf convert}. +Only the {\bf coerce} operations can be used by the interpreter to +change an object into an object of another type unless you explicitly +use a {\tt ::}. + +By now you will be quite familiar with what types and modes look like. +It is useful to think of a type or mode as a pattern for what you want +the result to be. + +Let's start with a square matrix of polynomials with complex rational +number coefficients. \index{SquareMatrix} +\spadcommand{m : SquareMatrix(2,POLY COMPLEX FRAC INT)} +\returnType{Type: Void} + +\spadcommand{m := matrix [ [x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5] ]} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} + +We first want to interchange the {\tt Complex} and {\tt Fraction} +layers. We do the conversion by doing the interchange in the type +expression. +\spadcommand{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \ i} \over 4}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{{3 \ i} \over 7} \ {y \sp 4}} -x} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Fraction Complex Integer)} + +Interchange the {\tt Polynomial} and the {\tt Fraction} levels. +\spadcommand{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{{3 \ i \ {y \sp 4}} -{7 \ x}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Polynomial Complex Integer)} + +Interchange the {\tt Polynomial} and the {\tt Complex} levels. +\spadcommand{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} + +All the entries have changed types, although in comparing the +last two results only the entry in the lower left corner looks different. +We did all the intermediate steps to show you what Axiom can do. + +In fact, we could have combined all these into one conversion. +\spadcommand{m :: SquareMatrix(2,FRAC COMPLEX POLY INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} + +There are times when Axiom is not be able to do the conversion in one +step. You may need to break up the transformation into several +conversions in order to get an object of the desired type. + +We cannot move either {\tt Fraction} or {\tt Complex} above (or to the +left of, depending on how you look at it) {\tt SquareMatrix} because +each of these levels requires that its argument type have commutative +multiplication, whereas {\tt SquareMatrix} does not. That is because +{\tt Fraction} requires that its argument belong to the category +{\tt IntegralDomain} and \index{category} {\tt Complex} requires that its +argument belong to {\tt CommutativeRing}. +See \ref{ugTypesBasic} on page~\pageref{ugTypesBasic} for a +brief discussion of categories. The {\tt Integer} level did not move +anywhere because it does not allow any arguments. We also did not +move the {\tt SquareMatrix} part anywhere, but we could have. + +Recall that $m$ looks like this. + +\spadcommand{m} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} + +If we want a polynomial with matrix coefficients rather than a matrix +with polynomial entries, we can just do the conversion. + +\spadcommand{m :: POLY SquareMatrix(2,COMPLEX FRAC INT)} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \over 7} \ i} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \over 4} \ i} & {1 \over 2} \\ +0 & {{12} -{{9 \over 5} \ i}} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} + +We have not yet used modes for any conversions. Modes are a great +shorthand for indicating the type of the object you want. Instead of +using the long type expression in the last example, we could have +simply said this. + +\spadcommand{m :: POLY ?} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \over 7} \ i} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \over 4} \ i} & {1 \over 2} \\ +0 & {{12} -{{9 \over 5} \ i}} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} + +We can also indicate more structure if we want the entries of the +matrices to be fractions. + +\spadcommand{m :: POLY SquareMatrix(2,FRAC ?)} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \ i} \over 7} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \ i} \over 4} & {1 \over 2} \\ +0 & {{{60} -{9 \ i}} \over 5} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Fraction Complex Integer)} + +\section{Subdomains Again} +\label{ugTypesSubdomains} + +A {\it subdomain} {\rm S} of a domain {\rm D} is a domain consisting of +\begin{enumerate} +\item those elements of {\rm D} that satisfy some +{\it predicate} (that is, a test that returns {\tt true} or {\tt false}) and +\item a subset of the operations of {\rm D}. +\end{enumerate} +Every domain is a subdomain of itself, trivially satisfying the +membership test: {\tt true}. + +Currently, there are only two system-defined subdomains in Axiom that +receive substantial use. {\tt PositiveInteger} and +{\tt NonNegativeInteger} are subdomains of {\tt Integer}. An element $x$ +of {\tt NonNegativeInteger} is an integer that is greater than or +equal to zero, that is, satisfies $x >= 0$. An element $x$ of +{\tt PositiveInteger} is a nonnegative integer that is, in fact, greater +than zero, that is, satisfies $x > 0$. Not all operations from +{\tt Integer} are available for these subdomains. For example, negation +and subtraction are not provided since the subdomains are not closed +under those operations. When you use an integer in an expression, +Axiom assigns to it the type that is the most specific subdomain whose +predicate is satisfied. + +This is a positive integer. +\spadcommand{5} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +This is a nonnegative integer. +\spadcommand{0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +This is neither of the above. +\spadcommand{-5} +$$ +-5 +$$ +\returnType{Type: Integer} + +Furthermore, unless you are assigning an integer to a declared variable +or using a conversion, any integer result has as type the most +specific subdomain. +\spadcommand{(-2) - (-3)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{0 :: Integer} +$$ +0 +$$ +\returnType{Type: Integer} + +\spadcommand{x : NonNegativeInteger := 5} +$$ +5 +$$ +\returnType{Type: NonNegativeInteger} + +When necessary, Axiom converts an integer object into one belonging to +a less specific subdomain. For example, in $3-2$, the arguments to +\spadopFrom{-}{Integer} are both elements of {\tt PositiveInteger}, +but this type does not provide a subtraction operation. Neither does +{\tt NonNegativeInteger}, so $3$ and $2$ are viewed as elements of +{\tt Integer}, where their difference can be calculated. The result +is $1$, which Axiom then automatically assigns the type +{\tt PositiveInteger}. + +Certain operations are very sensitive to the subdomains to which their +arguments belong. This is an element of {\tt PositiveInteger}. +\spadcommand{2 ** 2} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +This is an element of {\tt Fraction Integer}. +\spadcommand{2 ** (-2)} +$$ +1 \over 4 +$$ +\returnType{Type: Fraction Integer} + +It makes sense then that this is a list of elements of {\tt +PositiveInteger}. +\spadcommand{[10**i for i in 2..5]} +$$ +\left[ +{100}, {1000}, {10000}, {100000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +What should the type of {\tt [10**(i-1) for i in 2..5]} be? On one hand, +$i-1$ is always an integer greater than zero as $i$ ranges from $2$ to +$5$ and so $10**i$ is also always a positive integer. On the other, +$i-1$ is a very simple function of $i$. Axiom does not try to analyze +every such function over the index's range of values to determine +whether it is always positive or nowhere negative. For an arbitrary +Axiom function, this analysis is not possible. + +So, to be consistent no such analysis is done and we get this. +\spadcommand{[10**(i-1) for i in 2..5]} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List Fraction Integer} + +To get a list of elements of {\tt PositiveInteger} instead, you have +two choices. You can use a conversion. + +\spadcommand{[10**((i-1) :: PI) for i in 2..5]} +\begin{verbatim} +Compiling function G82696 with type Integer -> Boolean +Compiling function G82708 with type NonNegativeInteger -> Boolean +\end{verbatim} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Or you can use {\tt pretend}. \index{pretend} +\spadcommand{[10**((i-1) pretend PI) for i in 2..5]} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The operation {\tt pretend} is used to defeat the Axiom type system. +The expression {\tt object pretend D} means ``make a new object +(without copying) of type {\tt D} from {\tt object}.'' If +{\tt object} were an integer and you told Axiom to pretend it was a list, +you would probably see a message about a fatal error being caught and +memory possibly being damaged. Lists do not have the same internal +representation as integers! + +You use {\tt pretend} at your peril. \index{peril} + +Use $pretend$ with great care! Axiom trusts you that the value is of +the specified type. + +\spadcommand{(2/3) pretend Complex Integer} +$$ +2+{3 \ i} +$$ +\returnType{Type: Complex Integer} + +\section{Package Calling and Target Types} +\label{ugTypesPkgCall} + +Axiom works hard to figure out what you mean by an expression without +your having to qualify it with type information. Nevertheless, there +are times when you need to help it along by providing hints (or even +orders!) to get Axiom to do what you want. + +We saw in \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare} that +declarations using types and modes control the type of the results +produced. For example, we can either produce a complex object with +polynomial real and imaginary parts or a polynomial with complex +integer coefficients, depending on the declaration. + +Package calling is how you tell Axiom to use a particular function +from a particular part of the library. + +Use the \spadopFrom{/}{Fraction} from {\tt Fraction Integer} to create +a fraction of two integers. +\spadcommand{2/3} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Integer} + +If we wanted a floating point number, we can say ``use the +\spadopFrom{/}{Float} in {\tt Float}.'' +\spadcommand{(2/3)\$Float} +$$ +0.6666666666 6666666667 +$$ +\returnType{Type: Float} + +Perhaps we actually wanted a fraction of complex integers. +\spadcommand{(2/3)\$Fraction(Complex Integer)} +$$ +2 \over 3 +$$ +\returnType{Type: Float} + +In each case, AXIOM used the indicated operations, sometimes first +needing to convert the two integers into objects of the appropriate type. +In these examples, ``/'' is written as an infix operator. + +\boxed{4.6in}{ +\vskip 0.1cm +To use package calling with an infix operator, use the following syntax: +$$(\ arg_1{\rm \ op\ }arg_2\ )\$type$$ +} + +We used, for example, $(2/3)\${\rm Float}$. The expression $2+3+4$ +is equivalent to $(2+3)+4$. Therefore in the expression +$(2+3+4)\${\rm Float}$ the second ``+'' comes from the {\rm Float} +domain. The first ``+'' comes from {\rm Float} because the package +call causes AXIOM to convert $(2+3)$ and $4$ to type +{\rm Float}. Before the sum is converted, it is given a target type +of {\rm Float} by AXIOM and then evaluated. The target type causes the +``+'' from {\tt Float} to be used. + +\boxed{4.6in}{ +\vskip 0.1cm +For an operator written before its arguments, you must use parentheses +around the arguments (even if there is only one), and follow the closing +parenthesis by a ``\$'' and then the type. +$$ fun\ (\ arg_1, arg_2, \ldots, arg_N\ )\$type$$ +} + +For example, to call the ``minimum'' function from {\rm SmallFloat} on two +integers, you could write {\bf min}(4,89){\tt SmallFloat}. Another use of +package calling is to tell AXIOM to use a library function rather than a +function you defined. We discuss this in +Section \ref{ugUserUse} on page~\pageref{ugUserUse}. + +Sometimes rather than specifying where an operation comes from, you +just want to say what type the result should be. We say that you provide a +{\sl target type} for the expression. Instead of using a ``\$'', use a ``@'' +to specify the requested target type. Otherwise, the syntax is the same. +Note that giving a target type is not the same as explicitly doing a +conversion. The first says ``try to pick operations so that the result has +such-and-such a type.'' The second says ``compute the result and then convert +to an object of such-and-such a type.'' + +Sometimes it makes sense, as in this expression, to say ``choose the +operations in this expression so that the final result is {\rm Float}. +\spadcommand{(2/3)@Float} +$$ +0.6666666666 6666666667 +$$ +\returnType{Type: Float} + +Here we used ``{\tt @}'' to say that the target type of the left-hand side +was {\tt Float}. In this simple case, there was no real difference +between using ``{\tt \$}'' and ``{\tt @}''. +You can see the difference if you try the following. + +This says to try to choose ``{\tt +}'' so that the result is a string. +Axiom cannot do this. +\spadcommand{(2 + 3)@String} +\begin{verbatim} +An expression involving @ String actually evaluated to one of + type PositiveInteger . Perhaps you should use :: String . +\end{verbatim} + +This says to get the {\tt +} from {\tt String} and apply it to the two +integers. Axiom also cannot do this because there is no {\tt +} +exported by {\tt String}. +\spadcommand{(2 + 3)\$String} +\begin{verbatim} + The function + is not implemented in String . +\end{verbatim} + +(By the way, the operation \spadfunFrom{concat}{String} or juxtaposition +is used to concatenate two strings.) +\index{String} + +When we have more than one operation in an expression, the difference +is even more evident. The following two expressions show that Axiom +uses the target type to create different objects. +The ``{\tt +}'', ``{\tt *}'' and ``{\tt **}'' operations are all +chosen so that an object of the correct final type is created. + +This says that the operations should be chosen so that the result is a +{\tt Complex} object. +\spadcommand{((x + y * \%i)**2)@(Complex Polynomial Integer)} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +This says that the operations should be chosen so that the result is a +{\tt Polynomial} object. +\spadcommand{((x + y * \%i)**2)@(Polynomial Complex Integer)} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +What do you think might happen if we left off all target type and +package call information in this last example? +\spadcommand{(x + y * \%i)**2} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +We can convert it to {\tt Complex} as an afterthought. But this is +more work than just saying making what we want in the first place. +\spadcommand{\% :: Complex ?} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Finally, another use of package calling is to qualify fully an +operation that is passed as an argument to a function. + +Start with a small matrix of integers. +\spadcommand{h := matrix [ [8,6],[-4,9] ]} +$$ +\left[ +\begin{array}{cc} +8 & 6 \\ +-4 & 9 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +We want to produce a new matrix that has for entries the +multiplicative inverses of the entries of $h$. One way to do this is +by calling \spadfunFrom{map}{MatrixCategoryFunctions2} with the +\spadfunFrom{inv}{Fraction} function from {\tt Fraction (Integer)}. + +\spadcommand{map(inv\$Fraction(Integer),h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +We could have been a bit less verbose and used abbreviations. +\spadcommand{map(inv\$FRAC(INT),h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +As it turns out, Axiom is smart enough to know what we mean anyway. +We can just say this. +\spadcommand{map(inv,h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +\section{Resolving Types} +\label{ugTypesResolve} + +In this section we briefly describe an internal process by which +\index{resolve} Axiom determines a type to which two objects of +possibly different types can be converted. We do this to give you +further insight into how Axiom takes your input, analyzes it, and +produces a result. + +What happens when you enter $x + 1$ to Axiom? Let's look at what you +get from the two terms of this expression. + +This is a symbolic object whose type indicates the name. +\spadcommand{x} +$$ +x +$$ +\returnType{Type: Variable x} + +This is a positive integer. +\spadcommand{1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +There are no operations in {\tt PositiveInteger} that add positive +integers to objects of type {\tt Variable(x)} nor are there any in +{\tt Variable(x)}. Before it can add the two parts, Axiom must come +up with a common type to which both $x$ and $1$ can be converted. We +say that Axiom must {\it resolve} the two types into a common type. +In this example, the common type is {\tt Polynomial(Integer)}. + +Once this is determined, both parts are converted into polynomials, +and the addition operation from {\tt Polynomial(Integer)} is used to +get the answer. +\spadcommand{x + 1} +$$ +x+1 +$$ +\returnType{Type: Polynomial Integer} + +Axiom can always resolve two types: if nothing resembling the original +types can be found, then {\tt Any} is be used. \index{Any} This is +fine and useful in some cases. + +\spadcommand{["string",3.14159]} +$$ +\left[ +\mbox{\tt "string"} , {3.14159} +\right] +$$ +\returnType{Type: List Any} + +In other cases objects of type {\tt Any} can't be used by the +operations you specified. +\spadcommand{"string" + 3.14159} +\begin{verbatim} +There are 11 exposed and 5 unexposed library operations named + + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op + + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the + arguments will allow you to apply the operation. + +Cannot find a definition or applicable library operation named + + with argument type(s) + String + Float + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +Although this example was contrived, your expressions may need to be +qualified slightly to help Axiom resolve the types involved. You may +need to declare a few variables, do some package calling, provide some +target type information or do some explicit conversions. + +We suggest that you just enter the expression you want evaluated and +see what Axiom does. We think you will be impressed with its ability +to ``do what I mean.'' If Axiom is still being obtuse, give it some +hints. As you work with Axiom, you will learn where it needs a little +help to analyze quickly and perform your computations. + +\section{Exposing Domains and Packages} +\label{ugTypesExpose} + +In this section we discuss how Axiom makes some operations available +to you while hiding others that are meant to be used by developers or +only in rare cases. If you are a new user of Axiom, it is likely that +everything you need is available by default and you may want to skip +over this section on first reading. + +Every \index{constructor!exposed} domain and package in the Axiom +library \index{constructor!hidden} is \index{exposed!constructor} +either exposed (meaning that you can use its operations without doing +anything special) or it is {\it hidden} (meaning you have to either +package call (see \ref{ugTypesPkgCall} on +page~\pageref{ugTypesPkgCall}) the operations it contains or +explicitly expose it to use the operations). The initial exposure +status for a constructor is set in the file {\bf exposed.lsp} (see the +{\it Installer's Note} \index{exposed.lsp @{\bf exposed.lsp}} for +Axiom \index{file!exposed.lsp @{\bf exposed.lsp}} if you need to know +the location of this file). Constructors are collected together in +\index{group!exposure} {\it exposure groups}. \index{exposure!group} +Categories are all in the exposure group ``categories'' and the bulk +of the basic set of packages and domains that are exposed are in the +exposure group ``basic.'' Here is an abbreviated sample of the file +(without the Lisp parentheses): + +\begin{verbatim} +basic + AlgebraicNumber AN + AlgebraGivenByStructuralConstants ALGSC + Any ANY + AnyFunctions1 ANY1 + BinaryExpansion BINARY + Boolean BOOLEAN + CardinalNumber CARD + CartesianTensor CARTEN + Character CHAR + CharacterClass CCLASS + CliffordAlgebra CLIF + Color COLOR + Complex COMPLEX + ContinuedFraction CONTFRAC + DecimalExpansion DECIMAL + ... +\end{verbatim} +\begin{verbatim} +categories + AbelianGroup ABELGRP + AbelianMonoid ABELMON + AbelianMonoidRing AMR + AbelianSemiGroup ABELSG + Aggregate AGG + Algebra ALGEBRA + AlgebraicallyClosedField ACF + AlgebraicallyClosedFunctionSpace ACFS + ArcHyperbolicFunctionCategory AHYP + ... +\end{verbatim} + +For each constructor in a group, the full name and the abbreviation is +given. There are other groups in {\bf exposed.lsp} but initially only +the constructors in exposure groups ``basic'' ``categories'' +``naglink'' and ``anna'' are exposed. + +As an interactive user of Axiom, you do not need to modify this file. +Instead, use {\tt )set expose} to expose, hide or query the exposure +status of an individual constructor or exposure group. \index{set expose} +The reason for having exposure groups is to be able to expose +or hide multiple constructors with a single command. For example, you +might group together into exposure group ``quantum'' a number of +domains and packages useful for quantum mechanical computations. +These probably should not be available to every user, but you want an +easy way to make the whole collection visible to Axiom when it is +looking for operations to apply. + +If you wanted to hide all the basic constructors available by default, +you would issue {\tt )set expose drop group basic}. +\index{set expose drop group} We do not recommend that you do this. +If, however, you discover that you have hidden all the basic constructors, +you should issue {\tt )set expose add group basic} to restore your default +environment. \index{set expose add group} + +It is more likely that you would want to expose or hide individual +constructors. In \ref{ugUserTriangle} on +page~\pageref{ugUserTriangle} we use several operations from +{\tt OutputForm}, a domain usually hidden. To avoid package calling every +operation from {\tt OutputForm}, we expose the domain and let Axiom +conclude that those operations should be used. Use {\tt )set expose +add constructor} and {\tt )set expose drop constructor} to expose and +hide a constructor, respectively. \index{set expose drop constructor} +You should use the constructor name, not the abbreviation. The +{\tt )set expose} command guides you through these options. +\index{set expose add constructor} + +If you expose a previously hidden constructor, Axiom exhibits new +behavior (that was your intention) though you might not expect the +results that you get. {\tt OutputForm} is, in fact, one of the worst +offenders in this regard. \index{OutputForm} This domain is meant to +be used by other domains for creating a structure that Axiom knows how +to display. It has functions like \spadopFrom{+}{OutputForm} that +form output representations rather than do mathematical calculations. +Because of the order in which Axiom looks at constructors when it is +deciding what operation to apply, {\tt OutputForm} might be used +instead of what you expect. + +This is a polynomial. +\spadcommand{x + x} +$$ +2 \ x +$$ +\returnType{Type: Polynomial Integer} + +Expose {\tt OutputForm}. +\spadcommand{)set expose add constructor OutputForm } +\begin{verbatim} + OutputForm is now explicitly exposed in frame G82322 +\end{verbatim} + +This is what we get when {\tt OutputForm} is automatically available. +\spadcommand{x + x} +$$ +x+x +$$ +\returnType{Type: OutputForm} + +Hide {\tt OutputForm} so we don't run into problems with any later examples! +\spadcommand{)set expose drop constructor OutputForm } +\begin{verbatim} + OutputForm is now explicitly hidden in frame G82322 +\end{verbatim} + +Finally, exposure is done on a frame-by-frame basis. A {\it frame} +(see \ref{ugSysCmdframe} on page~\pageref{ugSysCmdframe}) +\index{frame!exposure and} is one of possibly several logical Axiom +workspaces within a physical one, each having its own environment (for +example, variables and function definitions). If you have several +Axiom workspace windows on your screen, they are all different frames, +automatically created for you by HyperDoc. Frames can be manually +created, made active and destroyed by the {\tt )frame} system command. +\index{frame} They do not share exposure information, so you need to +use {\tt )set expose} in each one to add or drop constructors from +view. + +\section{Commands for Snooping} +\label{ugAvailSnoop} + +To conclude this chapter, we introduce you to some system commands +that you can use for getting more information about domains, packages, +categories, and operations. The most powerful Axiom facility for +getting information about constructors and operations is the Browse +component of HyperDoc. This is discussed in Chapter +\ref{ugBrowse} on page~\pageref{ugBrowse}. + +Use the {\tt )what} system command to see lists of system objects +whose name contain a particular substring (uppercase or lowercase is +not significant). \index{what} + +Issue this to see a list of all operations with ``{\tt complex}'' in +their names. \index{what operation} +\spadcommand{)what operation complex} +\begin{verbatim} + +Operations whose names satisfy the above pattern(s): + +complex complex? +complexEigenvalues complexEigenvectors +complexElementary complexExpand +complexForm complexIntegrate +complexLimit complexNormalize +complexNumeric complexNumericIfCan +complexRoots complexSolve +complexZeros createLowComplexityNormalBasis +createLowComplexityTable doubleComplex? +drawComplex drawComplexVectorField +fortranComplex fortranDoubleComplex +pmComplexintegrate + +To get more information about an operation such as +complexZeros, issue the command )display op complexZeros +\end{verbatim} + +If you want to see all domains with ``{\tt matrix}'' in their names, +issue this. \index{what domain} +\spadcommand{)what domain matrix} +\begin{verbatim} +----------------------- Domains ----------------------- + +Domains with names matching patterns: + matrix + + DHMATRIX DenavitHartenbergMatrix + DPMM DirectProductMatrixModule + IMATRIX IndexedMatrix + LSQM LieSquareMatrix + M3D ThreeDimensionalMatrix + MATCAT- MatrixCategory& + MATRIX Matrix + RMATCAT- RectangularMatrixCategory& + RMATRIX RectangularMatrix + SMATCAT- SquareMatrixCategory& + SQMATRIX SquareMatrix +\end{verbatim} + +Similarly, if you wish to see all packages whose names contain ``{\tt +gauss}'', enter this. \index{what packages} +\spadcommand{)what package gauss} +\begin{verbatim} +---------------------- Packages ----------------------- + +Packages with names matching patterns: + gauss + + GAUSSFAC GaussianFactorizationPackage +\end{verbatim} + +This command shows all the operations that {\tt Any} provides. +Wherever {\tt \$} appears, it means ``{\tt Any}''. \index{show} +\spadcommand{)show Any} +\begin{verbatim} + Any is a domain constructor + Abbreviation for Any is ANY + This constructor is exposed in this frame. + Issue )edit /usr/local/axiom/mnt/algebra/any.spad + to see algebra source code for ANY + +--------------------- Operations ---------------------- + ?=? : (%,%) -> Boolean + any : (SExpression,None) -> % + coerce : % -> OutputForm + dom : % -> SExpression + domainOf : % -> OutputForm + hash : % -> SingleInteger + latex : % -> String + obj : % -> None + objectOf : % -> OutputForm + ?~=? : (%,%) -> Boolean + showTypeInOutput : Boolean -> String + +\end{verbatim} + +This displays all operations with the name {\tt complex}. +\index{display operation} +\spadcommand{)display operation complex} +\begin{verbatim} +There is one exposed function called complex : + [1] (D1,D1) -> D from D if D has COMPCAT D1 and D1 has COMRING +\end{verbatim} + +Let's analyze this output. + +First we find out what some of the abbreviations mean. +\spadcommand{)abbreviation query COMPCAT} +\begin{verbatim} + COMPCAT abbreviates category ComplexCategory +\end{verbatim} + +\spadcommand{)abbreviation query COMRING} +\begin{verbatim} + COMRING abbreviates category CommutativeRing +\end{verbatim} + +So if {\tt D1} is a commutative ring (such as the integers or floats) and +{\tt D} belongs to {\tt ComplexCategory D1}, then there is an operation +called {\bf complex} that takes two elements of {\tt D1} and creates an +element of {\tt D}. The primary example of a constructor implementing +domains belonging to {\tt ComplexCategory} is {\tt Complex}. See +\ref{Complex} on page~\pageref{Complex} for more information on that and see +\ref{ugUserDeclare} on page~\pageref{ugUserDeclare} +for more information on function types. + +\setcounter{chapter}{2} + +\chapter{Using HyperDoc} +\label{ugHyper} + +\begin{figure}[htbp] +\begin{picture}(324,260)%(-54,0) +\special{psfile=ps/h-root.ps} +\end{picture} +\caption{The HyperDoc root window page.} +\end{figure} + +HyperDoc is the gateway to Axiom. \index{HyperDoc} It's both an +on-line tutorial and an on-line reference manual. It also enables you +to use Axiom simply by using the mouse and filling in templates. +HyperDoc is available to you if you are running Axiom under the X +Window System. + +Pages usually have active areas, marked in {\bf this font} (bold +face). As you move the mouse pointer to an active area, the pointer +changes from a filled dot to an open circle. The active areas are +usually linked to other pages. When you click on an active area, you +move to the linked page. + +\section{Headings} +\label{ugHyperHeadings} +Most pages have a standard set of buttons at the top of the page. +This is what they mean: + +\begin{description} + +\item[\HelpBitmap] Click on this to get help. The button only appears +if there is specific help for the page you are viewing. You can get +{\it general} help for HyperDoc by clicking the help button on the +home page. + +\item[\UpBitmap] Click here to go back one page. +By clicking on this button repeatedly, you can go back several pages and +then take off in a new direction. + +\item[\ReturnBitmap] Go back to the home page, that is, the page on +which you started. Use HyperDoc to explore, to make forays into new +topics. Don't worry about how to get back. HyperDoc remembers where +you came from. Just click on this button to return. + +\item[\ExitBitmap] From the root window (the one that is displayed +when you start the system) this button leaves the HyperDoc program, +and it must be restarted if you want to use it again. From any other +HyperDoc window, it just makes that one window go away. You {\it must} +use this button to get rid of a window. If you use the window +manager ``Close'' button, then all of HyperDoc goes away. + +\end{description} + +The buttons are not displayed if they are not applicable to the page +you are viewing. For example, there is no \ReturnBitmap button on the +top-level menu. + +\section{Key Definitions} +\label{ugHyperKeys} + +The following keyboard definitions are in effect throughout HyperDoc. +See \ref{ugHyperScroll} on page~\pageref{ugHyperScroll} and +\ref{ugHyperInput} on page~\pageref{ugHyperInput} for some contextual key +definitions. + +\begin{description} +\item[F1] Display the main help page. +\item[F3] Same as \ExitBitmap{}, makes the window go away if you are not at the top-level window or quits the HyperDoc facility if you are at the top-level. +\item[F5] Rereads the HyperDoc database, if necessary (for system developers). +\item[F9] Displays this information about key definitions. +\item[F12] Same as {\bf F3}. +\item[Up Arrow] Scroll up one line. +\item[Down Arrow] Scroll down one line. +\item[Page Up] Scroll up one page. +\item[Page Down] Scroll down one page. +\end{description} + +\section{Scroll Bars} +\label{ugHyperScroll} + +Whenever there is too much text to fit on a page, a +{\it scroll \index{scroll bar} bar} +automatically appears along the right side. + +With a scroll bar, your page becomes an aperture, that is, a window +into a larger amount of text than can be displayed at one time. The +scroll bar lets you move up and down in the text to see different +parts. It also shows where the aperture is relative to the whole +text. The aperture is indicated by a strip on the scroll bar. + +Move the cursor with the mouse to the ``down-arrow'' at the bottom of +the scroll bar and click. See that the aperture moves down one line. +Do it several times. Each time you click, the aperture moves down one +line. Move the mouse to the ``up-arrow'' at the top of the scroll bar +and click. The aperture moves up one line each time you click. + +Next move the mouse to any position along the middle of the scroll bar +and click. HyperDoc attempts to move the top of the aperture to this +point in the text. + +You cannot make the aperture go off the bottom edge. When the +aperture is about half the size of text, the lowest you can move the +aperture is halfway down. + +To move up or down one screen at a time, use the \fbox{\bf PageUp} and +\fbox{\bf PageDown} keys on your keyboard. They move the visible part of the +region up and down one page each time you press them. + +If the HyperDoc page does not contain an input area (see +\ref{ugHyperInput} on page~\pageref{ugHyperInput}), you can also use +the \fbox{\bf Home} and \fbox{$\uparrow$} and \fbox{$\downarrow$} +arrow keys to navigate. When you press the \fbox{\bf Home} key, the +screen is positioned at the very top of the page. Use the +\fbox{$\uparrow$} and \fbox{$\downarrow$} arrow keys to move the +screen up and down one line at a time, respectively. + +\section{Input Areas} +\label{ugHyperInput} + +Input areas are boxes where you can put data. + +To enter characters, first move your mouse cursor to somewhere within +the HyperDoc page. Characters that you type are inserted in front of +the underscore. This means that when you type characters at your +keyboard, they go into this first input area. + +The input area grows to accommodate as many characters as you type. +Use the \fbox{\bf Backspace} key to erase characters to the left. To +modify what you type, use the right-arrow \fbox{$\rightarrow$} and +left-arrow keys \fbox{$\leftarrow$} and the keys \fbox{\bf Insert}, +\fbox{\bf Delete}, \fbox{\bf Home} and \fbox{\bf End}. These keys are +found immediately on the right of the standard IBM keyboard. + +If you press the \fbox{\bf Home} key, the cursor moves to the +beginning of the line and if you press the \fbox{\bf End} key, the +cursor moves to the end of the line. Pressing +\fbox{\bf Ctrl}--\fbox{\bf End} deletes all the text from the +cursor to the end of the line. + +A page may have more than one input area. Only one input area has an +underscore cursor. When you first see apage, the top-most input area +contains the cursor. To type information into another input area, use +the \fbox{\bf Enter} or \fbox{\bf Tab} key to move from one input area to +xanother. To move in the reverse order, use \fbox{\bf Shift}--\fbox{\bf Tab}. + +You can also move from one input area to another using your mouse. +Notice that each input area is active. Click on one of the areas. +As you can see, the underscore cursor moves to that window. + +\section{Radio Buttons and Toggles} +\label{ugHyperButtons} + +Some pages have {\it radio buttons} and {\it toggles}. +Radio buttons are a group of buttons like those on car radios: you can +select only one at a time. + +Once you have selected a button, it appears to be inverted and +contains a checkmark. To change the selection, move the cursor with +the mouse to a different radio button and click. + +A toggle is an independent button that displays some on/off state. +When ``on'', the button appears to be inverted and contains a +checkmark. When ``off'', the button is raised. + +Unlike radio buttons, you can set a group of them any way you like. +To change toggle the selection, move the cursor with the mouse to the +button and click. + +\section{Search Strings} +\label{ugHyperSearch} + +A {\it search string} is used for searching some database. To learn +about search strings, we suggest that you bring up the HyperDoc +glossary. To do this from the top-level page of HyperDoc: +\begin{enumerate} +\item Click on Reference, bringing up the Axiom Reference page. +\item Click on Glossary, bringing up the glossary. +\end{enumerate} + +The glossary has an input area at its bottom. We review the various +kinds of search strings you can enter to search the glossary. + +The simplest search string is a word, for example, {\tt operation}. A +word only matches an entry having exactly that spelling. Enter the +word {\tt operation} into the input area above then click on +{\bf Search}. As you can see, {\tt operation} matches only one entry, +namely with {\tt operation} itself. + +Normally matching is insensitive to whether the alphabetic characters +of your search string are in uppercase or lowercase. Thus +{\tt operation} and {\tt OperAtion} both have the same effect. +%If you prefer that matching be case-sensitive, issue the command +%{\tt set HHyperName mixedCase} command to the interpreter. + +You will very often want to use the wildcard ``{\tt *}'' in your search +string so as to match multiple entries in the list. The search key +``{\tt *}'' matches every entry in the list. You can also use ``{\tt *}'' +anywhere within a search string to match an arbitrary substring. Try +``{\tt cat*}'' for example: enter ``{\tt cat*}'' into the input area and click +on {\bf Search}. This matches several entries. + +You use any number of wildcards in a search string as long as they are +not adjacent. Try search strings such as ``{\tt *dom*}''. As you see, +this search string matches ``{\tt domain}'', ``{\tt domain constructor}'', +``{\tt subdomain}'', and so on. + +\subsection{Logical Searches} +\label{ugLogicalSearches} + +For more complicated searches, you can use ``{\tt and}'', ``{\tt or}'', and +``{\tt not}'' with basic search strings; write logical expressions using +these three operators just as in the Axiom language. For example, +{\tt domain or package} matches the two entries {\tt domain} and +{\tt package}. Similarly, ``{\tt dom* and *con*}'' matches +``{\tt domain constructor}'' and others. Also ``{\tt not *a*}'' matches +every entry that does not contain the letter ``{\tt a}'' somewhere. + +Use parentheses for grouping. For example, ``{\tt dom* and (not *con*)}'' +matches ``{\tt domain}'' but not ``{\tt domain constructor}''. + +There is no limit to how complex your logical expression can be. +For example, +\begin{center} +{\tt a* or b* or c* or d* or e* and (not *a*)} +\end{center} +is a valid expression. + +\section{Example Pages} +\label{ugHyperExample} + +Many pages have Axiom example commands. + +Each command has an active ``button'' along the left margin. When you +click on this button, the output for the command is ``pasted-in.'' +Click again on the button and you see that the pasted-in output +disappears. + +Maybe you would like to run an example? To do so, just click on any +part of its text! When you do, the example line is copied into a new +interactive Axiom buffer for this HyperDoc page. + +Sometimes one example line cannot be run before you run an earlier one. +Don't worry---HyperDoc automatically runs all the necessary +lines in the right order! + +The new interactive Axiom buffer disappears when you leave HyperDoc. +If you want to get rid of it beforehand, use the {\bf Cancel} button +of the X Window manager or issue the Axiom system command +{\tt )close.} \index{close} + +\section{X Window Resources for HyperDoc} +\label{ugHyperResources} + +You can control the appearance of HyperDoc while running under Version +11 \index{HyperDoc X Window System defaults} of the X Window System by +placing the following resources \index{X Window System} in the file +{\bf .Xdefaults} in your home directory. \index{file!.Xdefaults} +In what follows, {\it font} is any valid X11 font name +\index{font} (for example, {\tt Rom14}) and {\it color} is any valid +X11 color \index{color} specification (for example, {\tt NavyBlue}). +For more information about fonts and colors, refer to the X Window +documentation for your system. + +\begin{description} +\item[{\tt Axiom.hyperdoc.RmFont:} {\it font}] \ \newline +This is the standard text font. +The default value is {\tt Rom14} +\item[{\tt Axiom.hyperdoc.RmColor:} {\it color}] \ \newline +This is the standard text color. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.ActiveFont:} {\it font}] \ \newline +This is the font used for HyperDoc link buttons. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.ActiveColor:} {\it color}] \ \newline +This is the color used for HyperDoc link buttons. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.AxiomFont:} {\it font}] \ \newline +This is the font used for active Axiom commands. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.AxiomColor:} {\it color}] \ \newline +This is the color used for active Axiom commands. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.BoldFont:} {\it font}] \ \newline +This is the font used for bold face. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.BoldColor:} {\it color}] \ \newline +This is the color used for bold face. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.TtFont:} {\it font}] \ \newline +This is the font used for Axiom output in HyperDoc. +This font must be fixed-width. +The default value is {\tt Rom14} +\item[{\tt Axiom.hyperdoc.TtColor:} {\it color}] \ \newline +This is the color used for Axiom output in HyperDoc. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.EmphasizeFont:} {\it font}] \ \newline +This is the font used for italics. +The default value is {\tt Itl14} +\item[{\tt Axiom.hyperdoc.EmphasizeColor:} {\it color}] \ \newline +This is the color used for italics. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.InputBackground:} {\it color}] \ \newline +This is the color used as the background for input areas. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.InputForeground:} {\it color}] \ \newline +This is the color used as the foreground for input areas. +The default value is {\tt white} +\item[{\tt Axiom.hyperdoc.BorderColor:} {\it color}] \ \newline +This is the color used for drawing border lines. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.Background:} {\it color}] \ \newline +This is the color used for the background of all windows. +The default value is {\tt white} +\end{description} +\vfill +\eject + +\setcounter{chapter}{3} + +\chapter{Input Files and Output Styles} +\label{ugInOut} + +In this chapter we discuss how to collect Axiom statements +and commands into files and then read the contents into the +workspace. +We also show how to display the results of your computations in +several different styles including \TeX{}, FORTRAN and +monospace two-dimensional format.\footnote{\TeX{} is a +trademark of the American Mathematical Society.} + +The printed version of this book uses the Axiom \TeX{} output formatter. +When we demonstrate a particular output style, we will need to turn +\TeX{} formatting off and the output style on so that the correct output +is shown in the text. + +\section{Input Files} +\label{ugInOutIn} + +In this section we explain what an {\it input file} is and +\index{file!input} why you would want to know about it. We discuss +where Axiom looks for input files and how you can direct it to look +elsewhere. We also show how to read the contents of an input file +into the {\it workspace} and how to use the {\it history} facility to +generate an input file from the statements you have entered directly +into the workspace. + +An {\it input} file contains Axiom expressions and system commands. +Anything that you can enter directly to Axiom can be put into an input +file. This is how you save input functions and expressions that you +wish to read into Axiom more than one time. + +To read an input file into Axiom, use the {\tt )read} system command. +\index{read} For example, you can read a file in a particular +directory by issuing +\begin{verbatim} +)read /spad/src/input/matrix.input +\end{verbatim} + +The ``{\bf .input}'' is optional; this also works: +\begin{verbatim} +)read /spad/src/input/matrix +\end{verbatim} + +What happens if you just enter {\tt )read matrix.input} or even {\tt +)read matrix}? Axiom looks in your current working directory for +input files that are not qualified by a directory name. Typically, +this directory is the directory from which you invoked Axiom. + +To change the current working directory, use the {\tt )cd} system +command. The command {\tt {)cd}} by itself shows the current working +\index{directory!default for searching} directory. \index{cd} To +change it to \index{file!input!where found} the {\tt {src/input}} +subdirectory for user ``babar'', issue +\begin{verbatim} +)cd /u/babar/src/input +\end{verbatim} +Axiom looks first in this directory for an input file. If it is not +found, it looks in the system's directories, assuming you meant some +input file that was provided with Axiom. + +\boxed{4.6in}{ +\vskip 0.1cm +If you have the Axiom history facility turned on (which it is +by default), you can save all the lines you have entered into the +workspace by entering + +)history )write + +\index{history )write} + +Axiom tells you what input file to edit to see your statements. The +file is in your home directory or in the directory you specified with +\index{cd} {\tt {)cd}}.\\ +} + +In \ref{ugLangBlocks} on page~\pageref{ugLangBlocks} +we discuss using indentation in input files to +group statements into {\it blocks.} + +\section{The .axiom.input File} +\label{ugInOutSpadprof} + +When Axiom starts up, it tries to read the input file {\bf +.axiom.input}\footnote{{\bf.axiom.input} used to be called +{\bf axiom.input} in the NAG version} +from your home \index{start-up profile file} +directory. \index{file!start-up profile} It +there is no {\bf .axiom.input} in your home directory, it reads the +copy located in its own {\bf src/input} directory. +\index{file!.axiom.input @{\bf .axiom.input}} The file usually +contains system commands to personalize your Axiom environment. In +the remainder of this section we mention a few things that users +frequently place in their {\bf .axiom.input} files. + +In order to have FORTRAN output always produced from your +computations, place the system command {\tt )set output fortran on} in +{\bf .axiom.input}. \index{quit} If you do not want to be prompted +for confirmation when you issue the {\tt )quit} system command, place +{\tt )set quit unprotected} in {\bf .axiom.input}. +\index{set quit unprotected} +If you then decide that you do want to be prompted, issue +{\tt )set quit protected}. \index{set quit protected} This is the +default setting so that new users do not leave Axiom +inadvertently.\footnote{The system command {\tt )pquit} always +prompts you for confirmation.} + +To see the other system variables you can set, issue {\tt {)set}} +or use the HyperDoc {\bf Settings} facility to view and change +Axiom system variables. + +\section{Common Features of Using Output Formats} +\label{ugInOutOut} + +In this section we discuss how to start and stop the display +\index{output formats!common features} of the different output formats +and how to send the output to the screen or to a file. +\index{file!sending output to} To fix ideas, we use FORTRAN output +format for most of the examples. + +You can use the {\tt )set output} system \index{output +formats!starting} command to \index{output formats!stopping} toggle or +redirect the different kinds of output. \index{set output} The name +of the kind of output follows ``output'' in the command. The names are + +\begin{tabular}{@{}ll} +{\bf fortran} & for FORTRAN output. \\ +{\bf algebra} & for monospace two-dimensional mathematical output. \\ +{\bf tex} & for \TeX{} output. \\ +{\bf script} & for IBM Script Formula Format output. +\end{tabular} + +For example, issue {\tt {)set output fortran on}} to turn on FORTRAN +format and issue {\tt {)set output fortran off}} to turn it off. By +default, {\tt algebra} is {\tt on} and all others are {\tt off}. +\index{set output fortran} When output is started, it is sent to the +screen. To send the output to a file, give the file name without +\index{output formats!sending to file} directory or extension. Axiom +appends a file extension depending on the kind of output being +produced. + +Issue this to redirect FORTRAN output to, for example, the file +{\bf linalg.sfort}. +\spadcommand{)set output fortran linalg} +\begin{verbatim} + FORTRAN output will be written to file linalg.sfort . +\end{verbatim} + +You must {\it also} turn on the creation of FORTRAN output. +The above just says where it goes if it is created. +\spadcommand{)set output fortran on} + +In what directory is this output placed? It goes into the directory +from which you started Axiom, or if you have used the {\tt {)cd}} +system command, the one that you specified with {\tt {)cd}}. +\index{cd} You should use {\tt )cd} before you send the output to the file. + +You can always direct output back to the screen by issuing this. +\index{output formats!sending to screen} +\spadcommand{)set output fortran console} + +Let's make sure FORTRAN formatting is off so that nothing we +do from now on produces FORTRAN output. +\spadcommand{)set output fortran off} + +We also delete the demonstrated output file we created. +\spadcommand{)system rm linalg.sfort} + +You can abbreviate the words ``{\tt on},'' ``{\tt off},'' and +``{\tt console}'' to the minimal number of characters needed to distinguish +them. Because of this, you cannot send output to files called +{\bf on.sfort, off.sfort, of.sfort, console.sfort, consol.sfort} and so on. + +The width of the output on the page is set by \index{output +formats!line length} {\tt )set output length} for all formats except +FORTRAN. \index{set output length} Use {\tt )set fortran fortlength} +to change the FORTRAN line length from its default value of $72$. + +\section{Monospace Two-Dimensional Mathematical Format} +\label{ugInOutAlgebra} + +This is the default output format for Axiom. +It is usually on when you start the system. +\index{set output algebra} +\index{output formats!monospace 2D} +\index{monospace 2D output format} + +If it is not, issue this. +\spadcommand{)set output algebra on} + +Since the printed version of this book (as opposed to the HyperDoc +version) shows output produced by the \TeX{} output formatter, let us +temporarily turn off \TeX{} output. +\spadcommand{)set output tex off} + +Here is an example of what it looks like. +\spadcommand{matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]} +\begin{verbatim} + + + 3 3 2+ + |3%i y + x 3%i y + 2x | + (1) | | + | 4 4 2| + +4%i y + x 4%i y + 2x + +\end{verbatim} +\returnType{Type: Matrix Polynomial Complex Integer} + +Issue this to turn off this kind of formatting. +\spadcommand{)set output algebra off} + +Turn \TeX{} output on again. +\spadcommand{)set output tex on} + +The characters used for the matrix brackets above are rather ugly. +You get this character set when you issue \index{character set} +{\tt )set output characters plain}. \index{set output characters} This +character set should be used when you are running on a machine that +does not support the IBM extended ASCII character set. If you are +running on an IBM workstation, for example, issue +{\tt )set output characters default} to get better looking output. + +\section{TeX Format} +\label{ugInOutTeX} + +Axiom can produce \TeX{} output for your \index{output formats!TeX +@{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format} +The output is produced using macros from the \LaTeX{} document +preparation system by Leslie Lamport\cite{1}. The printed version +of this book was produced using this formatter. + +To turn on \TeX{} output formatting, issue this. +\index{set output tex} +\spadcommand{)set output tex on} + +Here is an example of its output. +\begin{verbatim} +matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4] + +$$ +\left[ +\begin{array}{cc} +{{3 \ i \ {y \sp 3}}+x} & +{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ +{{4 \ i \ {y \sp 4}}+x} & +{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} +\end{array} +\right] +$$ + +\end{verbatim} +This formats as +$$ +\left[ +\begin{array}{cc} +{{3 \ i \ {y \sp 3}}+x} & +{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ +{{4 \ i \ {y \sp 4}}+x} & +{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} +\end{array} +\right] +$$ + +To turn \TeX{} output formatting off, issue +{\tt {)set output tex off}}. +The \LaTeX macros in the output generated by Axiom +are all standard except for the following definitions: +\begin{verbatim} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} +\end{verbatim} + +\section{IBM Script Formula Format} +\label{ugInOutScript} + +Axiom can \index{output formats!IBM Script Formula Format} produce IBM +Script Formula Format output for your +\index{IBM Script Formula Format} expressions. + +To turn IBM Script Formula Format on, issue this. +\index{set output script} +\spadcommand{)set output script on} + +Here is an example of its output. +\begin{verbatim} +matrix [ [i*x**i + j*%i*y**j for i in 1..2] for j in 3..4] + +.eq set blank @ +:df. + >+x> here < <3 @@ %i @@ + >+<2 @@ > > > habove < < <4 @@ %i @@ + >+x> here < <4 @@ %i @@ >+<2 @@ + > > > > right rb> +:edf. +\end{verbatim} + +To turn IBM Script Formula Format output formatting off, issue this. +\spadcommand{)set output script off} + +\section{FORTRAN Format} +\label{ugInOutFortran} + +In addition to turning FORTRAN output on and off and stating where the +\index{output formats!FORTRAN} output should be placed, there are many +options that control the \index{FORTRAN output format} appearance of +the generated code. In this section we describe some of the basic +options. Issue {\tt )set fortran} to see a full list with their +current settings. + +The output FORTRAN expression usually begins in column 7. If the +expression needs more than one line, the ampersand character {\tt \&} +is used in column 6. Since some versions of FORTRAN have restrictions +on the number of lines per statement, Axiom breaks long expressions +into segments with a maximum of 1320 characters (20 lines of 66 +characters) per segment. \index{set fortran} If you want to change +this, say, to 660 characters, issue the system command +\index{set fortran explength} {\tt )set fortran explength 660}. +\index{FORTRAN output format!breaking into multiple statements} +You can turn off the line breaking by issuing {\tt )set fortran segment off}. +\index{set fortran segment} Various code optimization levels are available. + +FORTRAN output is produced after you issue this. +\index{set output fortran} +\spadcommand{)set output fortran on} + +For the initial examples, we set the optimization level to 0, which is the +lowest level. +\index{set fortran optlevel} +\spadcommand{)set fortran optlevel 0} + +The output is usually in columns 7 through 72, although fewer columns +are used in the following examples so that the output +\index{FORTRAN output format!line length} +fits nicely on the page. +\spadcommand{)set fortran fortlength 60} + +By default, the output goes to the screen and is displayed before the +standard Axiom two-dimensional output. In this example, an assignment +to the variable $R1$ was generated because this is the result of step 1. +\spadcommand{(x+y)**3} +\begin{verbatim} + R1=y**3+3*x*y*y+3*x*x*y+x**3 +\end{verbatim} +$$ +{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +Here is an example that illustrates the line breaking. +\spadcommand{(x+y+z)**3} +\begin{verbatim} + R2=z**3+(3*y+3*x)*z*z+(3*y*y+6*x*y+3*x*x)*z+y**3+3*x*y + &*y+3*x*x*y+x**3 +\end{verbatim} +$$ +{z \sp 3}+{{\left( {3 \ y}+{3 \ x} +\right)} +\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} +\right)} +\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +Note in the above examples that integers are generally converted to +\index{FORTRAN output format!integers vs. floats} floating point +numbers, except in exponents. This is the default behavior but can be +turned off by issuing {\tt )set fortran ints2floats off}. +\index{set fortran ints2floats} The rules governing when the conversion +is done are: +\begin{enumerate} +\item If an integer is an exponent, convert it to a floating point +number if it is greater than 32767 in absolute value, otherwise leave it +as an integer. +\item Convert all other integers in an expression to floating point numbers. +\end{enumerate} +These rules only govern integers in expressions. Numbers generated by +Axiom for $DIMENSION$ statements are also integers. + +To set the type of generated FORTRAN data, +\index{FORTRAN output format!data types} +use one of the following: +\begin{verbatim} +)set fortran defaulttype REAL +)set fortran defaulttype INTEGER +)set fortran defaulttype COMPLEX +)set fortran defaulttype LOGICAL +)set fortran defaulttype CHARACTER +\end{verbatim} + +When temporaries are created, they are given a default type of {\tt REAL.} +Also, the {\tt REAL} versions of functions are used by default. +\spadcommand{sin(x)} +\begin{verbatim} + R3=DSIN(x) +\end{verbatim} +$$ +\sin +\left( +{x} +\right) +$$ +\returnType{Type: Expression Integer} + +At optimization level 1, Axiom removes common subexpressions. +\index{FORTRAN output format!optimization level} +\index{set fortran optlevel} +\spadcommand{)set fortran optlevel 1} + +\spadcommand{(x+y+z)**3} +\begin{verbatim} + T2=y*y + T3=x*x + R4=z**3+(3*y+3*x)*z*z+(3*T2+6*x*y+3*T3)*z+y**3+3*x*T2+ + &3*T3*y+x**3 +\end{verbatim} +$$ +{z \sp 3}+{{\left( {3 \ y}+{3 \ x} +\right)} +\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} +\right)} +\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +This changes the precision to {\tt DOUBLE}. \index{set fortran +precision double} Substitute {\tt single} for {\tt double} +\index{FORTRAN output format!precision} to return to single precision. +\index{set fortran precision single} + +\spadcommand{)set fortran precision double} + +Complex constants display the precision. +\spadcommand{2.3 + 5.6*\%i } +\begin{verbatim} + R5=(2.3D0,5.6D0) +\end{verbatim} +$$ +{2.3}+{{5.6} \ i} +$$ +\returnType{Type: Complex Float} + +The function names that Axiom generates depend on the chosen precision. +\spadcommand{sin \%e} +%%NOTE: the book shows DSIN(DEXP(1.0D0)) +\begin{verbatim} + R6=DSIN(DEXP(1)) +\end{verbatim} +$$ +\sin +\left( +{e} +\right) +$$ +\returnType{Type: Expression Integer} + +Reset the precision to {\tt single} and look at these two examples again. +\spadcommand{)set fortran precision single} + +\spadcommand{2.3 + 5.6*\%i} +\begin{verbatim} + R7=(2.3,5.6) +\end{verbatim} +$$ +{2.3}+{{5.6} \ i} +$$ +\returnType{Type: Complex Float} + +\spadcommand{sin \%e} +%%NOTE: the book shows SIN(EXP(1.)) +\begin{verbatim} + R8=SIN(EXP(1)) +\end{verbatim} +$$ +\sin +\left( +{e} +\right) +$$ +\returnType{Type: Expression Integer} +Expressions that look like lists, streams, sets or matrices cause +array code to be generated. +\spadcommand{[x+1,y+1,z+1]} +\begin{verbatim} + T1(1)=x+1 + T1(2)=y+1 + T1(3)=z+1 + R9=T1 +\end{verbatim} +$$ +\left[ +{x+1}, {y+1}, {z+1} +\right] +$$ +\returnType{Type: List Polynomial Integer} + + +A temporary variable is generated to be the name of the array. +\index{FORTRAN output format!arrays} This may have to be changed in +your particular application. +\spadcommand{set[2,3,4,3,5]} +\begin{verbatim} + T1(1)=2 + T1(2)=3 + T1(3)=4 + T1(4)=5 + R10=T1 +\end{verbatim} +$$ +\left\{ +2, 3, 4, 5 +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +By default, the starting index for generated FORTRAN arrays is $0$. +\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} +\begin{verbatim} + T1(0,0)=2.3 + T1(0,1)=9.7 + T1(1,0)=0.0 + T1(1,1)=18.778 + T1 +\end{verbatim} +$$ +\left[ +\begin{array}{cc} +{2.3} & {9.7} \\ +{0.0} & {18.778} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +To change the starting index for generated FORTRAN arrays to be $1$, +\index{set fortran startindex} issue this. This value can only be $0$ +or $1$. +\spadcommand{)set fortran startindex 1} + +Look at the code generated for the matrix again. +\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} +\begin{verbatim} + T1(1,1)=2.3 + T1(1,2)=9.7 + T1(2,1)=0.0 + T1(2,2)=18.778 + T1 +\end{verbatim} +$$ +\left[ +\begin{array}{cc} +{2.3} & {9.7} \\ +{0.0} & {18.778} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + + +\setcounter{chapter}{4} + +\chapter{Overview of Interactive Language} +\label{ugLang} + +In this chapter we look at some of the basic components of the Axiom +language that you can use interactively. We show how to create a {\it +block} of expressions, how to form loops and list iterations, how to +modify the sequential evaluation of a block and how to use +{\tt if-then-else} to evaluate parts of your program conditionally. We +suggest you first read the boxed material in each section and then +proceed to a more thorough reading of the chapter. + +\section{Immediate and Delayed Assignments} +\label{ugLangAssign} + +A {\it variable} in Axiom refers to a value. A variable has a name +beginning with an uppercase or lowercase alphabetic character, +``{\tt \%}'', or ``{\tt !}''. Successive characters (if any) can be any of +the above, digits, or ``{\tt ?}''. Case is distinguished. The +following are all examples of valid, distinct variable names: + +\begin{verbatim} +a tooBig? a1B2c3%!? +A %j numberOfPoints +beta6 %J numberofpoints +\end{verbatim} + +The ``{\tt :=}'' operator is the immediate {\it assignment} operator. +\index{assignment!immediate} Use it to associate a value with a +variable. \index{immediate assignment} + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for immediate assignment for a single variable is +\begin{center} +{\it variable} $:=$ {\it expression} +\end{center} +The value returned by an immediate assignment is the value of +{\it expression}.\\ +} + +The right-hand side of the expression is evaluated, yielding $1$. +This value is then assigned to $a$. +\spadcommand{a := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The right-hand side of the expression is evaluated, yielding $1$. +This value is then assigned to $b$. Thus $a$ and $b$ both have the +value $1$ after the sequence of assignments. +\spadcommand{b := a} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +What is the value of $b$ if $a$ is assigned the value $2$? +\spadcommand{a := 2} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +As you see, the value of $b$ is left unchanged. +\spadcommand{b} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +This is what we mean when we say this kind of assignment is {\it +immediate}; $b$ has no dependency on $a$ after the initial assignment. +This is the usual notion of assignment found in programming languages +such as C, \index{C language!assignment} PASCAL +\index{PASCAL!assignment} and FORTRAN. \index{FORTRAN!assignment} + +Axiom provides delayed assignment with ``{\tt ==}''. +\index{assignment!delayed} This implements a \index{delayed +assignment} delayed evaluation of the right-hand side and dependency +checking. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for delayed assignment is +\begin{center} +{\it variable} $==$ {\it expression} +\end{center} +The value returned by a delayed assignment is the unique value of {\tt Void}.\\ +} + +Using $a$ and $b$ as above, these are the corresponding delayed assignments. +\spadcommand{a == 1} +\returnType{Type: Void} + +\spadcommand{b == a} +\returnType{Type: Void} + +The right-hand side of each delayed assignment is left unevaluated +until the variables on the left-hand sides are evaluated. Therefore +this evaluation and \ldots +\spadcommand{a} +\begin{verbatim} +Compiling body of rule a to compute value of type PositiveInteger +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +this evaluation seem the same as before. +\spadcommand{b} +\begin{verbatim} +Compiling body of rule b to compute value of type PositiveInteger +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +If we change $a$ to $2$ +\spadcommand{a == 2} +\begin{verbatim} + Compiled code for a has been cleared. + Compiled code for b has been cleared. + 1 old definition(s) deleted for function or rule a +\end{verbatim} +\returnType{Type: Void} + +then $a$ evaluates to $2$, as expected, but +\spadcommand{a} +\begin{verbatim} +Compiling body of rule a to compute value of type PositiveInteger + ++++ |*0;a;1;G82322| redefined +\end{verbatim} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +the value of $b$ reflects the change to $a$. +\spadcommand{b} +\begin{verbatim} +Compiling body of rule b to compute value of type PositiveInteger + ++++ |*0;b;1;G82322| redefined +\end{verbatim} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +It is possible to set several variables at the same time +\index{assignment!multiple immediate} by using \index{multiple +immediate assignment} a {\it tuple} of variables and a tuple of +expressions. Note that a {\it tuple} is a collection of things +separated by commas, often surrounded by parentheses. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for multiple immediate assignments is +\begin{center} +{\tt ( $\hbox{\it var}_{1}$, $\hbox{\it var}_{2}$, \ldots, +$\hbox{\it var}_{N}$ ) := ( $\hbox{\it expr}_{1}$, $\hbox{\it expr}_{2}$, +\ldots, $\hbox{\it expr}_{N}$ ) } +\end{center} +The value returned by an immediate assignment is the value of +$\hbox{\it expr}_{N}$.\\ +} + +This sets $x$ to $1$ and $y$ to $2$. +\spadcommand{(x,y) := (1,2)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Multiple immediate assigments are parallel in the sense that the +expressions on the right are all evaluated before any assignments on +the left are made. However, the order of evaluation of these +expressions is undefined. + +You can use multiple immediate assignment to swap the values held by +variables. +\spadcommand{(x,y) := (y,x)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +$x$ has the previous value of $y$. +\spadcommand{x} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +$y$ has the previous value of $x$. +\spadcommand{y} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +There is no syntactic form for multiple delayed assignments. See the +discussion in section \ref{ugUserDelay} on page~\pageref{ugUserDelay} +about how Axiom differentiates between delayed assignments and user +functions of no arguments. + +\section{Blocks} +\label{ugLangBlocks} + +A {\it block} is a sequence of expressions evaluated in the order that +they appear, except as modified by control expressions such as +{\tt break}, \index{break} {\tt return}, \index{return} {\tt iterate} and +\index{iterate} {\tt if-then-else} constructions. The value of a block is +the value of the expression last evaluated in the block. + +To leave a block early, use ``{\tt =>}''. For example, $i < 0 => x$. The +expression before the ``{\tt =>}'' must evaluate to {\tt true} or {\tt false}. +The expression following the ``{\tt =>}'' is the return value for the block. + +A block can be constructed in two ways: +\begin{enumerate} +\item the expressions can be separated by semicolons +and the resulting expression surrounded by parentheses, and +\item the expressions can be written on succeeding lines with each line +indented the same number of spaces (which must be greater than zero). +\index{indentation} +A block entered in this form is +called a {\it pile}. +\end{enumerate} +Only the first form is available if you are entering expressions +directly to Axiom. Both forms are available in {\bf .input} files. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for a simple block of expressions entered interactively is +\begin{center} +{\tt ( $\hbox{\it expression}_{1}$; $\hbox{\it expression}_{2}$; \ldots; +$\hbox{\it expression}_{N}$ )} +\end{center} +The value returned by a block is the value of an {\tt =>} expression, +or $\hbox{\it expression}_{N}$ if no {\tt =>} is encountered.\\ +} + +In {\bf .input} files, blocks can also be written using piles. The +examples throughout this book are assumed to come from {\bf .input} files. + +In this example, we assign a rational number to $a$ using a block +consisting of three expressions. This block is written as a pile. +Each expression in the pile has the same indentation, in this case two +spaces to the right of the first line. +\begin{verbatim} +a := + i := gcd(234,672) + i := 3*i**5 - i + 1 + 1 / i +\end{verbatim} +$$ +1 \over {23323} +$$ +\returnType{Type: Fraction Integer} + +Here is the same block written on one line. This is how you are +required to enter it at the input prompt. +\spadcommand{a := (i := gcd(234,672); i := 3*i**5 - i + 1; 1 / i)} +$$ +1 \over {23323} +$$ +\returnType{Type: Fraction Integer} + +Blocks can be used to put several expressions on one line. The value +returned is that of the last expression. +\spadcommand{(a := 1; b := 2; c := 3; [a,b,c])} +$$ +\left[ +1, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Axiom gives you two ways of writing a block and the preferred way in +an {\bf .input} file is to use a pile. \index{file!input} Roughly +speaking, a pile is a block whose constituent expressions are indented +the same amount. You begin a pile by starting a new line for the +first expression, indenting it to the right of the previous line. You +then enter the second expression on a new line, vertically aligning it +with the first line. And so on. If you need to enter an inner pile, +further indent its lines to the right of the outer pile. Axiom knows +where a pile ends. It ends when a subsequent line is indented to the +left of the pile or the end of the file. + +Blocks can be used to perform several steps before an assignment +(immediate or delayed) is made. +\begin{verbatim} +d := + c := a**2 + b**2 + sqrt(c * 1.3) +\end{verbatim} +$$ +2.5495097567 96392415 +$$ +\returnType{Type: Float} + +Blocks can be used in the arguments to functions. (Here $h$ is +assigned $2.1 + 3.5$.) +\begin{verbatim} +h := 2.1 + + 1.0 + 3.5 +\end{verbatim} +$$ +5.6 +$$ +\returnType{Type: Float} + +Here the second argument to {\bf eval} is $x = z$, where the value of +$z$ is computed in the first line of the block starting on the second +line. +\begin{verbatim} +eval(x**2 - x*y**2, + z := %pi/2.0 - exp(4.1) + x = z + ) +\end{verbatim} +$$ +{{58.7694912705 67072878} \ {y \sp 2}}+{3453.8531042012 59382} +$$ +\returnType{Type: Polynomial Float} + +Blocks can be used in the clauses of {\tt if-then-else} expressions +(see \ref{ugLangIf} on page~\pageref{ugLangIf}). + +\spadcommand{if h > 3.1 then 1.0 else (z := cos(h); max(z,0.5))} +$$ +1.0 +$$ +\returnType{Type: Float} + +This is the pile version of the last block. +\begin{verbatim} +if h > 3.1 then + 1.0 + else + z := cos(h) + max(z,0.5) +\end{verbatim} +$$ +1.0 +$$ +\returnType{Type: Float} + +Blocks can be nested. +\spadcommand{a := (b := factorial(12); c := (d := eulerPhi(22); factorial(d));b+c)} +$$ +482630400 +$$ +\returnType{Type: PositiveInteger} + +This is the pile version of the last block. +\begin{verbatim} +a := + b := factorial(12) + c := + d := eulerPhi(22) + factorial(d) + b+c +\end{verbatim} +$$ +482630400 +$$ +\returnType{Type: PositiveInteger} + +Since $c + d$ does equal $3628855$, $a$ has the value of $c$ and the +last line is never evaluated. +\begin{verbatim} +a := + c := factorial 10 + d := fibonacci 10 + c + d = 3628855 => c + d +\end{verbatim} +$$ +3628800 +$$ +\returnType{Type: PositiveInteger} + +\section{if-then-else} +\label{ugLangIf} + +Like many other programming languages, Axiom uses the three keywords +\index{if} {\tt if}, {\tt then} \index{then} and {\tt else} +\index{else} to form \index{conditional} conditional expressions. The +{\tt else} part of the conditional is optional. The expression +between the {\tt if} and {\tt then} keywords is a {\it predicate}: an +expression that evaluates to or is convertible to either {\tt true} or +{\tt false}, that is, a {\tt Boolean}. \index{Boolean} + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for conditional expressions is +\begin{center} +{\tt if\ }{\it predicate} +{\tt then\ }$\hbox{\it expression}_{1}$ +{\tt else\ }$\hbox{\it expression}_{2}$ +\end{center} +where the {\tt else} $\hbox{\it expression}_{2}$ part is optional. The +value returned from a conditional expression is +$\hbox{\it expression}_{1}$ if the predicate evaluates to {\tt true} and +$\hbox{\it expression}_{2}$ otherwise. If no {\tt else} clause is given, +the value is always the unique value of {\tt Void}.\\ +} + +An {\tt if-then-else} expression always returns a value. If the +{\tt else} clause is missing then the entire expression returns the unique +value of {\tt Void}. If both clauses are present, the type of the +value returned by {\tt if} is obtained by resolving the types of the +values of the two clauses. See \ref{ugTypesResolve} on +page~\pageref{ugTypesResolve} for more information. + +The predicate must evaluate to, or be convertible to, an object of +type {\tt Boolean}: {\tt true} or {\tt false}. By default, the equal +sign \spadopFrom{=}{Equation} creates \index{equation} an equation. + +This is an equation. \index{Equation} In particular, it is an object +of type {\tt Equation Polynomial Integer}. + +\spadcommand{x + 1 = y} +$$ +{x+1}=y +$$ +\returnType{Type: Equation Polynomial Integer} + +However, for predicates in {\tt if} expressions, Axiom \index{equality +testing} places a default target type of {\tt Boolean} on the +predicate and equality testing is performed. \index{Boolean} Thus you +need not qualify the ``{\tt =}'' in any way. In other contexts you +may need to tell Axiom that you want to test for equality rather than +create an equation. In those cases, use ``{\tt @}'' and a target type +of {\tt Boolean}. See section \ref{ugTypesPkgCall} on +page~\pageref{ugTypesPkgCall} for more information. + +The compound symbol meaning ``not equal'' in Axiom is +\index{inequality testing} ``{$\sim =$}''. \index{\_notequal@$\sim =$} +This can be used directly without a package call or a target +specification. The expression $a$~$\sim =$~$b$ is directly translated +into {\tt not}$(a = b)$. + +Many other functions have return values of type {\tt Boolean}. These +include ``{\tt <}'', ``{\tt <=}'', ``{\tt >}'', ``{\tt >=}'', +``{\tt $\sim$=}'' and ``{\bf member?}''. By convention, +operations with names ending in ``{\tt ?}'' return {\tt Boolean} values. + +The usual rules for piles are suspended for conditional expressions. +In {\bf .input} files, the {\tt then} and {\tt else} keywords can begin in the +same column as the corresponding {\tt if} but may also appear to the +right. Each of the following styles of writing {\tt if-then-else} +expressions is acceptable: +\begin{verbatim} +if i>0 then output("positive") else output("nonpositive") + +if i > 0 then output("positive") + else output("nonpositive") + +if i > 0 then output("positive") +else output("nonpositive") + +if i > 0 +then output("positive") +else output("nonpositive") + +if i > 0 + then output("positive") + else output("nonpositive") +\end{verbatim} + +A block can follow the {\tt then} or {\tt else} keywords. In the following +two assignments to {\tt a}, the {\tt then} and {\tt else} clauses each are +followed by two-line piles. The value returned in each is the value +of the second line. +\begin{verbatim} +a := + if i > 0 then + j := sin(i * pi()) + exp(j + 1/j) + else + j := cos(i * 0.5 * pi()) + log(abs(j)**5 + 1) + +a := + if i > 0 + then + j := sin(i * pi()) + exp(j + 1/j) + else + j := cos(i * 0.5 * pi()) + log(abs(j)**5 + 1) +\end{verbatim} + +These are both equivalent to the following: +\begin{verbatim} +a := + if i > 0 then (j := sin(i * pi()); exp(j + 1/j)) + else (j := cos(i * 0.5 * pi()); log(abs(j)**5 + 1)) +\end{verbatim} + +\section{Loops} +\label{ugLangLoops} + +A {\it loop} is an expression that contains another expression, +\index{loop} called the {\it loop body}, which is to be evaluated zero +or more \index{loop!body} times. All loops contain the {\tt repeat} +keyword and return the unique value of {\tt Void}. Loops can contain +inner loops to any depth. + +\boxed{4.6in}{ +\vskip 0.1cm +The most basic loop is of the form +\begin{center} +{\tt repeat\ }{\it loopBody} +\end{center} + +Unless {\it loopBody} contains a {\tt break} or {\tt return} expression, the +loop repeats forever. The value returned by the loop is the unique +value of {\tt Void}.\\ +} + +\subsection{Compiling vs. Interpreting Loops} +\label{ugLangLoopsCompInt} + +Axiom tries to determine completely the type of every object in a loop +and then to translate the loop body to LISP or even to machine code. +This translation is called compilation. + +If Axiom decides that it cannot compile the loop, it issues a +\index{loop!compilation} message stating the problem and then the +following message: +\begin{center} +{\bf We will attempt to step through and interpret the code.} +\end{center} + +It is still possible that Axiom can evaluate the loop but in {\it +interpret-code mode}. See section \ref{ugUserCompInt} on +page~\pageref{ugUserCompInt} where this is discussed in terms +\index{panic!avoiding} of compiling versus interpreting functions. + +\subsection{return in Loops} +\label{ugLangLoopsReturn} + +A {\tt return} expression is used to exit a function with +\index{loop!leaving via return} a particular value. In particular, if +a {\tt return} is in a loop within the \index{return} function, the loop +is terminated whenever the {\tt return} is evaluated. +%> This is a bug! The compiler should never accept allow +%> Void to be the return type of a function when it has to use +%> resolve to determine it. + +Suppose we start with this. +\begin{verbatim} +f() == + i := 1 + repeat + if factorial(i) > 1000 then return i + i := i + 1 +\end{verbatim} +\returnType{Type: Void} + +When {\tt factorial(i)} is big enough, control passes from inside the loop +all the way outside the function, returning the value of $i$ (or so we +think). +\spadcommand{f()} +\returnType{Type: Void} + +What went wrong? Isn't it obvious that this function should return an +integer? Well, Axiom makes no attempt to analyze the structure of a +loop to determine if it always returns a value because, in general, +this is impossible. So Axiom has this simple rule: the type of the +function is determined by the type of its body, in this case a block. +The normal value of a block is the value of its last expression, in +this case, a loop. And the value of every loop is the unique value of +{\tt Void}.! So the return type of {\bf f} is {\tt Void}. + +There are two ways to fix this. The best way is for you to tell Axiom +what the return type of $f$ is. You do this by giving $f$ a +declaration {\tt f:()~->~Integer} prior to calling for its value. This +tells Axiom: ``trust me---an integer is returned.'' We'll explain +more about this in the next chapter. Another clumsy way is to add a +dummy expression as follows. + +Since we want an integer, let's stick in a dummy final expression that is +an integer and will never be evaluated. +\begin{verbatim} +f() == + i := 1 + repeat + if factorial(i) > 1000 then return i + i := i + 1 + 0 +\end{verbatim} +\returnType{Type: Void} + +When we try {\bf f} again we get what we wanted. See +\ref{ugUserBlocks} on page~\pageref{ugUserBlocks} for more information. + +\spadcommand{f()} +\begin{verbatim} + Compiling function f with type () -> NonNegativeInteger +\end{verbatim} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +\subsection{break in Loops} +\label{ugLangLoopsBreak} + +The {\tt break} keyword is often more useful \index{break} in terminating +\index{loop!leaving via break} a loop. A {\tt break} causes control to +transfer to the expression immediately following the loop. As loops +always return the unique value of {\tt Void}., you cannot return a +value with {\tt break}. That is, {\tt break} takes no argument. + +This example is a modification of the last example in the previous +section \ref{ugLangLoopsReturn} on page~\pageref{ugLangLoopsReturn}. +Instead of using {\tt return}, we'll use {\tt break}. + +\begin{verbatim} +f() == + i := 1 + repeat + if factorial(i) > 1000 then break + i := i + 1 + i +\end{verbatim} +\begin{verbatim} + Compiled code for f has been cleared. + 1 old definition(s) deleted for function or rule f +\end{verbatim} +\returnType{Type: Void} + +The loop terminates when {\tt factorial(i)} gets big enough, the last line +of the function evaluates to the corresponding ``good'' value of $i$, +and the function terminates, returning that value. + +\spadcommand{f()} +\begin{verbatim} + Compiling function f with type () -> PositiveInteger + ++++ |*0;f;1;G82322| redefined +\end{verbatim} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +You can only use {\tt break} to terminate the evaluation of one loop. +Let's consider a loop within a loop, that is, a loop with a nested +loop. First, we initialize two counter variables. + +\spadcommand{(i,j) := (1, 1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Nested loops must have multiple {\tt break} \index{loop!nested} +expressions at the appropriate nesting level. How would you rewrite +this so {\tt (i + j) > 10} is only evaluated once? +\begin{verbatim} +repeat + repeat + if (i + j) > 10 then break + j := j + 1 + if (i + j) > 10 then break + i := i + 1 +\end{verbatim} +\returnType{Type: Void} + +\subsection{break vs. {\tt =>} in Loop Bodies} +\label{ugLangLoopsBreakVs} + +Compare the following two loops: +\begin{verbatim} +i := 1 i := 1 +repeat repeat + i := i + 1 i := i + 1 + i > 3 => i if i > 3 then break + output(i) output(i) +\end{verbatim} + +In the example on the left, the values $2$ and $3$ for $i$ are +displayed but then the ``{\tt =>}'' does not allow control to reach the +call to \spadfunFrom{output}{OutputForm} again. The loop will not +terminate until you run out of space or interrupt the execution. The +variable $i$ will continue to be incremented because the ``{\tt =>}'' only +means to leave the {\it block}, not the loop. + +In the example on the right, upon reaching $4$, the {\tt break} will be +executed, and both the block and the loop will terminate. This is one +of the reasons why both ``{\tt =>}'' and {\tt break} are provided. Using a +{\tt while} clause (see below) with the ``{\tt =>}'' \index{while} lets you +simulate the action of {\tt break}. + +\subsection{More Examples of break} +\label{ugLangLoopsBreakMore} + +Here we give four examples of {\tt repeat} loops that terminate when a +value exceeds a given bound. + +First, initialize $i$ as the loop counter. +\spadcommand{i := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +Here is the first loop. When the square of $i$ exceeds $100$, the +loop terminates. +\begin{verbatim} +repeat + i := i + 1 + if i**2 > 100 then break +\end{verbatim} +\returnType{Type: Void} + +Upon completion, $i$ should have the value $11$. +\spadcommand{i} +$$ +11 +$$ +\returnType{Type: NonNegativeInteger} + +Do the same thing except use ``{\tt =>}'' instead an {\tt if-then} expression. + +\spadcommand{i := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\begin{verbatim} +repeat + i := i + 1 + i**2 > 100 => break +\end{verbatim} +\returnType{Type: Void} + +\spadcommand{i} +$$ +11 +$$ +\returnType{Type: NonNegativeInteger} + +As a third example, we use a simple loop to compute $n!$. +\spadcommand{(n, i, f) := (100, 1, 1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Use $i$ as the iteration variable and $f$ to compute the factorial. +\begin{verbatim} +repeat + if i > n then break + f := f * i + i := i + 1 +\end{verbatim} +\returnType{Type: Void} + +Look at the value of $f$. +\spadcommand{f} +\begin{verbatim} + 93326215443944152681699238856266700490715968264381621468_ + 59296389521759999322991560894146397615651828625369792082_ + 7223758251185210916864000000000000000000000000 +\end{verbatim} +\returnType{Type: PositiveInteger} + +Finally, we show an example of nested loops. First define a four by +four matrix. +\spadcommand{m := matrix [ [21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13] ]} +$$ +\left[ +\begin{array}{cccc} +{21} & {37} & {53} & {14} \\ +8 & -{24} & {22} & -{16} \\ +2 & {10} & {15} & {14} \\ +{26} & {33} & {55} & -{13} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Next, set row counter $r$ and column counter $c$ to $1$. Note: if we +were writing a function, these would all be local variables rather +than global workspace variables. +\spadcommand{(r, c) := (1, 1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Also, let {\tt lastrow} and {\tt lastcol} be the final row and column index. + +\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Scan the rows looking for the first negative element. We remark that +you can reformulate this example in a better, more concise form by +using a {\tt for} clause with {\tt repeat}. See +\ref{ugLangLoopsForIn} on page~\pageref{ugLangLoopsForIn} for more +information. + +\begin{verbatim} +repeat + if r > lastrow then break + c := 1 + repeat + if c > lastcol then break + if elt(m,r,c) < 0 then + output [r, c, elt(m,r,c)] + r := lastrow + break -- don't look any further + c := c + 1 + r := r + 1 + + [2,2,- 24] +\end{verbatim} +\returnType{Type: Void} + +\subsection{iterate in Loops} +\label{ugLangLoopsIterate} + +Axiom provides an {\tt iterate} expression that \index{iterate} skips over +the remainder of a loop body and starts the next loop iteration. + +We first initialize a counter. + +\spadcommand{i := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +Display the even integers from $2$ to $5$. +\begin{verbatim} +repeat + i := i + 1 + if i > 5 then break + if odd?(i) then iterate + output(i) + + 2 + 4 +\end{verbatim} +\returnType{Type: Void} + +\subsection{while Loops} +\label{ugLangLoopsWhile} + +The {\tt repeat} in a loop can be modified by adding one or more {\tt while} +clauses. \index{while} Each clause contains a {\it predicate} +immediately following the {\tt while} keyword. The predicate is tested +{\it before} the evaluation of the body of the loop. The loop body is +evaluated whenever the predicates in a {\tt while} clause are all {\tt true}. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for a simple loop using {\tt while} is +\begin{center} +{\tt while} {\it predicate} {\tt repeat} {\it loopBody} +\end{center} +The {\it predicate} is evaluated before {\it loopBody} is evaluated. +A {\tt while} loop terminates immediately when {\it predicate} evaluates +to {\tt false} or when a {\tt break} or {\tt return} expression is evaluated in +{\it loopBody}. The value returned by the loop is the unique value of +{\tt Void}.\\ +} + +Here is a simple example of using {\tt while} in a loop. We first +initialize the counter. +\spadcommand{i := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The steps involved in computing this example are\\ +(1) set $i$ to $1$,\\ +(2) test the condition $i < 1$ and determine that it is not {\tt true}, and\\ +(3) do not evaluate the loop body and therefore do not display $"hello"$. +\begin{verbatim} +while i < 1 repeat + output "hello" + i := i + 1 +\end{verbatim} +\returnType{Type: Void} + +If you have multiple predicates to be tested use the logical {\tt and} +operation to separate them. Axiom evaluates these predicates from +left to right. +\spadcommand{(x, y) := (1, 1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 and y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] +\end{verbatim} +\returnType{Type: Void} + +A {\tt break} expression can be included in a loop body to terminate a +loop even if the predicate in any {\tt while} clauses are not {\tt false}. +\spadcommand{(x, y) := (1, 1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +This loop has multiple {\tt while} clauses and the loop terminates +before any one of their conditions evaluates to {\tt false}. +\begin{verbatim} +while x < 4 while y < 10 repeat + if x + y > 7 then break + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] +\end{verbatim} +\returnType{Type: Void} + +Here's a different version of the nested loops that looked for the +first negative element in a matrix. +\spadcommand{m := matrix [ [21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13] ]} +$$ +\left[ +\begin{array}{cccc} +{21} & {37} & {53} & {14} \\ +8 & -{24} & {22} & -{16} \\ +2 & {10} & {15} & {14} \\ +{26} & {33} & {55} & -{13} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Initialized the row index to $1$ and get the number of rows and +columns. If we were writing a function, these would all be local +variables. +\spadcommand{r := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Scan the rows looking for the first negative element. +\begin{verbatim} +while r <= lastrow repeat + c := 1 -- index of first column + while c <= lastcol repeat + if elt(m,r,c) < 0 then + output [r, c, elt(m,r,c)] + r := lastrow + break -- don't look any further + c := c + 1 + r := r + 1 + + [2,2,- 24] +\end{verbatim} +\returnType{Type: Void} + +\subsection{for Loops} +\label{ugLangLoopsForIn} + +Axiom provides the {\tt for} \index{for} and {\tt in\ } \index{in} keywords in +{\tt repeat} loops, allowing you to iterate across all \index{iteration} +elements of a list, or to have a variable take on integral values from +a lower bound to an upper bound. We shall refer to these modifying +clauses of {\tt repeat} loops as {\tt for} clauses. These clauses can be +present in addition to {\tt while} clauses. As with all other types of +{\tt repeat} loops, {\tt break} can \index{break} be used to prematurely +terminate the evaluation of the loop. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for a simple loop using {\tt for} is +\begin{center} +{\tt for} {\it iterator} {\tt repeat} {\it loopBody} +\end{center} + +The {\it iterator} has several forms. Each form has an end test which +is evaluated before {\it loopBody} is evaluated. A {\tt for} loop +terminates immediately when the end test succeeds (evaluates to +{\tt true}) or when a {\tt break} or {\tt return} expression is evaluated +in {\it loopBody}. The value returned by the loop is the unique value +of {\tt Void}.\\ } + +\subsection{for i in n..m repeat} +\label{ugLangLoopsForInNM} + +If {\tt for} \index{for} is followed by a variable name, the {\tt in\ } +\index{in} keyword and then an integer segment of the form $n..m$, +\index{segment} the end test for this loop is the predicate $i > m$. +The body of the loop is evaluated $m-n+1$ times if this number is +greater than 0. If this number is less than or equal to 0, the loop +body is not evaluated at all. + +The variable $i$ has the value $n, n+1, ..., m$ for successive iterations +of the loop body.The loop variable is a {\it local variable} +within the loop body: its value is not available outside the loop body +and its value and type within the loop body completely mask any outer +definition of a variable with the same name. + +This loop prints the values of +${10}^3$, ${11}^3$, and $12^3$: +\spadcommand{for i in 10..12 repeat output(i**3)} +\begin{verbatim} + 1000 + 1331 + 1728 +\end{verbatim} +\returnType{Type: Void} + +Here is a sample list. +\spadcommand{a := [1,2,3]} +$$ +\left[ +1, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Iterate across this list, using ``{\tt .}'' to access the elements of +a list and the ``{\bf \#}'' operation to count its elements. + +\spadcommand{for i in 1..\#a repeat output(a.i)} +\begin{verbatim} + 1 + 2 + 3 +\end{verbatim} +\returnType{Type: Void} + +This type of iteration is applicable to anything that uses ``{\tt .}''. +You can also use it with functions that use indices to extract elements. + +Define $m$ to be a matrix. +\spadcommand{m := matrix [ [1,2],[4,3],[9,0] ]} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +4 & 3 \\ +9 & 0 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Display the rows of $m$. +\spadcommand{for i in 1..nrows(m) repeat output row(m,i)} +\begin{verbatim} + [1,2] + [4,3] + [9,0] +\end{verbatim} +\returnType{Type: Void} + +You can use {\tt iterate} with {\tt for}-loops.\index{iterate} + +Display the even integers in a segment. +\begin{verbatim} +for i in 1..5 repeat + if odd?(i) then iterate + output(i) + + 2 + 4 +\end{verbatim} +\returnType{Type: Void} + +See section \ref{SegmentXmpPage} on page~\pageref{SegmentXmpPage} for +more information about segments. + +\subsection{for i in n..m by s repeat} +\label{ugLangLoopsForInNMS} + +By default, the difference between values taken on by a variable in +loops such as {\tt for i in n..m repeat ...} is $1$. It is possible to +supply another, possibly negative, step value by using the {\tt by} +\index{by} keyword along with {\tt for} and {\tt in\ }. Like the upper and +lower bounds, the step value following the {\tt by} keyword must be an +integer. Note that the loop {\tt for i in 1..2 by 0 repeat output(i)} +will not terminate by itself, as the step value does not change the +index from its initial value of $1$. + +This expression displays the odd integers between two bounds. +\spadcommand{for i in 1..5 by 2 repeat output(i)} +\begin{verbatim} + 1 + 3 + 5 +\end{verbatim} +\returnType{Type: Void} + +Use this to display the numbers in reverse order. +\spadcommand{for i in 5..1 by -2 repeat output(i)} +\begin{verbatim} + 5 + 3 + 1 +\end{verbatim} +\returnType{Type: Void} + +\subsection{for i in n.. repeat} +\label{ugLangLoopsForInN} + +If the value after the ``{\tt ..}'' is omitted, the loop has no end test. +A potentially infinite loop is thus created. The variable is given +the successive values ${n}, {n+1}, {n+2}, ...$ and the loop is terminated +only if a {\tt break} or {\tt return} expression is evaluated in the loop +body. However you may also add some other modifying clause on the +{\tt repeat} (for example, a {\tt while} clause) to stop the loop. + +This loop displays the integers greater than or equal to $15$ +and less than the first prime greater than $15$. +\spadcommand{for i in 15.. while not prime?(i) repeat output(i)} +\begin{verbatim} + 15 + 16 +\end{verbatim} +\returnType{Type: Void} + +\subsection{for x in l repeat} +\label{ugLangLoopsForInXL} + +Another variant of the {\tt for} loop has the form: +\begin{center} +{\it {\tt for} x {\tt in\ } list {\tt repeat} loopBody} +\end{center} + +This form is used when you want to iterate directly over the elements +of a list. In this form of the {\tt for} loop, the variable {\tt x} takes on +the value of each successive element in {\tt l}. The end test is most +simply stated in English: ``are there no more {\tt x} in {\tt l}?'' + +If {\tt l} is this list, +\spadcommand{l := [0,-5,3]} +$$ +\left[ +0, -5, 3 +\right] +$$ +\returnType{Type: List Integer} + +display all elements of {\tt l}, one per line. +\spadcommand{for x in l repeat output(x)} +\begin{verbatim} + 0 + - 5 + 3 +\end{verbatim} +\returnType{Type: Void} + +Since the list constructing expression {\bf expand}{\tt [n..m]} creates the +list $[n, {n+1}, ..., m]$. Note that this list is empty if $n > m$. You +might be tempted to think that the loops +\begin{verbatim} +for i in n..m repeat output(i) +\end{verbatim} + +and +\begin{verbatim} +for x in expand [n..m] repeat output(x) +\end{verbatim} + +are equivalent. The second form first creates the list {\bf +expand}{\tt [n..m]} (no matter how large it might be) and then does +the iteration. The first form potentially runs in much less space, as +the index variable $i$ is simply incremented once per loop and the +list is not actually created. Using the first form is much more +efficient. + +Of course, sometimes you really want to iterate across a specific list. +This displays each of the factors of $2400000$. +\spadcommand{for f in factors(factor(2400000)) repeat output(f)} +\begin{verbatim} + [factor= 2,exponent= 8] + [factor= 3,exponent= 1] + [factor= 5,exponent= 5] +\end{verbatim} +\returnType{Type: Void} + +\subsection{``Such that'' Predicates} +\label{ugLangLoopsForInPred} + +A {\tt for} loop can be followed by a ``{\tt |}'' and then a predicate. The +predicate qualifies the use of the values from the iterator following +the {\tt for}. Think of the vertical bar ``{\tt |}'' as the phrase ``such +that.'' + +This loop expression prints out the integers $n$ in the given segment +such that $n$ is odd. +\spadcommand{for n in 0..4 | odd? n repeat output n} +\begin{verbatim} + 1 + 3 +\end{verbatim} +\returnType{Type: Void} + +\boxed{4.6in}{ +\vskip 0.1cm +A {\tt for} loop can also be written +$$ +{\rm for} {\it \ iterator\ } | {\it \ predicate\ } +{\rm repeat} {\it \ loopBody\ } +$$ + +which is equivalent to: +$$ +{\rm for} {\it \ iterator\ } {\rm repeat\ if} +{\it \ predicate\ } {\rm then} {\it \ loopBody\ } {\rm else\ }iterate +$$ +} + +The predicate need not refer only to the variable in the {\tt for} clause: +any variable in an outer scope can be part of the predicate. + +In this example, the predicate on the inner {\tt for} loop uses $i$ from +the outer loop and the $j$ from the {\tt for} \index{iteration!nested} +clause that it directly modifies. +\begin{verbatim} +for i in 1..50 repeat + for j in 1..50 | factorial(i+j) < 25 repeat + output [i,j] + + [1,1] + [1,2] + [1,3] + [2,1] + [2,2] + [3,1] +\end{verbatim} +\returnType{Type: Void} + +\subsection{Parallel Iteration} +\label{ugLangLoopsPar} + +The last example of the previous section +\ref{ugLangLoopsForInPred} on page~\pageref{ugLangLoopsForInPred} +gives an example of {\it nested iteration}: a loop is contained +\index{iteration!nested} in another loop. \index{iteration!parallel} +Sometimes you want to iterate across two lists in parallel, or perhaps +you want to traverse a list while incrementing a variable. + +\boxed{4.6in}{ +\vskip 0.1cm +The general syntax of a repeat loop is +$$ +iterator_1\ iterator_2\ \ldots\ iterator_N {\rm \ repeat\ } loopBody +$$ +where each {\it iterator} is either a {\tt for} or a {\tt while} clause. The +loop terminates immediately when the end test of any {\it iterator} +succeeds or when a {\tt break} or {\tt return} expression is evaluated in {\it +loopBody}. The value returned by the loop is the unique value of {\tt +Void}.\\ +} + +Here we write a loop to iterate across two lists, computing the sum of +the pairwise product of elements. Here is the first list. +\spadcommand{l := [1,3,5,7]} +$$ +\left[ +1, 3, 5, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +And the second. +\spadcommand{m := [100,200]} +$$ +\left[ +{100}, {200} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The initial value of the sum counter. +\spadcommand{sum := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +The last two elements of $l$ are not used in the calculation because +$m$ has two fewer elements than $l$. +\begin{verbatim} +for x in l for y in m repeat + sum := sum + x*y +\end{verbatim} +\returnType{Type: Void} + +Display the ``dot product.'' +\spadcommand{sum} +$$ +700 +$$ +\returnType{Type: NonNegativeInteger} + +Next, we write a loop to compute the sum of the products of the loop +elements with their positions in the loop. +\spadcommand{l := [2,3,5,7,11,13,17,19,23,29,31,37]} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +{31}, {37} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The initial sum. +\spadcommand{sum := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +Here looping stops when the list $l$ is exhausted, even though +the $for i in 0..$ specifies no terminating condition. + +\spadcommand{for i in 0.. for x in l repeat sum := i * x} +\returnType{Type: Void} + +Display this weighted sum. +\spadcommand{sum} +$$ +407 +$$ +\returnType{Type: NonNegativeInteger} + +When ``{\tt |}'' is used to qualify any of the {\tt for} clauses in a parallel +iteration, the variables in the predicates can be from an outer scope +or from a {\tt for} clause in or to the left of a modified clause. + +This is correct: +% output from following is too long to show +\begin{verbatim} +for i in 1..10 repeat + for j in 200..300 | odd? (i+j) repeat + output [i,j] +\end{verbatim} + +This is not correct since the variable $j$ has not been defined +outside the inner loop. +\begin{verbatim} +for i in 1..10 | odd? (i+j) repeat -- wrong, j not defined + for j in 200..300 repeat + output [i,j] +\end{verbatim} + +\subsection{Mixing Loop Modifiers} +\label{ugLangLoopsMix} + +This example shows that it is possible to mix several of the +\index{loop!mixing modifiers} forms of {\tt repeat} modifying clauses on a loop. +\begin{verbatim} +for i in 1..10 + for j in 151..160 | odd? j + while i + j < 160 repeat + output [i,j] + + [1,151] + [3,153] +\end{verbatim} +\returnType{Type: Void} + +Here are useful rules for composing loop expressions: +\begin{enumerate} +\item {\tt while} predicates can only refer to variables that +are global (or in an outer scope) +or that are defined in {\tt for} clauses to the left of the +predicate. +\item A ``such that'' predicate (something following ``{\tt |}'') +must directly follow a {\tt for} clause and can only refer to +variables that are global (or in an outer scope) +or defined in the modified {\tt for} clause +or any {\tt for} clause to the left. +\end{enumerate} + +\section{Creating Lists and Streams with Iterators} +\label{ugLangIts} + +All of what we did for loops in +\ref{ugLangLoops} on page~\pageref{ugLangLoops} \index{iteration} +can be transformed into expressions that create lists +\index{list!created by iterator} and streams. \index{stream!created +by iterator} The {\tt repeat}, {\tt break} or {\tt iterate} words are not used but +all the other ideas carry over. Before we give you the general rule, +here are some examples which give you the idea. + +This creates a simple list of the integers from $1$ to $10$. +\spadcommand{list := [i for i in 1..10]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Create a stream of the integers greater than or equal to $1$. +\spadcommand{stream := [i for i in 1..]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +This is a list of the prime integers between $1$ and $10$, inclusive. +\spadcommand{[i for i in 1..10 | prime? i]} +$$ +\left[ +2, 3, 5, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +This is a stream of the prime integers greater than or equal to $1$. +\spadcommand{[i for i in 1.. | prime? i]} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +This is a list of the integers between $1$ and $10$, inclusive, whose +squares are less than $700$. +\spadcommand{[i for i in 1..10 while i*i < 700]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +This is a stream of the integers greater than or equal to $1$ +whose squares are less than $700$. +\spadcommand{[i for i in 1.. while i*i < 700]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +Here is the general rule. +\index{collection} + +\boxed{4.6in}{ +\vskip 0.1cm +The general syntax of a collection is +\begin{center} +{\tt [ {\it collectExpression} $\hbox{\it iterator}_{1}$ +$\hbox{\it iterator}_{2}$ \ldots $\hbox{\it iterator}_{N}$ ]} +\end{center} + +where each $\hbox{\it iterator}_{i}$ is either a {\tt for} or a {\tt while} +clause. The loop terminates immediately when the end test of any +$\hbox{\it iterator}_{i}$ succeeds or when a {\tt return} expression is +evaluated in {\it collectExpression}. The value returned by the +collection is either a list or a stream of elements, one for each +iteration of the {\it collectExpression}.\\ +} + +Be careful when you use {\tt while} +\index{stream!using while @{using {\tt while}}} +to create a stream. By default, Axiom tries to compute and +display the first ten elements of a stream. If the {\tt while} condition +is not satisfied quickly, Axiom can spend a long (possibly infinite) +time trying to compute \index{stream!number of elements computed} the +elements. Use {\tt )set streams calculate} to change the default to +something else. \index{set streams calculate} This also affects the +number of terms computed and displayed for power series. For the +purposes of this book, we have used this system command to display +fewer than ten terms. + +Use nested iterators to create lists of \index{iteration!nested} lists +which can then be given as an argument to {\bf matrix}. +\spadcommand{matrix [ [x**i+j for i in 1..3] for j in 10..12]} +$$ +\left[ +\begin{array}{ccc} +{x+{10}} & {{x \sp 2}+{10}} & {{x \sp 3}+{10}} \\ +{x+{11}} & {{x \sp 2}+{11}} & {{x \sp 3}+{11}} \\ +{x+{12}} & {{x \sp 2}+{12}} & {{x \sp 3}+{12}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +You can also create lists of streams, streams of lists and streams of +streams. Here is a stream of streams. +\spadcommand{[ [i/j for i in j+1..] for j in 1..]} +$$ +\begin{array}{@{}l} +\left[ +{\left[ 2, 3, 4, 5, 6, 7, 8, 9, {10}, {11}, +\ldots +\right]}, + {\left[ {3 \over 2}, 2, {5 \over 2}, 3, {7 \over 2}, 4, +{9 \over 2}, 5, {{11} \over 2}, 6, \ldots +\right]}, +\right. +\\ +\\ +\displaystyle + {\left[ {4 \over 3}, {5 \over 3}, 2, {7 \over 3}, {8 \over 3}, + 3, {{10} \over 3}, {{11} \over 3}, 4, {{13} \over 3}, +\ldots +\right]}, + {\left[ {5 \over 4}, {3 \over 2}, {7 \over 4}, 2, {9 \over 4}, + {5 \over 2}, {{11} \over 4}, 3, {{13} \over 4}, {7 \over 2}, + \ldots +\right]}, +\\ +\\ +\displaystyle + {\left[ {6 \over 5}, {7 \over 5}, {8 \over 5}, {9 \over 5}, 2, + {{11} \over 5}, {{12} \over 5}, {{13} \over 5}, {{14} \over 5}, + 3, \ldots +\right]}, + {\left[ {7 \over 6}, {4 \over 3}, {3 \over 2}, {5 \over 3}, +{{11} \over 6}, 2, {{13} \over 6}, {7 \over 3}, {5 \over 2}, +{8 \over 3}, \ldots +\right]}, +\\ +\\ +\displaystyle + {\left[ {8 \over 7}, {9 \over 7}, {{10} \over 7}, {{11} \over 7}, + {{12} \over 7}, {{13} \over 7}, 2, {{15} \over 7}, {{16} \over +7}, {{17} \over 7}, \ldots +\right]}, + {\left[ {9 \over 8}, {5 \over 4}, {{11} \over 8}, {3 \over 2}, +{{13} \over 8}, {7 \over 4}, {{15} \over 8}, 2, {{17} \over 8}, + {9 \over 4}, \ldots +\right]}, +\\ +\\ +\displaystyle + {\left[ {{10} \over 9}, {{11} \over 9}, {4 \over 3}, {{13} \over +9}, {{14} \over 9}, {5 \over 3}, {{16} \over 9}, {{17} \over 9}, + 2, {{19} \over 9}, \ldots +\right]}, +\\ +\\ +\displaystyle +\left. + {\left[ {{11} \over {10}}, {6 \over 5}, {{13} \over {10}}, {7 +\over 5}, {3 \over 2}, {8 \over 5}, {{17} \over {10}}, {9 \over +5}, {{19} \over {10}}, 2, \ldots +\right]}, + \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream Stream Fraction Integer} + +You can use parallel iteration across lists and streams to create +\index{iteration!parallel} new lists. +\spadcommand{[i/j for i in 3.. by 10 for j in 2..]} +$$ +\left[ +{3 \over 2}, {{13} \over 3}, {{23} \over 4}, {{33} \over 5}, +{{43} \over 6}, {{53} \over 7}, {{63} \over 8}, {{73} \over 9}, +{{83} \over {10}}, {{93} \over {11}}, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +Iteration stops if the end of a list or stream is reached. +\spadcommand{[i**j for i in 1..7 for j in 2.. ]} +$$ +\left[ +1, 8, {81}, {1024}, {15625}, {279936}, {5764801} +\right] +$$ +\returnType{Type: Stream Integer} + +%or a while condition fails. +%\spadcommand{[i**j for i in 1.. for j in 2.. while i + j < 5 ]} +%tpdhere +% There are no library operations named swhile +% Use HyperDoc Browse or issue +% )what op swhile +% to learn if there is any operation containing " swhile " in its +% name. +% +% Cannot find a definition or applicable library operation named +% swhile with argument type(s) +% (Record(part1: PositiveInteger,part2: PositiveInteger) -> Boolean) +% InfiniteTuple Record(part1: PositiveInteger,part2: PositiveInteger) +% +% Perhaps you should use "@" to indicate the required return type, +% or "$" to specify which version of the function you need. + +As with loops, you can combine these modifiers to make very +complicated conditions. +\spadcommand{[ [ [i,j] for i in 10..15 | prime? i] for j in 17..22 | j = squareFreePart j]} +$$ +\left[ +{\left[ {\left[ {11}, {17} +\right]}, + {\left[ {13}, {17} +\right]} +\right]}, + {\left[ {\left[ {11}, {19} +\right]}, + {\left[ {13}, {19} +\right]} +\right]}, + {\left[ {\left[ {11}, {21} +\right]}, + {\left[ {13}, {21} +\right]} +\right]}, + {\left[ {\left[ {11}, {22} +\right]}, + {\left[ {13}, {22} +\right]} +\right]} +\right] +$$ +\returnType{Type: List List List PositiveInteger} + +See List +(section \ref{ListXmpPage} on page~\pageref{ListXmpPage}) and Stream +(section \ref{StreamXmpPage} on page~\pageref{StreamXmpPage}) +for more information on creating and +manipulating lists and streams, respectively. + +\section{An Example: Streams of Primes} +\label{ugLangStreamsPrimes} + +We conclude this chapter with an example of the creation and +manipulation of infinite streams of prime integers. This might be +useful for experiments with numbers or other applications where you +are using sequences of primes over and over again. As for all +streams, the stream of primes is only computed as far out as you need. +Once computed, however, all the primes up to that point are saved for +future reference. + +Two useful operations provided by the Axiom library are +\spadfunFrom{prime?}{IntegerPrimesPackage} and +\spadfunFrom{nextPrime}{IntegerPrimesPackage}. A straight-forward way +to create a stream of prime numbers is to start with the stream of +positive integers $[2,..]$ and filter out those that are prime. + +Create a stream of primes. +\spadcommand{primes : Stream Integer := [i for i in 2.. | prime? i]} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +A more elegant way, however, is to use the +\spadfunFrom{generate}{Stream} operation from {\tt Stream}. Given an +initial value $a$ and a function $f$, \spadfunFrom{generate}{Stream} +constructs the stream $[a, f(a), f(f(a)), ...]$. This function gives +you the quickest method of getting the stream of primes. + +This is how you use \spadfunFrom{generate}{Stream} to generate an +infinite stream of primes. +\spadcommand{primes := generate(nextPrime,2)} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Once the stream is generated, you might only be interested in primes +starting at a particular value. +\spadcommand{smallPrimes := [p for p in primes | p > 1000]} +$$ +\left[ +{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, +{1049}, {1051}, {1061}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Here are the first 11 primes greater than 1000. +\spadcommand{[p for p in smallPrimes for i in 1..11]} +$$ +\left[ +{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, +{1049}, {1051}, {1061}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Here is a stream of primes between 1000 and 1200. +\spadcommand{[p for p in smallPrimes while p < 1200]} +$$ +\left[ +{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, +{1049}, {1051}, {1061}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +To get these expanded into a finite stream, you call +\spadfunFrom{complete}{Stream} on the stream. +\spadcommand{complete \%} +$$ +\left[ +{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, +{1049}, {1051}, {1061}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Twin primes are consecutive odd number pairs which are prime. +Here is the stream of twin primes. +\spadcommand{twinPrimes := [ [p,p+2] for p in primes | prime?(p + 2)]} +$$ +\begin{array}{@{}l} +\left[ +{\left[ 3, 5 \right]}, +{\left[ 5, 7 \right]}, +{\left[ {11}, {13} \right]}, +{\left[ {17}, {19} \right]}, +{\left[ {29}, {31} \right]}, +{\left[ {41}, {43} \right]}, +{\left[ {59}, {61} \right]}, +{\left[ {71}, {73} \right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ {101}, {103} \right]}, +{\left[ {107}, {109} \right]}, + \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream List Integer} + +Since we already have the primes computed we can avoid the call to +\spadfunFrom{prime?}{IntegerPrimesPackage} by using a double +iteration. This time we'll just generate a stream of the first of the +twin primes. +\spadcommand{firstOfTwins:= [p for p in primes for q in rest primes | q=p+2]} +$$ +\left[ +3, 5, {11}, {17}, {29}, {41}, {59}, {71}, {101}, +{107}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Let's try to compute the infinite stream of triplet primes, the set of +primes $p$ such that $[p,p+2,p+4]$ are primes. For example, $[3,5,7]$ +is a triple prime. We could do this by a triple {\tt for} iteration. A +more economical way is to use {\bf firstOfTwins}. This time however, +put a semicolon at the end of the line. + +Create the stream of firstTriplets. Put a semicolon at the end so +that no elements are computed. +\spadcommand{firstTriplets := [p for p in firstOfTwins for q in rest firstOfTwins | q = p+2];} +\returnType{Type: Stream Integer} + +What happened? As you know, by default Axiom displays the first ten +elements of a stream when you first display it. And, therefore, it +needs to compute them! If you want {\it no} elements computed, just +terminate the expression by a semicolon (``{\tt ;}''). The semi-colon +prevents the display of the result of evaluating the expression. +Since no stream elements are needed for display (or anything else, so +far), none are computed. + +Compute the first triplet prime. +\spadcommand{firstTriplets.1} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +If you want to compute another, just ask for it. But wait a second! +Given three consecutive odd integers, one of them must be divisible by +$3$. Thus there is only one triplet prime. But suppose that you did not +know this and wanted to know what was the tenth triplet prime. +\begin{verbatim} +firstTriples.10 +\end{verbatim} + +To compute the tenth triplet prime, Axiom first must compute the +second, the third, and so on. But since there isn't even a second +triplet prime, Axiom will compute forever. Nonetheless, this effort +can produce a useful result. After waiting a bit, hit \fbox{\bf Ctrl-c}. +The system responds as follows. +\begin{verbatim} + >> System error: + Console interrupt. + You are being returned to the top level of + the interpreter. +\end{verbatim} + +If you want to know how many primes have been computed, type: +\begin{verbatim} +numberOfComputedEntries primes +\end{verbatim} + +and, for this discussion, let's say that the result is $2045$. +How big is the $2045$-th prime? +\spadcommand{primes.2045} +$$ +17837 +$$ +\returnType{Type: PositiveInteger} + +What you have learned is that there are no triplet primes between 5 +and 17837. Although this result is well known (some might even say +trivial), there are many experiments you could make where the result +is not known. What you see here is a paradigm for testing of +hypotheses. Here our hypothesis could have been: ``there is more than +one triplet prime.'' We have tested this hypothesis for 17837 cases. +With streams, you can let your machine run, interrupt it to see how +far it has progressed, then start it up and let it continue from where +it left off. + +\setcounter{chapter}{5} + +\chapter{User-Defined Functions, Macros and Rules} +\label{ugUser} + +In this chapter we show you how to write functions and macros, +and we explain how Axiom looks for and applies them. +We show some simple one-line examples of functions, together +with larger ones that are defined piece-by-piece or through the use of +piles. + +\section{Functions vs. Macros} +\label{ugUserFunMac} + +A function is a program to perform some \index{function!vs. macro} +computation. \index{macro!vs. function} Most functions have names so +that it is easy to refer to them. A simple example of a function is +one named \spadfunFrom{abs}{Integer} which computes the absolute value +of an integer. + +This is a use of the ``absolute value'' library function for integers. +\spadcommand{abs(-8)} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +This is an unnamed function that does the same thing, using the +``maps-to'' syntax {\tt +->} that we discuss in +section \ref{ugUserAnon} on page~\pageref{ugUserAnon}. +\spadcommand{(x +-> if x < 0 then -x else x)(-8)} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +Functions can be used alone or serve as the building blocks for larger +programs. Usually they return a value that you might want to use in +the next stage of a computation, but not always (for example, see +\ref{ExitXmpPage} on page~\pageref{ExitXmpPage} and \ref{VoidXmpPage} +on page~\pageref{VoidXmpPage}). They may also read data from your +keyboard, move information from one place to another, or format and +display results on your screen. + +In Axiom, as in mathematics, functions \index{function!parameters} are +usually parameterized. Each time you {\it call} (some people say {\it +apply} or invoke) a function, you give \index{parameters to a +function} values to the parameters (variables). Such a value is +called an {\it argument} of \index{function!arguments} the function. +Axiom uses the arguments for the computation. In this way you get +different results depending on what you ``feed'' the function. + +Functions can have local variables or refer to global variables in the +workspace. Axiom can often compile functions so that they execute +very efficiently. Functions can be passed as arguments to other +functions. + +Macros are textual substitutions. They are used to clarify the +meaning of constants or expressions and to be templates for frequently +used expressions. Macros can be parameterized but they are not +objects that can be passed as arguments to functions. In effect, +macros are extensions to the Axiom expression parser. + +\section{Macros} +\label{ugUserMacros} + +A {\it macro} provides general textual substitution of \index{macro} +an Axiom expression for a name. You can think of a macro as being a +generalized abbreviation. You can only have one macro in your +workspace with a given name, no matter how many arguments it has. + +\boxed{4.6in}{ +\vskip 0.1cm +The two general forms for macros are +\begin{center} +{\tt macro} {\it name} {\tt ==} {\it body} \\ +{\tt macro} {\it name(arg1,...)} {\tt ==} {\it body} +\end{center} +where the body of the macro can be any Axiom expression.\\ +} + +For example, suppose you decided that you like to use {\tt df} for +{\tt D}. You define the macro {\tt df} like this. +\spadcommand{macro df == D} +\returnType{Type: Void} + +Whenever you type {\tt df}, the system expands it to {\tt D}. +\spadcommand{df(x**2 + x + 1,x)} +$$ +{2 \ x}+1 +$$ +\returnType{Type: Polynomial Integer} + +Macros can be parameterized and so can be used for many different +kinds of objects. +\spadcommand{macro ff(x) == x**2 + 1} +\returnType{Type: Void} + +Apply it to a number, a symbol, or an expression. +\spadcommand{ff z} +$$ +{z \sp 2}+1 +$$ +\returnType{Type: Polynomial Integer} + +Macros can also be nested, but you get an error message if you +run out of space because of an infinite nesting loop. +\spadcommand{macro gg(x) == ff(2*x - 2/3)} +\returnType{Type: Void} + +This new macro is fine as it does not produce a loop. +\spadcommand{gg(1/w)} +$$ +{{{13} \ {w \sp 2}} -{{24} \ w}+{36}} \over {9 \ {w \sp 2}} +$$ +\returnType{Type: Fraction Polynomial Integer} + +This, however, loops since {\tt gg} is defined in terms of {\tt ff}. +\spadcommand{macro ff(x) == gg(-x)} +\returnType{Type: Void} + +The body of a macro can be a block. +\spadcommand{macro next == (past := present; present := future; future := past + present)} +\returnType{Type: Void} + +Before entering {\tt next}, we need values for {\tt present} and {\tt future}. +\spadcommand{present : Integer := 0} +$$ +0 +$$ +\returnType{Type: Integer} + +\spadcommand{future : Integer := 1} +$$ +1 +$$ +\returnType{Type: Integer} + +Repeatedly evaluating {\tt next} produces the next Fibonacci number. +\spadcommand{next} +$$ +1 +$$ +\returnType{Type: Integer} + +And the next one. +\spadcommand{next} +$$ +2 +$$ +\returnType{Type: Integer} + +Here is the infinite stream of the rest of the Fibonacci numbers. +\spadcommand{[next for i in 1..]} +$$ +\left[ +3, 5, 8, {13}, {21}, {34}, {55}, {89}, {144}, +{233}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Bundle all the above lines into a single macro. +\begin{verbatim} +macro fibStream == + present : Integer := 1 + future : Integer := 1 + [next for i in 1..] where + macro next == + past := present + present := future + future := past + present +\end{verbatim} +\returnType{Type: Void} + +Use \spadfunFrom{concat}{Stream} to start with the first two +\index{Fibonacci numbers} Fibonacci numbers. +\spadcommand{concat([0,1],fibStream)} +$$ +\left[ +0, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +The library operation {\bf fibonacci} is an easier way to compute +these numbers. + +\spadcommand{[fibonacci i for i in 1..]} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\section{Introduction to Functions} +\label{ugUserIntro} + +Each name in your workspace can refer to a single object. This may be +any kind of object including a function. You can use interactively +any function from the library or any that you define in the workspace. +In the library the same name can have very many functions, but you can +have only one function with a given name, although it can have any +number of arguments that you choose. + +If you define a function in the workspace that has the same name and +number of arguments as one in the library, then your definition takes +precedence. In fact, to get the library function you must +{\sl package-call} it +(see section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}). + +To use a function in Axiom, you apply it to its arguments. Most +functions are applied by entering the name of the function followed by +its argument or arguments. +\spadcommand{factor(12)} +$$ +{2 \sp 2} \ 3 +$$ +\returnType{Type: Factored Integer} + +Some functions like ``{\tt +}'' have {\it infix} {\it operators} as names. +\spadcommand{3 + 4} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +The function ``{\tt +}'' has two arguments. When you give it more than +two arguments, Axiom groups the arguments to the left. This +expression is equivalent to $(1 + 2) + 7$. +\spadcommand{1 + 2 + 7} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +All operations, including infix operators, can be written in prefix +form, that is, with the operation name followed by the arguments in +parentheses. For example, $2 + 3$ can alternatively be written as +$+(2,3)$. But $+(2,3,4)$ is an error since {\tt +} takes only two +arguments. + +Prefix operations are generally applied before the infix operation. +Thus the form ${\bf factorial\ } 3 + 1$ means ${\bf factorial}(3) + 1$ +producing $7$, and $-2 + 5$ means $(-2) + 5$ producing $3$. An +example of a prefix operator is prefix ``{\tt -}''. For example, $- 2 + +5$ converts to $(- 2) + 5$ producing the value $3$. Any prefix +function taking two arguments can be written in an infix manner by +putting an ampersand ``{\tt \&}'' before the name. Thus ${\tt D}(2*x,x)$ can +be written as $2*x\ {\tt \&D} x$ returning $2$. + +Every function in Axiom is identified by a {\it name} and +{\it type}. (An exception is an ``anonymous function'' discussed in +\ref{ugUserAnon} on page~\pageref{ugUserAnon}.) +The type of a function is always a mapping of the +form \spadsig{Source}{Target} where {\tt Source} and {\tt Target} are types. +To enter a type from the keyboard, enter the arrow by using a hyphen +``{\tt -}'' followed by a greater-than sign ``{\tt >}'', e.g. +{\tt Integer -> Integer}. + +Let's go back to ``{\tt +}''. There are many ``{\tt +}'' functions in the +Axiom library: one for integers, one for floats, another for rational +numbers, and so on. These ``{\tt +}'' functions have different types and +thus are different functions. You've seen examples of this +{\it overloading} before---using the same name for different functions. +Overloading is the rule rather than the exception. You can add two +integers, two polynomials, two matrices or two power series. These +are all done with the same function name but with different functions. + +\section{Declaring the Type of Functions} +\label{ugUserDeclare} + +In \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare} we discussed +how to declare a variable to restrict the kind of values that can be +assigned to it. In this section we show how to declare a variable +that refers to function objects. + +\boxed{4.6in}{ +\vskip 0.1cm +A function is an object of type +\begin{center} +{\sf Source $\rightarrow$ Type} +\end{center} + +where {\tt Source} and {\tt Target} can be any type. A common type +for {\tt Source} is {\tt Tuple}($\hbox{\it T}_{1}$, \ldots, +$\hbox{\it T}_{n}$), usually written ($\hbox{\it T}_{1}$, \ldots, +$\hbox{\it T}_{n}$), to indicate a function of $n$ arguments.\\ +} + +If $g$ takes an {\tt Integer}, a {\tt Float} and another {\tt Integer}, +and returns a {\tt String}, the declaration is written: +\spadcommand{g: (Integer,Float,Integer) -> String} +\returnType{Type: Void} + +The types need not be written fully; using abbreviations, the above +declaration is: +\spadcommand{g: (INT,FLOAT,INT) -> STRING} +\returnType{Type: Void} + +It is possible for a function to take no arguments. If $ h$ takes no +arguments but returns a {\tt Polynomial} {\tt Integer}, any of the +following declarations is acceptable. +\spadcommand{h: () -> POLY INT} +\returnType{Type: Void} + +\spadcommand{h: () -> Polynomial INT} +\returnType{Type: Void} + +\spadcommand{h: () -> POLY Integer} +\returnType{Type: Void} + +\boxed{4.6in}{ +\vskip 0.1cm +Functions can also be declared when they are being defined. +The syntax for combined declaration/definition is: +\begin{center} +\frenchspacing{\tt {\it functionName}($\hbox{\it parm}_{1}$: +$\hbox{\it parmType}_{1}$, \ldots, $\hbox{\it parm}_{N}$: +$\hbox{\it parmType}_{N}$): {\it functionReturnType}} +\end{center} +{\ }%force a blank line +} + +The following definition fragments show how this can be done for +the functions $g$ and $h$ above. +\begin{verbatim} +g(arg1: INT, arg2: FLOAT, arg3: INT): STRING == ... + +h(): POLY INT == ... +\end{verbatim} + +A current restriction on function declarations is that they must +involve fully specified types (that is, cannot include modes involving +explicit or implicit ``{\tt ?}''). For more information on declaring +things in general, see \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}. + +\section{One-Line Functions} +\label{ugUserOne} + +As you use Axiom, you will find that you will write many short +functions \index{function!one-line definition} to codify sequences of +operations that you often perform. In this section we write some +simple one-line functions. + +This is a simple recursive factorial function for positive integers. +\spadcommand{fac n == if n < 3 then n else n * fac(n-1)} +\returnType{Type: Void} + +\spadcommand{fac 10} +$$ +3628800 +$$ +\returnType{Type: PositiveInteger} + +This function computes $1 + 1/2 + 1/3 + ... + 1/n$. +\spadcommand{s n == reduce(+,[1/i for i in 1..n])} +\returnType{Type: Void} + +\spadcommand{s 50} +$$ +{13943237577224054960759} \over {3099044504245996706400} +$$ +\returnType{Type: Fraction Integer} + +This function computes a Mersenne number, several of which are prime. +\index{Mersenne number} +\spadcommand{mersenne i == 2**i - 1} +\returnType{Type: Void} + +If you type {\tt mersenne}, Axiom shows you the function definition. +\spadcommand{mersenne} +$$ +mersenne \ i \ == \ {{2 \sp i} -1} +$$ +\returnType{Type: FunctionCalled mersenne} + +Generate a stream of Mersenne numbers. +\spadcommand{[mersenne i for i in 1..]} +$$ +\left[ +1, 3, 7, {15}, {31}, {63}, {127}, {255}, {511}, +{1023}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Create a stream of those values of $i$ such that {\tt mersenne(i)} is prime. +\spadcommand{mersenneIndex := [n for n in 1.. | prime?(mersenne(n))]} +\begin{verbatim} + Compiling function mersenne with type PositiveInteger -> Integer +\end{verbatim} +$$ +\left[ +2, 3, 5, 7, {13}, {17}, {19}, {31}, {61}, {89}, +\ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +Finally, write a function that returns the $n$-th Mersenne prime. +\spadcommand{mersennePrime n == mersenne mersenneIndex(n)} +\returnType{Type: Void} + +\spadcommand{mersennePrime 5} +$$ +8191 +$$ +\returnType{Type: PositiveInteger} + +\section{Declared vs. Undeclared Functions} +\label{ugUserDecUndec} + +If you declare the type of a function, you can apply it to any data +that can be converted to the source type of the function. + +Define {\bf f} with type {\sf Integer $\rightarrow$ Integer}. +\spadcommand{f(x: Integer): Integer == x + 1} +\begin{verbatim} + Function declaration f : Integer -> Integer has been added to + workspace. +\end{verbatim} +\returnType{Type: Void} + +The function {\bf f} can be applied to integers, \ldots +\spadcommand{f 9} +\begin{verbatim} + Compiling function f with type Integer -> Integer +\end{verbatim} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +and to values that convert to integers, \ldots +\spadcommand{f(-2.0)} +$$ +-1 +$$ +\returnType{Type: Integer} + +but not to values that cannot be converted to integers. +\spadcommand{f(2/3)} +\begin{verbatim} + Conversion failed in the compiled user function f . + + Cannot convert from type Fraction Integer to Integer for value + 2 + - + 3 +\end{verbatim} + +To make the function over a wide range of types, do not declare its type. +Give the same definition with no declaration. +\spadcommand{g x == x + 1} +\returnType{Type: Void} + +If $x + 1$ makes sense, you can apply {\bf g} to $x$. +\spadcommand{g 9} +\begin{verbatim} + Compiling function g with type PositiveInteger -> PositiveInteger +\end{verbatim} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +A version of {\bf g} with different argument types get compiled for +each new kind of argument used. +\spadcommand{g(2/3)} +\begin{verbatim} + Compiling function g with type Fraction Integer -> Fraction Integer +\end{verbatim} +$$ +5 \over 3 +$$ +\returnType{Type: Fraction Integer} + +Here $x+1$ for $x = "axiom"$ makes no sense. +\spadcommand{g("axiom")} +\begin{verbatim} + There are 11 exposed and 5 unexposed library operations named + + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op + + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + Cannot find a definition or applicable library operation named + + with argument type(s) + String + PositiveInteger + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. + AXIOM will attempt to step through and interpret the code. + There are 11 exposed and 5 unexposed library operations named + + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op + + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named + + with argument type(s) + String + PositiveInteger + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +As you will see in Chapter \ref{ugCategories} on +page~\pageref{ugCategories}, Axiom has a formal idea of categories for +what ``makes sense.'' + +\section{Functions vs. Operations} +\label{ugUserDecOpers} + +A function is an object that you can create, manipulate, pass to, and +return from functions (for some interesting examples of library +functions that manipulate functions, see \ref{MappingPackage1XmpPage} +on page~\pageref{MappingPackage1XmpPage}). Yet, we often seem to use +the term {\it operation} and {\it function} interchangeably in Axiom. What +is the distinction? + +First consider values and types associated with some variable $n$ in +your workspace. You can make the declaration {\tt n : Integer}, then +assign $n$ an integer value. You then speak of the integer $n$. +However, note that the integer is not the name $n$ itself, but the +value that you assign to $n$. + +Similarly, you can declare a variable $f$ in your workspace to have +type {\sf Integer $\rightarrow$ Integer}, then assign $f$, through a +definition or an assignment of an anonymous function. You then speak +of the function $f$. However, the function is not $f$, but the value +that you assign to $f$. + +A function is a value, in fact, some machine code for doing something. +Doing what? Well, performing some {\it operation}. Formally, an +operation consists of the constituent parts of $f$ in your workspace, +excluding the value; thus an operation has a name and a type. An +operation is what domains and packages export. Thus {\tt Ring} +exports one operation ``{\tt +}''. Every ring also exports this +operation. Also, the author of every ring in the system is obliged +under contract (see \ref{ugPackagesAbstract} on +page~\pageref{ugPackagesAbstract}) to provide an implementation for +this operation. + +This chapter is all about functions---how you create them +interactively and how you apply them to meet your needs. In Chapter +\ref{ugPackages} on page~\pageref{ugPackages} you will learn how to +create them for the Axiom library. Then in Chapter \ref{ugCategories} +on page~\pageref{ugCategories}, you will learn about categories and +exported operations. + +\section{Delayed Assignments vs. Functions with No Arguments} +\label{ugUserDelay} + +In \ref{ugLangAssign} on page~\pageref{ugLangAssign} we discussed the +difference between immediate and \index{function!with no arguments} +delayed assignments. In this section we show the difference between +delayed assignments and functions of no arguments. + +A function of no arguments is sometimes called a {\it nullary function.} +\spadcommand{sin24() == sin(24.0)} +\returnType{Type: Void} + +You must use the parentheses ``{\tt ()}'' to evaluate it. Like a +delayed assignment, the right-hand-side of a function evaluation is +not evaluated until the left-hand-side is used. +\spadcommand{sin24()} +\begin{verbatim} + Compiling function sin24 with type () -> Float +\end{verbatim} +$$ +-{0.9055783620\ 0662384514} +$$ +\returnType{Type: Float} + +If you omit the parentheses, you just get the function definition. +\spadcommand{sin24} +$$ +sin24 \ {\left( +\right)} +\ == \ {\sin +\left( +{{24.0}} +\right)} +$$ +\returnType{Type: FunctionCalled sin24} + +You do not use the parentheses ``{\tt ()}'' in a delayed assignment\ldots + +\spadcommand{cos24 == cos(24.0)} +\returnType{Type: Void} + +nor in the evaluation. + +\spadcommand{cos24} +\begin{verbatim} + Compiling body of rule cos24 to compute value of type Float +\end{verbatim} +$$ +0.4241790073\ 3699697594 +$$ +\returnType{Type: Float} + +The only syntactic difference between delayed assignments +and nullary functions is that you use ``{\tt ()}'' in the latter case. + +\section{How Axiom Determines What Function to Use} +\label{ugUserUse} + +What happens if you define a function that has the same name as a +library function? Well, if your function has the same name and number +of arguments (we sometimes say {\it arity}) as another function in the +library, then your function covers up the library function. If you +want then to call the library function, you will have to {\sl package-call} +it. Axiom can use both the functions you write and those that come +from the library. Let's do a simple example to illustrate this. + +Suppose you (wrongly!) define {\bf sin} in this way. +\spadcommand{sin x == 1.0} +\returnType{Type: Void} + +The value $1.0$ is returned for any argument. +\spadcommand{sin 4.3} +\begin{verbatim} + Compiling function sin with type Float -> Float +\end{verbatim} +$$ +1.0 +$$ +\returnType{Type: Float} + +If you want the library operation, we have to package-call it +(see \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall} +for more information). +\spadcommand{sin(4.3)\$Float} +$$ +-{0.9161659367 4945498404} +$$ +\returnType{Type: Float} + +\spadcommand{sin(34.6)\$Float} +$$ +-{0.0424680347 1695010154 3} +$$ +\returnType{Type: Float} + +Even worse, say we accidentally used the same name as a library +function in the function. +\spadcommand{sin x == sin x} +\begin{verbatim} + Compiled code for sin has been cleared. + 1 old definition(s) deleted for function or rule sin +\end{verbatim} +\returnType{Type: Void} + +Then Axiom definitely does not understand us. +\spadcommand{sin 4.3} +\begin{verbatim} +AXIOM cannot determine the type of sin because it cannot analyze + the non-recursive part, if that exists. This may be remedied + by declaring the function. +\end{verbatim} + +Again, we could package-call the inside function. +\spadcommand{sin x == sin(x)\$Float} +\begin{verbatim} + 1 old definition(s) deleted for function or rule sin +\end{verbatim} +\returnType{Type: Void} + +\spadcommand{sin 4.3} +\begin{verbatim} + Compiling function sin with type Float -> Float + ++++ |*1;sin;1;G82322| redefined +\end{verbatim} +$$ +-{0.9161659367 4945498404} +$$ +\returnType{Type: Float} + +Of course, you are unlikely to make such obvious errors. It is more +probable that you would write a function and in the body use a +function that you think is a library function. If you had also +written a function by that same name, the library function would be +invisible. + +How does Axiom determine what library function to call? It very much +depends on the particular example, but the simple case of creating the +polynomial $x + 2/3$ will give you an idea. +\begin{enumerate} +\item The $x$ is analyzed and its default type is +{\tt Variable(x)}. +\item The $2$ is analyzed and its default type is +{\tt PositiveInteger}. +\item The $3$ is analyzed and its default type is +{\tt PositiveInteger}. +\item Because the arguments to ``{\tt /}'' are integers, Axiom +gives the expression $2/3$ a default target type of +{\tt Fraction(Integer)}. +\item Axiom looks in {\tt PositiveInteger} for ``{\tt /}''. +It is not found. +\item Axiom looks in {\tt Fraction(Integer)} for ``{\tt /}''. +It is found for arguments of type {\tt Integer}. +\item The $2$ and $3$ are converted to objects of type +{\tt Integer} (this is trivial) and ``{\tt /}'' is applied, +creating an object of type {\tt Fraction(Integer)}. +\item No ``{\tt +}'' for arguments of types {\tt Variable(x)} and +{\tt Fraction(Integer)} are found in either domain. +\item Axiom resolves +\index{resolve} +(see \ref{ugTypesResolve} on page~\pageref{ugTypesResolve}) +the types and gets {\tt Polynomial (Fraction (Integer))}. +\item The $x$ and the $2/3$ are converted to objects of this +type and {\tt +} is applied, yielding the answer, an object of type +{\tt Polynomial (Fraction (Integer))}. +\end{enumerate} + +\section{Compiling vs. Interpreting} +\label{ugUserCompInt} + +When possible, Axiom completely determines the type of every object in +a function, then translates the function definition to Common Lisp or +to machine code (see the next section). This translation, +\index{function!compiler} called compilation, happens the first time +you call the function and results in a computational delay. +Subsequent function calls with the same argument types use the +compiled version of the code without delay. + +If Axiom cannot determine the type of everything, the function may +still be executed \index{function!interpretation} but +\index{interpret-code mode} in interpret-code mode: each statement in +the function is analyzed and executed as the control flow indicates. +This process is slower than executing a compiled function, but it +allows the execution of code that may involve objects whose types +change. + +\boxed{4.6in}{ +\vskip 0.1cm +If Axiom decides that it cannot compile the code, it issues a message +stating the problem and then the following message: +\begin{center} +{\bf We will attempt to step through and interpret the code.} +\end{center} + +This is not a time to panic. \index{panic!avoiding} Rather, it just +means that what you gave to Axiom is somehow ambiguous: either it is +not specific enough to be analyzed completely, or it is beyond Axiom's +present interactive compilation abilities.\\ +} + +This function runs in interpret-code mode, but it does not compile. +\begin{verbatim} +varPolys(vars) == + for var in vars repeat + output(1 :: UnivariatePolynomial(var,Integer)) +\end{verbatim} +\returnType{Type: Void} + +For $vars$ equal to $['x, 'y, 'z]$, this function displays $1$ three times. +\spadcommand{varPolys ['x,'y,'z]} +\begin{verbatim} +Cannot compile conversion for types involving local variables. + In particular, could not compile the expression involving :: + UnivariatePolynomial(var,Integer) + AXIOM will attempt to step through and interpret the code. + 1 + 1 + 1 +\end{verbatim} +\returnType{Type: Void} + +The type of the argument to {\bf output} changes in each iteration, so +Axiom cannot compile the function. In this case, even the inner loop +by itself would have a problem: +\begin{verbatim} +for var in ['x,'y,'z] repeat + output(1 :: UnivariatePolynomial(var,Integer)) +\end{verbatim} +\begin{verbatim} +Cannot compile conversion for types involving local variables. + In particular, could not compile the expression involving :: + UnivariatePolynomial(var,Integer) + AXIOM will attempt to step through and interpret the code. + 1 + 1 + 1 +\end{verbatim} +\returnType{Type: Void} + +Sometimes you can help a function to compile by using an extra +conversion or by using $pretend$. \index{pretend} See +\ref{ugTypesSubdomains} on page~\pageref{ugTypesSubdomains} for details. + +When a function is compilable, you have the choice of whether it is +compiled to Common Lisp and then interpreted by the Common Lisp +interpreter or then further compiled from Common Lisp to machine code. +\index{machine code} The option is controlled via +{\tt )set functions compile}. +\index{set function compile} Issue {\tt )set functions compile on} +to compile all the way to machine code. With the default +setting {\tt )set functions compile off}, Axiom has its Common Lisp +code interpreted because the overhead of further compilation is larger +than the run-time of most of the functions our users have defined. +You may find that selectively turning this option on and off will +\index{performance} give you the best performance in your particular +application. For example, if you are writing functions for graphics +applications where hundreds of points are being computed, it is almost +certainly true that you will get the best performance by issuing +{\tt )set functions compile on}. + +\section{Piece-Wise Function Definitions} +\label{ugUserPiece} + +To move beyond functions defined in one line, we introduce in this +section functions that are defined piece-by-piece. That is, we say +``use this definition when the argument is such-and-such and use this +other definition when the argument is that-and-that.'' + +\subsection{A Basic Example} +\label{ugUserPieceBasic} + +There are many other ways to define a factorial function for +nonnegative integers. You might +\index{function!piece-wise definition} +say \index{piece-wise function definition} factorial of +$0$ is $1$, otherwise factorial of $n$ is $n$ times factorial of +$n-1$. Here is one way to do this in Axiom. + +Here is the value for $n = 0$. +\spadcommand{fact(0) == 1} +\returnType{Type: Void} + +Here is the value for $n > 0$. The vertical bar ``{\tt |}'' means ``such +that''. \index{such that} +\spadcommand{fact(n | n > 0) == n * fact(n - 1)} +\returnType{Type: Void} + +What is the value for $n = 3$? +\spadcommand{fact(3)} +\begin{verbatim} + Compiling function fact with type Integer -> Integer + Compiling function fact as a recurrence relation. +\end{verbatim} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +What is the value for $n = -3$? +\spadcommand{fact(-3)} +\begin{verbatim} + You did not define fact for argument -3 . +\end{verbatim} + +Now for a second definition. Here is the value for $n = 0$. +\spadcommand{facto(0) == 1} +\returnType{Type: Void} + +Give an error message if $n < 0$. +\spadcommand{facto(n | n < 0) == error "arguments to facto must be non-negative"} +\returnType{Type: Void} + +Here is the value otherwise. +\spadcommand{facto(n) == n * facto(n - 1)} +\returnType{Type: Void} + +What is the value for $n = 7$? +\spadcommand{facto(3)} +\begin{verbatim} + Compiling function facto with type Integer -> Integer +\end{verbatim} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +What is the value for $n = -7$? +\spadcommand{facto(-7)} +\begin{verbatim} + Error signalled from user code in function facto: + arguments to facto must be non-negative +\end{verbatim} +\returnType{Type: PositiveInteger} + +To see the current piece-wise definition of a function, use +{\tt )display value}. +\spadcommand{)display value facto} +\begin{verbatim} + Definition: + facto 0 == 1 + facto (n | n < 0) == + error(arguments to facto must be non-negative) + facto n == n facto(n - 1) +\end{verbatim} + +In general a {\it piece-wise definition} of a function consists of two +or more parts. Each part gives a ``piece'' of the entire definition. +Axiom collects the pieces of a function as you enter them. When you +ask for a value of the function, it then ``glues'' the pieces together +to form a function. + +The two piece-wise definitions for the factorial function are examples +of recursive functions, that is, functions that are defined in terms +of themselves. Here is an interesting doubly-recursive function. +This function returns the value $11$ for all positive integer +arguments. + +Here is the first of two pieces. +\spadcommand{eleven(n | n < 1) == n + 11} +\returnType{Type: Void} + +And the general case. +\spadcommand{eleven(m) == eleven(eleven(m - 12))} +\returnType{Type: Void} + +Compute $elevens$, the infinite stream of values of $eleven$. +\spadcommand{elevens := [eleven(i) for i in 0..]} +$$ +\left[ +{11}, {11}, {11}, {11}, {11}, {11}, {11}, {11}, {11}, + {11}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +What is the value at $n = 200$? +\spadcommand{elevens 200} +$$ +11 +$$ +\returnType{Type: PositiveInteger} + +What is the Axiom's definition of $eleven$? +\spadcommand{)display value eleven} +\begin{verbatim} + Definition: + eleven (m | m < 1) == m + 11 + eleven m == eleven(eleven(m - 12)) +\end{verbatim} + +\subsection{Picking Up the Pieces} +\label{ugUserPiecePicking} + +Here are the details about how Axiom creates a function from its +pieces. Axiom converts the $i$-th piece of a function definition +into a conditional expression of the form: +{\tt if} $\hbox{\it pred}_{i}$ {\tt then} $\hbox{\it expression}_{i}$. +If any new piece has a $\hbox{\it pred}_{i}$ that is +identical (after all variables are uniformly named) to +an earlier $\hbox{\it pred}_{j}$, the earlier piece is removed. +Otherwise, the new piece is always added at the end. + +\boxed{4.6in}{ +\vskip 0.1cm +If there are $n$ pieces to a function definition for $f$, the function +defined $f$ is: \newline +\hspace*{3pc} +{\tt if} $\hbox{\it pred}_{1}$ {\tt then} $\hbox{\it expression}_{1}$ {\tt else}\newline +\hspace*{6pc}. . . \newline +\hspace*{3pc} +{\tt if} $\hbox{\it pred}_{n}$ {\tt then} $\hbox{\it expression}_{n}$ {\tt else}\newline +\hspace*{3pc} +{\tt error "You did not define f for argument ."}\\ +} + +You can give definitions of any number of mutually recursive function +definitions, piece-wise or otherwise. No computation is done until +you ask for a value. When you do ask for a value, all the relevant +definitions are gathered, analyzed, and translated into separate +functions and compiled. + +Let's recall the definition of {\bf eleven} from +the previous section. +\spadcommand{eleven(n | n < 1) == n + 11} +\returnType{Type: Void} + +\spadcommand{eleven(m) == eleven(eleven(m - 12))} +\returnType{Type: Void} + +A similar doubly-recursive function below produces $-11$ for all +negative positive integers. If you haven't worked out why or how +{\bf eleven} works, the structure of this definition gives a clue. + +This definition we write as a block. +\begin{verbatim} +minusEleven(n) == + n >= 0 => n - 11 + minusEleven (5 + minusEleven(n + 7)) +\end{verbatim} +\returnType{Type: Void} + +Define $s(n)$ to be the sum of plus and minus ``eleven'' functions +divided by $n$. Since $11 - 11 = 0$, we define $s(0)$ to be $1$. +\spadcommand{s(0) == 1} +\returnType{Type: Void} + +And the general term. +\spadcommand{s(n) == (eleven(n) + minusEleven(n))/n} +\returnType{Type: Void} + +What are the first ten values of $s$? +\spadcommand{[s(n) for n in 0..]} +$$ +\left[ +1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +%% interpreter puts the rule at the end - should fix + +% Oops! Evidently $s(0)$ should be $1$. +% Let's check the current definition of {\bf s} using {\tt )display}. + +% \spadcommand{)display value s} + +% Change the value at $n = 0$. + +% \spadcommand{s(0) == 1} + +% Now, what is the definition of {\bf s}? +% Note: {\it you can only replace a given piece if you give exactly the same +% predicate!} + +% \spadcommand{)display value s} + +Axiom can create infinite streams in the positive direction (for +example, for index values $0,1, \ldots$) or negative direction (for +example, for $0,-1,-2, \ldots$). Here we would like a +stream of values of $s(n)$ that is infinite in both directions. The +function $t(n)$ below returns the $n$-th term of the infinite stream +$$[s(0), s(1), s(-1), s(2), s(-2), \ldots]$$ +Its definition has three pieces. + +Define the initial term. +\spadcommand{t(1) == s(0)} +\returnType{Type: Void} + +The even numbered terms are the $s(i)$ for positive $i$. We use +``{\tt quo}'' rather than ``{\tt /}'' since we want the result to be +an integer. + +\spadcommand{t(n | even?(n)) == s(n quo 2)} +\returnType{Type: Void} + +Finally, the odd numbered terms are the $s(i)$ for negative $i$. In +piece-wise definitions, you can use different variables to define +different pieces. Axiom will not get confused. +\spadcommand{t(p) == s(- p quo 2)} +\returnType{Type: Void} + +Look at the definition of $t$. In the first piece, the variable $n$ +was used; in the second piece, $p$. Axiom always uses your last +variable to display your definitions back to you. +\spadcommand{)display value t} +\begin{verbatim} + Definition: + t 1 == s(0) + t (p | even?(p)) == s(p quo 2) + t p == s(- p quo 2) +\end{verbatim} + +Create a series of values of $s$ applied to +alternating positive and negative arguments. +\spadcommand{[t(i) for i in 1..]} +\begin{verbatim} + Compiling function s with type Integer -> Fraction Integer + Compiling function t with type PositiveInteger -> Fraction Integer +\end{verbatim} +$$ +\left[ +1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +Evidently $t(n) = 1$ for all $i$. Check it at $n= 100$. + +\spadcommand{t(100)} +$$ +1 +$$ +\returnType{Type: Fraction Integer} + +\subsection{Predicates} +\label{ugUserPiecePred} + +We have already seen some examples of \index{function!predicate} +predicates \index{predicate!in function definition} +(\ref{ugUserPieceBasic} on page~\pageref{ugUserPieceBasic}). +Predicates are {\tt Boolean}-valued expressions and Axiom uses them +for filtering collections (see \ref{ugLangIts} on +page~\pageref{ugLangIts}) and for placing constraints on function +arguments. In this section we discuss their latter usage. + +The simplest use of a predicate is one you don't see at all. +\spadcommand{opposite 'right == 'left} +\returnType{Type: Void} + +Here is a longer way to give the ``opposite definition.'' +\spadcommand{opposite (x | x = 'left) == 'right} +\returnType{Type: Void} + +Try it out. +\spadcommand{for x in ['right,'left,'inbetween] repeat output opposite x} +\begin{verbatim} +Compiling function opposite with type + OrderedVariableList [right, left,inbetween] -> Symbol + left + right + +The function opposite is not defined for the given argument(s). +\end{verbatim} + +Explicit predicates tell Axiom that the given function definition +piece is to be applied if the predicate evaluates to {\tt true} for +the arguments to the function. You can use such ``constant'' +arguments for integers, \index{function!constant argument} strings, +and quoted symbols. \index{constant function argument} The {\tt +Boolean} values {\tt true} and {\tt false} can also be used if qualified with +``$@$'' or ``$\$$'' and {\tt Boolean}. The following are all valid +function definition fragments using constant arguments. +\begin{verbatim} +a(1) == ... +b("unramified") == ... +c('untested) == ... +d(true@Boolean) == ... +\end{verbatim} + +If a function has more than one argument, each argument can have its +own predicate. However, if a predicate involves two or more +arguments, it must be given {\it after} all the arguments mentioned in +the predicate have been given. You are always safe to give a single +predicate at the end of the argument list. + +A function involving predicates on two arguments. +\spadcommand{inFirstHalfQuadrant(x | x > 0,y | y < x) == true} +\returnType{Type: Void} + +This is incorrect as it gives a predicate on $y$ before the argument +$y$ is given. +\spadcommand{inFirstHalfQuadrant(x | x > 0 and y < x,y) == true} +\begin{verbatim} + 1 old definition(s) deleted for function or rule inFirstHalfQuadrant +\end{verbatim} +\returnType{Type: Void} + +It is always correct to write the predicate at the end. +\spadcommand{inFirstHalfQuadrant(x,y | x > 0 and y < x) == true} +\begin{verbatim} + 1 old definition(s) deleted for function or rule inFirstHalfQuadrant +\end{verbatim} +\returnType{Type: Void} + +Here is the rest of the definition. +\spadcommand{inFirstHalfQuadrant(x,y) == false} +\returnType{Type: Void} + +Try it out. +\spadcommand{[inFirstHalfQuadrant(i,3) for i in 1..5]} +\begin{verbatim} + Compiling function inFirstHalfQuadrant with type (PositiveInteger, + PositiveInteger) -> Boolean +\end{verbatim} +$$ +\left[ +{\tt false}, {\tt false}, {\tt false}, {\tt true}, {\tt true} +\right] +$$ +\returnType{Type: List Boolean} + +\section{Caching Previously Computed Results} +\label{ugUserCache} + +By default, Axiom does not save the values of any function. +\index{function!caching values} You can cause it to save values and +not to recompute unnecessarily \index{remembering function values} by +using {\tt )set functions cache}. \index{set functions cache} This +should be used before the functions are defined or, at least, before +they are executed. The word following ``cache'' should be $0$ to turn +off caching, a positive integer $n$ to save the last $n$ computed +values or ``all'' to save all computed values. If you then give a +list of names of functions, the caching only affects those functions. +Use no list of names or ``all'' when you want to define the default +behavior for functions not specifically mentioned in other +{\tt )set functions cache} statements. If you give no list of names, all +functions will have the caching behavior. If you explicitly turn on +caching for one or more names, you must explicitly turn off caching +for those names when you want to stop saving their values. + +This causes the functions {\bf f} and {\bf g} to have the last three +computed values saved. +\spadcommand{)set functions cache 3 f g} +\begin{verbatim} + function f will cache the last 3 values. + function g will cache the last 3 values. +\end{verbatim} + +This is a sample definition for {\bf f}. +\spadcommand{f x == factorial(2**x)} +\returnType{Type: Void} + +A message is displayed stating what {\bf f} will cache. +\spadcommand{f(4)} +\begin{verbatim} + Compiling function f with type PositiveInteger -> Integer + f will cache 3 most recently computed value(s). + ++++ |*1;f;1;G82322| redefined +\end{verbatim} +$$ +20922789888000 +$$ +\returnType{Type: PositiveInteger} + +This causes all other functions to have all computed values saved by default. +\spadcommand{)set functions cache all} +\begin{verbatim} + In general, interpreter functions will cache all values. +\end{verbatim} + +This causes all functions that have not been specifically cached in some way +to have no computed values saved. +\spadcommand{)set functions cache 0} +\begin{verbatim} + In general, functions will cache no returned values. +\end{verbatim} + +We also make {\bf f} and {\bf g} uncached. +\spadcommand{)set functions cache 0 f g} +\begin{verbatim} + Caching for function f is turned off + Caching for function g is turned off +\end{verbatim} + +\boxed{4.6in}{ +\vskip 0.1cm +Be careful about caching functions that have side effects. Such a +function might destructively modify the elements of an array or issue +a {\bf draw} command, for example. A function that you expect to +execute every time it is called should not be cached. Also, it is +highly unlikely that a function with no arguments should be cached.\\ +} + +You should also be careful about caching functions that depend on free +variables. See \ref{ugUserFreeLocal} on +page~\pageref{ugUserFreeLocal} for an example. + +\section{Recurrence Relations} +\label{ugUserRecur} + +One of the most useful classes of function are those defined via a +``recurrence relation.'' A {\it recurrence relation} makes each +successive \index{recurrence relation} value depend on some or all of +the previous values. A simple example is the ordinary ``factorial'' function: +\begin{verbatim} +fact(0) == 1 +fact(n | n > 0) == n * fact(n-1) +\end{verbatim} + +The value of $fact(10)$ depends on the value of $fact(9)$, $fact(9)$ +on $fact(8)$, and so on. Because it depends on only one previous +value, it is usually called a {\it first order recurrence relation.} +You can easily imagine a function based on two, three or more previous +values. The Fibonacci numbers are probably the most famous function +defined by a \index{Fibonacci numbers} second order recurrence relation. + +The library function {\bf fibonacci} computes Fibonacci numbers. +It is obviously optimized for speed. +\spadcommand{[fibonacci(i) for i in 0..]} +$$ +\left[ +0, 1, 1, 2, 3, 5, 8, {13}, {21}, {34}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Define the Fibonacci numbers ourselves using a piece-wise definition. +\spadcommand{fib(1) == 1} +\returnType{Type: Void} + +\spadcommand{fib(2) == 1} +\returnType{Type: Void} + +\spadcommand{fib(n) == fib(n-1) + fib(n-2)} +\returnType{Type: Void} + +As defined, this recurrence relation is obviously doubly-recursive. +To compute $fib(10)$, we need to compute $fib(9)$ and $fib(8)$. And +to $fib(9)$, we need to compute $fib(8)$ and $fib(7)$. And so on. It +seems that to compute $fib(10)$ we need to compute $fib(9)$ once, +$fib(8)$ twice, $fib(7)$ three times. Look familiar? The number of +function calls needed to compute {\it any} second order recurrence +relation in the obvious way is exactly $fib(n)$. These numbers grow! +For example, if Axiom actually did this, then $fib(500)$ requires more +than $10^{104}$ function calls. And, given all +this, our definition of {\bf fib} obviously could not be used to +calculate the five-hundredth Fibonacci number. + +Let's try it anyway. +\spadcommand{fib(500)} +\begin{verbatim} + Compiling function fib with type Integer -> PositiveInteger + Compiling function fib as a recurrence relation. + +13942322456169788013972438287040728395007025658769730726410_ +8962948325571622863290691557658876222521294125 +\end{verbatim} + +\returnType{Type: PositiveInteger} + +Since this takes a short time to compute, it obviously didn't do as +many as $10^{104}$ operations! By default, Axiom transforms any +recurrence relation it recognizes into an iteration. Iterations are +efficient. To compute the value of the $n$-th term of a recurrence +relation using an iteration requires only $n$ function calls. Note +that if you compare the speed of our {\bf fib} function to the library +function, our version is still slower. This is because the library +\spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} uses a +``powering algorithm'' with a computing time proportional to +$\log^3(n)$ to compute {\tt fibonacci(n)}. + +To turn off this special recurrence relation compilation, issue +\index{set function recurrence} +\begin{verbatim} +)set functions recurrence off +\end{verbatim} +To turn it back on, substitute ``{\tt on}'' for ``{\tt off}''. + +The transformations that Axiom uses for {\bf fib} caches the last two +values. For a more general $k$-th order recurrence relation, Axiom +caches the last $k$ values. If, after computing a value for {\bf +fib}, you ask for some larger value, Axiom picks up the cached values +and continues computing from there. See \ref{ugUserFreeLocal} on +page~\pageref{ugUserFreeLocal} for an example of a function definition +that has this same behavior. Also see \ref{ugUserCache} on +page~\pageref{ugUserCache} for a more general discussion of how you +can cache function values. + +Recurrence relations can be used for defining recurrence relations +involving polynomials, rational functions, or anything you like. +Here we compute the infinite stream of Legendre polynomials. + +The Legendre polynomial of degree $0.$ +\spadcommand{p(0) == 1} +\returnType{Type: Void} + +The Legendre polynomial of degree $1.$ +\spadcommand{p(1) == x} +\returnType{Type: Void} + +The Legendre polynomial of degree $n$. +\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1)*p(n-2))/n} +\returnType{Type: Void} + +Compute the Legendre polynomial of degree $6.$ +\spadcommand{p(6)} +\begin{verbatim} + Compiling function p with type Integer -> Polynomial Fraction + Integer + Compiling function p as a recurrence relation. +\end{verbatim} +$$ +{{{231} \over {16}} \ {x \sp 6}} -{{{315} \over {16}} \ {x \sp 4}}+{{{105} +\over {16}} \ {x \sp 2}} -{5 \over {16}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +\section{Making Functions from Objects} +\label{ugUserMake} + +There are many times when you compute a complicated expression and +then wish to use that expression as the body of a function. Axiom +provides an operation called {\bf function} to do \index{function!from +an object} this. \index{function!made by function @{made by +{\bf function}}} It creates a function object and places it into the +workspace. There are several versions, depending on how many +arguments the function has. The first argument to {\bf function} is +always the expression to be converted into the function body, and the +second is always the name to be used for the function. For more +information, see section \ref{MakeFunctionXmpPage} on +page~\pageref{MakeFunctionXmpPage}. + +Start with a simple example of a polynomial in three variables. +\spadcommand{p := -x + y**2 - z**3} +$$ +-{z \sp 3}+{y \sp 2} -x +$$ +\returnType{Type: Polynomial Integer} + +To make this into a function of no arguments that simply returns the +polynomial, use the two argument form of {\bf function}. +\spadcommand{function(p,'f0)} +$$ +f0 +$$ +\returnType{Type: Symbol} + +To avoid possible conflicts (see below), it is a good idea to +quote always this second argument. +\spadcommand{f0} +$$ +f0 \ {\left( +\right)} +\ == \ {-{z \sp 3}+{y \sp 2} -x} +$$ +\returnType{Type: FunctionCalled f0} + +This is what you get when you evaluate the function. +\spadcommand{f0()} +$$ +-{z \sp 3}+{y \sp 2} -x +$$ +\returnType{Type: Polynomial Integer} + +To make a function in $x$, use a version of {\bf function} that takes +three arguments. The last argument is the name of the variable to use +as the parameter. Typically, this variable occurs in the expression +and, like the function name, you should quote it to avoid possible confusion. +\spadcommand{function(p,'f1,'x)} +$$ +f1 +$$ +\returnType{Type: Symbol} + +This is what the new function looks like. +\spadcommand{f1} +$$ +f1 \ x \ == \ {-{z \sp 3}+{y \sp 2} -x} +$$ +\returnType{Type: FunctionCalled f1} + +This is the value of {\bf f1} at $x = 3$. Notice that the return type +of the function is {\tt Polynomial (Integer)}, the same as $p$. +\spadcommand{f1(3)} +\begin{verbatim} + Compiling function f1 with type PositiveInteger -> Polynomial + Integer +\end{verbatim} +$$ +-{z \sp 3}+{y \sp 2} -3 +$$ +\returnType{Type: Polynomial Integer} + +To use $x$ and $y$ as parameters, use the four argument form of {\bf function}. +\spadcommand{function(p,'f2,'x,'y)} +$$ +f2 +$$ +\returnType{Type: Symbol} + +\spadcommand{f2} +$$ +f2 \ {\left( x, y +\right)} +\ == \ {-{z \sp 3}+{y \sp 2} -x} +$$ +\returnType{Type: FunctionCalled f2} + +Evaluate $f2$ at $x = 3$ and $y = 0$. The return type of {\bf f2} is +still {\tt Polynomial(Integer)} because the variable $z$ is still +present and not one of the parameters. +\spadcommand{f2(3,0)} +$$ +-{z \sp 3} -3 +$$ +\returnType{Type: Polynomial Integer} + +Finally, use all three variables as parameters. There is no five +argument form of {\bf function}, so use the one with three arguments, +the third argument being a list of the parameters. +\spadcommand{function(p,'f3,['x,'y,'z])} +$$ +f3 +$$ +\returnType{Type: Symbol} + +Evaluate this using the same values for $x$ and $y$ as above, but let +$z$ be $-6$. The result type of {\bf f3} is {\tt Integer}. +\spadcommand{f3} +$$ +f3 \ {\left( x, y, z +\right)} +\ == \ {-{z \sp 3}+{y \sp 2} -x} +$$ +\returnType{Type: FunctionCalled f3} + +\spadcommand{f3(3,0,-6)} +\begin{verbatim} + Compiling function f3 with type (PositiveInteger,NonNegativeInteger, + Integer) -> Integer +\end{verbatim} +$$ +213 +$$ +\returnType{Type: PositiveInteger} + +The four functions we have defined via $p$ have been undeclared. To +declare a function whose body is to be generated by +\index{function!declaring} {\bf function}, issue the declaration +{\it before} the function is created. +\spadcommand{g: (Integer, Integer) -> Float} +\returnType{Type: Void} + +\spadcommand{D(sin(x-y)/cos(x+y),x)} +$$ +{-{{\sin +\left( +{{y -x}} +\right)} +\ {\sin +\left( +{{y+x}} +\right)}}+{{\cos +\left( +{{y -x}} +\right)} +\ {\cos +\left( +{{y+x}} +\right)}}} +\over {{\cos +\left( +{{y+x}} +\right)} +\sp 2} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{function(\%,'g,'x,'y)} +$$ +g +$$ +\returnType{Type: Symbol} + +\spadcommand{g} +$$ +g \ {\left( x, y +\right)} +\ == \ {{-{{\sin +\left( +{{y -x}} +\right)} +\ {\sin +\left( +{{y+x}} +\right)}}+{{\cos +\left( +{{y -x}} +\right)} +\ {\cos +\left( +{{y+x}} +\right)}}} +\over {{\cos +\left( +{{y+x}} +\right)} +\sp 2}} +$$ +\returnType{Type: FunctionCalled g} + +It is an error to use $g$ without the quote in the penultimate +expression since $g$ had been declared but did not have a value. +Similarly, since it is common to overuse variable names like $x$, $y$, +and so on, you avoid problems if you always quote the variable names +for {\bf function}. In general, if $x$ has a value and you use $x$ +without a quote in a call to {\bf function}, then Axiom does not know +what you are trying to do. + +What kind of object is allowable as the first argument to +{\bf function}? Let's use the Browse facility of HyperDoc to find out. +\index{Browse@Browse} At the main Browse menu, enter the string +{\tt function} and then click on {\bf Operations.} The exposed operations +called {\bf function} all take an object whose type belongs to +category {\tt ConvertibleTo InputForm}. What domains are those? Go +back to the main Browse menu, erase {\tt function}, enter +{\tt ConvertibleTo} in the input area, and click on {\bf categories} on the +{\tt Constructors} line. At the bottom of the page, enter +{\tt InputForm} in the input area following {\bf S =}. Click on +{\tt Cross Reference} and then on {\tt Domains}. +The list you see contains over forty domains that belong to the +category {\tt ConvertibleTo InputForm}. Thus you can use {\bf function} +for {\tt Integer}, {\tt Float}, {\tt String}, {\tt Complex}, +{\tt Expression}, and so on. + +\section{Functions Defined with Blocks} +\label{ugUserBlocks} + +You need not restrict yourself to functions that only fit on one line +or are written in a piece-wise manner. The body of the function can +be a block, as discussed in \ref{ugLangBlocks} on page~\pageref{ugLangBlocks}. + +Here is a short function that swaps two elements of a list, array or vector. +\begin{verbatim} +swap(m,i,j) == + temp := m.i + m.i := m.j + m.j := temp +\end{verbatim} +\returnType{Type: Void} + +The significance of {\bf swap} is that it has a destructive +effect on its first argument. +\spadcommand{k := [1,2,3,4,5]} +$$ +\left[ +1, 2, 3, 4, 5 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{swap(k,2,4)} +\begin{verbatim} + Compiling function swap with type (List PositiveInteger, + PositiveInteger,PositiveInteger) -> PositiveInteger +\end{verbatim} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +You see that the second and fourth elements are interchanged. +\spadcommand{k} +$$ +\left[ +1, 4, 3, 2, 5 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Using this, we write a couple of different sort functions. First, a +simple bubble sort. \index{sort!bubble} The operation +\spadopFrom{\#}{List} returns the number of elements in an aggregate. +\begin{verbatim} +bubbleSort(m) == + n := #m + for i in 1..(n-1) repeat + for j in n..(i+1) by -1 repeat + if m.j < m.(j-1) then swap(m,j,j-1) + m +\end{verbatim} +\returnType{Type: Void} + +Let this be the list we want to sort. +\spadcommand{m := [8,4,-3,9]} +$$ +\left[ +8, 4, -3, 9 +\right] +$$ +\returnType{Type: List Integer} + +This is the result of sorting. +\spadcommand{bubbleSort(m)} +\begin{verbatim} + Compiling function swap with type (List Integer,Integer,Integer) -> + Integer + ++++ |*3;swap;1;G82322| redefined + Compiling function bubbleSort with type List Integer -> List Integer +\end{verbatim} +$$ +\left[ +-3, 4, 8, 9 +\right] +$$ +\returnType{Type: List Integer} + +Moreover, $m$ is destructively changed to be the sorted version. +\spadcommand{m} +$$ +\left[ +-3, 4, 8, 9 +\right] +$$ +\returnType{Type: List Integer} + +This function implements an insertion sort. \index{sort!insertion} +The basic idea is to traverse the list and insert the $i$-th element +in its correct position among the $i-1$ previous elements. Since we +start at the beginning of the list, the list elements before the +$i$-th element have already been placed in ascending order. +\begin{verbatim} +insertionSort(m) == + for i in 2..#m repeat + j := i + while j > 1 and m.j < m.(j-1) repeat + swap(m,j,j-1) + j := j - 1 + m +\end{verbatim} +\returnType{Type: Void} + +As with our bubble sort, this is a destructive function. +\spadcommand{m := [8,4,-3,9]} +$$ +\left[ +8, 4, -3, 9 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{insertionSort(m)} +\begin{verbatim} + Compiling function insertionSort with type List Integer -> List + Integer +\end{verbatim} +$$ +\left[ +-3, 4, 8, 9 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{m} +$$ +\left[ +-3, 4, 8, 9 +\right] +$$ +\returnType{Type: List Integer} + +Neither of the above functions is efficient for sorting large lists +since they reference elements by asking for the $j$-th element of the +structure $m$. + +Here is a more efficient bubble sort for lists. +\begin{verbatim} +bubbleSort2(m: List Integer): List Integer == + null m => m + l := m + while not null (r := l.rest) repeat + r := bubbleSort2 r + x := l.first + if x < r.first then + l.first := r.first + r.first := x + l.rest := r + l := l.rest + m + + Function declaration bubbleSort2 : List Integer -> List Integer has + been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Try it out. +\spadcommand{bubbleSort2 [3,7,2]} +$$ +\left[ +7, 3, 2 +\right] +$$ +\returnType{Type: List Integer} + +This definition is both recursive and iterative, and is tricky! +Unless you are {\it really} curious about this definition, we suggest +you skip immediately to the next section. + +Here are the key points in the definition. First notice that if you +are sorting a list with less than two elements, there is nothing to +do: just return the list. This definition returns immediately if +there are zero elements, and skips the entire {\tt while} loop if there is +just one element. + +The second point to realize is that on each outer iteration, the +bubble sort ensures that the minimum element is propagated leftmost. +Each iteration of the {\tt while} loop calls {\bf bubbleSort2} recursively +to sort all but the first element. When finished, the minimum element +is either in the first or second position. The conditional expression +ensures that it comes first. If it is in the second, then a swap +occurs. In any case, the {\bf rest} of the original list must be +updated to hold the result of the recursive call. + +\section{Free and Local Variables} +\label{ugUserFreeLocal} + +When you want to refer to a variable that is not local to your +function, use a ``{\tt free}'' declaration. \index{free} Variables +declared to be {\tt free} \index{free variable} are assumed to be defined +globally \index{variable!free} in the \index{variable!global} +workspace. \index{global variable} + +This is a global workspace variable. +\spadcommand{counter := 0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +This function refers to the global $counter$. +\begin{verbatim} +f() == + free counter + counter := counter + 1 +\end{verbatim} +\returnType{Type: Void} + +The global $counter$ is incremented by $1$. +\spadcommand{f()} +\begin{verbatim} + Compiling function f with type () -> NonNegativeInteger + ++++ |*0;f;1;G82322| redefined +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{counter} +$$ +1 +$$ +\returnType{Type: NonNegativeInteger} + + +Usually Axiom can tell that you mean to refer to a global variable and +so {\tt free} isn't always necessary. However, for clarity and the sake +of self-documentation, we encourage you to use it. + +Declare a variable to be ``{\tt local}'' when you do not want to refer to +\index{variable!local} a global variable by the same name. +\index{local variable} + +This function uses $counter$ as a local variable. +\begin{verbatim} +g() == + local counter + counter := 7 +\end{verbatim} +\returnType{Type: Void} + +Apply the function. +\spadcommand{g()} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +Check that the global value of $counter$ is unchanged. +\spadcommand{counter} +$$ +1 +$$ +\returnType{Type: NonNegativeInteger} + +Parameters to a function are local variables in the function. Even if +you issue a {\tt free} declaration for a parameter, it is still local. + +What happens if you do not declare that a variable $x$ in the body of +your function is {\tt local} or {\tt free}? Well, Axiom decides on this basis: +\begin{enumerate} +\item Axiom scans your function line-by-line, from top-to-bottom. +The right-hand side of an assignment is looked at before the left-hand +side. +\item If $x$ is referenced before it is assigned a value, it is a +{\tt free} (global) variable. +\item If $x$ is assigned a value before it is referenced, it is a +{\tt local} variable. +\end{enumerate} + +Set two global variables to 1. +\spadcommand{a := b := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Refer to $a$ before it is assigned a value, but assign a value to $b$ +before it is referenced. +\begin{verbatim} +h() == + b := a + 1 + a := b + a +\end{verbatim} +\returnType{Type: Void} + +Can you predict this result? +\spadcommand{h()} +\begin{verbatim} + Compiling function h with type () -> PositiveInteger + ++++ |*0;h;1;G82322| redefined +\end{verbatim} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +How about this one? +\spadcommand{[a, b]} +$$ +\left[ +3, 1 +\right] +$$ +\returnType{Type: List PositiveInteger} + +What happened? In the first line of the function body for $h$, $a$ is +referenced on the right-hand side of the assignment. Thus $a$ is a +free variable. The variable $b$ is not referenced in that line, but +it is assigned a value. Thus $b$ is a local variable and is given the +value $a + 1 = 2$. In the second line, the free variable $a$ is +assigned the value $b + a$ which equals $2 + 1 = 3.$ This is the value +returned by the function. Since $a$ was free in {\bf h}, the global +variable $a$ has value $3.$ Since $b$ was local in {\bf h}, the global +variable $b$ is unchanged---it still has the value $1$. + +It is good programming practice always to declare global variables. +However, by far the most common situation is to have local variables +in your functions. No declaration is needed for this situation, but +be sure to initialize their values. + +Be careful if you use free variables and you cache the value of your +function (see \ref{ugUserCache} on page~\pageref{ugUserCache}). +Caching {\it only} checks if the values of the function arguments are +the same as in a function call previously seen. It does not check if +any of the free variables on which the function depends have changed +between function calls. + +Turn on caching for {\bf p}. +\spadcommand{)set fun cache all p} +\begin{verbatim} + function p will cache all values. +\end{verbatim} + +Define {\bf p} to depend on the free variable $N$. +\spadcommand{p(i,x) == ( free N; reduce( + , [ (x-i)**n for n in 1..N ] ) )} +\returnType{Type: Void} + +Set the value of $N$. +\spadcommand{N := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Evaluate {\bf p} the first time. +\spadcommand{p(0, x)} +$$ +x +$$ +\returnType{Type: Polynomial Integer} + +Change the value of $N$. +\spadcommand{N := 2} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Evaluate {\bf p} the second time. +\spadcommand{p(0, x)} +$$ +x +$$ +\returnType{Type: Polynomial Integer} + +If caching had been turned off, the second evaluation would have +reflected the changed value of $N$. + +Turn off caching for {\bf p}. +\spadcommand{)set fun cache 0 p} +\begin{verbatim} + Caching for function p is turned off +\end{verbatim} + +Axiom does not allow {\it fluid variables}, that is, variables +\index{variable!fluid} bound by a function $f$ that can be referenced +by functions called by $f$. \index{fluid variable} + +Values are passed to functions by {\it reference}: a pointer to the +value is passed rather than a copy of the value or a pointer to a +copy. + +This is a global variable that is bound to a record object. +\spadcommand{r : Record(i : Integer) := [1]} +$$ +\left[ +{i=1} +\right] +$$ +\returnType{Type: Record(i: Integer)} + +This function first modifies the one component of its record argument +and then rebinds the parameter to another record. +\begin{verbatim} +resetRecord rr == + rr.i := 2 + rr := [10] +\end{verbatim} +\returnType{Type: Void} + +Pass $r$ as an argument to {\bf resetRecord}. +\spadcommand{resetRecord r} +$$ +\left[ +{i={10}} +\right] +$$ +\returnType{Type: Record(i: Integer)} + +The value of $r$ was changed by the expression $rr.i := 2$ but not by +$rr := [10]$. +\spadcommand{r} +$$ +\left[ +{i=2} +\right] +$$ +\returnType{Type: Record(i: Integer)} + +To conclude this section, we give an iterative definition of +\index{Fibonacci numbers} a function that computes Fibonacci numbers. +This definition approximates the definition into which Axiom +transforms the recurrence relation definition of {\bf fib} in +\ref{ugUserRecur} on page~\pageref{ugUserRecur}. + +Global variables {\tt past} and {\tt present} are used to hold the last +computed Fibonacci numbers. +\spadcommand{past := present := 1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Global variable $index$ gives the current index of $present$. +\spadcommand{index := 2} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Here is a recurrence relation defined in terms of these three global +variables. +\begin{verbatim} +fib(n) == + free past, present, index + n < 3 => 1 + n = index - 1 => past + if n < index-1 then + (past,present) := (1,1) + index := 2 + while (index < n) repeat + (past,present) := (present, past+present) + index := index + 1 + present +\end{verbatim} +\returnType{Type: Void} + +Compute the infinite stream of Fibonacci numbers. +\spadcommand{fibs := [fib(n) for n in 1..]} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +What is the 1000th Fibonacci number? +\spadcommand{fibs 1000} +\begin{verbatim} + 434665576869374564356885276750406258025646605173717804024_ + 8172908953655541794905189040387984007925516929592259308_ + 0322634775209689623239873322471161642996440906533187938_ + 298969649928516003704476137795166849228875 +\end{verbatim} +\returnType{Type: PositiveInteger} + +As an exercise, we suggest you write a function in an iterative style +that computes the value of the recurrence relation +$p(n) = p(n-1) - 2 \, p(n-2) + 4 \, p(n-3)$ +having the initial values +$p(1) = 1,\, p(2) = 3 \hbox{ and } p(3) = 9.$ +How would you write the function using an element {\tt OneDimensionalArray} +or {\tt Vector} to hold the previously computed values? + +\section{Anonymous Functions} +\label{ugUserAnon} + +\boxed{4.6in}{ +\vskip 0.1cm +An {\it anonymous function} is a function that is +\index{function!anonymous} defined \index{anonymous function} by +giving a list of parameters, the ``maps-to'' compound +\index{+-> @{\tt +->}} symbol ``{\tt +->}'' +(from the mathematical symbol $\mapsto$), and +by an expression involving the parameters, the evaluation of which +determines the return value of the function. + +\begin{center} +{\tt ( $\hbox{\it parm}_{1}$, $\hbox{\it parm}_{2}$, \ldots, +$\hbox{\it parm}_{N}$ ) {\tt +->} {\it expression}} +\end{center} +{\ }%force a blank line +} + +You can apply an anonymous function in several ways. +\begin{enumerate} +\item Place the anonymous function definition in parentheses +directly followed by a list of arguments. +\item Assign the anonymous function to a variable and then +use the variable name when you would normally use a function name. +\item Use ``{\tt ==}'' to use the anonymous function definition as +the arguments and body of a regular function definition. +\item Have a named function contain a declared anonymous function and +use the result returned by the named function. +\end{enumerate} + +\subsection{Some Examples} +\label{ugUserAnonExamp} + +Anonymous functions are particularly useful for defining functions +``on the fly.'' That is, they are handy for simple functions that are +used only in one place. In the following examples, we show how to +write some simple anonymous functions. + +This is a simple absolute value function. +\spadcommand{x +-> if x < 0 then -x else x} +$$ +x \mapsto {if \ {x<0} \ {\begin{array}{l} {then \ -x} \\ +{else \ x} +\end{array} +}} +$$ +\returnType{Type: AnonymousFunction} + +\spadcommand{abs1 := \%} +$$ +x \mapsto {if \ {x<0} \ {\begin{array}{l} {then \ -x} \\ +{else \ x} +\end{array} +}} +$$ +\returnType{Type: AnonymousFunction} + +This function returns {\tt true} if the absolute value of +the first argument is greater than the absolute value of the +second, {\tt false} otherwise. + +\spadcommand{(x,y) +-> abs1(x) > abs1(y)} +$$ +{\left( x, y +\right)} +\mapsto {{abs1 +\left( +{y} +\right)}<{abs1 +\left( +{x} +\right)}} +$$ +\returnType{Type: AnonymousFunction} + +We use the above function to ``sort'' a list of integers. +\spadcommand{sort(\%,[3,9,-4,10,-3,-1,-9,5])} +$$ +\left[ +{10}, -9, 9, 5, -4, -3, 3, -1 +\right] +$$ +\returnType{Type: List Integer} + +This function returns $1$ if $i + j$ is even, $-1$ otherwise. +\spadcommand{ev := ( (i,j) +-> if even?(i+j) then 1 else -1)} +$$ +{\left( i, j +\right)} +\mapsto {if \ {even? +\left( +{{i+j}} +\right)} +\ {\begin{array}{l} {then \ 1} \\ +{else \ -1} +\end{array} +}} +$$ +\returnType{Type: AnonymousFunction} + +We create a four-by-four matrix containing $1$ or $-1$ depending on +whether the row plus the column index is even or not. +\spadcommand{matrix([ [ev(row,col) for row in 1..4] for col in 1..4])} +$$ +\left[ +\begin{array}{cccc} +1 & -1 & 1 & -1 \\ +-1 & 1 & -1 & 1 \\ +1 & -1 & 1 & -1 \\ +-1 & 1 & -1 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +This function returns {\tt true} if a polynomial in $x$ has multiple +roots, {\tt false} otherwise. It is defined and applied in the same +expression. +\spadcommand{( p +-> not one?(gcd(p,D(p,x))) )(x**2+4*x+4)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +This and the next expression are equivalent. +\spadcommand{g(x,y,z) == cos(x + sin(y + tan(z)))} +\returnType{Type: Void} + +The one you use is a matter of taste. +\spadcommand{g == (x,y,z) +-> cos(x + sin(y + tan(z)))} +\begin{verbatim} + 1 old definition(s) deleted for function or rule g +\end{verbatim} +\returnType{Type: Void} + +\subsection{Declaring Anonymous Functions} +\label{ugUserAnonDeclare} + +If you declare any of the arguments you must declare all of them. Thus, +\begin{verbatim} +(x: INT,y): FRAC INT +-> (x + 2*y)/(y - 1) +\end{verbatim} +is not legal. + +This is an example of a fully declared anonymous function. +\index{function!declaring} \index{function!anonymous!declaring} The +output shown just indicates that the object you created is a +particular kind of map, that is, function. +\spadcommand{(x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)} +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: ((Integer,Integer) {\tt ->} Fraction Integer)} + +Axiom allows you to declare the arguments and not declare +the return type. +\spadcommand{(x: INT,y: INT) +-> (x + 2*y)/(y - 1)} +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: ((Integer,Integer) {\tt ->} Fraction Integer)} + +The return type is computed from the types of the arguments and the +body of the function. You cannot declare the return type if you do +not declare the arguments. Therefore, +\begin{verbatim} +(x,y): FRAC INT +-> (x + 2*y)/(y - 1) +\end{verbatim} + +is not legal. This and the next expression are equivalent. +\spadcommand{h(x: INT,y: INT): FRAC INT == (x + 2*y)/(y - 1)} +\begin{verbatim} + Function declaration h : (Integer,Integer) -> Fraction Integer + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +The one you use is a matter of taste. +\spadcommand{h == (x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)} +\begin{verbatim} + Function declaration h : (Integer,Integer) -> Fraction Integer + has been added to workspace. + 1 old definition(s) deleted for function or rule h +\end{verbatim} +\returnType{Type: Void} + +When should you declare an anonymous function? +\begin{enumerate} +\item If you use an anonymous function and Axiom can't figure out what +you are trying to do, declare the function. +\item If the function has nontrivial argument types or a nontrivial +return type that Axiom may be able to determine eventually, but you +are not willing to wait that long, declare the function. +\item If the function will only be used for arguments of specific types +and it is not too much trouble to declare the function, do so. +\item If you are using the anonymous function as an argument to another +function (such as {\bf map} or {\bf sort}), consider declaring the function. +\item If you define an anonymous function inside a named function, +you {\it must} declare the anonymous function. +\end{enumerate} + +This is an example of a named function for integers that returns a +function. +\spadcommand{addx x == ((y: Integer): Integer +-> x + y)} +\returnType{Type: Void} + +We define {\bf g} to be a function that adds $10$ to its +argument. +\spadcommand{g := addx 10} +\begin{verbatim} + Compiling function addx with type + PositiveInteger -> (Integer -> Integer) +\end{verbatim} +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (Integer {\tt ->} Integer)} + +Try it out. +\spadcommand{g 3} +$$ +13 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{g(-4)} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +\index{function!anonymous!restrictions} +An anonymous function cannot be recursive: since it does not have a +name, you cannot even call it within itself! If you place an +anonymous function inside a named function, the anonymous function +must be declared. + +\section{Example: A Database} +\label{ugUserDatabase} + +This example shows how you can use Axiom to organize a database of +lineage data and then query the database for relationships. + +The database is entered as ``assertions'' that are really pieces of a +function definition. +\spadcommand{children("albert") == ["albertJr","richard","diane"]} +\returnType{Type: Void} + +Each piece $children(x) == y$ means ``the children of $x$ are $y$''. +\spadcommand{children("richard") == ["douglas","daniel","susan"]} +\returnType{Type: Void} + +This family tree thus spans four generations. +\spadcommand{children("douglas") == ["dougie","valerie"]} +\returnType{Type: Void} + +Say ``no one else has children.'' +\spadcommand{children(x) == []} +\returnType{Type: Void} + +We need some functions for computing lineage. Start with {\tt childOf}. +\spadcommand{childOf(x,y) == member?(x,children(y))} +\returnType{Type: Void} + +To find the {\tt parentOf} someone, you have to scan the database of +people applying {\tt children}. +\begin{verbatim} +parentOf(x) == + for y in people repeat + (if childOf(x,y) then return y) + "unknown" +\end{verbatim} +\returnType{Type: Void} + +And a grandparent of $x$ is just a parent of a parent of $x$. +\spadcommand{grandParentOf(x) == parentOf parentOf x} +\returnType{Type: Void} + +The grandchildren of $x$ are the people $y$ such that $x$ is a +grandparent of $y$. +\spadcommand{grandchildren(x) == [y for y in people | grandParentOf(y) = x]} +\returnType{Type: Void} + +Suppose you want to make a list of all great-grandparents. Well, a +great-grandparent is a grandparent of a person who has children. + +\begin{verbatim} +greatGrandParents == [x for x in people | + reduce(_or, + [not empty? children(y) for y in grandchildren(x)],false)] +\end{verbatim} +\returnType{Type: Void} + +Define {\tt descendants} to include the parent as well. +\begin{verbatim} +descendants(x) == + kids := children(x) + null kids => [x] + concat(x,reduce(concat,[descendants(y) + for y in kids],[])) +\end{verbatim} +\returnType{Type: Void} + +Finally, we need a list of people. Since all people are descendants +of ``albert'', let's say so. +\spadcommand{people == descendants "albert"} +\returnType{Type: Void} + +We have used ``{\tt ==}'' to define the database and some functions to +query the database. But no computation is done until we ask for some +information. Then, once and for all, the functions are analyzed and +compiled to machine code for run-time efficiency. Notice that no +types are given anywhere in this example. They are not needed. + +Who are the grandchildren of ``richard''? +\spadcommand{grandchildren "richard"} +\begin{verbatim} +Compiling function children with type String -> List String +Compiling function descendants with type String -> List String +Compiling body of rule people to compute value of type List String +Compiling function childOf with type (String,String) -> Boolean +Compiling function parentOf with type String -> String +Compiling function grandParentOf with type String -> String +Compiling function grandchildren with type String -> List String +\end{verbatim} +$$ +\left[ +\mbox{\tt "dougie"} , \mbox{\tt "valerie"} +\right] +$$ +\returnType{Type: List String} + +Who are the great-grandparents? +\spadcommand{greatGrandParents} +\begin{verbatim} +Compiling body of rule greatGrandParents to compute value of + type List String +\end{verbatim} +$$ +\left[ +\mbox{\tt "albert"} +\right] +$$ +\returnType{Type: List String} + +\section{Example: A Famous Triangle} +\label{ugUserTriangle} + +In this example we write some functions that display Pascal's +triangle. \index{Pascal's triangle} It demonstrates the use of +piece-wise definitions and some output operations you probably haven't +seen before. + +To make these output operations available, we have to {\it expose} the +domain {\tt OutputForm}. \index{OutputForm} See +\ref{ugTypesExpose} on page~\pageref{ugTypesExpose} +for more information about exposing domains and packages. +\spadcommand{)set expose add constructor OutputForm} +\begin{verbatim} + OutputForm is now explicitly exposed in frame G82322 +\end{verbatim} + +Define the values along the first row and any column $i$. +\spadcommand{pascal(1,i) == 1} +\returnType{Type: Void} + +Define the values for when the row and column index $i$ are equal. +Repeating the argument name indicates that the two index values are equal. +\spadcommand{pascal(n,n) == 1} +\returnType{Type: Void} + +\begin{verbatim} +pascal(i,j | 1 < i and i < j) == + pascal(i-1,j-1)+pascal(i,j-1) +\end{verbatim} +\returnType{Type: Void} + +Now that we have defined the coefficients in Pascal's triangle, let's +write a couple of one-liners to display it. + +First, define a function that gives the $n$-th row. +\spadcommand{pascalRow(n) == [pascal(i,n) for i in 1..n]} +\returnType{Type: Void} + +Next, we write the function {\bf displayRow} to display the row, +separating entries by blanks and centering. +\spadcommand{displayRow(n) == output center blankSeparate pascalRow(n)} +\returnType{Type: Void} + +Here we have used three output operations. Operation +\spadfunFrom{output}{OutputForm} displays the printable form of +objects on the screen, \spadfunFrom{center}{OutputForm} centers a +printable form in the width of the screen, and +\spadfunFrom{blankSeparate}{OutputForm} takes a list of nprintable +forms and inserts a blank between successive elements. + +Look at the result. +\spadcommand{for i in 1..7 repeat displayRow i} +\begin{verbatim} + Compiling function pascal with type (Integer,Integer) -> + PositiveInteger + Compiling function pascalRow with type PositiveInteger -> List + PositiveInteger + Compiling function displayRow with type PositiveInteger -> Void + + + 1 + 1 1 + 1 2 1 + 1 3 3 1 + 1 4 6 4 1 + 1 5 10 10 5 1 + 1 6 15 20 15 6 1 +\end{verbatim} +\returnType{Type: Void} + +Being purists, we find this less than satisfactory. Traditionally, +elements of Pascal's triangle are centered between the left and right +elements on the line above. + +To fix this misalignment, we go back and redefine {\bf pascalRow} to +right adjust the entries within the triangle within a width of four +characters. + +\spadcommand{pascalRow(n) == [right(pascal(i,n),4) for i in 1..n]} +\begin{verbatim} + Compiled code for pascalRow has been cleared. + Compiled code for displayRow has been cleared. + 1 old definition(s) deleted for function or rule pascalRow +\end{verbatim} +\returnType{Type: Void} + +Finally let's look at our purely reformatted triangle. +\spadcommand{for i in 1..7 repeat displayRow i} +\begin{verbatim} + Compiling function pascalRow with type PositiveInteger -> List + OutputForm + ++++ |*1;pascalRow;1;G82322| redefined + Compiling function displayRow with type PositiveInteger -> Void + ++++ |*1;displayRow;1;G82322| redefined + 1 + 1 1 + 1 2 1 + 1 3 3 1 + 1 4 6 4 1 + 1 5 10 10 5 1 + 1 6 15 20 15 6 1 +\end{verbatim} +\returnType{Type: Void} + +Unexpose {\tt OutputForm} so we don't get unexpected results later. +\spadcommand{)set expose drop constructor OutputForm} +\begin{verbatim} + OutputForm is now explicitly hidden in frame G82322 +\end{verbatim} + +\section{Example: Testing for Palindromes} +\label{ugUserPal} + +In this section we define a function {\bf pal?} that tests whether its +\index{palindrome} argument is a {\it palindrome}, that is, something +that reads the same backwards and forwards. For example, the string +``Madam I'm Adam'' is a palindrome (excluding blanks and punctuation) +and so is the number $123454321$. The definition works for any +datatype that has $n$ components that are accessed by the indices +$1\ldots n$. + +Here is the definition for {\bf pal?}. It is simply a call to an +auxiliary function called {\bf palAux?}. We are following the +convention of ending a function's name with {\tt ?} if the function +returns a {\tt Boolean} value. +\spadcommand{pal? s == palAux?(s,1,\#s)} +\returnType{Type: Void} + +Here is {\bf palAux?}. It works by comparing elements that are +equidistant from the start and end of the object. +\begin{verbatim} +palAux?(s,i,j) == + j > i => + (s.i = s.j) and palAux?(s,i+1,i-1) + true +\end{verbatim} +\returnType{Type: Void} + +Try {\bf pal?} on some examples. First, a string. +\spadcommand{pal? "Oxford"} +\begin{verbatim} + Compiling function palAux? with type (String,Integer,Integer) -> + Boolean + Compiling function pal? with type String -> Boolean +\end{verbatim} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +A list of polynomials. +\spadcommand{pal? [4,a,x-1,0,x-1,a,4]} +\begin{verbatim} + Compiling function palAux? with type (List Polynomial Integer, + Integer,Integer) -> Boolean + Compiling function pal? with type List Polynomial Integer -> Boolean +\end{verbatim} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +A list of integers from the example in the last section. +\spadcommand{pal? [1,6,15,20,15,6,1]} +\begin{verbatim} + Compiling function palAux? with type (List PositiveInteger,Integer, + Integer) -> Boolean + Compiling function pal? with type List PositiveInteger -> Boolean +\end{verbatim} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +To use {\bf pal?} on an integer, first convert it to a string. +\spadcommand{pal?(1441::String)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Compute an infinite stream of decimal numbers, each of which is an +obvious palindrome. +\spadcommand{ones := [reduce(+,[10**j for j in 0..i]) for i in 1..]} +$$ +\begin{array}{@{}l} +\left[ +{11}, {111}, {1111}, {11111}, {111111}, {1111111}, +\right. +\\ +\\ +\displaystyle +\left. +{11111111}, {111111111}, {1111111111}, {11111111111}, \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream PositiveInteger} + +\spadcommand{)set streams calculate 9} + +How about their squares? +\spadcommand{squares := [x**2 for x in ones]} +$$ +\begin{array}{@{}l} +\left[ +{121}, {12321}, {1234321}, {123454321}, {12345654321}, {1234567654321}, +\right. +\\ +\\ +\displaystyle +{123456787654321}, {12345678987654321}, {1234567900987654321}, +\\ +\\ +\displaystyle +\left. +{123456790120987654321}, \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream PositiveInteger} + +Well, let's test them all. +\spadcommand{[pal?(x::String) for x in squares]} +$$ +\left[ +{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt true}, +{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt true}, \ldots +\right] +$$ +\returnType{Type: Stream Boolean} + +\spadcommand{)set streams calculate 7} + +\section{Rules and Pattern Matching} +\label{ugUserRules} + +A common mathematical formula is +$$ \log(x) + \log(y) = \log(x y) \quad\forall \, x \hbox{\ and\ } y$$ +The presence of ``$\forall$'' indicates that $x$ and $y$ can stand for +arbitrary mathematical expressions in the above formula. You can use +such mathematical formulas in Axiom to specify ``rewrite rules''. +Rewrite rules are objects in Axiom that can be assigned to variables +for later use, often for the purpose of simplification. Rewrite rules +look like ordinary function definitions except that they are preceded +by the reserved word $rule$. \index{rule} For example, a rewrite rule +for the above formula is: +\begin{verbatim} +rule log(x) + log(y) == log(x * y) +\end{verbatim} + +Like function definitions, no action is taken when a rewrite rule is +issued. Think of rewrite rules as functions that take one argument. +When a rewrite rule $A = B$ is applied to an argument $f$, its meaning +is: ``rewrite every subexpression of $f$ that {\it matches} $A$ by +$B.$'' The left-hand side of a rewrite rule is called a {\it pattern}; +its right-side side is called its {\it substitution}. + +Create a rewrite rule named {\bf logrule}. The generated symbol +beginning with a ``{\tt \%}'' is a place-holder for any other terms that +might occur in the sum. +\spadcommand{logrule := rule log(x) + log(y) == log(x * y)} +$$ +{{\log +\left( +{y} +\right)}+{\log +\left( +{x} +\right)}+ + \%C} \mbox{\rm == } {{\log +\left( +{{x \ y}} +\right)}+ + \%C} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Create an expression with logarithms. +\spadcommand{f := log sin x + log x} +$$ +{\log +\left( +{{\sin +\left( +{x} +\right)}} +\right)}+{\log +\left( +{x} +\right)} +$$ +\returnType{Type: Expression Integer} + +Apply {\bf logrule} to $f$. +\spadcommand{logrule f} +$$ +\log +\left( +{{x \ {\sin +\left( +{x} +\right)}}} +\right) +$$ +\returnType{Type: Expression Integer} + +The meaning of our example rewrite rule is: ``for all expressions $x$ +and $y$, rewrite $log(x) + log(y)$ by $log(x * y)$.'' Patterns +generally have both operation names (here, {\bf log} and ``{\tt +}'') and +variables (here, $x$ and $y$). By default, every operation name +stands for itself. Thus {\bf log} matches only ``$log$'' and not any +other operation such as {\bf sin}. On the other hand, variables do +not stand for themselves. Rather, a variable denotes a {\it pattern +variable} that is free to match any expression whatsoever. +\index{pattern!variables} + +When a rewrite rule is applied, a process called +{\it pattern matching} goes to work by systematically scanning +\index{pattern!matching} the subexpressions of the argument. When a +subexpression is found that ``matches'' the pattern, the subexpression +is replaced by the right-hand side of the rule. The details of what +happens will be covered later. + +The customary Axiom notation for patterns is actually a shorthand for +a longer, more general notation. Pattern variables can be made +explicit by using a percent ``{\tt \%}'' as the first character of the +variable name. To say that a name stands for itself, you can prefix +that name with a quote operator ``{\tt '}''. Although the current Axiom +parser does not let you quote an operation name, this more general +notation gives you an alternate way of giving the same rewrite rule: +\begin{verbatim} +rule log(%x) + log(%y) == log(x * y) +\end{verbatim} + +This longer notation gives you patterns that the standard notation +won't handle. For example, the rule +\begin{verbatim} +rule %f(c * 'x) == c*%f(x) +\end{verbatim} +means ``for all $f$ and $c$, replace $f(y)$ by $c * f(x)$ when $y$ is +the product of $c$ and the explicit variable $x$.'' + +Thus the pattern can have several adornments on the names that appear there. +Normally, all these adornments are dropped in the substitution on the +right-hand side. + +To summarize: + +\boxed{4.6in}{ +\vskip 0.1cm +To enter a single rule in Axiom, use the following syntax: \index{rule} +\begin{center} +{\tt rule {\it leftHandSide} == {\it rightHandSide}} +\end{center} + +The {\it leftHandSide} is a pattern to be matched and the {\it +rightHandSide} is its substitution. The rule is an object of type +{\tt RewriteRule} that can be assigned to a variable and applied to +expressions to transform them.\\ +} + +Rewrite rules can be collected +into rulesets so that a set of rules can be applied at once. +Here is another simplification rule for logarithms. +$$y \log(x) = \log(x^y) \quad\forall \, x \hbox{\ and\ } y$$ +If instead of giving a single rule following the reserved word $rule$ +you give a ``pile'' of rules, you create what is called a {\it +ruleset.} \index{ruleset} Like rules, rulesets are objects in Axiom +and can be assigned to variables. You will find it useful to group +commonly used rules into input files, and read them in as needed. + +Create a ruleset named $logrules$. +\begin{verbatim} +logrules := rule + log(x) + log(y) == log(x * y) + y * log x == log(x ** y) +\end{verbatim} +$$ +\left\{ +{{{\log +\left( +{y} +\right)}+{\log +\left( +{x} +\right)}+ + \%B} \mbox{\rm == } {{\log +\left( +{{x \ y}} +\right)}+ + \%B}}, {{y \ {\log +\left( +{x} +\right)}} +\mbox{\rm == } {\log +\left( +{{x \sp y}} +\right)}} +\right\} +$$ +\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} + +Again, create an expression $f$ containing logarithms. +\spadcommand{f := a * log(sin x) - 2 * log x} +$$ +{a \ {\log +\left( +{{\sin +\left( +{x} +\right)}} +\right)}} +-{2 \ {\log +\left( +{x} +\right)}} +$$ +\returnType{Type: Expression Integer} + +Apply the ruleset {\bf logrules} to $f$. +\spadcommand{logrules f} +$$ +\log +\left( +{{{{\sin +\left( +{x} +\right)} +\sp a} \over {x \sp 2}}} +\right) +$$ +\returnType{Type: Expression Integer} + + +We have allowed pattern variables to match arbitrary expressions in +the above examples. Often you want a variable only to match +expressions satisfying some predicate. For example, we may want to +apply the transformation +$$y \log(x) = \log(x^y)$$ +only when $y$ is an integer. + +The way to restrict a pattern variable $y$ by a predicate $f(y)$ +\index{pattern!variable!predicate} is by using a vertical bar ``{\tt |}'', +which means ``such that,'' in \index{such that} much the same way it +is used in function definitions. \index{predicate!on a pattern +variable} You do this only once, but at the earliest (meaning deepest +and leftmost) part of the pattern. + +This restricts the logarithmic rule to create integer exponents only. +\begin{verbatim} +logrules2 := rule + log(x) + log(y) == log(x * y) + (y | integer? y) * log x == log(x ** y) +\end{verbatim} +$$ +\left\{ +{{{\log +\left( +{y} +\right)}+{\log +\left( +{x} +\right)}+ + \%D} \mbox{\rm == } {{\log +\left( +{{x \ y}} +\right)}+ + \%D}}, {{y \ {\log +\left( +{x} +\right)}} +\mbox{\rm == } {\log +\left( +{{x \sp y}} +\right)}} +\right\} +$$ +\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} + +Compare this with the result of applying the previous set of rules. +\spadcommand{f} +$$ +{a \ {\log +\left( +{{\sin +\left( +{x} +\right)}} +\right)}} +-{2 \ {\log +\left( +{x} +\right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{logrules2 f} +$$ +{a \ {\log +\left( +{{\sin +\left( +{x} +\right)}} +\right)}}+{\log +\left( +{{1 \over {x \sp 2}}} +\right)} +$$ +\returnType{Type: Expression Integer} + +You should be aware that you might need to apply a function like +{\tt integer} within your predicate expression to actually apply the test +function. + +Here we use {\tt integer} because $n$ has type {\tt Expression +Integer} but {\bf even?} is an operation defined on integers. +\spadcommand{evenRule := rule cos(x)**(n | integer? n and even? integer n)==(1-sin(x)**2)**(n/2)} +$$ +{{\cos +\left( +{x} +\right)} +\sp n} \mbox{\rm == } {{\left( -{{\sin +\left( +{x} +\right)} +\sp 2}+1 +\right)} +\sp {n \over 2}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Here is the application of the rule. +\spadcommand{evenRule( cos(x)**2 )} +$$ +-{{\sin +\left( +{x} +\right)} +\sp 2}+1 +$$ +\returnType{Type: Expression Integer} + +This is an example of some of the usual identities involving products of +sines and cosines. +\begin{verbatim} +sinCosProducts == rule + sin(x) * sin(y) == (cos(x-y) - cos(x + y))/2 + cos(x) * cos(y) == (cos(x-y) + cos(x+y))/2 + sin(x) * cos(y) == (sin(x-y) + sin(x + y))/2 +\end{verbatim} +\returnType{Type: Void} + +\spadcommand{g := sin(a)*sin(b) + cos(b)*cos(a) + sin(2*a)*cos(2*a)} +$$ +{{\sin +\left( +{a} +\right)} +\ {\sin +\left( +{b} +\right)}}+{{\cos +\left( +{{2 \ a}} +\right)} +\ {\sin +\left( +{{2 \ a}} +\right)}}+{{\cos +\left( +{a} +\right)} +\ {\cos +\left( +{b} +\right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{sinCosProducts g} +\begin{verbatim} + Compiling body of rule sinCosProducts to compute value of type + Ruleset(Integer,Integer,Expression Integer) +\end{verbatim} +$$ +{{\sin +\left( +{{4 \ a}} +\right)}+{2 +\ {\cos +\left( +{{b -a}} +\right)}}} +\over 2 +$$ +\returnType{Type: Expression Integer} + +Another qualification you will often want to use is to allow a pattern to +match an identity element. +Using the pattern $x + y$, for example, neither $x$ nor $y$ +matches the expression $0$. +Similarly, if a pattern contains a product $x*y$ or an exponentiation +$x**y$, then neither $x$ or $y$ matches $1$. + +If identical elements were matched, pattern matching would generally loop. +Here is an expansion rule for exponentials. +\spadcommand{exprule := rule exp(a + b) == exp(a) * exp(b)} +$$ +{e \sp {\left( b+a +\right)}} +\mbox{\rm == } {{e \sp a} \ {e \sp b}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +This rule would cause infinite rewriting on this if either $a$ or +$b$ were allowed to match $0$. +\spadcommand{exprule exp x} +$$ +e \sp x +$$ +\returnType{Type: Expression Integer} + +There are occasions when you do want a pattern variable in a sum or +product to match $0$ or $1$. If so, prefix its name +with a ``{\tt ?}'' whenever it appears in a left-hand side of a rule. +For example, consider the following rule for the exponential integral: +$$\int \left(\frac{y+e^x}{x}\right) dx = +\int \frac{y}{x} dx + \hbox{\rm Ei}(x) \quad\forall \, x \hbox{\ and\ } y$$ +This rule is valid for $y = 0$. One solution is to create a {\tt +Ruleset} with two rules, one with and one without $y$. A better +solution is to use an ``optional'' pattern variable. + +Define rule {\tt eirule} with +a pattern variable $?y$ to indicate +that an expression may or may not occur. +\spadcommand{eirule := rule integral((?y + exp x)/x,x) == integral(y/x,x) + Ei x} +$$ +{\int \sp{\displaystyle x} {{{{e \sp \%M}+y} \over \%M} \ {d \%M}}} +\mbox{\rm == } {{{{\tt '}integral} +\left( +{{y \over x}, x} +\right)}+{{{\tt +'}Ei} +\left( +{x} +\right)}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Apply rule {\tt eirule} to an integral without this term. +\spadcommand{eirule integral(exp u/u, u)} +$$ +Ei +\left( +{u} +\right) +$$ +\returnType{Type: Expression Integer} + +Apply rule {\tt eirule} to an integral with this term. +\spadcommand{eirule integral(sin u + exp u/u, u)} +$$ +{\int \sp{\displaystyle u} {{\sin +\left( +{ \%M} +\right)} +\ {d \%M}}}+{Ei +\left( +{u} +\right)} +$$ +\returnType{Type: Expression Integer} + +Here is one final adornment you will find useful. When matching a +pattern of the form $x + y$ to an expression containing a long sum of +the form $a +\ldots+ b$, there is no way to predict in advance which +subset of the sum matches $x$ and which matches $y$. Aside from +efficiency, this is generally unimportant since the rule holds for any +possible combination of matches for $x$ and $y$. In some situations, +however, you may want to say which pattern variable is a sum (or +product) of several terms, and which should match only a single term. +To do this, put a prefix colon ``{\tt :}'' before the pattern variable +that you want to match multiple terms. +\index{pattern!variable!matching several terms} + +The remaining rules involve operators $u$ and $v$. \index{operator} +\spadcommand{u := operator 'u} +$$ +u +$$ +\returnType{Type: BasicOperator} + +These definitions tell Axiom that $u$ and $v$ are formal operators to +be used in expressions. +\spadcommand{v := operator 'v} +$$ +v +$$ +\returnType{Type: BasicOperator} + +First define {\tt myRule} with no restrictions on the pattern variables +$x$ and $y$. +\spadcommand{myRule := rule u(x + y) == u x + v y} +$$ +{u +\left( +{{y+x}} +\right)} +\mbox{\rm == } {{{{\tt '}v} +\left( +{y} +\right)}+{{{\tt +'}u} +\left( +{x} +\right)}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Apply {\tt myRule} to an expression. +\spadcommand{myRule u(a + b + c + d)} +$$ +{v +\left( +{{d+c+b}} +\right)}+{u +\left( +{a} +\right)} +$$ +\returnType{Type: Expression Integer} + +Define {\tt myOtherRule} to match several terms so that the rule gets +applied recursively. +\spadcommand{myOtherRule := rule u(:x + y) == u x + v y} +$$ +{u +\left( +{{y+x}} +\right)} +\mbox{\rm == } {{{{\tt '}v} +\left( +{y} +\right)}+{{{\tt +'}u} +\left( +{x} +\right)}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Apply {\tt myOtherRule} to the same expression. +\spadcommand{myOtherRule u(a + b + c + d)} +$$ +{v +\left( +{c} +\right)}+{v +\left( +{b} +\right)}+{v +\left( +{a} +\right)}+{u +\left( +{d} +\right)} +$$ +\returnType{Type: Expression Integer} + +\boxed{4.6in}{ +\vskip 0.1cm +Summary of pattern variable adornments: +\vskip .5\baselineskip +\begin{tabular}{@{}ll} +{\tt (x | predicate?(x))} & + means that the substutution $s$ for $x$\\ & + must satisfy {\tt predicate(s) = true.} \\ +{\tt ?x} & + means that $x$ can match an identity \\ & element (0 or 1). \\ +{\tt :x} & + means that $x$ can match several terms \\ & in a sum. +\end{tabular}\\ +} + +Here are some final remarks on pattern matching. Pattern matching +provides a very useful paradigm for solving certain classes of +problems, namely, those that involve transformations of one form to +another and back. However, it is important to recognize its +limitations. \index{pattern!matching!caveats} + +First, pattern matching slows down as the number of rules you have to +apply increases. Thus it is good practice to organize the sets of +rules you use optimally so that irrelevant rules are never included. + +Second, careless use of pattern matching can lead to wrong answers. +You should avoid using pattern matching to handle hidden algebraic +relationships that can go undetected by other programs. As a simple +example, a symbol such as ``J'' can easily be used to represent the +square root of $-1$ or some other important algebraic quantity. Many +algorithms branch on whether an expression is zero or not, then divide +by that expression if it is not. If you fail to simplify an +expression involving powers of $J$ to $-1,$ algorithms may incorrectly +assume an expression is non-zero, take a wrong branch, and produce a +meaningless result. + +Pattern matching should also not be used as a substitute for a domain. +In Axiom, objects of one domain are transformed to objects of other +domains using well-defined {\bf coerce} operations. Pattern matching +should be used on objects that are all the same type. Thus if your +application can be handled by type {\tt Expression} in Axiom and you +think you need pattern matching, consider this choice carefully. +\index{Expression} You may well be better served by extending an +existing domain or by building a new domain of objects for your +application. + +\setcounter{chapter}{6} + +\chapter{Graphics} +\label{ugGraph} + +\begin{figure}[htbp] +\includegraphics{ps/torusknot.ps} +\caption{Torus knot of type (15,17).} +\end{figure} + +This chapter shows how to use the Axiom graphics facilities +\index{graphics} under the X Window System. Axiom has +two-di\-men\-sion\-al and three-di\-men\-sion\-al drawing and +rendering packages that allow the drawing, coloring, transforming, +mapping, clipping, and combining of graphic output from Axiom +computations. This facility is particularly useful for investigating +problems in areas such as topology. The graphics package is capable +of plotting functions of one or more variables or plotting parametric +surfaces and curves. Various coordinate systems are also available, +such as polar and spherical. + +A graph is displayed in a viewport window and it has a +\index{viewport} control-panel that uses interactive mouse commands. +PostScript and other output forms are available so that Axiom +\index{PostScript} images can be printed or used by other programs. + +\section{Two-Dimensional Graphics} +\label{ugGraphTwoD} + +The Axiom two-di\-men\-sion\-al graphics package provides the ability +to \index{graphics!two-dimensional} display +\begin{itemize} +\item curves defined by functions of a single real variable +\item curves defined by parametric equations +\item implicit non-singular curves defined by polynomial equations +\item planar graphs generated from lists of point components. +\end{itemize} + +These graphs can be modified by specifying various options, such as +calculating points in the polar coordinate system or changing the size +of the graph viewport window. + +\subsection{Plotting Two-Dimensional Functions of One Variable} +\label{ugGraphTwoDPlot} + +\index{curve!one variable function} The first kind of +two-di\-men\-sion\-al graph is that of a curve defined by a function +$y = f(x)$ over a finite interval of the $x$ axis. + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a function defined by a formula $f(x)$ is: +\begin{center} +{\tt draw(f(x), x = a..b, {\it options})} +\end{center} + +where $a..b$ defines the range of $x$, and where {\it options} +prescribes zero or more options as described in +\ref{ugGraphTwoDOptions} on page~\pageref{ugGraphTwoDOptions}. An +example of an option is $curveColor == bright red().$ An alternative +format involving functions $f$ and $g$ is also available.\\ +} + +A simple way to plot a function is to use a formula. The first +argument is the formula. For the second argument, write the name of +the independent variable (here, $x$), followed by an ``{\tt =}'', and the +range of values. + +Display this formula over the range $0 \leq x \leq 6$. +Axiom converts your formula to a compiled function so that the +results can be computed quickly and efficiently. + +\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)} +%\begin{figure}[htbp] +%{\rm draw(sin(tan(x)) - tan(sin(x)),x = 0..6)\\} +%\vskip 0.2cm +%\includegraphics{ps/2d1vara.ps} +%\caption{$sin(tan(x)) - tan(sin(x))\ \ \ x = 0 \ldots6$} +%\end{figure} +%{\rm +%\par +%Notice that Axiom compiled the function before the graph was put +%on the screen. +% +%Here is the same graph on a different interval.\\ +% +%draw(sin(tan(x)) - tan(sin(x)),x = 10..16)\\ +%} +%\vskip 0.2cm +%\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16)} +%\begin{figure}[htbp] +%\includegraphics{ps/2d1varb.ps} +%\caption{$sin(tan(x)) - tan(sin(x))\ \ \ x = 10 \ldots16$} +%\end{figure} + +Once again the formula is converted to a compiled function before any +points were computed. If you want to graph the same function on +several intervals, it is a good idea to define the function first so +that the function has to be compiled only once. + +This time we first define the function. +\spadcommand{f(x) == (x-1)*(x-2)*(x-3) } +%\begin{figure} +%\tpd{0 0 300 300 ps/2d1vard.ps} +%\end{figure} +\returnType{Type: Void} +%\epsffile[14 14 188 199]{ps/2d1vard.ps} +%\hspace*{\baseLeftSkip}\special{psfile=ps/2dctrl.ps} + +To draw the function, the first argument is its name and the second is +just the range with no independent variable. +\spadgraph{draw(f, 0..4) } +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2d1vard.ps} + +\subsection{Plotting Two-Dimensional Parametric Plane Curves} +\label{ugGraphTwoDPar} + +The second kind of two-di\-men\-sion\-al graph is that of +\index{parametric plane curve} curves produced by parametric +equations. \index{curve!parametric plane} Let $x = f(t)$ and +$y = g(t)$ be formulas or two functions $f$ and $g$ as the parameter $t$ +ranges over an interval $[a,b]$. The function {\bf curve} takes the +two functions $f$ and $g$ as its parameters. + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a two-di\-men\-sion\-al plane curve defined by +parametric formulas $x = f(t)$ and $y = g(t)$ is: +\begin{center} +{\tt draw(curve(f(t), g(t)), t = a..b, {\it options})} +\end{center} + +where $a..b$ defines the range of the independent variable $t$, and +where {\it options} prescribes zero or more options as described in +\ref{ugGraphThreeDOptions} on page~\pageref{ugGraphThreeDOptions}. An +example of an option is $curveColor == bright red().$\\ } + +Here's an example: + +Define a parametric curve using a range involving $\%pi$, Axiom's way +of saying $\pi$. For parametric curves, Axiom +compiles two functions, one for each of the functions $f$ and $g$. +\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)} +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2dppca.ps} + +The title may be an arbitrary string and is an optional argument to +the {\bf draw} command. +\spadgraph{draw(curve(cos(t), sin(t)), t = 0..2*\%pi)} +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2dppcb.ps} + +If you plan on plotting $x = f(t)$, $y = g(t)$ as $t$ ranges over +several intervals, you may want to define functions $f$ and $g$ first, +so that they need not be recompiled every time you create a new graph. +Here's an example: + +As before, you can first define the functions you wish to draw. +\spadcommand{f(t:DFLOAT):DFLOAT == sin(3*t/4) } +\begin{verbatim} + Function declaration f : DoubleFloat -> DoubleFloat has been + added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Axiom compiles them to map {\tt DoubleFloat} values to {\tt DoubleFloat} +values. +\spadcommand{g(t:DFLOAT):DFLOAT == sin(t) } +\begin{verbatim} + Function declaration f : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +Give to {\tt curve} the names of the functions, then write the range +without the name of the independent variable. +\spadgraph{draw(curve(f,g),0..\%pi) } +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2dppcc.ps} + +Here is another look at the same curve but over a different +range. Notice that $f$ and $g$ are not recompiled. Also note that +Axiom provides a default title based on the first function specified +in {\bf curve}. +\spadgraph{draw(curve(f,g),-4*\%pi..4*\%pi) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2dppce.ps} + +\subsection{Plotting Plane Algebraic Curves} +\label{ugGraphTwoDPlane} + +A third kind of two-di\-men\-sion\-al graph is a non-singular +``solution curve'' \index{curve!plane algebraic} in a rectangular +region of the plane. A solution curve is a curve defined by a +polynomial equation $p(x,y) = 0$. \index{plane algebraic curve} +Non-singular means that the curve is ``smooth'' in that it does not +cross itself or come to a point (cusp). Algebraically, this means +that for any point $(x,y)$ on the curve, that is, a point such that +$p(x,y) = 0$, the partial derivatives +${{\partial p}\over{\partial x}}(x,y)$ and +${{\partial p}\over{\partial y}}(x,y)$ are not both zero. +\index{curve!smooth} \index{curve!non-singular} \index{smooth curve} +\index{non-singular curve} + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a non-singular solution curve given by +a polynomial of the form $p(x,y) = 0$ is: +\begin{center} +{\tt draw(p(x,y) = 0, x, y, range == [a..b, c..d], {\it options})} +\end{center} + +where the second and third arguments name the first and second +independent variables of $p$. A {\tt range} option is always given to +designate a bounding rectangular region of the plane +$a \leq x \leq b, c \leq y \leq d$. +Zero or more additional options as described in +\ref{ugGraphTwoDOptions} on page~\pageref{ugGraphTwoDOptions} may be given.\\ +} + +We require that the polynomial has rational or integral coefficients. +Here is an algebraic curve example (``Cartesian ovals''): +\index{Cartesian!ovals} +\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1)-4*x-1) } +$$ +{y \sp 4}+{{\left( {2 \ {x \sp 2}} -{{16} \ x} -6 +\right)} +\ {y \sp 2}}+{x \sp 4} -{{16} \ {x \sp 3}}+{{58} \ {x \sp 2}} -{{12} \ x} +-6 +$$ +\returnType{Type: Polynomial Integer} + +The first argument is always expressed as an equation of the form $p = 0$ +where $p$ is a polynomial. +\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7]) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2dpaca.ps} + +\subsection{Two-Dimensional Options} +\label{ugGraphTwoDOptions} + +The {\bf draw} commands take an optional list of options, such as {\tt +title} shown above. Each option is given by the syntax: +{\it name} {\tt ==} {\it value}. +Here is a list of the available options in the +order that they are described below. + +\begin{tabular}{llll} +adaptive&clip&unit\\ +clip&curveColor&range\\ +toScale&pointColor&coordinates\\ +\end{tabular} + +The $adaptive$ option turns adaptive plotting on or off. +\index{adaptive plotting} Adaptive plotting uses an algorithm that +traverses a graph and computes more points for those parts of the +graph with high curvature. The higher the curvature of a region is, +the more points the algorithm computes. +\index{graphics!2D options!adaptive} + +The {\tt adaptive} option is normally on. Here we turn it off. +\spadgraph{draw(sin(1/x),x=-2*\%pi..2*\%pi, adaptive == false)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptad.ps} + +The {\tt clip} option turns clipping on or off. +\index{graphics!2D options!clipping} +If on, large values are cut off according to +\spadfunFrom{clipPointsDefault}{GraphicsDefaults}. + +\spadgraph{draw(tan(x),x=-2*\%pi..2*\%pi, clip == true)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptcp.ps} + +Option {\tt toScale} does plotting to scale if {\tt true} or uses the +entire viewport if {\tt false}. The default can be determined using +\spadfunFrom{drawToScale}{GraphicsDefaults}. +\index{graphics!2D options!to scale} + +\spadgraph{draw(sin(x),x=-\%pi..\%pi, toScale == true, unit == [1.0,1.0])} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptsc.ps} + +Option {\tt clip} with a range sets point clipping of a graph within +the \index{graphics!2D options!clip in a range} ranges specified in +the list $[x range,y range]$. \index{clipping} If only one range is +specified, clipping applies to the y-axis. +\spadgraph{draw(sec(x),x=-2*\%pi..2*\%pi, clip == [-2*\%pi..2*\%pi,-\%pi..\%pi], unit == [1.0,1.0])} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptcpr.ps} + +Option {\tt curveColor} sets the color of the graph curves or lines to +be the \index{graphics!2D options!curve color} indicated palette color +\index{curve!color} (see \ref{ugGraphColor} on page~\pageref{ugGraphColor} and +\ref{ugGraphColorPalette} on page~\pageref{ugGraphColorPalette}). +\index{color!curve} + +\spadgraph{draw(sin(x),x=-\%pi..\%pi, curveColor == bright red())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptcvc.ps} + +Option {\tt pointColor} sets the color of the graph points to the +indicated \index{graphics!2D options!point color} palette color (see +\ref{ugGraphColor} on page~\pageref{ugGraphColor} and +\ref{ugGraphColorPalette} on page~\pageref{ugGraphColorPalette}). +\index{color!point} +\spadgraph{draw(sin(x),x=-\%pi..\%pi, pointColor == pastel yellow())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptptc.ps} + +Option {\tt unit} sets the intervals at which the axis units are +plotted \index{graphics!2D options!set units} according to the +indicated steps [$x$ interval, $y$ interval]. +\spadgraph{draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*\%pi..4*\%pi, unit == [2.0,1.0])} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptut.ps} + +Option {\tt range} sets the range of variables in a graph to be within +the ranges \index{graphics!2D options!range} for solving plane +algebraic curve plots. +\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptrga.ps} + +A second example of a solution plot. +\spadgraph{draw(x**2 + y**2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptrgb.ps} + +Option $coordinates$ indicates the coordinate system in which the +graph \index{graphics!2D options!coordinates} is plotted. The default +is to use the Cartesian coordinate system. +\index{Cartesian!coordinate system} For more details, see +\ref{ugGraphCoord} on page~\pageref{ugGraphCoord} +{or {\tt CoordinateSystems}.} +\index{coordinate system!Cartesian} + +\spadgraph{draw(curve(sin(5*t),t),t=0..2*\%pi, coordinates == polar)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/2doptplr.ps} + +\subsection{Color} +\label{ugGraphColor} + +The domain {\tt Color} \index{Color} provides operations for +manipulating \index{graphics!color} colors in two-di\-men\-sion\-al +graphs. \index{color} Colors are objects of {\tt Color}. Each color +has a {\it hue} and a {\it weight}. \index{hue} Hues are represented +by integers that range from $1$ to the +\spadfunFrom{numberOfHues()}{Color}, normally +\index{graphics!color!number of hues} $27$. \index{weight} Weights +are floats and have the value $1.0$ by default. + +\begin{description} + +\item[{\bf color}]\funArgs{integer} +creates a color of hue {\it integer} and weight $1.0$. + +\item[{\bf hue}]\funArgs{color} +returns the hue of {\it color} as an integer. +\index{graphics!color!hue function} + +\item[{\bf red}]\funArgs{} +\funSyntax{blue}{}, +\funSyntax{green}{}, and \funSyntax{yellow}{} +\index{graphics!color!primary color functions} +create colors of that hue with weight $1.0$. + +\item[$\hbox{\it color}_{1}$ {\tt +} $\hbox{\it color}_{2}$] returns the +color that results from additively combining the indicated +$\hbox{\it color}_{1}$ and $\hbox{\it color}_{2}$. +Color addition is not commutative: changing the order of the arguments +produces different results. + +\item[{\it integer} {\tt *} {\it color}] +changes the weight of {\it color} by {\it integer} +without affecting its hue. +\index{graphics!color!multiply function} +For example, +$red() + 3*yellow()$ produces a color closer to yellow than to red. +Color multiplication is not associative: changing the order of grouping +\index{color!multiplication} +produces different results. +\end{description} + +These functions can be used to change the point and curve colors +for two- and three-di\-men\-sion\-al graphs. +Use the {\tt pointColor} option for points. + +\spadgraph{draw(x**2,x=-1..1,pointColor == green())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/23dcola.ps} + + +Use the {\tt curveColor} option for curves. + +\spadgraph{draw(x**2,x=-1..1,curveColor == color(13) + 2*blue())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/23dcolb.ps} + +\subsection{Palette} +\label{ugGraphColorPalette} +\index{graphics!palette} + +Domain {\tt Palette} is the domain of shades of colors: +{\bf dark}, {\bf dim}, {\bf bright}, {\bf pastel}, and {\bf light}, +designated by the integers $1$ through $5$, respectively. +\index{Palette} + +Colors are normally ``bright.'' + +\spadcommand{shade red()} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +To change the shade of a color, apply the name of a shade to it. +\index{color!shade} +\index{shade} + +\spadcommand{myFavoriteColor := dark blue() } +$$ +[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Dark \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +The expression $shade(color)$ +returns the value of a shade of $color$. + +\spadcommand{shade myFavoriteColor } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The expression $hue(color)$ returns its hue. + +\spadcommand{hue myFavoriteColor } +$$ +\mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0} +$$ +\returnType{Type: Color} + +Palettes can be used in specifying colors in two-di\-men\-sion\-al graphs. + +\spadgraph{draw(x**2,x=-1..1,curveColor == dark blue())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/23dpal.ps} + + +\subsection{Two-Dimensional Control-Panel} +\label{ugGraphTwoDControl} + +\index{graphics!2D control-panel} +Once you have created a viewport, move your mouse to the viewport and click +with your left mouse button to display a control-panel. +The panel is displayed on the side of the viewport closest to +where you clicked. Each of the buttons which toggle on and off show the +current state of the graph. + +%\begin{texonly} +%\typeout{2D control-panel.} +\begin{figure}[htbp] +%{\epsfverbosetrue\epsfxsize=2in% +%\def\epsfsize#1#2{\epsfxsize}\hspace*{\baseLeftSkip}% +%%\epsffile[0 0 144 289]{ps/2dctrl.ps}} +%\begin{picture}(147,252)%(-143,0) +%\hspace*{\baseLeftSkip}\special{psfile=ps/2dctrl.ps} +%\end{picture} +\caption{Two-dimensional control-panel.} +\end{figure} +%\end{texonly} + +\subsubsection{Transformations} +\index{graphics!2D control-panel!transformations} + +Object transformations are executed from the control-panel by mouse-activated +potentiometer windows. +% +\begin{description} +% +\item[Scale:] To scale a graph, click on a mouse button +\index{graphics!2D control-panel!scale} +within the {\bf Scale} window in the upper left corner of the control-panel. +The axes along which the scaling is to occur are indicated by setting the +toggles above the arrow. +With {\tt X On} and {\tt Y On} appearing, both axes are selected and scaling +is uniform. +If either is not selected, for example, if {\tt X Off} appears, scaling is +non-uniform. +% +\item[Translate:] To translate a graph, click the mouse in the +\index{graphics!2D control-panel!translate} +{\bf Translate} window in the direction you wish the graph to move. +This window is located in the upper right corner of the control-panel. +Along the top of the {\bf Translate} window are two buttons for selecting +the direction of translation. +Translation along both coordinate axes results when {\tt X On} and {\tt Y +On} appear or along one axis when one is on, for example, {\tt X On} and +{\tt Y Off} appear. +\end{description} + +\subsubsection{Messages} +\index{graphics!2D control-panel!messages} + +The window directly below the transformation potentiometer windows is +used to display system messages relating to the viewport and the control-panel. +The following format is displayed: \newline +% +\begin{center} +[scaleX, scaleY] $>$graph$<$ [translateX, translateY] \newline +\end{center} +The two values to the left show the scale factor along the {\tt X} and +{\tt Y} coordinate axes. The two values to the right show the distance of +translation from the center in the {\tt X} and {\tt Y} directions. The number +in the center shows which graph in the viewport this data pertains to. +When multiple graphs exist in the same viewport, +the graph must be selected (see ``Multiple Graphs,'' below) in +order for its transformation data to be shown, otherwise the number +is 1. + +\subsubsection{Multiple Graphs} + +\index{graphics!2D control-panel!multiple graphs} +The {\bf Graphs} window contains buttons that allow the placement +of two-di\-men\-sion\-al graphs into one of nine available slots in any other +two-di\-men\-sion\-al viewport. +In the center of the window are numeral buttons from one to nine +that show whether a graph is displayed in the viewport. +Below each number button is a button showing whether a graph +that is present is selected for application of some +transformation. +When the caret symbol is displayed, then the graph in that slot +will be manipulated. +Initially, the graph for which the viewport is created occupies +the first slot, is displayed, and is selected. +% +% +\begin{description} +% +\item[Clear:] The {\bf Clear} button deselects every viewport graph slot. +\index{graphics!2D control-panel!clear} +A graph slot is reselected by selecting the button below its number. +% +\item[Query:] The {\bf Query} button is used to display the scale and +\index{graphics!2D control-panel!query} +translate data for the indicated graph. When this button is selected the +message ``Click on the graph to query'' appears. Select a slot +number button from the {\bf Graphs} window. The scaling factor and translation +offset of the graph are then displayed in the message window. +% +\item[Pick:] The {\bf Pick} button is used to select a graph +\index{graphics!2D control-panel!pick} +to be placed or dropped into the indicated viewport. When this button is +selected, the message ``Click on the graph to pick'' appears. +Click on the slot with the graph number of the desired +graph. The graph information is held waiting for +you to execute a {\bf Drop} in some other graph. +% +\item[Drop:] Once a graph has been picked up using the {\bf Pick} button, +\index{graphics!2D control-panel!drop} +the {\bf Drop} button places it into a new viewport slot. +The message ``Click on the graph to drop'' appears in the message +window when the {\bf Drop} button is selected. +By selecting one of the slot number buttons in the {\bf Graphs} +window, the graph currently being held is dropped into this slot +and displayed. +\end{description} + +\subsubsection{Buttons} +\index{graphics!2D control-panel!buttons} + +% +\begin{description} +% +\item[Axes] turns the coordinate axes on or off. +\index{graphics!2D control-panel!axes} +% +\item[Units] turns the units along the {\tt x} +and {\tt y} axis on or off. +\index{graphics!2D control-panel!units} +% +\item[Box] encloses the area of the viewport graph +in a bounding box, or removes the box if already enclosed. +\index{graphics!2D control-panel!box} +% +\item[Pts] turns on or off the display of points. +\index{graphics!2D control-panel!points} +% +\item[Lines] turns on or off the display +of lines connecting points. +\index{graphics!2D control-panel!lines} +% +\item[PS] writes the current viewport contents to +\index{graphics!2D control-panel!ps} +a file {\bf axiom2d.ps} or to a name specified in the user's {\bf +\index{graphics!.Xdefaults!PostScript file name} +.Xdefaults} file. +\index{file!.Xdefaults @{\bf .Xdefaults}} +The file is placed in the directory from which Axiom or the {\bf +viewalone} program was invoked. +\index{PostScript} +% +\item[Reset] resets the object transformation +characteristics and attributes back to their initial states. +\index{graphics!2D control-panel!reset} +% +\item[Hide] makes the control-panel disappear. +\index{graphics!2D control-panel!hide} +% +\item[Quit] queries whether the current viewport +\index{graphics!2D control-panel!quit} +session should be terminated. +\end{description} + +\subsection{Operations for Two-Dimensional Graphics} +\label{ugGraphTwoDops} + +Here is a summary of useful Axiom operations for two-di\-men\-sion\-al +graphics. +Each operation name is followed by a list of arguments. +Each argument is written as a variable informally named according +to the type of the argument (for example, {\it integer}). +If appropriate, a default value for an argument is given in +parentheses immediately following the name. + +% +\begin{description} +% +\item[{\bf adaptive}]\funArgs{\optArg{boolean\argDef{true}}} +\index{adaptive plotting} +sets or indicates whether graphs are plotted +\index{graphics!set 2D defaults!adaptive} +according to the adaptive refinement algorithm. +% +\item[{\bf axesColorDefault}]\funArgs{\optArg{color\argDef{dark blue()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!axes color} +axes in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf clipPointsDefault}]\funArgs{\optArg{boolean\argDef{false}}} +sets or +indicates whether point clipping is +\index{graphics!set 2D defaults!clip points} +to be applied as the default for graph plots. +% +\item[{\bf drawToScale}]\funArgs{\optArg{boolean\argDef{false}}} +sets or +indicates whether the plot of a graph +\index{graphics!set 2D defaults!to scale} +is ``to scale'' or uses the entire viewport space as the default. +% +\item[{\bf lineColorDefault}]\funArgs{\optArg{color\argDef{pastel yellow()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!line color} +lines or curves in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf maxPoints}]\funArgs{\optArg{integer\argDef{500}}} +sets or indicates +the default maximum number of +\index{graphics!set 2D defaults!max points} +possible points to be used when constructing a two-di\-men\-sion\-al graph. +% +\item[{\bf minPoints}]\funArgs{\optArg{integer\argDef{21}}} +sets or indicates the default minimum number of +\index{graphics!set 2D defaults!min points} +possible points to be used when constructing a two-di\-men\-sion\-al graph. +% +\item[{\bf pointColorDefault}]\funArgs{\optArg{color\argDef{bright red()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!point color} +points in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf pointSizeDefault}]\funArgs{\optArg{integer\argDef{5}}} +sets or indicates the default size of the +\index{graphics!set 2D defaults!point size} +dot used to plot points in a two-di\-men\-sion\-al graph. +% +\item[{\bf screenResolution}]\funArgs{\optArg{integer\argDef{600}}} +sets or indicates the default screen +\index{graphics!set 2D defaults!screen resolution} +resolution constant used in setting the computation limit of adaptively +\index{adaptive plotting} +generated curve plots. +% +\item[{\bf unitsColorDefault}]\funArgs{\optArg{color\argDef{dim green()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!units color} +unit labels in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf viewDefaults}]\funArgs{} +resets the default settings for the following +\index{graphics!set 2D defaults!reset viewport} +attributes: point color, line color, axes color, units color, point size, +viewport upper left-hand corner position, and the viewport size. +% +\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[100,100]}}} +sets or indicates the default position of the +\index{graphics!set 2D defaults!viewport position} +upper left-hand corner of a two-di\-men\-sion\-al viewport, relative to the +display root window. +The upper left-hand corner of the display is considered to be at the +(0, 0) position. +% +\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[200,200]}}} +sets or +indicates the default size in which two +\index{graphics!set 2D defaults!viewport size} +dimensional viewport windows are shown. +It is defined by a width and then a height. +% +\item[{\bf viewWriteAvailable}] +\funArgs{\optArg{list\argDef{["pixmap","bitmap", "postscript", "image"]}}} +indicates the possible file types +\index{graphics!2D defaults!available viewport writes} +that can be created with the \spadfunFrom{write}{TwoDimensionalViewport} function. +% +\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} +sets or indicates the default types of files, in +\index{graphics!set 2D defaults!write viewport} +addition to the {\bf data} file, that are created when a +{\bf write} function is executed on a viewport. +% +\item[{\bf units}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} +turns the units on or off for the graph with index {\it integer}. +% +\item[{\bf axes}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +turns the axes on +\index{graphics!2D commands!axes} +or off for the graph with index {\it integer}. +% +\item[{\bf close}]\funArgs{viewport} +closes {\it viewport}. +\index{graphics!2D commands!close} +% +\item[{\bf connect}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +declares whether lines +\index{graphics!2D commands!connect} +connecting the points are displayed or not. +% +\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} +declares +whether the two-di\-men\-sion\-al control-panel is automatically displayed +or not. +% +\item[{\bf graphs}]\funArgs{viewport} +returns a list +\index{graphics!2D commands!graphs} +describing the state of each graph. +If the graph state is not being used this is shown by {\tt "undefined"}, +otherwise a description of the graph's contents is shown. +% +\item[{\bf graphStates}]\funArgs{viewport} +displays +\index{graphics!2D commands!state of graphs} +a list of all the graph states available for {\it viewport}, giving the +values for every property. +% +\item[{\bf key}]\funArgs{viewport} +returns the process +\index{graphics!2D commands!key} +ID number for {\it viewport}. +% +\item[{\bf move}]\funArgs{viewport, +$integer_{x}$(viewPosDefault), +$integer_{y}$(viewPosDefault)} +moves {\it viewport} on the screen so that the +\index{graphics!2D commands!move} +upper left-hand corner of {\it viewport} is at the position {\it (x,y)}. +% +\item[{\bf options}]\funArgs{\it viewport} +returns a list +\index{graphics!2D commands!options} +of all the {\tt DrawOption}s used by {\it viewport}. +% +\item[{\bf points}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +specifies whether the graph points for graph {\it integer} are +\index{graphics!2D commands!points} +to be displayed or not. +% +\item[{\bf region}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} +declares whether graph {\it integer} is or is not to be displayed +with a bounding rectangle. +% +\item[{\bf reset}]\funArgs{viewport} +resets all the properties of {\it viewport}. +% +\item[{\bf resize}]\funArgs{viewport, +$integer_{width}$,$integer_{height}$} +\index{graphics!2D commands!resize} +resizes {\it viewport} with a new {\it width} and {\it height}. +% +\item[{\bf scale}]\funArgs{viewport, $integer_{n}$\argDef{1}, +$integer_{x}$\argDef{0.9}, $integer_{y}$\argDef{0.9}} +scales values for the +\index{graphics!2D commands!scale} +{\it x} and {\it y} coordinates of graph {\it n}. +% +\item[{\bf show}]\funArgs{viewport, $integer_{n}$\argDef{1}, +string\argDef{"on"}} +indicates if graph {\it n} is shown or not. +% +\item[{\bf title}]\funArgs{viewport, string\argDef{"Axiom 2D"}} +designates the title for {\it viewport}. +% +\item[{\bf translate}]\funArgs{viewport, +$integer_{n}$\argDef{1}, +$float_{x}$\argDef{0.0}, $float_{y}$\argDef{0.0}} +\index{graphics!2D commands!translate} +causes graph {\it n} to be moved {\it x} and {\it y} units in the respective directions. +% +\item[{\bf write}]\funArgs{viewport, $string_{directory}$, +\optArg{strings}} +if no third argument is given, writes the {\bf data} file onto the directory +with extension {\bf data}. +The third argument can be a single string or a list of strings with some or +all the entries {\tt "pixmap"}, {\tt "bitmap"}, {\tt "postscript"}, and +{\tt "image"}. +\end{description} + +\subsection{Addendum: Building Two-Dimensional Graphs} +\label{ugGraphTwoDbuild} + +In this section we demonstrate how to create two-di\-men\-sion\-al graphs from +lists of points and give an example showing how to read the lists +of points from a file. + +\subsubsection{Creating a Two-Dimensional Viewport from a List of Points} + +Axiom creates lists of points in a two-di\-men\-sion\-al viewport by utilizing +the {\tt GraphImage} and {\tt TwoDimensionalViewport} domains. +In this example, the \spadfunFrom{makeGraphImage}{GraphImage} +function takes a list of lists of points parameter, a list of colors for +each point in the graph, a list of colors for each line in the graph, and +a list of sizes for each point in the graph. +% + +The following expressions create a list of lists of points which will be read +by Axiom and made into a two-di\-men\-sion\-al viewport. + +\spadcommand{p1 := point [1,1]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p2 := point [0,1]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p3 := point [0,0]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p4 := point [1,0]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p5 := point [1,.5]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p6 := point [.5,0]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p7 := point [0,0.5]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p8 := point [.5,1]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p9 := point [.25,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p10 := point [.25,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p11 := point [.75,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p12 := point [.75,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +Finally, here is the list. + +\spadcommand{llp := [ [p1,p2], [p2,p3], [p3,p4], [p4,p1], [p5,p6], [p6,p7], [p7,p8], [p8,p5], [p9,p10], [p10,p11], [p11,p12], [p12,p9] ] } +$$ +\left[ +{\left[ {\left[ {1.0}, {1.0} +\right]}, + {\left[ {0.0}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {0.0}, {1.0} +\right]}, + {\left[ {0.0}, {0.0} +\right]} +\right]}, + {\left[ {\left[ {0.0}, {0.0} +\right]}, + {\left[ {1.0}, {0.0} +\right]} +\right]}, + {\left[ {\left[ {1.0}, {0.0} +\right]}, + {\left[ {1.0}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {1.0}, {0.5} +\right]}, + {\left[ {0.5}, {0.0} +\right]} +\right]}, + {\left[ {\left[ {0.5}, {0.0} +\right]}, + {\left[ {0.0}, {0.5} +\right]} +\right]}, + {\left[ {\left[ {0.0}, {0.5} +\right]}, + {\left[ {0.5}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {0.5}, {1.0} +\right]}, + {\left[ {1.0}, {0.5} +\right]} +\right]}, + {\left[ {\left[ {0.25}, {0.25} +\right]}, + {\left[ {0.25}, {0.75} +\right]} +\right]}, + {\left[ {\left[ {0.25}, {0.75} +\right]}, + {\left[ {0.75}, {0.75} +\right]} +\right]}, + {\left[ {\left[ {0.75}, {0.75} +\right]}, + {\left[ {0.75}, {0.25} +\right]} +\right]}, + {\left[ {\left[ {0.75}, {0.25} +\right]}, + {\left[ {0.25}, {0.25} +\right]} +\right]} +\right] +$$ +\returnType{Type: List List Point DoubleFloat} + +Now we set the point sizes for all components of the graph. + +\spadcommand{size1 := 6::PositiveInteger } +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{size2 := 8::PositiveInteger } +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{size3 := 10::PositiveInteger } + +\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] } +$$ +\left[ +6, 6, 6, 6, 8, 8, 8, 8, size3, size3, size3, +size3 +\right] +$$ +\returnType{Type: List Polynomial Integer} + +Here are the colors for the points. + +\spadcommand{pc1 := pastel red() } +$$ +[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{pc2 := dim green() } +$$ +[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Dim \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{pc3 := pastel yellow() } +$$ +[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] } +$$ +\left[ +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } +{1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm +Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm +palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] +from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} +\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm palette} }, + {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the +} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } +{1.0}} \mbox{\rm ] from the } Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: +} {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm +palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm +] from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} +\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} +}, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from +the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm +Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} } +\right] +$$ +\returnType{Type: List Palette} + +Here are the colors for the lines. + +\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] } +$$ +\left[ +{[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } +{1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, {[{ \mbox{\rm +Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm +palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] +from the } Bright \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} +\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, + {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the +} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } +{1.0}} \mbox{\rm ] from the } Bright \mbox{\rm palette} }, {[{ \mbox{\rm +Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dark \mbox{\rm +palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] +from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {22} +\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, + {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the +} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } +{1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} } +\right] +$$ +\returnType{Type: List Palette} + +Now the {\tt GraphImage} is created according to the component +specifications indicated above. + +\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE } + +The \spadfunFrom{makeViewport2D}{TwoDimensionalViewport} function now +creates a {\tt TwoDimensionalViewport} for this graph according to the +list of options specified within the brackets. + +\spadgraph{makeViewport2D(g,[title("Lines")])\$VIEW2D } + +%See Figure #.#. + +This example demonstrates the use of the {\tt GraphImage} functions +\spadfunFrom{component}{GraphImage} and \spadfunFrom{appendPoint}{GraphImage} +in adding points to an empty {\tt GraphImage}. + +\spadcommand{)clear all } + +\spadcommand{g := graphImage()\$GRIMAGE } +$$ +\mbox{\rm Graph with } 0 \mbox{\rm point lists} +$$ +\returnType{Type: GraphImage} + +\spadcommand{p1 := point [0,0]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p2 := point [.25,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p3 := point [.5,.5]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p4 := point [.75,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p5 := point [1,1]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{component(g,p1)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{component(g,p2)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p3)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p4)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p5)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{g1 := makeGraphImage(g)\$GRIMAGE } + +Here is the graph. + +\spadgraph{makeViewport2D(g1,[title("Graph Points")])\$VIEW2D } + +% +%See Figure #.#. +% + +A list of points can also be made into a {\tt GraphImage} by using +the operation \spadfunFrom{coerce}{GraphImage}. It is equivalent to adding +each point to $g2$ using \spadfunFrom{component}{GraphImage}. + +\spadcommand{g2 := coerce([ [p1],[p2],[p3],[p4],[p5] ])\$GRIMAGE } + +Now, create an empty {\tt TwoDimensionalViewport}. + +\spadcommand{v := viewport2D()\$VIEW2D } + +\spadcommand{options(v,[title("Just Points")])\$VIEW2D } + +Place the graph into the viewport. + +\spadcommand{putGraph(v,g2,1)\$VIEW2D } + +Take a look. + +\spadgraph{makeViewport2D(v)\$VIEW2D } + +%See Figure #.#. + +\subsubsection{Creating a Two-Dimensional Viewport of a List of Points from a File} + +The following three functions read a list of points from a +file and then draw the points and the connecting lines. The +points are stored in the file in readable form as floating point numbers +(specifically, {\tt DoubleFloat} values) as an alternating +stream of $x$- and $y$-values. For example, +\begin{verbatim} +0.0 0.0 1.0 1.0 2.0 4.0 +3.0 9.0 4.0 16.0 5.0 25.0 +\end{verbatim} + +\begin{verbatim} +drawPoints(lp:List Point DoubleFloat):VIEW2D == + g := graphImage()$GRIMAGE + for p in lp repeat + component(g,p,pointColorDefault(),lineColorDefault(), + pointSizeDefault()) + gi := makeGraphImage(g)$GRIMAGE + makeViewport2D(gi,[title("Points")])$VIEW2D + +drawLines(lp:List Point DoubleFloat):VIEW2D == + g := graphImage()$GRIMAGE + component(g, lp, pointColorDefault(), lineColorDefault(), + pointSizeDefault())$GRIMAGE + gi := makeGraphImage(g)$GRIMAGE + makeViewport2D(gi,[title("Points")])$VIEW2D + +plotData2D(name, title) == + f:File(DFLOAT) := open(name,"input") + lp:LIST(Point DFLOAT) := empty() + while ((x := readIfCan!(f)) case DFLOAT) repeat + y : DFLOAT := read!(f) + lp := cons(point [x,y]$(Point DFLOAT), lp) + lp + close!(f) + drawPoints(lp) + drawLines(lp) +\end{verbatim} +% +This command will actually create the viewport and the graph if +the point data is in the file $"file.data"$. +\begin{verbatim} +plotData2D("file.data", "2D Data Plot") +\end{verbatim} + +\subsection{Addendum: Appending a Graph to a Viewport Window Containing a Graph} +\label{ugGraphTwoDappend} + +This section demonstrates how to append a two-di\-men\-sion\-al graph to a viewport +already containing other graphs. +The default {\bf draw} command places a graph into the first +{\tt GraphImage} slot position of the {\tt TwoDimensionalViewport}. + +This graph is in the first slot in its viewport. + +\spadcommand{v1 := draw(sin(x),x=0..2*\%pi) } + +So is this graph. + +\spadcommand{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red()) } + +The operation \spadfunFrom{getGraph}{TwoDimensionalViewport} +retrieves the {\tt GraphImage} $g1$ from the first slot position +in the viewport $v1$. + +\spadcommand{g1 := getGraph(v1,1) } + +Now \spadfunFrom{putGraph}{TwoDimensionalViewport} +places $g1$ into the the second slot position of $v2$. + +\spadcommand{putGraph(v2,g1,2) } + +Display the new {\tt TwoDimensionalViewport} containing both graphs. + +\spadgraph{makeViewport2D(v2) } + +% +%See Figure #.#. +% + +\section{Three-Dimensional Graphics} +\label{ugGraphThreeD} + +% +The Axiom three-di\-men\-sion\-al graphics package provides the ability to +\index{graphics!three-dimensional} +% +\begin{itemize} +% +\item generate surfaces defined by a function of two real variables +% +\item generate space curves and tubes defined by parametric equations +% +\item generate surfaces defined by parametric equations +\end{itemize} +These graphs can be modified by using various options, such as calculating +points in the spherical coordinate system or changing the polygon grid size +of a surface. + +\subsection{Plotting Three-Dimensional Functions of Two Variables} +\label{ugGraphThreeDPlot} + +\index{surface!two variable function} +The simplest three-di\-men\-sion\-al graph is that of a surface defined by a function +of two variables, $z = f(x,y)$. + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a surface defined by a formula $f(x,y)$ +of two variables $x$ and $y$ is: +% +\begin{center} +{\tt draw(f(x,y), x = a..b, y = c..d, {\it options})} +\end{center} +where $a..b$ and $c..d$ define the range of $x$ +and $y$, and where {\it options} prescribes zero or more +options as described in \ref{ugGraphThreeDOptions} +on page~\pageref{ugGraphThreeDOptions}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving a function $f$ is also +available.\\ +} + +The simplest way to plot a function of two variables is to use a formula. +With formulas you always precede the range specifications with +the variable name and an {\tt =} sign. + +\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3d2vara.ps} + +If you intend to use a function more than once, +or it is long and complex, then first +give its definition to Axiom. + +\spadcommand{f(x,y) == sin(x)*cos(y) } +\returnType{Type: Void} + +To draw the function, just give its name and drop the variables +from the range specifications. +Axiom compiles your function for efficient computation +of data for the graph. +Notice that Axiom uses the text of your function as a +default title. + +\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3d2varb.ps} + +\subsection{Plotting Three-Dimensional Parametric Space Curves} +\label{ugGraphThreeDParm} + + +A second kind of three-di\-men\-sion\-al graph is a three-di\-men\-sion\-al space curve +\index{curve!parametric space} +defined by the parametric equations for $x(t)$, $y(t)$, +\index{parametric space curve} +and $z(t)$ as a function of an independent variable $t$. + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a three-di\-men\-sion\-al space curve defined by +parametric formulas $x = f(t)$, $y = g(t)$, and +$z = h(t)$ is: +% +\begin{center} +{\tt draw(curve(f(t),g(t),h(t)), t = a..b, {\it options})} +\end{center} +where $a..b$ defines the range of the independent variable +$t$, and where {\it options} prescribes zero or more options +as described in \ref{ugGraphThreeDOptions} +on page~\pageref{ugGraphThreeDOptions}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving functions $f$, $g$ and +$h$ is also available.\\ +} + +If you use explicit formulas to draw a space curve, always precede +the range specification with the variable name and an +{\tt =} sign. + +\spadgraph{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dpsca.ps} + +Alternatively, you can draw space curves by referring to functions. + +\spadcommand{i1(t:DFLOAT):DFLOAT == sin(t)*cos(3*t/5) } +\begin{verbatim} + Function declaration i1 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +This is useful if the functions are to be used more than once \ldots + +\spadcommand{i2(t:DFLOAT):DFLOAT == cos(t)*cos(3*t/5) } +\begin{verbatim} + Function declaration i2 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +or if the functions are long and complex. + +\spadcommand{i3(t:DFLOAT):DFLOAT == cos(t)*sin(3*t/5) } +\begin{verbatim} + Function declaration i3 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +Give the names of the functions and +drop the variable name specification in the second argument. +Again, Axiom supplies a default title. + +\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dpscb.ps} + +\subsection{Plotting Three-Dimensional Parametric Surfaces} +\label{ugGraphThreeDPar} + +\index{surface!parametric} +A third kind of three-di\-men\-sion\-al graph is a surface defined by +\index{parametric surface} +parametric equations for $x(u,v)$, $y(u,v)$, and +$z(u,v)$ of two independent variables $u$ and $v$. + + +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a three-di\-men\-sion\-al graph defined by +parametric formulas $x = f(u,v)$, $y = g(u,v)$, +and $z = h(u,v)$ is: +% +\begin{center} +{\tt draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d, {\it options})} +\end{center} +where $a..b$ and $c..d$ define the range of the +independent variables $u$ and $v$, and where +{\it options} prescribes zero or more options as described in +\ref{ugGraphThreeDOptions} on page~\pageref{ugGraphThreeDOptions}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving functions $f$, $g$ and +$h$ is also available.\\ +} + +This example draws a graph of a surface plotted using the +parabolic cylindrical coordinate system option. +\index{coordinate system!parabolic cylindrical} +The values of the functions supplied to {\bf surface} are +\index{parabolic cylindrical coordinate system} +interpreted in coordinates as given by a {\tt coordinates} option, +here as parabolic cylindrical coordinates (see +\ref{ugGraphCoord} on page~\pageref{ugGraphCoord}). + +\spadgraph{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dpsa.ps} + +Again, you can graph these parametric surfaces using functions, +if the functions are long and complex. + +Here we declare the types of arguments and values to be of type +{\tt DoubleFloat}. + +\spadcommand{n1(u:DFLOAT,v:DFLOAT):DFLOAT == u*cos(v) } +\begin{verbatim} + Function declaration n1 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +As shown by previous examples, these declarations are necessary. + +\spadcommand{n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v) } +\begin{verbatim} + Function declaration n2 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +In either case, Axiom compiles the functions +when needed to graph a result. + +\spadcommand{n3(u:DFLOAT,v:DFLOAT):DFLOAT == u } +\begin{verbatim} + Function declaration n3 : DoubleFloat -> DoubleFloat has been added + to workspace. +\end{verbatim} +\returnType{Type: Void} + +Without these declarations, you have to suffix floats +with $@DFLOAT$ to get a {\tt DoubleFloat} result. +However, a call here with an unadorned float produces a {\tt DoubleFloat}. + +\spadcommand{n3(0.5,1.0)} +\begin{verbatim} + Compiling function n3 with type (DoubleFloat,DoubleFloat) -> + DoubleFloat +\end{verbatim} +\returnType{Type: DoubleFloat} + +Draw the surface by referencing the function names, this time +choosing the toroidal coordinate system. +\index{coordinate system!toroidal} +\index{toroidal coordinate system} + +\spadgraph{draw(surface(n1,n2,n3), 1..4, 1..2*\%pi, coordinates == toroidal(1\$DFLOAT)) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dpsb.ps} + +\subsection{Three-Dimensional Options} +\label{ugGraphThreeDOptions} + +\index{graphics!3D options} +The {\bf draw} commands optionally take an optional list of options such +as {\tt coordinates} as shown in the last example. +Each option is given by the syntax: $name$ {\tt ==} $value$. +Here is a list of the available options in the order that they are +described below: + +\begin{tabular}{llll} +title&coordinates&var1Steps\\ +style&tubeRadius&var2Steps\\ +colorFunction&tubePoints&space\\ +\end{tabular} + +The option $title$ gives your graph a title. +\index{graphics!3D options!title} + +\spadgraph{draw(cos(x*y),x=0..2*\%pi,y=0..\%pi,title == "Title of Graph") } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptttl.ps} + +The $style$ determines which of four rendering algorithms is used for +\index{rendering} +the graph. +The choices are +{\tt "wireMesh"}, {\tt "solid"}, {\tt "shade"}, and {\tt "smooth"}. + +\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="smooth", title=="Smooth Option")} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptsty.ps} + +In all but the wire-mesh style, polygons in a surface or tube plot +are normally colored in a graph according to their +$z$-coordinate value. Space curves are colored according to their +parametric variable value. +\index{graphics!3D options!color function} +To change this, you can give a coloring function. +\index{function!coloring} +The coloring function is sampled across the range of its arguments, then +normalized onto the standard Axiom colormap. + +A function of one variable makes the color depend on the +value of the parametric variable specified for a tube plot. + +\spadcommand{color1(t) == t } +\returnType{Type: Void} + +\spadgraph{draw(curve(sin(t), cos(t),0), t=0..2*\%pi, tubeRadius == .3, colorFunction == color1) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptcf1.ps} + +A function of two variables makes the color depend on the +values of the independent variables. + +\spadcommand{color2(u,v) == u**2 - v**2 } +\returnType{Type: Void} + +Use the option {\tt colorFunction} for special coloring. + +\spadgraph{draw(cos(u*v), u=-3..3, v=-3..3, colorFunction == color2) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptcf2.ps} + +With a three variable function, the +color also depends on the value of the function. + +\spadcommand{color3(x,y,fxy) == sin(x*fxy) + cos(y*fxy) } +\returnType{Type: Void} + +\spadgraph{draw(cos(x*y), x=-3..3, y=-3..3, colorFunction == color3) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptcf3.ps} + +Normally the Cartesian coordinate system is used. +\index{Cartesian!coordinate system} +To change this, use the {\tt coordinates} option. +\index{coordinate system!Cartesian} +For details, see \ref{ugGraphCoord} on page~\pageref{ugGraphCoord}. + +\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } +\begin{verbatim} + Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Use the spherical +\index{spherical coordinate system} +coordinate system. +\index{coordinate system!spherical} + +\spadgraph{draw(m, 0..2*\%pi,0..\%pi, coordinates == spherical, style=="shade") } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptcrd.ps} + +Space curves may be displayed as tubes with polygonal cross sections. +\index{tube} +Two options, {\tt tubeRadius} and {\tt tubePoints}, control the size and +shape of this cross section. +% + +The {\tt tubeRadius} option specifies the radius of the tube that +\index{tube!radius} +encircles the specified space curve. + +\spadgraph{draw(curve(sin(t),cos(t),0),t=0..2*\%pi, style=="shade", tubeRadius == .3)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptrad.ps} + +The {\tt tubePoints} option specifies the number of vertices +\index{tube!points in polygon} +defining the polygon that is used to create a tube around the +specified space curve. +The larger this number is, the more cylindrical the tube becomes. + +\spadgraph{draw(curve(sin(t), cos(t), 0), t=0..2*\%pi, style=="shade", tubeRadius == .25, tubePoints == 3)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptpts.ps} + +\index{graphics!3D options!variable steps} +% + +Options \spadfunFrom{var1Steps}{DrawOption} and +\spadfunFrom{var2Steps}{DrawOption} specify the number of intervals into +which the grid defining a surface plot is subdivided with respect to the +first and second parameters of the surface function(s). + +\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="shade", var1Steps == 30, var2Steps == 30)} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3doptvb.ps} + +The {\tt space} option +of a {\bf draw} command lets you build multiple graphs in three space. +To use this option, first create an empty three-space object, +then use the {\tt space} option thereafter. +There is no restriction as to the number or kinds +of graphs that can be combined this way. + +Create an empty three-space object. + +\spadcommand{s := create3Space()\$(ThreeSpace DFLOAT) } +$$ +{3-Space with }0 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } +\begin{verbatim} + Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Add a graph to this three-space object. +The new graph destructively inserts the graph +into $s$. + +\spadgraph{draw(m,0..\%pi,0..2*\%pi, coordinates == spherical, space == s) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dmult1a.ps} + +Add a second graph to $s$. + +\spadgraph{v := draw(curve(1.5*sin(t), 1.5*cos(t),0), t=0..2*\%pi, tubeRadius == .25, space == s) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dmult1b.ps} + +A three-space object can also be obtained from an existing three-di\-men\-sion\-al viewport +using the \spadfunFrom{subspace}{ThreeSpace} command. +You can then use {\bf makeViewport3D} to create a viewport window. + +Assign to $subsp$ the three-space object in viewport $v$. + +\spadcommand{subsp := subspace v } + +Reset the space component of $v$ to the value of $subsp$. + +\spadcommand{subspace(v, subsp) } + +Create a viewport window from a three-space object. + +\spadgraph{makeViewport3D(subsp,"Graphs") } + +\subsection{The makeObject Command} +\label{ugGraphMakeObject} + +An alternate way to create multiple graphs is to use +{\bf makeObject}. +The {\bf makeObject} command is similar to the {\bf draw} +command, except that it returns a three-space object rather than a +{\tt ThreeDimensionalViewport}. +In fact, {\bf makeObject} is called by the {\bf draw} +command to create the {\tt ThreeSpace} then +\spadfunFrom{makeViewport3D}{ThreeDimensionalViewport} to create a +viewport window. + +\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } +\begin{verbatim} + Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Do the last example a new way. +First use {\bf makeObject} to +create a three-space object $sph$. + +\spadcommand{sph := makeObject(m, 0..\%pi, 0..2*\%pi, coordinates==spherical)} +\begin{verbatim} + Compiling function m with type (DoubleFloat,DoubleFloat) -> + DoubleFloat +\end{verbatim} +$$ +{3-Space with }1 \mbox{\rm component} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +Add a second object to $sph$. + +\spadcommand{makeObject(curve(1.5*sin(t), 1.5*cos(t), 0), t=0..2*\%pi, space == sph, tubeRadius == .25) } +\begin{verbatim} + Compiling function %D with type DoubleFloat -> DoubleFloat + Compiling function %F with type DoubleFloat -> DoubleFloat + Compiling function %H with type DoubleFloat -> DoubleFloat +\end{verbatim} +$$ +{3-Space with }2 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +Create and display a viewport +containing $sph$. + +\spadgraph{makeViewport3D(sph,"Multiple Objects") } + +Note that an undefined {\tt ThreeSpace} parameter declared in a +{\bf makeObject} or {\bf draw} command results in an error. +Use the \spadfunFrom{create3Space}{ThreeSpace} function to define a +{\tt ThreeSpace}, or obtain a {\tt ThreeSpace} that has been +previously generated before including it in a command line. + +\subsection{Building Three-Dimensional Objects From Primitives} +\label{ugGraphThreeDBuild} + +Rather than using the {\bf draw} and {\bf makeObject} commands, +\index{graphics!advanced!build 3D objects} +you can create three-di\-men\-sion\-al graphs from primitives. +Operation \spadfunFrom{create3Space}{ThreeSpace} creates a +three-space object to which points, curves and polygons +can be added using the operations from the {\tt ThreeSpace} +domain. +The resulting object can then be displayed in a viewport using +\spadfunFrom{makeViewport3D}{ThreeDimensionalViewport}. + +Create the empty three-space object $space$. + +\spadcommand{space := create3Space()\$(ThreeSpace DFLOAT) } +$$ +{3-Space with }0 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +Objects can be sent to this $space$ using the operations +exported by the {\tt ThreeSpace} domain. +\index{ThreeSpace} +The following examples place curves into $space$. + +Add these eight curves to the space. + +\spadcommand{closedCurve(space,[ [0,30,20], [0,30,30], [0,40,30], [0,40,100], [0,30,100],[0,30,110], [0,60,110], [0,60,100], [0,50,100], [0,50,30], [0,60,30], [0,60,20] ]) } +$$ +{3-Space with }1 \mbox{\rm component} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35] ]) } +$$ +{3-Space with }2 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20], [80,0,30] ]) } +$$ +{3-Space with }3 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [0,70,20], [0,70,110], [0,110,110], [0,120,100], [0,120,70], [0,115,65], [0,120,60], [0,120,30], [0,110,20], [0,80,20], [0,80,30], [0,80,20] ]) } +$$ +{3-Space with }4 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [0,105,30], [0,110,35], [0,110,55], [0,105,60], [0,80,60], [0,80,70], [0,105,70], [0,110,75], [0,110,95], [0,105,100], [0,80,100], [0,80,20], [0,80,30] ]) } +$$ +{3-Space with }5 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [140,0,20], [140,0,110], [130,0,110], [90,0,20], [101,0,20],[114,0,50], [130,0,50], [130,0,60], [119,0,60], [130,0,85], [130,0,20] ]) } +$$ +{3-Space with }6 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [0,140,20], [0,140,110], [0,150,110], [0,170,50], [0,190,110], [0,200,110], [0,200,20], [0,190,20], [0,190,75], [0,175,35], [0,165,35],[0,150,75], [0,150,20] ]) } +$$ +{3-Space with }7 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{closedCurve(space,[ [200,0,20], [200,0,110], [189,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [161,0,20], [190,0,85], [190,0,20] ]) } +$$ +{3-Space with }8 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +Create and display the viewport using {\bf makeViewport3D}. +Options may also be given but here are displayed as a list with values +enclosed in parentheses. + +\spadgraph{makeViewport3D(space, title == "Letters") } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dbuilda.ps} + +\subsubsection{Cube Example} + +As a second example of the use of primitives, we generate a cube using a +polygon mesh. +It is important to use a consistent orientation of the polygons for +correct generation of three-di\-men\-sion\-al objects. + +Again start with an empty three-space object. + +\spadcommand{spaceC := create3Space()\$(ThreeSpace DFLOAT) } +$$ +{3-Space with }0 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +For convenience, +give {\tt DoubleFloat} values $+1$ and $-1$ names. + +\spadcommand{x: DFLOAT := 1 } +$$ +1.0 +$$ +\returnType{Type: DoubleFloat} + +\spadcommand{y: DFLOAT := -1 } +$$ +-{1.0} +$$ +\returnType{Type: DoubleFloat} + +Define the vertices of the cube. + +\spadcommand{a := point [x,x,y,1::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0}, -{1.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{b := point [y,x,y,4::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +-{1.0}, {1.0}, -{1.0}, {4.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{c := point [y,x,x,8::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +-{1.0}, {1.0}, {1.0}, {8.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{d := point [x,x,x,12::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0}, {1.0}, {12.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{e := point [x,y,y,16::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, -{1.0}, -{1.0}, {16.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{f := point [y,y,y,20::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +-{1.0}, -{1.0}, -{1.0}, {20.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{g := point [y,y,x,24::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +-{1.0}, -{1.0}, {1.0}, {24.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{h := point [x,y,x,27::DFLOAT]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, -{1.0}, {1.0}, {27.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +Add the faces of the cube as polygons to the space using a +consistent orientation. + +\spadcommand{polygon(spaceC,[d,c,g,h]) } +$$ +{3-Space with }1 \mbox{\rm component} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{polygon(spaceC,[d,h,e,a]) } +$$ +{3-Space with }2 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{polygon(spaceC,[c,d,a,b]) } +$$ +{3-Space with }3 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{polygon(spaceC,[g,c,b,f]) } +$$ +{3-Space with }4 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{polygon(spaceC,[h,g,f,e]) } +$$ +{3-Space with }5 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +\spadcommand{polygon(spaceC,[e,f,b,a]) } +$$ +{3-Space with }6 \mbox{\rm components} +$$ +\returnType{Type: ThreeSpace DoubleFloat} + +Create and display the viewport. + +\spadgraph{makeViewport3D(spaceC, title == "Cube") } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/3dbuildb.ps} + +\subsection{Coordinate System Transformations} +\label{ugGraphCoord} +\index{graphics!advanced!coordinate systems} + +The {\tt CoordinateSystems} package provides coordinate transformation +functions that map a given data point from the coordinate system specified +into the Cartesian coordinate system. +\index{CoordinateSystems} +The default coordinate system, given a triplet $(f(u,v), u, v)$, assumes +that $z = f(u, v)$, $x = u$ and $y = v$, +that is, reads the coordinates in $(z, x, y)$ order. + +\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == u**2 } +\begin{verbatim} + Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Graph plotted in default coordinate system. + +\spadgraph{draw(m,0..3,0..5) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/defcoord.ps} + +The $z$ coordinate comes first since the first argument of +the {\bf draw} command gives its values. +In general, the coordinate systems Axiom provides, or any +that you make up, must provide a map to an $(x, y, z)$ triplet in +order to be compatible with the +\spadfunFrom{coordinates}{DrawOption} {\tt DrawOption}. +\index{DrawOption} +Here is an example. + +Define the identity function. + +\spadcommand{cartesian(point:Point DFLOAT):Point DFLOAT == point } +\begin{verbatim} + Function declaration cartesian : Point DoubleFloat -> Point + DoubleFloat has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Pass $cartesian$ as the \spadfunFrom{coordinates}{DrawOption} +parameter to the {\bf draw} command. + +\spadgraph{draw(m,0..3,0..5,coordinates==cartesian) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/cartcoord.ps} + +What happened? The option {\tt coordinates == cartesian} directs +Axiom to treat the dependent variable $m$ defined by $m=u^2$ as the +$x$ coordinate. Thus the triplet of values $(m, u, v)$ is transformed +to coordinates $(x, y, z)$ and so we get the graph of $x=y^2$. + +Here is another example. +The \spadfunFrom{cylindrical}{CoordinateSystems} transform takes +\index{coordinate system!cylindrical} +input of the form $(w,u,v)$, interprets it in the order +\index{cylindrical coordinate system} +($r$,$\theta$,$z$) +and maps it to the Cartesian coordinates +$x=r\cos(\theta)$, $y=r\sin(\theta)$, $z=z$ +in which +$r$ is the radius, +$\theta$ is the angle and +$z$ is the z-coordinate. + +An example using the \spadfunFrom{cylindrical}{CoordinateSystems} +coordinates for the constant $r = 3$. + +\spadcommand{f(u:DFLOAT,v:DFLOAT):DFLOAT == 3 } +\begin{verbatim} + Function declaration f : (DoubleFloat,DoubleFloat) -> DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +Graph plotted in cylindrical coordinates. + +\spadgraph{draw(f,0..\%pi,0..6,coordinates==cylindrical) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/cylcoord.ps} + +Suppose you would like to specify $z$ as a function of +$r$ and $\theta$ instead of just $r$? +Well, you still can use the {\bf cylindrical} Axiom +transformation but we have to reorder the triplet before +passing it to the transformation. + +First, let's create a point to +work with and call it $pt$ with some color $col$. + +\spadcommand{col := 5 } +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{pt := point[1,2,3,col]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {2.0}, {3.0}, {5.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +The reordering you want is +$(z,r, \theta)$ to +$(r, \theta,z)$ +so that the first element is moved to the third element, while the second +and third elements move forward and the color element does not change. + +Define a function {\bf reorder} to reorder the point elements. + +\spadcommand{reorder(p:Point DFLOAT):Point DFLOAT == point[p.2, p.3, p.1, p.4] } +\begin{verbatim} + Function declaration reorder : Point DoubleFloat -> Point + DoubleFloat has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +The function moves the second and third elements +forward but the color does not change. + +\spadcommand{reorder pt } +$$ +\left[ +{2.0}, {3.0}, {1.0}, {5.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +The function {\bf newmap} converts our reordered version of +the cylindrical coordinate system to the standard +$(x,y,z)$ Cartesian system. + +\spadcommand{newmap(pt:Point DFLOAT):Point DFLOAT == cylindrical(reorder pt) } +\begin{verbatim} + Function declaration newmap : Point DoubleFloat -> Point DoubleFloat + has been added to workspace. +\end{verbatim} +\returnType{Type: Void} + +\spadcommand{newmap pt } +$$ +\left[ +-{1.9799849932008908}, {0.28224001611973443}, {1.0}, {5.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +Graph the same function $f$ using the coordinate mapping of the function +$newmap$, so it is now interpreted as +$z=3$: + +\spadgraph{draw(f,0..3,0..2*\%pi,coordinates==newmap) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/newmap.ps} + +% I think this is good to say here: it shows a lot of depth. RSS +%{\sloppy +The {\tt CoordinateSystems} package exports the following +\index{coordinate system} +operations: +{\bf bipolar}, +{\bf bipolarCylindrical}, +{\bf cartesian}, +{\bf conical}, +{\bf cylindrical}, +{\bf elliptic}, +{\bf ellipticCylindrical}, +{\bf oblateSpheroidal}, +{\bf parabolic}, +{\bf parabolicCylindrical}, +{\bf paraboloidal}, +{\bf polar}, +{\bf prolateSpheroidal}, +{\bf spherical}, and +{\bf toroidal}. +Use Browse or the {\tt )show} system command +\index{show} +to get more information. + +\subsection{Three-Dimensional Clipping} +\label{ugGraphClip} + +A three-di\-men\-sion\-al graph can be explicitly clipped within the {\bf draw} +\index{graphics!advanced!clip} +command by indicating a minimum and maximum threshold for the +\index{clipping} +given function definition. +These thresholds can be defined using the Axiom {\bf min} +and {\bf max} functions. + +\begin{verbatim} +gamma(x,y) == + g := Gamma complex(x,y) + point [x, y, max( min(real g, 4), -4), argument g] +\end{verbatim} + +Here is an example that clips +the gamma function in order to eliminate the extreme divergence it creates. + +\spadgraph{draw(gamma,-\%pi..\%pi,-\%pi..\%pi,var1Steps==50,var2Steps==50) } + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/clipgamma.ps} + +\subsection{Three-Dimensional Control-Panel} +\label{ugGraphThreeDControl} + +\index{graphics!3D control-panel} +Once you have created a viewport, move your mouse to the viewport +and click with your left mouse button. +This displays a control-panel on the side of the viewport +that is closest to where you clicked. + +\begin{figure}[htbp] +%{\epsfverbosetrue\epsfxsize=2in% +%\def\epsfsize#1#2{\epsfxsize}\hspace*{\baseLeftSkip}% +%%\epsffile[0 0 144 289]{ps/3dctrl.ps}} +\begin{picture}(183,252)%(-125,0) +\hspace*{\baseLeftSkip}\special{psfile=ps/3dctrl.ps} +\end{picture} +\caption{Three-dimensional control-panel.} +\end{figure} + +\subsubsection{Transformations} + +We recommend you first select the {\bf Bounds} button while +\index{graphics!3D control-panel!transformations} +executing transformations since the bounding box displayed +indicates the object's position as it changes. +% +\begin{description} +% +\item[Rotate:] A rotation transformation occurs by clicking the mouse +\index{graphics!3D control-panel!rotate} +within the {\bf Rotate} window in the upper left corner of the +control-panel. +The rotation is computed in spherical coordinates, using the +horizontal mouse position to increment or decrement the value of +the longitudinal angle $\theta$ within the +range of 0 to 2$\pi$ and the vertical mouse position +to increment or decrement the value of the latitudinal angle +$\phi$ within the range of -$\pi$ +to $\pi$. +The active mode of rotation is displayed in green on a color +monitor or in clear text on a black and white monitor, while the +inactive mode is displayed in red for color display or a mottled +pattern for black and white. +% +\begin{description} +% +\item[origin:] The {\bf origin} button indicates that the +rotation is to occur with respect to the origin of the viewing space, that is +indicated by the axes. +% +\item[object:] The {\bf object} button indicates that the +rotation is to occur with respect to the center of volume of the object, +independent of the axes' origin position. +\end{description} +% +\item[Scale:] A scaling transformation occurs by clicking the mouse +\index{graphics!3D control-panel!scale} +within the {\bf Scale} window in the upper center of the +control-panel, containing a zoom arrow. +The axes along which the scaling is to occur are indicated by +selecting the appropriate button above the zoom arrow window. +The selected axes are displayed in green on a color monitor or in +clear text on a black and white monitor, while the unselected axes +are displayed in red for a color display or a mottled pattern for +black and white. +% +\begin{description} +% +\item[uniform:] Uniform scaling along the {\tt x}, {\tt y} +and {\tt z} axes occurs when all the axes buttons are selected. +% +\item[non-uniform:] If any of the axes buttons are +not selected, non-uniform scaling occurs, that is, scaling occurs only in the +direction of the axes that are selected. +\end{description} +% +\item[Translate:] Translation occurs by indicating with the mouse in the +\index{graphics!3D control-panel!translate} +{\bf Translate} window the direction you want the graph to move. +This window is located in the upper right corner of the +control-panel and contains a potentiometer with crossed arrows +pointing up, down, left and right. +Along the top of the {\bf Translate} window are three buttons +({\bf XY}, +{\bf XZ}, and {\bf YZ}) indicating the three orthographic projection planes. +Each orientates the group as a view into that plane. +Any translation of the graph occurs only along this plane. +\end{description} + +\subsubsection{Messages} + +\index{graphics!3D control-panel!messages} + +The window directly below the potentiometer windows for transformations is +used to display system messages relating to the viewport, the control-panel +and the current graph displaying status. + +\subsubsection{Colormap} + +\index{graphics!3D control-panel!color map} + +Directly below the message window is the colormap range indicator +window. +\index{colormap} +The Axiom Colormap shows a sampling of the spectrum from +which hues can be drawn to represent the colors of a surface. +The Colormap is composed of five shades for each of the hues along +this spectrum. +By moving the markers above and below the Colormap, the range of +hues that are used to color the existing surface are set. +The bottom marker shows the hue for the low end of the color range +and the top marker shows the hue for the upper end of the range. +Setting the bottom and top markers at the same hue results in +monochromatic smooth shading of the graph when {\bf Smooth} mode is selected. +At each end of the Colormap are {\bf +} and {\bf -} buttons. +When clicked on, these increment or decrement the top or bottom +marker. + +\subsubsection{Buttons} +\index{graphics!3D control-panel!buttons} + +Below the Colormap window and to the left are located various +buttons that determine the characteristics of a graph. +The buttons along the bottom and right hand side all have special +meanings; the remaining buttons in the first row indicate the mode +or style used to display the graph. +The second row are toggles that turn on or off a property of the +graph. +On a color monitor, the property is on if green (clear text, on a +monochrome monitor) and off if red (mottled pattern, on a +monochrome monitor). +Here is a list of their functions. +% +\begin{description} +% +\item[Wire] displays surface and tube plots as a +\index{graphics!3D control-panel!wire} +wireframe image in a single color (blue) with no hidden surfaces removed, +or displays space curve plots in colors based upon their parametric variables. +This is the fastest mode for displaying a graph. +This is very useful when you +want to find a good orientation of your graph. +% +\item[Solid] displays the graph with hidden +\index{graphics!3D control-panel!solid} +surfaces removed, drawing each polygon beginning with the furthest +from the viewer. +The edges of the polygons are displayed in the hues specified by +the range in the Colormap window. +% +\item[Shade] displays the graph with hidden +\index{graphics!3D control-panel!shade} +surfaces removed and with the polygons shaded, drawing each +polygon beginning with the furthest from the viewer. +Polygons are shaded in the hues specified by the range in the +Colormap window using the Phong illumination model. +\index{Phong!illumination model} +% +\item[Smooth] displays the graph using a +\index{graphics!3D control-panel!smooth} +renderer that computes the graph one line at a time. +The location and color of the graph at each visible point on the +screen are determined and displayed using the Phong illumination +\index{Phong!illumination model} +model. +Smooth shading is done in one of two ways, depending on the range +selected in the colormap window and the number of colors available +from the hardware and/or window manager. +When the top and bottom markers of the colormap range are set to +different hues, the graph is rendered by dithering between the +\index{dithering} +transitions in color hue. +When the top and bottom markers of the colormap range are set to +the same hue, the graph is rendered using the Phong smooth shading +model. +\index{Phong!smooth shading model} +However, if enough colors cannot be allocated for this purpose, +the renderer reverts to the color dithering method until a +sufficient color supply is available. +For this reason, it may not be possible to render multiple Phong +smooth shaded graphs at the same time on some systems. +% +\item[Bounds] encloses the entire volume of the +viewgraph within a bounding box, or removes the box if previously selected. +\index{graphics!3D control-panel!bounds} +The region that encloses the entire volume of the viewport graph is displayed. +% +\item[Axes] displays Cartesian +\index{graphics!3D control-panel!axes} +coordinate axes of the space, or turns them off if previously selected. +% +\item[Outline] causes +\index{graphics!3D control-panel!outline} +quadrilateral polygons forming the graph surface to be outlined in black when +the graph is displayed in {\bf Shade} mode. +% +\item[BW] converts a color viewport to black and white, or vice-versa. +\index{graphics!3D control-panel!bw} +When this button is selected the +control-panel and viewport switch to an immutable colormap composed of a range +of grey scale patterns or tiles that are used wherever shading is necessary. +% +\item[Light] takes you to a control-panel described below. +% +\item[ViewVolume] takes you to another control-panel as described below. +\index{graphics!3D control-panel!save} +% +\item[Save] creates a menu of the possible file types that can +be written using the control-panel. +The {\bf Exit} button leaves the save menu. +The {\bf Pixmap} button writes an Axiom pixmap of +\index{graphics!3D control-panel!pixmap} +the current viewport contents. The file is called {\bf axiom3D.pixmap} and is +located in the directory from which Axiom or {\bf viewalone} was +started. +The {\bf PS} button writes the current viewport contents to +\index{graphics!3D control-panel!ps} +PostScript output rather than to the viewport window. +By default the file is called {\bf axiom3D.ps}; however, if a file +\index{file!.Xdefaults @{\bf .Xdefaults}} +name is specified in the user's {\bf .Xdefaults} file it is +\index{graphics!.Xdefaults!PostScript file name} +used. +The file is placed in the directory from which the Axiom or +{\bf viewalone} session was begun. +See also the \spadfunFrom{write}{ThreeDimensionalViewport} +function. +\index{PostScript} +% +\item[Reset] returns the object transformation +\index{graphics!3D control-panel!reset} +characteristics back to their initial states. +% +\item[Hide] causes the control-panel for the +\index{graphics!3D control-panel!hide} +corresponding viewport to disappear from the screen. +% +\item[Quit] queries whether the current viewport +\index{graphics!3D control-panel!quit} +session should be terminated. +\end{description} + +\subsubsection{Light} + +\index{graphics!3D control-panel!light} + +%>>>\begin{figure}[htbp] +%>>>\begin{picture}(183,252)(-125,0) +%>>>\special{psfile=ps/3dlight.ps} +%>>>\end{picture} +%>>>\caption{Three-Dimensional Lighting Panel.} +%>>>\end{figure} + +The {\bf Light} button changes the control-panel into the +{\bf Lighting Control-Panel}. At the top of this panel, the three axes +are shown with the same orientation as the object. A light vector from +the origin of the axes shows the current position of the light source +relative to the object. At the bottom of the panel is an {\bf Abort} +button that cancels any changes to the lighting that were made, and a +{\bf Return} button that carries out the current set of lighting changes +on the graph. +% +\begin{description} +% +\item[XY:] The {\bf XY} lighting axes window is below the +\index{graphics!3D control-panel!move xy} +{\bf Lighting Control-Panel} title and to the left. +This changes the light vector within the {\bf XY} view plane. +% +\item[Z:] The {\bf Z} lighting axis window is below the +\index{graphics!3D control-panel!move z} +{\bf Lighting Control-Panel} title and in the center. This +changes the {\bf Z} +location of the light vector. +% +\item[Intensity:] +Below the {\bf Lighting Control-Panel} title +\index{graphics!3D control-panel!intensity} +and to the right is the light intensity meter. +Moving the intensity indicator down decreases the amount of +light emitted from the light source. +When the indicator is at the top of the meter the light source is +emitting at 100\% intensity. +At the bottom of the meter the light source is emitting at a level +slightly above ambient lighting. +\end{description} + +\subsubsection{View Volume} + +\index{graphics!3D control-panel!view volume} + +The {\bf View Volume} button changes the control-panel into +the {\bf Viewing Volume Panel}. +At the bottom of the viewing panel is an {\bf Abort} button that +cancels any changes to the viewing volume that were made and a +{\it Return} button that carries out the current set of +viewing changes to the graph. +% +%>>>\begin{figure}[htbp] +%>>>\begin{picture}(183,252)(-125,0) +%>>>\special{psfile=ps/3dvolume.ps} +%>>>\end{picture} +%>>>\caption{Three-Dimensional Volume Panel.} +%>>>\end{figure} + +\begin{description} + +\item[Eye Reference:] At the top of this panel is the +\index{graphics!3D control-panel!eye reference} +{\bf Eye Reference} window. +It shows a planar projection of the viewing pyramid from the eye +of the viewer relative to the location of the object. +This has a bounding region represented by the rectangle on the +left. +Below the object rectangle is the {\bf Hither} window. +By moving the slider in this window the hither clipping plane sets +\index{hither clipping plane} +the front of the view volume. +As a result of this depth clipping all points of the object closer +to the eye than this hither plane are not shown. +The {\bf Eye Distance} slider to the right of the {\bf Hither} +slider is used to change the degree of perspective in the image. +% +\item[Clip Volume:] The {\bf Clip Volume} window is at the +\index{graphics!3D control-panel!clip volume} +bottom of the {\bf Viewing Volume Panel}. +On the right is a {\bf Settings} menu. +In this menu are buttons to select viewing attributes. +Selecting the {\bf Perspective} button computes the image using +perspective projection. +\index{graphics!3D control-panel!perspective} +The {\bf Show Region} button indicates whether the clipping region +of the +\index{graphics!3D control-panel!show clip region} +volume is to be drawn in the viewport and the {\bf Clipping On} +button shows whether the view volume clipping is to be in effect +when the image +\index{graphics!3D control-panel!clipping on} +is drawn. +The left side of the {\bf Clip Volume} window shows the clipping +\index{graphics!3D control-panel!clip volume} +boundary of the graph. +Moving the knobs along the {\bf X}, {\bf Y}, and {\bf Z} sliders +adjusts the volume of the clipping region accordingly. +\end{description} + +\subsection{Operations for Three-Dimensional Graphics} +\label{ugGraphThreeDops} + + +Here is a summary of useful Axiom operations for three-di\-men\-sion\-al +graphics. +Each operation name is followed by a list of arguments. +Each argument is written as a variable informally named according +to the type of the argument (for example, {\it integer}). +If appropriate, a default value for an argument is given in +parentheses immediately following the name. + +% +\bgroup\hbadness = 10001\sloppy +\begin{description} +% +\item[{\bf adaptive3D?}]\funArgs{} +tests whether space curves are to be plotted +\index{graphics!plot3d defaults!adaptive} +according to the +\index{adaptive plotting} +adaptive refinement algorithm. + +% +\item[{\bf axes}]\funArgs{viewport, string\argDef{"on"}} +turns the axes on and off. +\index{graphics!3D commands!axes} + +% +\item[{\bf close}]\funArgs{viewport} +closes the viewport. +\index{graphics!3D commands!close} + +% +\item[{\bf colorDef}]\funArgs{viewport, +$\hbox{\it color}_{1}$\argDef{1}, $\hbox{\it color}_{2}$\argDef{27}} +sets the colormap +\index{graphics!3D commands!define color} +range to be from +$\hbox{\it color}_{1}$ to $\hbox{\it color}_{2}$. + +% +\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} +declares whether the +\index{graphics!3D commands!control-panel} +control-panel for the viewport is to be displayed or not. + +% +\item[{\bf diagonals}]\funArgs{viewport, string\argDef{"off"}} +declares whether the +\index{graphics!3D commands!diagonals} +polygon outline includes the diagonals or not. + +% +\item[{\bf drawStyle}]\funArgs{viewport, style} +selects which of four drawing styles +\index{graphics!3D commands!drawing style} +are used: {\tt "wireMesh", "solid", "shade",} or {\tt "smooth".} + +% +\item[{\bf eyeDistance}]\funArgs{viewport,float\argDef{500}} +sets the distance of the eye from the origin of the object +\index{graphics!3D commands!eye distance} +for use in the \spadfunFrom{perspective}{ThreeDimensionalViewport}. + +% +\item[{\bf key}]\funArgs{viewport} +returns the operating +\index{graphics!3D commands!key} +system process ID number for the viewport. + +% +\item[{\bf lighting}]\funArgs{viewport, +$float_{x}$\argDef{-0.5}, +$float_{y}$\argDef{0.5}, $float_{z}$\argDef{0.5}} +sets the Cartesian +\index{graphics!3D commands!lighting} +coordinates of the light source. + +% +\item[{\bf modifyPointData}]\funArgs{viewport,integer,point} +replaces the coordinates of the point with +\index{graphics!3D commands!modify point data} +the index {\it integer} with {\it point}. + +% +\item[{\bf move}]\funArgs{viewport, +$integer_{x}$\argDef{viewPosDefault}, +$integer_{y}$\argDef{viewPosDefault}} +moves the upper +\index{graphics!3D commands!move} +left-hand corner of the viewport to screen position +\allowbreak +({\small $integer_{x}$, $integer_{y}$}). + +% +\item[{\bf options}]\funArgs{viewport} +returns a list of all current draw options. + +% +\item[{\bf outlineRender}]\funArgs{viewport, string\argDef{"off"}} +turns polygon outlining +\index{graphics!3D commands!outline} +off or on when drawing in {\tt "shade"} mode. + +% +\item[{\bf perspective}]\funArgs{viewport, string\argDef{"on"}} +turns perspective +\index{graphics!3D commands!perspective} +viewing on and off. + +% +\item[{\bf reset}]\funArgs{viewport} +resets the attributes of a viewport to their +\index{graphics!3D commands!reset} +initial settings. + +% +\item[{\bf resize}]\funArgs{viewport, +$integer_{width}$ \argDef{viewSizeDefault}, +$integer_{height}$ \argDef{viewSizeDefault}} +resets the width and height +\index{graphics!3D commands!resize} +values for a viewport. + +% +\item[{\bf rotate}]\funArgs{viewport, +$number_{\theta}$\argDef{viewThetaDefapult}, +$number_{\phi}$\argDef{viewPhiDefault}} +rotates the viewport by rotation angles for longitude +({\it $\theta$}) and +latitude ({\it $\phi$}). +Angles designate radians if given as floats, or degrees if given +\index{graphics!3D commands!rotate} +as integers. + +% +\item[{\bf setAdaptive3D}]\funArgs{boolean\argDef{true}} +sets whether space curves are to be plotted +\index{graphics!plot3d defaults!set adaptive} +according to the adaptive +\index{adaptive plotting} +refinement algorithm. + +% +\item[{\bf setMaxPoints3D}]\funArgs{integer\argDef{1000}} + sets the default maximum number of possible +\index{graphics!plot3d defaults!set max points} +points to be used when constructing a three-di\-men\-sion\-al space curve. + +% +\item[{\bf setMinPoints3D}]\funArgs{integer\argDef{49}} +sets the default minimum number of possible +\index{graphics!plot3d defaults!set min points} +points to be used when constructing a three-di\-men\-sion\-al space curve. + +% +\item[{\bf setScreenResolution3D}]\funArgs{integer\argDef{49}} +sets the default screen resolution constant +\index{graphics!plot3d defaults!set screen resolution} +used in setting the computation limit of adaptively +\index{adaptive plotting} +generated three-di\-men\-sion\-al space curve plots. + +% +\item[{\bf showRegion}]\funArgs{viewport, string\argDef{"off"}} +declares whether the bounding +\index{graphics!3D commands!showRegion} +box of a graph is shown or not. +% +\item[{\bf subspace}]\funArgs{viewport} +returns the space component. +% +\item[{\bf subspace}]\funArgs{viewport, subspace} +resets the space component +\index{graphics!3D commands!subspace} +to {\it subspace}. + +% +\item[{\bf title}]\funArgs{viewport, string} +gives the viewport the +\index{graphics!3D commands!title} +title {\it string}. + +% +\item[{\bf translate}]\funArgs{viewport, +$float_{x}$\argDef{viewDeltaXDefault}, +$float_{y}$\argDef{viewDeltaYDefault}} +translates +\index{graphics!3D commands!translate} +the object horizontally and vertically relative to the center of the viewport. + +% +\item[{\bf intensity}]\funArgs{viewport,float\argDef{1.0}} +resets the intensity {\it I} of the light source, +\index{graphics!3D commands!intensity} +$0 \le I \le 1.$ + +% +\item[{\bf tubePointsDefault}]\funArgs{\optArg{integer\argDef{6}}} +sets or indicates the default number of +\index{graphics!3D defaults!tube points} +vertices defining the polygon that is used to create a tube around +a space curve. + +% +\item[{\bf tubeRadiusDefault}]\funArgs{\optArg{float\argDef{0.5}}} +sets or indicates the default radius of +\index{graphics!3D defaults!tube radius} +the tube that encircles a space curve. + +% +\item[{\bf var1StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} +sets or indicates the default number of +\index{graphics!3D defaults!var1 steps} +increments into which the grid defining a surface plot is subdivided with +respect to the first parameter declared in the surface function. + +% +\item[{\bf var2StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} +sets or indicates the default number of +\index{graphics!3D defaults!var2 steps} +increments into which the grid defining a surface plot is subdivided with +respect to the second parameter declared in the surface function. + +% +\item[{\bf viewDefaults}]\funArgs{{\tt [}$integer_{point}$, +$integer_{line}$, $integer_{axes}$, +$integer_{units}$, $float_{point}$, +\allowbreak$list_{position}$, +$list_{size}${\tt ]}} +resets the default settings for the +\index{graphics!3D defaults!reset viewport defaults} +point color, line color, axes color, units color, point size, +viewport upper left-hand corner position, and the viewport size. + +% +\item[{\bf viewDeltaXDefault}]\funArgs{\optArg{float\argDef{0}}} +resets the default horizontal offset +\index{graphics!3D commands!deltaX default} +from the center of the viewport, or returns the current default offset if no argument is given. + +% +\item[{\bf viewDeltaYDefault}]\funArgs{\optArg{float\argDef{0}}} +resets the default vertical offset +\index{graphics!3D commands!deltaY default} +from the center of the viewport, or returns the current default offset if no argument is given. + +% +\item[{\bf viewPhiDefault}]\funArgs{\optArg{float\argDef{-$\pi$/4}}} +resets the default latitudinal view angle, +or returns the current default angle if no argument is given. +\index{graphics!3D commands!phi default} +$\phi$ is set to this value. + +% +\item[{\bf viewpoint}]\funArgs{viewport, $float_{x}$, +$float_{y}$, $float_{z}$} +sets the viewing position in Cartesian coordinates. + +% +\item[{\bf viewpoint}]\funArgs{viewport, +$float_{\theta}$, +$Float_{\phi}$} +sets the viewing position in spherical coordinates. + +% +\item[{\bf viewpoint}]\funArgs{viewport, +$Float_{\theta}$, +$Float_{\phi}$, +$Float_{scaleFactor}$, +$Float_{xOffset}$, $Float_{yOffset}$} +sets the viewing position in spherical coordinates, +the scale factor, and offsets. +\index{graphics!3D commands!viewpoint} +$\theta$ (longitude) and +$\phi$ (latitude) are in radians. + +% +\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[0,0]}}} +sets or indicates the position of the upper +\index{graphics!3D defaults!viewport position} +left-hand corner of a two-di\-men\-sion\-al viewport, relative to the display root +window (the upper left-hand corner of the display is $[0, 0]$). + +% +\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[400,400]}}} +sets or indicates the width and height dimensions +\index{graphics!3D defaults!viewport size} +of a viewport. + +% +\item[{\bf viewThetaDefault}]\funArgs{\optArg{float\argDef{$\pi$/4}}} +resets the default longitudinal view angle, +or returns the current default angle if no argument is given. +\index{graphics!3D commands!theta default} +When a parameter is specified, the default longitudinal view angle +$\theta$ is set to this value. + +% +\item[{\bf viewWriteAvailable}]\funArgs{\optArg{list\argDef{["pixmap", +"bitmap", "postscript", "image"]}}} +indicates the possible file types +\index{graphics!3D defaults!available viewport writes} +that can be created with the \spadfunFrom{write}{ThreeDimensionalViewport} function. + +% +\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} +sets or indicates the default types of files +that are created in addition to the {\bf data} file when a +\spadfunFrom{write}{ThreeDimensionalViewport} command +\index{graphics!3D defaults!viewport writes} +is executed on a viewport. + +% +\item[{\bf viewScaleDefault}]\funArgs{\optArg{float}} +sets the default scaling factor, or returns +\index{graphics!3D commands!scale default} +the current factor if no argument is given. + +% +\item[{\bf write}]\funArgs{viewport, directory, \optArg{option}} +writes the file {\bf data} for {\it viewport} +in the directory {\it directory}. +An optional third argument specifies a file type (one of {\tt +pixmap}, {\tt bitmap}, {\tt postscript}, or {\tt image}), or a +list of file types. +An additional file is written for each file type listed. + +% +\item[{\bf scale}]\funArgs{viewport, float\argDef{2.5}} +specifies the scaling factor. +\index{graphics!3D commands!scale} +\index{scaling graphs} +\end{description} +\egroup + +\subsection{Customization using .Xdefaults} +\label{ugXdefaults} + +\index{graphics!.Xdefaults} + +Both the two-di\-men\-sion\-al and three-di\-men\-sion\-al drawing facilities consult +the {\bf .Xdefaults} file for various defaults. +\index{file!.Xdefaults @{\bf .Xdefaults}} +The list of defaults that are recognized by the graphing routines +is discussed in this section. +These defaults are preceded by {\tt Axiom.3D.} +for three-di\-men\-sion\-al viewport defaults, {\tt Axiom.2D.} +for two-di\-men\-sion\-al viewport defaults, or {\tt Axiom*} (no dot) for +those defaults that are acceptable to either viewport type. + +% +\begin{description} +% +\item[{\tt Axiom*buttonFont:\ \it font}] \ \newline +This indicates which +\index{graphics!.Xdefaults!button font} +font type is used for the button text on the control-panel. +{\bf Rom11} +% +\item[{\tt Axiom.2D.graphFont:\ \it font}] \quad (2D only) \newline +This indicates +\index{graphics!.Xdefaults!graph number font} +which font type is used for displaying the graph numbers and +slots in the {\bf Graphs} section of the two-di\-men\-sion\-al control-panel. +{\bf Rom22} +% +\item[{\tt Axiom.3D.headerFont:\ \it font}] \ \newline +This indicates which +\index{graphics!.Xdefaults!graph label font} +font type is used for the axes labels and potentiometer +header names on three-di\-men\-sion\-al viewport windows. +This is also used for two-di\-men\-sion\-al control-panels for indicating +which font type is used for potentionmeter header names and +multiple graph title headers. +%for example, {\tt Axiom.2D.headerFont: 8x13}. +{\bf Itl14} +% +\item[{\tt Axiom*inverse:\ \it switch}] \ \newline +This indicates whether the +\index{graphics!.Xdefaults!inverting background} +background color is to be inverted from white to black. +If {\tt on}, the graph viewports use black as the background +color. +If {\tt off} or no declaration is made, the graph viewports use a +white background. +{\bf off} +% +\item[{\tt Axiom.3D.lightingFont:\ \it font}] \quad (3D only) \newline +This indicates which font type is used for the {\bf x}, +\index{graphics!.Xdefaults!lighting font} +{\bf y}, and {\bf z} labels of the two lighting axes potentiometers, and for +the {\bf Intensity} title on the lighting control-panel. +{\bf Rom10} +% +\item[{\tt Axiom.2D.messageFont, Axiom.3D.messageFont:\ \it font}] \ \newline +These indicate the font type +\index{graphics!.Xdefaults!message font} +to be used for the text in the control-panel message window. +{\bf Rom14} +% +\item[{\tt Axiom*monochrome:\ \it switch}] \ \newline +This indicates whether the +\index{graphics!.Xdefaults!monochrome} +graph viewports are to be displayed as if the monitor is black and +white, that is, a 1 bit plane. +If {\tt on} is specified, the viewport display is black and white. +If {\tt off} is specified, or no declaration for this default is +given, the viewports are displayed in the normal fashion for the +monitor in use. +{\bf off} +% +\item[{\tt Axiom.2D.postScript:\ \it filename}] \ \newline +This specifies +\index{graphics!.Xdefaults!PostScript file name} +the name of the file that is generated when a 2D PostScript graph +\index{PostScript} +is saved. +{\bf axiom2d.ps} +% +\item[{\tt Axiom.3D.postScript:\ \it filename}] \ \newline +This specifies +\index{graphics!.Xdefaults!PostScript file name} +the name of the file that is generated when a 3D PostScript graph +\index{PostScript} +is saved. +{\bf axiom3D.ps} +% +\item[{\tt Axiom*titleFont \it font}] \ \newline +This +\index{graphics!.Xdefaults!title font} +indicates which font type is used +for the title text and, for three-di\-men\-sion\-al graphs, +in the lighting and viewing-volume control-panel windows. +\index{graphics!Xdefaults!2d} +{\bf Rom14} +% +\item[{\tt Axiom.2D.unitFont:\ \it font}] \quad (2D only) \newline +This indicates +\index{graphics!.Xdefaults!unit label font} +which font type is used for displaying the unit labels on +two-di\-men\-sion\-al viewport graphs. +{\bf 6x10} +% +\item[{\tt Axiom.3D.volumeFont:\ \it font}] \quad (3D only) \newline +This indicates which font type is used for the {\bf x}, +\index{graphics!.Xdefaults!volume label font} +{\bf y}, and {\bf z} labels of the clipping region sliders; for the +{\bf Perspective}, {\bf Show Region}, and {\bf Clipping On} buttons under +{\bf Settings}, and above the windows for the {\bf Hither} and +{\bf Eye Distance} sliders in the {\bf Viewing Volume Panel} of the +three-di\-men\-sion\-al control-panel. +{\bf Rom8} +\end{description} + +\setcounter{chapter}{7} % Chapter 8 +% viewSizeDefault [300,300] + +\chapter{Advanced Problem Solving} +\label{ugProblem} + +In this chapter we describe techniques useful in solving advanced problems +with Axiom. + +\section{Numeric Functions} +\label{ugProblemNumeric} + +% +Axiom provides two basic floating-point types: {\tt Float} and +{\tt DoubleFloat}. This section describes how to use numerical +\index{function!numeric} +operations defined on these types and the related complex types. +\index{numeric operations} +% +As we mentioned in Chapter +\ref{ugIntro} on page~\pageref{ugIntro}, the {\tt Float} type is a software +implementation of floating-point numbers in which the exponent and the +\index{floating-point number} +significand may have any number of digits. +\index{number!floating-point} +See +\ref{FloatXmpPage} on page~\pageref{FloatXmpPage} +for detailed information about this domain. +The {\tt DoubleFloat} (see \ref{DoubleFloatXmpPage} on +page~\pageref{DoubleFloatXmpPage}) is usually a hardware implementation +of floating point numbers, corresponding to machine double +precision. +The types {\tt Complex Float} and {\tt Complex DoubleFloat} are +\index{floating-point number!complex} +the corresponding software implementations of complex floating-point numbers. +\index{complex!floating-point number} +In this section the term {\it floating-point type} means any of these +\index{number!complex floating-point} +four types. +% +The floating-point types implement the basic elementary functions. +These include (where {\tt \$} means +{\tt DoubleFloat}, +{\tt Float}, +{\tt Complex DoubleFloat}, or +{\tt Complex Float}): + +\noindent +{\bf exp}, {\bf log}: $\$ -> \$$ \newline +{\bf sin}, {\bf cos}, {\bf tan}, {\bf cot}, {\bf sec}, {\bf csc}: $\$ -> \$$ \newline +{\bf sin}, {\bf cos}, {\bf tan}, {\bf cot}, {\bf sec}, {\bf csc}: $\$ -> \$$ \newline +{\bf asin}, {\bf acos}, {\bf atan}, {\bf acot}, {\bf asec}, {\bf acsc}: $\$ -> \$$ \newline +{\bf sinh}, {\bf cosh}, {\bf tanh}, {\bf coth}, {\bf sech}, {\bf csch}: $\$ -> \$$ \newline +{\bf asinh}, {\bf acosh}, {\bf atanh}, {\bf acoth}, {\bf asech}, {\bf acsch}: $\$ -> \$$ \newline +{\bf pi}: $() -> \$$ \newline +{\bf sqrt}: $\$ -> \$$ \newline +{\bf nthRoot}: $(\$, Integer) -> \$$ \newline +\spadfunFrom{**}{Float}: $(\$, Fraction Integer) -> \$$ \newline +\spadfunFrom{**}{Float}: $(\$,\$) -> \$$ \newline + +The handling of roots depends on whether the floating-point type +\index{root!numeric approximation} +is real or complex: for the real floating-point types, +{\tt DoubleFloat} and {\tt Float}, if a real root exists +the one with the same sign as the radicand is returned; for the +complex floating-point types, the principal value is returned. +\index{principal value} +Also, for real floating-point types the inverse functions +produce errors if the results are not real. +This includes cases such as $asin(1.2)$, $log(-3.2)$, +$sqrt(-1.1)$. +% + +The default floating-point type is {\tt Float} so to evaluate +functions using {\tt Float} or {\tt Complex Float}, just use +normal decimal notation. + +\spadcommand{exp(3.1)} +$$ +22.1979512814 41633405 +$$ +\returnType{Type: Float} + +\spadcommand{exp(3.1 + 4.5 * \%i)} +$$ +-{4.6792348860 969899118} -{{21.6991659280 71731864} \ i} +$$ +\returnType{Type: Complex Float} + +To evaluate functions using {\tt DoubleFloat} +or {\tt Complex DoubleFloat}, +a declaration or conversion is required. + +\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} +$$ +-{4.6792348860969906} -{{21.699165928071732} \ i} +$$ +\returnType{Type: Complex DoubleFloat} + +\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)} +$$ +-{4.6792348860969906} -{{21.699165928071732} \ i} +$$ +\returnType{Type: Complex DoubleFloat} + +A number of special functions are provided by the package +{\tt DoubleFloatSpecialFunctions} for the machine-precision +\index{special functions} +floating-point types. +\index{DoubleFloatSpecialFunctions} +The special functions provided are listed below, where $F$ stands for +the types {\tt DoubleFloat} and {\tt Complex DoubleFloat}. +The real versions of the functions yield an error if the result is not real. +\index{function!special} + +\noindent +{\bf Gamma}: $F -> F$\hfill\newline +$Gamma(z)$ is the Euler gamma function, +\index{function!Gamma} + $\Gamma(z)$, + defined by +\index{Euler!gamma function} + $$\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt.$$ + + +\noindent +{\bf Beta}: $F -> F$\hfill\newline + $Beta(u, v)$ is the Euler Beta function, +\index{function!Euler Beta} + $Beta(u,v)$, defined by +\index{Euler!Beta function} + $$Beta(u,v) = \int_{0}^{1} t^{u-1} (1-t)^{v-1} dt.$$ + + This is related to $\Gamma(z)$ by + $$Beta(u,v) = \frac{\Gamma(u) \Gamma(v)}{\Gamma(u + v)}.$$ + +\noindent +{\bf logGamma}: $F -> F$\hfill\newline + $logGamma(z)$ is the natural logarithm of +$\Gamma(z)$. + This can often be computed even if $\Gamma(z)$ +cannot. +% + +\noindent +{\bf digamma}: $F -> F$\hfill\newline + $digamma(z)$, also called $psi(z)$, +\index{psi @ $\psi$} +is the function $\psi(z)$, +\index{function!digamma} + defined by $$\psi(z) = \Gamma'(z)/\Gamma(z).$$ + +\noindent +{\bf polygamma}: $(NonNegativeInteger, F) -> F$\hfill\newline + $polygamma(n, z)$ is the $n$-th derivative of +\index{function!polygamma} + $\psi(z)$, written $\psi^{(n)}(z)$. + +\noindent +{\bf E1}: $(DoubleFloat) -> OnePointCompletion DoubleFloat$\hfill\newline + E1(x) is the Exponential Integral function + The current implementation is a piecewise approximation + involving one poly from $-4..4$ and a second poly for $x > 4$ +\index{function!E1} + +\noindent +{\bf En}: $(PI, DFLOAT) -> OnePointCompletion DoubleFloat$\hfill\newline + En(PI,R) is the nth Exponential Integral +\index{function!En} + +\noindent +{\bf Ei}: $(OnePointCompletion DFLOAT) -> OnePointCompletion DFLOAT$ +\hfill\newline + Ei is the Exponential Integral function + This is computed using a 6 part piecewise approximation. + DoubleFloat can only preserve about 16 digits but the + Chebyshev approximation used can give 30 digits. +\index{function!Ei} + +\noindent +{\bf Ei1}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei1 is the first approximation of Ei where the result is + $x*e^-x*Ei(x)$ from -infinity to -10 (preserves digits) +\index{function!Ei1} + +\noindent +{\bf Ei2}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei2 is the first approximation of Ei where the result is + $x*e^-x*Ei(x)$ from -10 to -4 (preserves digits) +\index{function!Ei2} + +\noindent +{\bf Ei3}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei3 is the first approximation of Ei where the result is + $(Ei(x)-log |x| - gamma)/x$ from -4 to 4 (preserves digits) +\index{function!Ei3} + +\noindent +{\bf Ei4}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei4 is the first approximation of Ei where the result is + $x*e^-x*Ei(x)$ from 4 to 12 (preserves digits) +\index{function!Ei4} + +\noindent +{\bf Ei5}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei5 is the first approximation of Ei where the result is + $x*e^-x*Ei(x)$ from 12 to 32 (preserves digits) +\index{function!Ei5} + +\noindent +{\bf Ei6}: $(DoubleFloat) -> DoubleFloat$\hfill\newline + Ei6 is the first approximation of Ei where the result is + $x*e^-x*Ei(x)$ from 32 to infinity (preserves digits) +\index{function!Ei6} + +\noindent +{\bf besselJ}: $(F,F) -> F$\hfill\newline + $besselJ(v,z)$ is the Bessel function of the first kind, +\index{function!Bessel} + $J_\nu (z)$. + This function satisfies the differential equation + $$z^2 w''(z) + z w'(z) + (z^2-\nu^2)w(z) = 0.$$ + +\noindent +{\bf besselY}: $(F,F) -> F$\hfill\newline + $besselY(v,z)$ is the Bessel function of the second kind, +\index{function!Bessel} + $Y_\nu (z)$. + This function satisfies the same differential equation as + {\bf besselJ}. + The implementation simply uses the relation + $$Y_\nu (z) = \frac{J_\nu (z) \cos(\nu \pi) - J_{-\nu} (z)}{\sin(\nu \pi)}.$$ + +\noindent +{\bf besselI}: $(F,F) -> F$\hfill\newline + $besselI(v,z)$ is the modified Bessel function of the first kind, +\index{function!Bessel} + $I_\nu (z)$. + This function satisfies the differential equation + $$z^2 w''(z) + z w'(z) - (z^2+\nu^2)w(z) = 0.$$ + +\noindent +{\bf besselK}: $(F,F) -> F$\hfill\newline + $besselK(v,z)$ is the modified Bessel function of the second kind, +\index{function!Bessel} + $K_\nu (z)$. + This function satisfies the same differential equation as {\bf besselI}. +\index{Bessel function} + The implementation simply uses the relation + $$K_\nu (z) = \pi \frac{I_{-\nu} (z) - I_{\nu} (z)}{2 \sin(\nu \pi)}.$$ + + +\noindent +{\bf airyAi}: $F -> F$\hfill\newline + $airyAi(z)$ is the Airy function $Ai(z)$. +\index{function!Airy Ai} + This function satisfies the differential equation + $w''(z) - z w(z) = 0.$ + The implementation simply uses the relation + $$Ai(-z) = \frac{1}{3}\sqrt{z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) + J_{1/3} (\frac{2}{3}z^{3/2}) ).$$ + +\noindent +{\bf airyBi}: $F -> F$\hfill\newline + $airyBi(z)$ is the Airy function $Bi(z)$. +\index{function!Airy Bi} + This function satisfies the same differential equation as {\bf airyAi}. +\index{Airy function} + The implementation simply uses the relation + $$Bi(-z) = \frac{1}{3}\sqrt{3 z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) - J_{1/3} (\frac{2}{3}z^{3/2}) ).$$ + +\noindent +{\bf hypergeometric0F1}: $(F,F) -> F$\hfill\newline + $hypergeometric0F1(c,z)$ is the hypergeometric function +\index{function!hypergeometric} + ${}_0 F_1 ( ; c; z)$. + +The above special functions are defined only for small floating-point types. +If you give {\tt Float} arguments, they are converted to +{\tt DoubleFloat} by Axiom. + +\spadcommand{Gamma(0.5)**2} +$$ +3.14159265358979 +$$ +\returnType{Type: DoubleFloat} + +\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} +$$ +{2.489481690673867} -{{2.365846713181643} \ i} +$$ +\returnType{Type: Complex DoubleFloat} + +A number of additional operations may be used to compute numerical values. +These are special polynomial functions that can be evaluated for values in +any commutative ring $R$, and in particular for values in any +floating-point type. +The following operations are provided by the package +{\tt OrthogonalPolynomialFunctions}: +\index{OrthogonalPolynomialFunctions} + +\noindent +{\bf chebyshevT}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $chebyshevT(n,z)$ is the $n$-th Chebyshev polynomial of the first + kind, $T_n (z)$. These are defined by + $$\frac{1-t z}{1-2 t z+t^2} = \sum_{n=0}^{\infty} T_n (z) t^n.$$ + +\noindent +{\bf chebyshevU}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $chebyshevU(n,z)$ is the $n$-th Chebyshev polynomial of the second + kind, $U_n (z)$. These are defined by + $$\frac{1}{1-2 t z+t^2} = \sum_{n=0}^{\infty} U_n (z) t^n.$$ + +\noindent +{\bf hermiteH}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $hermiteH(n,z)$ is the $n$-th Hermite polynomial, + $H_n (z)$. + These are defined by + $$e^{2 t z - t^2} = \sum_{n=0}^{\infty} H_n (z) \frac{t^n}{n!}.$$ + +\noindent +{\bf laguerreL}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $laguerreL(n,z)$ is the $n$-th Laguerre polynomial, + $L_n (z)$. + These are defined by + $$\frac{e^{-\frac{t z}{1-t}}}{1-t} = \sum_{n=0}^{\infty} L_n (z) \frac{t^n}{n!}.$$ + +\noindent +{\bf laguerreL}: $(NonNegativeInteger, NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $laguerreL(m,n,z)$ is the associated Laguerre polynomial, + $L^m_n (z)$. + This is the $m$-th derivative of $L_n (z)$. + +\noindent +{\bf legendreP}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $legendreP(n,z)$ is the $n$-th Legendre polynomial, + $P_n (z)$. These are defined by + $$\frac{1}{\sqrt{1-2 t z+t^2}} = \sum_{n=0}^{\infty} P_n (z) t^n.$$ + +These operations require non-negative integers for the indices, but otherwise +the argument can be given as desired. + +\spadcommand{[chebyshevT(i, z) for i in 0..5]} +$$ +\left[ +1, z, {{2 \ {z \sp 2}} -1}, {{4 \ {z \sp 3}} -{3 \ z}}, {{8 \ +{z \sp 4}} -{8 \ {z \sp 2}}+1}, {{{16} \ {z \sp 5}} -{{20} \ {z \sp +3}}+{5 \ z}} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +The expression $chebyshevT(n,z)$ evaluates to the $n$-th Chebyshev +\index{polynomial!Chebyshev!of the first kind} +polynomial of the first kind. + +\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)} +$$ +-{1675.0}+{{918.0} \ i} +$$ +\returnType{Type: Complex Float} + +\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} +$$ +485.0 +$$ +\returnType{Type: DoubleFloat} + +The expression $chebyshevU(n,z)$ evaluates to the $n$-th Chebyshev +\index{polynomial!Chebyshev!of the second kind} +polynomial of the second kind. + +\spadcommand{[chebyshevU(i, z) for i in 0..5]} +$$ +\left[ +1, {2 \ z}, {{4 \ {z \sp 2}} -1}, {{8 \ {z \sp 3}} -{4 \ z}}, +{{{16} \ {z \sp 4}} -{{12} \ {z \sp 2}}+1}, {{{32} \ {z \sp 5}} -{{32} +\ {z \sp 3}}+{6 \ z}} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +\spadcommand{chebyshevU(3, 0.2)} +$$ +-{0.736} +$$ +\returnType{Type: Float} + +The expression $hermiteH(n,z)$ evaluates to the $n$-th Hermite +\index{polynomial!Hermite} +polynomial. + +\spadcommand{[hermiteH(i, z) for i in 0..5]} +$$ +\left[ +1, {2 \ z}, {{4 \ {z \sp 2}} -2}, {{8 \ {z \sp 3}} -{{12} \ z}}, + {{{16} \ {z \sp 4}} -{{48} \ {z \sp 2}}+{12}}, {{{32} \ {z \sp 5}} +-{{160} \ {z \sp 3}}+{{120} \ z}} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +\spadcommand{hermiteH(100, 1.0)} +$$ +-{0.1448706729 337934088 E 93} +$$ +\returnType{Type: Float} + +The expression $laguerreL(n,z)$ evaluates to the $n$-th Laguerre +\index{polynomial!Laguerre} +polynomial. + +\spadcommand{[laguerreL(i, z) for i in 0..4]} +$$ +\left[ +1, {-z+1}, {{z \sp 2} -{4 \ z}+2}, {-{z \sp 3}+{9 \ {z \sp 2}} +-{{18} \ z}+6}, {{z \sp 4} -{{16} \ {z \sp 3}}+{{72} \ {z \sp 2}} +-{{96} \ z}+{24}} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +\spadcommand{laguerreL(4, 1.2)} +$$ +-{13.0944} +$$ +\returnType{Type: Float} + +\spadcommand{[laguerreL(j, 3, z) for j in 0..4]} +$$ +\left[ +{-{z \sp 3}+{9 \ {z \sp 2}} -{{18} \ z}+6}, {-{3 \ {z \sp 2}}+{{18} \ +z} -{18}}, {-{6 \ z}+{18}}, -6, 0 +\right] +$$ +\returnType{Type: List Polynomial Integer} + +\spadcommand{laguerreL(1, 3, 2.1)} +$$ +6.57 +$$ +\returnType{Type: Float} + +The expression +\index{polynomial!Legendre} +$legendreP(n,z)$ evaluates to the $n$-th Legendre polynomial, + +\spadcommand{[legendreP(i,z) for i in 0..5]} +$$ +\left[ +1, z, {{{3 \over 2} \ {z \sp 2}} -{1 \over 2}}, {{{5 \over 2} \ {z +\sp 3}} -{{3 \over 2} \ z}}, {{{{35} \over 8} \ {z \sp 4}} -{{{15} \over +4} \ {z \sp 2}}+{3 \over 8}}, {{{{63} \over 8} \ {z \sp 5}} -{{{35} +\over 4} \ {z \sp 3}}+{{{15} \over 8} \ z}} +\right] +$$ +\returnType{Type: List Polynomial Fraction Integer} + +\spadcommand{legendreP(3, 3.0*\%i)} +$$ +-{{72.0} \ i} +$$ +\returnType{Type: Complex Float} + +Finally, three number-theoretic polynomial operations may be evaluated. +\index{number theory} +The following operations are provided by the package +{\tt NumberTheoreticPolynomialFunctions}. +\index{NumberTheoreticPolynomialFunctions}. + +\noindent +{\bf bernoulliB}: $(NonNegativeInteger, R) -> R$ \hbox{}\hfill\newline + $bernoulliB(n,z)$ is the $n$-th Bernoulli polynomial, +\index{polynomial!Bernoulli} + $B_n (z)$. These are defined by + $$\frac{t e^{z t}}{e^t - 1} = \sum_{n=0}^{\infty} B_n (z) \frac{t^n}{n!}.$$ + +\noindent +{\bf eulerE}: $(NonNegativeInteger, R) -> R$ \hbox{}\hfill\newline + $eulerE(n,z)$ is the $n$-th Euler polynomial, +\index{Euler!polynomial} + $E_n (z)$. These are defined by +\index{polynomial!Euler} + $$\frac{2 e^{z t}}{e^t + 1} = \sum_{n=0}^{\infty} E_n (z) \frac{t^n}{n!}.$$ + +\noindent +{\bf cyclotomic}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline + $cyclotomic(n,z)$ is the $n$-th cyclotomic polynomial + $\Phi_n (z)$. This is the polynomial whose + roots are precisely the primitive $n$-th roots of unity. +\index{Euler!totient function} + This polynomial has degree given by the Euler totient function +\index{function!totient} + $\phi(n)$. + +The expression $bernoulliB(n,z)$ evaluates to the $n$-th Bernoulli +\index{polynomial!Bernouilli} +polynomial. + +\spadcommand{bernoulliB(3, z)} +$$ +{z \sp 3} -{{3 \over 2} \ {z \sp 2}}+{{1 \over 2} \ z} +$$ +\returnType{Type: Polynomial Fraction Integer} + +\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)} +$$ +-{0.138} -{{0.116} \ i} +$$ +\returnType{Type: Complex Float} + +The expression +\index{polynomial!Euler} +$eulerE(n,z)$ evaluates to the $n$-th Euler polynomial. + +\spadcommand{eulerE(3, z)} +$$ +{z \sp 3} -{{3 \over 2} \ {z \sp 2}}+{1 \over 4} +$$ +\returnType{Type: Polynomial Fraction Integer} + +\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)} +$$ +-{0.238} -{{0.316} \ i} +$$ +\returnType{Type: Complex Float} + +The expression +\index{polynomial!cyclotomic} +$cyclotomic(n,z)$ evaluates to the $n$-th cyclotomic polynomial. +\index{cyclotomic polynomial} + +\spadcommand{cyclotomic(3, z)} +$$ +{z \sp 2}+z+1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))} +$$ +0.0 +$$ +\returnType{Type: Complex Float} + +Drawing complex functions in Axiom is presently somewhat +awkward compared to drawing real functions. +It is necessary to use the {\bf draw} operations that operate +on functions rather than expressions. + +This is the complex exponential function (rotated interactively). +\index{function!complex exponential} +When this is displayed in color, the height is the value of the real part of +the function and the color is the imaginary part. +Red indicates large negative imaginary values, green indicates imaginary +values near zero and blue/violet indicates large positive imaginary values. + +\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")} + +%\epsffile[0 0 295 295]{ps/compexp.ps} + +This is the complex arctangent function. +\index{function!complex arctangent} +Again, the height is the real part of the function value but here +the color indicates the function value's phase. +The position of the branch cuts are clearly visible and one can +see that the function is real only for a real argument. + +\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp} + +%\epsffile[0 0 295 295]{ps/compatan.ps} + +This is the complex Gamma function. + +\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)} + +%\epsffile[0 0 295 295]{ps/compgamm.ps} + +This shows the real Beta function near the origin. + +\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)} + +%\epsffile[0 0 295 295]{ps/realbeta.ps} + +This is the Bessel function $J_\alpha (x)$ +for index $\alpha$ in the range $-6..4$ and +argument $x$ in the range $2..14$. + +\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)} + +%\epsffile[0 0 295 295]{ps/bessel.ps} + +This is the modified Bessel function +$I_\alpha (x)$ +evaluated for various real values of the index $\alpha$ +and fixed argument $x = 5$. + +\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} + +%\epsffile[0 0 295 295]{ps/modbess.ps} + +This is similar to the last example +except the index $\alpha$ +takes on complex values in a $6 x 6$ rectangle centered on the origin. + +\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")} + +%\epsffile[0 0 295 295]{ps/modbessc.ps} + +\section{Polynomial Factorization} +\label{ugProblemFactor} + +% +The Axiom polynomial factorization +\index{polynomial!factorization} +facilities are available for all polynomial types and a wide variety of +coefficient domains. +\index{factorization} +Here are some examples. + +\subsection{Integer and Rational Number Coefficients} +\label{ugProblemFactorIntRat} + +Polynomials with integer +\index{polynomial!factorization!integer coefficients} +coefficients can be be factored. + +\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12) } +$$ +-{2 \ {x \sp 3} \ {y \sp 3}}+{{\left( {{24} \ {x \sp 5}}+{24} +\right)} +\ {y \sp 2}}+{{\left( -{4 \ {x \sp 6}} -{x \sp 3} +\right)} +\ y}+{{48} \ {x \sp 8}}+{{12} \ {x \sp 5}}+{{48} \ {x \sp 3}}+{12} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{factor v } +$$ +-{{\left( {{x \sp 3} \ y} -{{12} \ {x \sp 5}} -{12} +\right)} +\ {\left( {2 \ {y \sp 2}}+{4 \ {x \sp 3}}+1 +\right)}} +$$ +\returnType{Type: Factored Polynomial Integer} + +Also, Axiom can factor polynomials with +\index{polynomial!factorization!rational number coefficients} +rational number coefficients. + +\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12) } +$$ +{{48} \ {x \sp 8}}+{8 \ {x \sp 7}} -{2 \ {x \sp 6}}+{{{35} \over 3} \ {x +\sp 5}}+{{{95} \over 2} \ {x \sp 3}}+{8 \ {x \sp 2}}+{12} +$$ +\returnType{Type: Polynomial Fraction Integer} + +\spadcommand{factor w } +$$ +{48} \ {\left( {x \sp 3}+{{1 \over 6} \ {x \sp 2}}+{1 \over 4} +\right)} +\ {\left( {x \sp 5} -{{1 \over {24}} \ {x \sp 3}}+1 +\right)} +$$ +\returnType{Type: Factored Polynomial Fraction Integer} + +\subsection{Finite Field Coefficients} +\label{ugProblemFactorFF} + +Polynomials with coefficients in a finite field +\index{polynomial!factorization!finite field coefficients} +can be also be factored. +\index{finite field!factoring polynomial with coefficients in} + +\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18 } +$$ +{3 \ {x \sp 4}}+{2 \ {x \sp 2}}+{{15} \ x}+{18} +$$ +\returnType{Type: Polynomial PrimeField 19} + +These include the integers mod $p$, where $p$ is prime, and +extensions of these fields. + +\spadcommand{factor u } +$$ +3 \ {\left( x+{18} +\right)} +\ {\left( {x \sp 3}+{x \sp 2}+{8 \ x}+{13} +\right)} +$$ +\returnType{Type: Factored Polynomial PrimeField 19} + +Convert this to have coefficients in the finite +field with $19^3$ elements. +See \ref{ugProblemFinite} on page~\pageref{ugProblemFinite} +for more information about finite fields. + +\spadcommand{factor(u :: POLY FFX(PF 19,3)) } +$$ +3 \ {\left( x+{18} +\right)} +\ {\left( x+{5 \ { \%I \sp 2}}+{3 \ \%I}+{13} +\right)} +\ {\left( x+{{16} \ { \%I \sp 2}}+{{14} \ \%I}+{13} +\right)} +\ {\left( x+{{17} \ { \%I \sp 2}}+{2 \ \%I}+{13} +\right)} +$$ +\returnType{Type: Factored Polynomial FiniteFieldExtension(PrimeField 19,3)} + +\subsection{Simple Algebraic Extension Field Coefficients} +\label{ugProblemFactorAlg} + +Polynomials with coefficients in simple algebraic extensions +\index{polynomial!factorization!algebraic extension field coefficients} +of the rational numbers can be factored. +\index{algebraic number} +\index{number!algebraic} + +Here, $aa$ and $bb$ are symbolic roots of polynomials. + +\spadcommand{aa := rootOf(aa**2+aa+1) } +$$ +aa +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2 } +$$ +\begin{array}{@{}l} +{{\left( -aa -1 \right)} \ {y \sp 5}}+ +{{\left( {{\left( -aa -1 \right)}\ {x \sp 3}}+ +{aa \ x} \right)}\ {y \sp 4}}+ +\\ +\\ +\displaystyle +{{\left( {{\left( -{2 \ aa} -2 \right)}\ {x \sp 2}}+ +{{\left( -{2 \ aa} -2 \right)}\ x} \right)}\ {y \sp 3}}+ +\\ +\\ +\displaystyle +{{\left( {{\left( -{2 \ aa} -2 \right)}\ {x \sp 5}}+ +{{\left( -{2 \ aa} -2 \right)}\ {x \sp 4}}+ +{2 \ aa \ {x \sp 3}}+{2 \ aa \ {x \sp 2}} \right)}\ {y \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( {{\left( -aa -1 \right)}\ {x \sp 4}}+ +{{\left( -{2 \ aa} -2 \right)}\ {x \sp 3}}+ +{{\left( -aa -1 \right)}\ {x \sp 2}} \right)}\ y}+ +\\ +\\ +\displaystyle +{{\left( -aa -1 \right)}\ {x \sp 7}}+ +{{\left( -{2 \ aa} -2 \right)}\ {x \sp 6}} - +{x \sp 5}+ +{2 \ aa \ {x \sp 4}}+ +{aa \ {x \sp 3}} +\end{array} +$$ +\returnType{Type: Polynomial AlgebraicNumber} + +Note that the second argument to factor can be a list of +algebraic extensions to factor over. + +\spadcommand{factor(p,[aa]) } +%Note: this answer differs from the book but is equivalent. +$$ +{\left( -aa -1 +\right)} +\ {\left( y+{x \sp 3}+{{\left( -aa -1 +\right)} +\ x} +\right)} +\ {{\left( {y \sp 2}+{x \sp 2}+x +\right)} +\sp 2} +$$ +\returnType{Type: Factored Polynomial AlgebraicNumber} + +This factors $x**2+3$ over the integers. + +\spadcommand{factor(x**2+3)} +$$ +{x \sp 2}+3 +$$ +\returnType{Type: Factored Polynomial Integer} + +Factor the same polynomial over the field obtained by adjoining +$aa$ to the rational numbers. + +\spadcommand{factor(x**2+3,[aa]) } +$$ +{\left( x -{2 \ aa} -1 +\right)} +\ {\left( x+{2 \ aa}+1 +\right)} +$$ +\returnType{Type: Factored Polynomial AlgebraicNumber} + +Factor $x**6+108$ over the same field. + +\spadcommand{factor(x**6+108,[aa]) } +$$ +{\left( {x \sp 3} -{{12} \ aa} -6 +\right)} +\ {\left( {x \sp 3}+{{12} \ aa}+6 +\right)} +$$ +\returnType{Type: Factored Polynomial AlgebraicNumber} + +\spadcommand{bb:=rootOf(bb**3-2) } +$$ +bb +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{factor(x**6+108,[bb]) } +$$ +{\left( {x \sp 2} -{3 \ bb \ x}+{3 \ {bb \sp 2}} +\right)} +\ {\left( {x \sp 2}+{3 \ {bb \sp 2}} +\right)} +\ {\left( {x \sp 2}+{3 \ bb \ x}+{3 \ {bb \sp 2}} +\right)} +$$ +\returnType{Type: Factored Polynomial AlgebraicNumber} + +Factor again over the field obtained by adjoining both $aa$ +and $bb$ to the rational numbers. + +\spadcommand{factor(x**6+108,[aa,bb]) } +$$ +\begin{array}{@{}l} +{\left( x+{{\left( -{2 \ aa} -1 \right)}\ bb} \right)} +\ {\left( x+{{\left( -aa -2 \right)}\ bb} \right)} +\ {\left( x+{{\left( -aa+1 \right)}\ bb} \right)} +\\ +\\ +\displaystyle +\ {\left( x+{{\left( aa -1 \right)}\ bb} \right)} +\ {\left( x+{{\left( aa+2 \right)}\ bb} \right)} +\ {\left( x+{{\left( {2 \ aa}+1 \right)}\ bb} \right)} +\end{array} +$$ +\returnType{Type: Factored Polynomial AlgebraicNumber} + +\subsection{Factoring Rational Functions} +\label{ugProblemFactorRatFun} + +Since fractions of polynomials form a field, every element (other than zero) +\index{rational function!factoring} +divides any other, so there is no useful notion of irreducible factors. +Thus the {\bf factor} operation is not very useful for fractions +of polynomials. + +There is, instead, a specific operation {\bf factorFraction} +that separately factors the numerator and denominator and returns +a fraction of the factored results. + +\spadcommand{factorFraction((x**2-4)/(y**2-4))} +$$ +{{\left( x -2 +\right)} +\ {\left( x+2 +\right)}} +\over {{\left( y -2 +\right)} +\ {\left( y+2 +\right)}} +$$ +\returnType{Type: Fraction Factored Polynomial Integer} + +You can also use {\bf map}. This expression +applies the {\bf factor} operation +to the numerator and denominator. + +\spadcommand{map(factor,(x**2-4)/(y**2-4))} +$$ +{{\left( x -2 +\right)} +\ {\left( x+2 +\right)}} +\over {{\left( y -2 +\right)} +\ {\left( y+2 +\right)}} +$$ +\returnType{Type: Fraction Factored Polynomial Integer} + +\section{Manipulating Symbolic Roots of a Polynomial} +\label{ugProblemSymRoot} + +% +In this section we show you how to work with one root or all roots +\index{root!symbolic} +of a polynomial. +These roots are represented symbolically (as opposed to being +numeric approximations). +See \ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol} and +\ref{ugxProblemPolSys} on page~\pageref{ugxProblemPolSys} for +information about solving for the roots of one or more +polynomials. + +\subsection{Using a Single Root of a Polynomial} +\label{ugxProblemSymRootOne} + +Use {\bf rootOf} to get a symbolic root of a polynomial: +$rootOf(p, x)$ returns a root of $p(x)$. + +This creates an algebraic number $a$. +\index{algebraic number} +\index{number!algebraic} + +\spadcommand{a := rootOf(a**4+1,a) } +$$ +a +$$ +\returnType{Type: Expression Integer} + +To find the algebraic relation that defines $a$, +use {\bf definingPolynomial}. + +\spadcommand{definingPolynomial a } +$$ +{a \sp 4}+1 +$$ +\returnType{Type: Expression Integer} + +You can use $a$ in any further expression, +including a nested {\bf rootOf}. + +\spadcommand{b := rootOf(b**2-a-1,b) } +$$ +b +$$ +\returnType{Type: Expression Integer} + +Higher powers of the roots are automatically reduced during +calculations. + +\spadcommand{a + b } +$$ +b+a +$$ +\returnType{Type: Expression Integer} + +\spadcommand{\% ** 5 } +$$ +{{\left( {{10} \ {a \sp 3}}+{{11} \ {a \sp 2}}+{2 \ a} -4 +\right)} +\ b}+{{15} \ {a \sp 3}}+{{10} \ {a \sp 2}}+{4 \ a} -{10} +$$ +\returnType{Type: Expression Integer} + +The operation {\bf zeroOf} is similar to {\bf rootOf}, +except that it may express the root using radicals in some cases. +\index{radical} + +\spadcommand{rootOf(c**2+c+1,c)} +$$ +c +$$ +\returnType{Type: Expression Integer} + +\spadcommand{zeroOf(d**2+d+1,d)} +$$ +{{\sqrt {-3}} -1} \over 2 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{rootOf(e**5-2,e)} +$$ +e +$$ +\returnType{Type: Expression Integer} + +\spadcommand{zeroOf(f**5-2,f)} +$$ +\root {5} \of {2} +$$ +\returnType{Type: Expression Integer} + +\subsection{Using All Roots of a Polynomial} +\label{ugxProblemSymRootAll} + +Use {\bf rootsOf} to get all symbolic roots of a polynomial: +$rootsOf(p, x)$ returns a +list of all the roots of $p(x)$. +If $p(x)$ has a multiple root of order $n$, then that root +\index{root!multiple} +appears $n$ times in the list. +\typeout{Make sure these variables are x0 etc} + +Compute all the roots of $x**4 + 1$. + +\spadcommand{l := rootsOf(x**4+1,x) } +$$ +\left[ + \%x0, { \%x0 \ \%x1}, - \%x0, -{ \%x0 \ \%x1} +\right] +$$ +\returnType{Type: List Expression Integer} + +As a side effect, the variables $\%x0, \%x1$ and $\%x2$ are bound +to the first three roots of $x**4+1$. + +\spadcommand{\%x0**5 } +$$ +- \%x0 +$$ +\returnType{Type: Expression Integer} + +Although they all satisfy $x**4 + 1 = 0, \%x0, \%x1,$ +and $\%x2$ are different algebraic numbers. +To find the algebraic relation that defines each of them, +use {\bf definingPolynomial}. + +\spadcommand{definingPolynomial \%x0 } +$$ +{ \%x0 \sp 4}+1 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{definingPolynomial \%x1 } +$$ +{ \%x1 \sp 2}+1 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{definingPolynomial \%x2 } +$$ +- \%x2+ \%\%var +$$ +\returnType{Type: Expression Integer} + +We can check that the sum and product of the roots of $x**4+1$ are +its trace and norm. + +\spadcommand{x3 := last l } +$$ +-{ \%x0 \ \%x1} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{\%x0 + \%x1 + \%x2 + x3 } +$$ +{{\left( - \%x0+1 +\right)} +\ \%x1}+ \%x0+ \%x2 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{\%x0 * \%x1 * \%x2 * x3 } +$$ + \%x2 \ { \%x0 \sp 2} +$$ +\returnType{Type: Expression Integer} + +Corresponding to the pair of operations +{\bf rootOf}/{\bf zeroOf} in +\ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol}, there is +an operation {\bf zerosOf} that, like {\bf rootsOf}, +computes all the roots +of a given polynomial, but which expresses some of them in terms of +radicals. + +\spadcommand{zerosOf(y**4+1,y) } +$$ +\left[ +{{{\sqrt {-1}}+1} \over {\sqrt {2}}}, {{{\sqrt {-1}} -1} \over {\sqrt +{2}}}, {{-{\sqrt {-1}} -1} \over {\sqrt {2}}}, {{-{\sqrt {-1}}+1} \over +{\sqrt {2}}} +\right] +$$ +\returnType{Type: List Expression Integer} + +As you see, only one implicit algebraic number was created +($\%y1$), and its defining equation is this. +The other three roots are expressed in radicals. + +\spadcommand{definingPolynomial \%y1 } +$$ +{ \%\%var \sp 2}+1 +$$ +\returnType{Type: Expression Integer} + +\section{Computation of Eigenvalues and Eigenvectors} +\label{ugProblemEigen} +% +In this section we show you +some of Axiom's facilities for computing and +\index{eigenvalue} +manipulating eigenvalues and eigenvectors, also called +\index{eigenvector} +characteristic values and characteristic vectors, +\index{characteristic!value} +respectively. +\index{characteristic!vector} + +\vskip 4pc + +Let's first create a matrix with integer entries. + +\spadcommand{m1 := matrix [ [1,2,1],[2,1,-2],[1,-2,4] ] } +$$ +\left[ +\begin{array}{ccc} +1 & 2 & 1 \\ +2 & 1 & -2 \\ +1 & -2 & 4 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +To get a list of the {\it rational} eigenvalues, +use the operation {\bf eigenvalues}. + +\spadcommand{leig := eigenvalues(m1) } +$$ +\left[ +5, {\left( \%K \mid {{ \%K \sp 2} - \%K -5} +\right)} +\right] +$$ +\returnType{Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer))} + +Given an explicit eigenvalue, {\bf eigenvector} computes the eigenvectors +corresponding to it. + +\spadcommand{eigenvector(first(leig),m1) } +$$ +\left[ +{\left[ +\begin{array}{c} +0 \\ +-{1 \over 2} \\ +1 +\end{array} +\right]} +\right] +$$ +\returnType{Type: List Matrix Fraction Polynomial Fraction Integer} + +The operation {\bf eigenvectors} returns a list of pairs of values and +vectors. When an eigenvalue is rational, Axiom gives you +the value explicitly; otherwise, its minimal polynomial is given, +(the polynomial of lowest degree with the eigenvalues as roots), +together with a parametric representation of the eigenvector using the +eigenvalue. +This means that if you ask Axiom to {\bf solve} +the minimal polynomial, then you can substitute these roots +\index{polynomial!minimal} +into the parametric form of the corresponding eigenvectors. +\index{minimal polynomial} + +You must be aware that unless an exact eigenvalue has been computed, +the eigenvector may be badly in error. + +\spadcommand{eigenvectors(m1) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {eigval=5}, {eigmult=1}, {eigvec= +{\left[ +{\left[ +\begin{array}{c} +0 \\ +-{1 \over 2} \\ +1 +\end{array} +\right]} +\right]}} +\right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ +{eigval={\left( \%L \mid {{ \%L \sp 2} - \%L -5} \right)}}, +{eigmult=1}, {eigvec= +{\left[ +{\left[ +\begin{array}{c} + \%L \\ +2 \\ +1 +\end{array} +\right]} +\right]}} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)} + +Another possibility is to use the operation +{\bf radicalEigenvectors} +tries to compute explicitly the eigenvectors +in terms of radicals. +\index{radical} + +\spadcommand{radicalEigenvectors(m1) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {radval={{{\sqrt {{21}}}+1} \over 2}}, {radmult=1}, +{radvect={\left[ {\left[ +\begin{array}{c} +{{{\sqrt {{21}}}+1} \over 2} \\ +2 \\ +1 +\end{array} +\right]} +\right]}} +\right]}, +\right. +\\ +\\ +\displaystyle + \left[ {radval={{-{\sqrt {{21}}}+1} \over 2}}, {radmult=1}, +{radvect={\left[ {\left[ +\begin{array}{c} +{{-{\sqrt {{21}}}+1} \over 2} \\ +2 \\ +1 +\end{array} +\right]} +\right]}} +\right], +\\ +\\ +\displaystyle +\left. + \left[ {radval=5}, {radmult=1}, +{radvect={\left[ {\left[ +\begin{array}{c} +0 \\ +-{1 \over 2} \\ +1 +\end{array} +\right]} +\right]}} +\right] +\right] +\end{array} +$$ +\returnType{Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer)} + +Alternatively, Axiom can compute real or complex approximations to the +\index{approximation} +eigenvectors and eigenvalues using the operations {\bf realEigenvectors} +or {\bf complexEigenvectors}. +They each take an additional argument $\epsilon$ +to specify the ``precision'' required. +\index{precision} +In the real case, this means that each approximation will be within +$\pm\epsilon$ of the actual +result. +In the complex case, this means that each approximation will be within +$\pm\epsilon$ of the actual result +in each of the real and imaginary parts. + +The precision can be specified as a {\tt Float} if the results are +desired in floating-point notation, or as {\tt Fraction Integer} if the +results are to be expressed using rational (or complex rational) numbers. + +\spadcommand{realEigenvectors(m1,1/1000) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {outval=5}, {outmult=1}, {outvect={\left[ {\left[ +\begin{array}{c} +0 \\ +-{1 \over 2} \\ +1 +\end{array} +\right]} +\right]}} +\right]}, +\right. +\\ +\\ +\displaystyle + {\left[ {outval={{5717} \over {2048}}}, {outmult=1}, +{outvect={\left[ {\left[ +\begin{array}{c} +{{5717} \over {2048}} \\ +2 \\ +1 +\end{array} +\right]} +\right]}} +\right]}, +\\ +\\ +\displaystyle +\left. + {\left[ {outval=-{{3669} \over {2048}}}, {outmult=1}, +{outvect={\left[ {\left[ +\begin{array}{c} +-{{3669} \over {2048}} \\ +2 \\ +1 +\end{array} +\right]} +\right]}} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List Record(outval: Fraction Integer,outmult: Integer,outvect: List Matrix Fraction Integer)} + +If an $n$ by $n$ matrix has $n$ distinct eigenvalues (and +therefore $n$ eigenvectors) the operation {\bf eigenMatrix} +gives you a matrix of the eigenvectors. + +\spadcommand{eigenMatrix(m1) } +$$ +\left[ +\begin{array}{ccc} +{{{\sqrt {{21}}}+1} \over 2} & {{-{\sqrt {{21}}}+1} \over 2} & 0 \\ +2 & 2 & -{1 \over 2} \\ +1 & 1 & 1 +\end{array} +\right] +$$ +\returnType{Type: Union(Matrix Expression Integer,...)} + +\spadcommand{m2 := matrix [ [-5,-2],[18,7] ] } +$$ +\left[ +\begin{array}{cc} +-5 & -2 \\ +{18} & 7 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\spadcommand{eigenMatrix(m2) } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +If a symmetric matrix +\index{matrix!symmetric} +has a basis of orthonormal eigenvectors, then +\index{basis!orthonormal} +{\bf orthonormalBasis} computes a list of these vectors. +\index{orthonormal basis} + +\spadcommand{m3 := matrix [ [1,2],[2,1] ] } +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +2 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\spadcommand{orthonormalBasis(m3) } +$$ +\left[ +{\left[ +\begin{array}{c} +-{1 \over {\sqrt {2}}} \\ +{1 \over {\sqrt {2}}} +\end{array} +\right]}, + {\left[ +\begin{array}{c} +{1 \over {\sqrt {2}}} \\ +{1 \over {\sqrt {2}}} +\end{array} +\right]} +\right] +$$ +\returnType{Type: List Matrix Expression Integer} + +\section{Solution of Linear and Polynomial Equations} +\label{ugProblemLinPolEqn} +% +In this section we discuss the Axiom facilities for solving +systems of linear equations, finding the roots of polynomials and +\index{linear equation} +solving systems of polynomial equations. +For a discussion of the solution of differential equations, see +\ref{ugProblemDEQ} on page~\pageref{ugProblemDEQ}. + +\subsection{Solution of Systems of Linear Equations} +\label{ugxProblemLinSys} + +You can use the operation {\bf solve} to solve systems of linear equations. +\index{equation!linear!solving} + +The operation {\bf solve} takes two arguments, the list of equations and the +list of the unknowns to be solved for. +A system of linear equations need not have a unique solution. + +To solve the linear system: +$$ +\begin{array}{rcrcrcr} + x &+& y &+& z &=& 8 \\ +3 x &-& 2 y &+& z &=& 0 \\ + x &+& 2 y &+& 2 z &=& 17 +\end{array} +$$ +evaluate this expression. + +\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} +$$ +\left[ +{\left[ {x=-1}, {y=2}, {z=7} +\right]} +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +Parameters are given as new variables starting with a percent sign and +{\tt \%} and +the variables are expressed in terms of the parameters. +If the system has no solutions then the empty list is returned. + +When you solve the linear system +$$ +\begin{array}{rcrcrcr} + x&+&2 y&+&3 z&=&2 \\ +2 x&+&3 y&+&4 z&=&2 \\ +3 x&+&4 y&+&5 z&=&2 +\end{array} +$$ +with this expression +you get a solution involving a parameter. + +\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])} +$$ +\left[ +{\left[ {x={ \%Q -2}}, {y={-{2 \ \%Q}+2}}, {z= \%Q} +\right]} +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +The system can also be presented as a matrix and a vector. +The matrix contains the coefficients of the linear equations and +the vector contains the numbers appearing on the right-hand sides +of the equations. +You may input the matrix as a list of rows and the vector as a +list of its elements. + +To solve the system: +$$ +\begin{array}{rcrcrcr} + x&+& y&+& z&=&8 \\ +3 x&-&2 y&+& z&=&0 \\ + x&+&2 y&+&2 z&=&17 +\end{array} +$$ +in matrix form you would evaluate this expression. + +\spadcommand{solve([ [1,1,1],[3,-2,1],[1,2,2] ],[8,0,17])} +$$ +\left[ +{particular={\left[ -1, 2, 7\right]}}, +{basis={\left[ {\left[ 0, 0, 0\right]}\right]}} +\right] +$$ +\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), +basis: List Vector Fraction Integer)} + +The solutions are presented as a {\tt Record} with two +components: the component {\it particular} contains a particular solution of the given system or +the item {\tt "failed"} if there are no solutions, the component +{\it basis} contains a list of vectors that +are a basis for the space of solutions of the corresponding +homogeneous system. +If the system of linear equations does not have a unique solution, +then the {\it basis} component contains +non-trivial vectors. + +This happens when you solve the linear system +$$ +\begin{array}{rcrcrcr} + x&+&2 y&+&3 z&=&2 \\ +2 x&+&3 y&+&4 z&=&2 \\ +3 x&+&4 y&+&5 z&=&2 +\end{array} +$$ +with this command. + +\spadcommand{solve([ [1,2,3],[2,3,4],[3,4,5] ],[2,2,2])} +$$ +\left[ +{particular={\left[ -2, 2, 0 \right]}}, +{basis={\left[ {\left[ 1, -2, 1 \right]}\right]}} +\right] +$$ +\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), +basis: List Vector Fraction Integer)} + + +All solutions of this system are obtained by adding the particular +solution with a linear combination of the {\it basis} vectors. + +When no solution exists then {\tt "failed"} is returned as the +{\it particular} component, as follows: + +\spadcommand{solve([ [1,2,3],[2,3,4],[3,4,5] ],[2,3,2])} +$$ +\left[ +{particular= \mbox{\tt "failed"} }, +{basis={\left[ {\left[ 1, -2, 1\right]}\right]}} +\right] +$$ +\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), +basis: List Vector Fraction Integer)} + +When you want to solve a system of homogeneous equations (that is, +a system where the numbers on the right-hand sides of the +\index{nullspace} +equations are all zero) in the matrix form you can omit the second +argument and use the {\bf nullSpace} operation. + +This computes the solutions of the following system of equations: +$$ +\begin{array}{rcrcrcr} + x&+&2 y&+&3 z&=&0 \\ +2 x&+&3 y&+&4 z&=&0 \\ +3 x&+&4 y&+&5 z&=&0 +\end{array} +$$ +The result is given as a list of vectors and +these vectors form a basis for the solution space. + +\spadcommand{nullSpace([ [1,2,3],[2,3,4],[3,4,5] ])} +$$ +\left[ +{\left[ 1, -2, 1 \right]} +\right] +$$ +\returnType{Type: List Vector Integer} + +\subsection{Solution of a Single Polynomial Equation} +\label{ugxProblemOnePol} + +Axiom can solve polynomial equations producing either approximate +\index{polynomial!root finding} +or exact solutions. +\index{equation!polynomial!solving} +Exact solutions are either members of the ground +field or can be presented symbolically as roots of irreducible polynomials. + +This returns the one rational root along with an irreducible +polynomial describing the other solutions. + +\spadcommand{solve(x**3 = 8,x)} +$$ +\left[ +{x=2}, {{{x \sp 2}+{2 \ x}+4}=0} +\right] +$$ +\returnType{Type: List Equation Fraction Polynomial Integer} + +If you want solutions expressed in terms of radicals you would use this +instead. +\index{radical} + +\spadcommand{radicalSolve(x**3 = 8,x)} +$$ +\left[ +{x={-{\sqrt {-3}} -1}}, {x={{\sqrt {-3}} -1}}, {x=2} +\right] +$$ +\returnType{Type: List Equation Expression Integer} + +The {\bf solve} command always returns a value but +{\bf radicalSolve} returns only the solutions that it is +able to express in terms of radicals. +\index{radical} + +If the polynomial equation has rational coefficients +you can ask for approximations to its real roots by calling +solve with a second argument that specifies the ``precision'' +\index{precision} +$\epsilon$. +This means that each approximation will be within +$\pm\epsilon$ of the actual +result. + +Notice that the type of second argument controls the type of the result. + +\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)} +$$ +\left[ +{x={3.6180114746 09375}}, {x={1.3819885253 90625}} +\right] +$$ +\returnType{Type: List Equation Polynomial Float} + +If you give a floating-point precision you get a floating-point result; +if you give the precision as a rational number you get a rational result. + +\spadcommand{solve(x**3-2,1/1000)} +$$ +\left[ +{x={{2581} \over {2048}}} +\right] +$$ +\returnType{Type: List Equation Polynomial Fraction Integer} + +If you want approximate complex results you should use the +\index{approximation} +command {\bf complexSolve} that takes the same precision argument +$\epsilon$. + +\spadcommand{complexSolve(x**3-2,.0001)} +$$ +\begin{array}{@{}l} +\left[ +{x={1.2599182128 90625}}, +\right. +\\ +\\ +\displaystyle +{x={-{0.6298943279 5395613131} -{{1.0910949707 03125} \ i}}}, +\\ +\\ +\displaystyle +\left. +{x={-{0.6298943279 5395613131}+{{1.0910949707 03125} \ +i}}} +\right] +\end{array} +$$ +\returnType{Type: List Equation Polynomial Complex Float} + +Each approximation will be within +$\pm\epsilon$ of the actual result +in each of the real and imaginary parts. + +\spadcommand{complexSolve(x**2-2*\%i+1,1/100)} +$$ +\left[ +{x={-{{13028925} \over {16777216}} -{{{325} \over {256}} \ i}}}, +{x={{{13028925} \over {16777216}}+{{{325} \over {256}} \ i}}} +\right] +$$ +\returnType{Type: List Equation Polynomial Complex Fraction Integer} + +Note that if you omit the {\tt =} from the first argument +Axiom generates an equation by equating the first argument to zero. +Also, when only one variable is present in the equation, you +do not need to specify the variable to be solved for, that is, +you can omit the second argument. + +Axiom can also solve equations involving rational functions. +Solutions where the denominator vanishes are discarded. + +\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)} +$$ +\left[ +{x={{-{\sqrt {-3}} -1} \over 2}}, {x={{{\sqrt {-3}} -1} \over 2}} +\right] +$$ +\returnType{Type: List Equation Expression Integer} + +\subsection{Solution of Systems of Polynomial Equations} +\label{ugxProblemPolSys} + +Given a system of equations of rational functions with exact coefficients: +\index{equation!polynomial!solving} +\vskip 0.1cm +$$ +\begin{array}{c} +p_1(x_1, \ldots, x_n) \\ \vdots \\ p_m(x_1,\ldots,x_n) +\end{array} +$$ + +Axiom can find +numeric or symbolic solutions. +The system is first split into irreducible components, then for +each component, a triangular system of equations is found that reduces +the problem to sequential solution of univariate polynomials resulting +from substitution of partial solutions from the previous stage. +$$ +\begin{array}{c} +q_1(x_1, \ldots, x_n) \\ \vdots \\ q_m(x_n) +\end{array} +$$ + +Symbolic solutions can be presented using ``implicit'' algebraic numbers +defined as roots of irreducible polynomials or in terms of radicals. +Axiom can also find approximations to the real or complex roots +of a system of polynomial equations to any user-specified accuracy. + +The operation {\bf solve} for systems is used in a way similar +to {\bf solve} for single equations. +Instead of a polynomial equation, one has to give a list of +equations and instead of a single variable to solve for, a list of +variables. +For solutions of single equations see +\ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol}. + +Use the operation {\bf solve} if you want implicitly presented +solutions. + +\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])} +$$ +\left[ +{\left[ {x=-1}, {y=2} \right]}, +{\left[ {{{x \sp 2} -x+1}=0}, {y=2} \right]}, +{\left[ {{{3 \ {x \sp 3}} -1}=0}, {y=-2} \right]} +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])} +$$ +\left[ +{\left[ {x={z \over 3}}, +{y={{{3 \ {z \sp 2}}+z+9} \over 3}}, +{{{9 \ {z \sp 4}}+{6 \ {z \sp 3}}+{{55} \ {z \sp 2}}+{{15} \ z} -{90}}=0} +\right]} +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +Use {\bf radicalSolve} if you want your solutions expressed +in terms of radicals. + +\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])} +$$ +\begin{array}{@{}l} +\left[ +{\left[ {x={{{\sqrt {-3}}+1} \over 2}}, {y=2} \right]}, +{\left[ {x={{-{\sqrt {-3}}+1} \over 2}}, {y=2} \right]}, +\right. +\\ +\\ +\displaystyle +{\left[ {x={{-{{\sqrt {-1}} \ {\sqrt {3}}} -1} \over {2 \ {\root {3} \of +{3}}}}}, {y=-2} \right]}, +{\left[ {x={{{{\sqrt {-1}} \ {\sqrt {3}}} -1} \over {2 \ {\root {3} \of +{3}}}}}, {y=-2} \right]}, +\\ +\\ +\displaystyle +\left. +{\left[ {x={1 \over {\root {3} \of {3}}}}, {y=-2} \right]}, +{\left[ {x=-1}, {y=2} \right]} +\right] +\end{array} +$$ +\returnType{Type: List List Equation Expression Integer} + +To get numeric solutions you only need to give the list of +equations and the precision desired. +The list of variables would be redundant information since there +can be no parameters for the numerical solver. + +If the precision is expressed as a floating-point number you get +results expressed as floats. + +\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)} +$$ +\left[ +{\left[ {y={1.5859375}}, {x={0.79296875}} \right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Float} + +To get complex numeric solutions, use the operation {\bf complexSolve}, +which takes the same arguments as in the real case. + +\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)} +$$ +\begin{array}{@{}l} +\left[ +{\left[ {y={{1625} \over {1024}}}, {x={{1625} \over {2048}}} \right]}, +\right. +\\ +\\ +\displaystyle +{\left[ {y={-{{435445573689} \over {549755813888}} -{{{1407} \over {1024}} +\ i}}}, +{x={-{{435445573689} \over {1099511627776}} -{{{1407} \over {2048}} \ i}}} +\right]}, +\\ +\\ +\displaystyle +\left. +{\left[ {y={-{{435445573689} \over {549755813888}}+{{{1407} \over {1024}} +\ i}}}, +{x={-{{435445573689} \over {1099511627776}}+{{{1407} \over {2048}} \ i}}} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List List Equation Polynomial Complex Fraction Integer} + + +It is also possible to solve systems of equations in rational functions +over the rational numbers. +Note that $[x = 0.0, a = 0.0]$ is not returned as a solution since +the denominator vanishes there. + +\spadcommand{solve([x**2/a = a,a = a*x],.001)} +$$ +\left[ +{\left[ {x={1.0}}, {a=-{1.0}} \right]}, +{\left[ {x={1.0}}, {a={1.0}} \right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Float} + + +When solving equations with +denominators, all solutions where the denominator vanishes are +discarded. + +\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])} +$$ +\begin{array}{@{}l} +\left[ +{\left[ {x=-{\sqrt {{{-{a \sp 4}+{2 \ {a \sp 3}}+{3 \ {a \sp 2}}+{3 \ +a}+1} \over {a \sp 2}}}}}, {y={{-a -1} \over a}} +\right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ {x={\sqrt {{{-{a \sp 4}+{2 \ {a \sp 3}}+{3 \ {a \sp 2}}+{3 \ +a}+1} \over {a \sp 2}}}}}, {y={{-a -1} \over a}} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List List Equation Expression Integer} + +\section{Limits} +\label{ugProblemLimits} + +% +To compute a limit, you must specify a functional expression, +\index{limit} +a variable, and a limiting value for that variable. +If you do not specify a direction, Axiom attempts to +compute a two-sided limit. + +Issue this to compute the limit +$$\lim_{x \rightarrow 1}{{\displaystyle x^2 - 3x + +2}\over{\displaystyle x^2 - 1}}.$$ + +\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} +$$ +-{1 \over 2} +$$ +\returnType{Type: Union(OrderedCompletion Fraction Polynomial Integer,...)} + +Sometimes the limit when approached from the left is different from +the limit from the right and, in this case, you may wish to ask for a +one-sided limit. Also, if you have a function that is only defined on +one side of a particular value, \index{limit!one-sided vs. two-sided} +you can compute a one-sided limit. + +The function $log(x)$ is only defined to the right of zero, that is, +for $x > 0$. Thus, when computing limits of functions involving +$log(x)$, you probably want a ``right-hand'' limit. + +\spadcommand{limit(x * log(x),x = 0,"right")} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +When you do not specify ``$right$'' or ``$left$'' as the optional fourth +argument, {\bf limit} tries to compute a two-sided limit. Here the +limit from the left does not exist, as Axiom indicates when you try to +take a two-sided limit. + +\spadcommand{limit(x * log(x),x = 0)} +$$ +\left[ +{leftHandLimit= \mbox{\tt "failed"} }, {rightHandLimit=0} +\right] +$$ +\returnType{Type: Union(Record(leftHandLimit: +Union(OrderedCompletion Expression Integer,"failed"), +rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} + +A function can be defined on both sides of a particular value, but +tend to different limits as its variable approaches that value from +the left and from the right. We can construct an example of this as +follows: Since $\sqrt{y^2}$ is simply the absolute value of $y$, the +function $\sqrt{y^2} / y$ is simply the sign ($+1$ or $-1$) of the +nonzero real number $y$. Therefore, $\sqrt{y^2} / y = -1$ for $y < 0$ +and $\sqrt{y^2} / y = +1$ for $y > 0$. + +This is what happens when we take the limit at $y = 0$. +The answer returned by Axiom gives both a +``left-hand'' and a ``right-hand'' limit. + +\spadcommand{limit(sqrt(y**2)/y,y = 0)} +$$ +\left[ +{leftHandLimit=-1}, {rightHandLimit=1} +\right] +$$ +\returnType{Type: Union(Record(leftHandLimit: +Union(OrderedCompletion Expression Integer,"failed"), +rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} + +Here is another example, this time using a more complicated function. + +\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} +$$ +\left[ +{leftHandLimit=-{1 \over {\sqrt {2}}}}, +{rightHandLimit={1 \over {\sqrt {2}}}} +\right] +$$ +\returnType{Type: Union(Record(leftHandLimit: +Union(OrderedCompletion Expression Integer,"failed"), +rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} + +You can compute limits at infinity by passing either +\index{limit!at infinity} $+\infty$ or $-\infty$ as the third +argument of {\bf limit}. + +To do this, use the constants $\%plusInfinity$ and $\%minusInfinity$. + +\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} +$$ +{\sqrt {3}} \over 5 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} +$$ +-{{\sqrt {3}} \over 5} +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +You can take limits of functions with parameters. +\index{limit!of function with parameters} +As you can see, the limit is expressed in terms of the parameters. + +\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} +$$ +a \over b +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +When you use {\bf limit}, you are taking the limit of a real +function of a real variable. + +When you compute this, Axiom returns $0$ because, as a function of a +real variable, $sin(1/z)$ is always between $-1$ and $1$, so +$z * sin(1/z)$ tends to $0$ as $z$ tends to $0$. + +\spadcommand{limit(z * sin(1/z),z = 0)} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +However, as a function of a {\it complex} variable, $sin(1/z)$ is badly +\index{limit!real vs. complex} +behaved near $0$ (one says that $sin(1/z)$ has an +\index{essential singularity} +{\it essential singularity} at $z = 0$). +\index{singularity!essential} + +When viewed as a function of a complex variable, $z * sin(1/z)$ +does not approach any limit as $z$ tends to $0$ in the complex plane. +Axiom indicates this when we call {\bf complexLimit}. + +\spadcommand{complexLimit(z * sin(1/z),z = 0)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +Here is another example. +As $x$ approaches $0$ along the real axis, $exp(-1/x**2)$ +tends to $0$. +\spadcommand{limit(exp(-1/x**2),x = 0)} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +However, if $x$ is allowed to approach $0$ along any path in the +complex plane, the limiting value of $exp(-1/x**2)$ depends on the +path taken because the function has an essential singularity at $x=0$. +This is reflected in the error message returned by the function. +\spadcommand{complexLimit(exp(-1/x**2),x = 0)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +You can also take complex limits at infinity, that is, limits of a +function of $z$ as $z$ approaches infinity on the Riemann sphere. Use +the symbol $\%infinity$ to denote ``complex infinity.'' + +As above, to compute complex limits rather than real limits, use +{\bf complexLimit}. + +\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} +$$ +-1 +$$ +\returnType{Type: OnePointCompletion Fraction Polynomial Integer} + +In many cases, a limit of a real function of a real variable exists +when the corresponding complex limit does not. This limit exists. + +\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +But this limit does not. + +\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\section{Laplace Transforms} +\label{ugProblemLaplace} + +Axiom can compute some forward Laplace transforms, mostly +\index{Laplace transform} of elementary \index{function!elementary} +functions \index{transform!Laplace} not involving logarithms, although +some cases of special functions are handled. + +To compute the forward Laplace transform of $F(t)$ with respect to +$t$ and express the result as $f(s)$, issue the command +$laplace(F(t), t, s)$. + +\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} +$$ +{4 \ {a \sp 3}} \over {{s \sp 4}+{4 \ {a \sp 4}}} +$$ +\returnType{Type: Expression Integer} + +Here are some other non-trivial examples. + +\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} +$$ +-{\log \left({{s -a}} \right)}+{\log\left({{s -b}} \right)} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} +$$ +{\log \left({{{s \sp 2}+{a \sp 2}}} \right)}-{2 \ {\log \left({s} \right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)} +$$ +1 \over {{b \ {s \sp 2}}+{2 \ a \ b \ s}+{b \sp 3}+{{a \sp 2} \ b}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} +$$ +{{\log \left({{{s \sp 2}+{b \sp 2}}} \right)}- +{\log \left({{{s \sp 2}+{a \sp 2}}} \right)}} +\over 2 +$$ +\returnType{Type: Expression Integer} + +Axiom also knows about a few special functions. + +\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} +$$ +{{e \sp b} \ {\log \left({{{s+c -a} \over c}} \right)}}\over {s -a} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} +$$ +{{a \ {\log \left({{{{s \sp 2}+{b \sp 2}} \over {b \sp 2}}} \right)}}+ +{2\ c \ {\arctan \left({{d \over s}} \right)}}}\over {2 \ s} +$$ +\returnType{Type: Expression Integer} + +When Axiom does not know about a particular transform, +it keeps it as a formal transform in the answer. + +\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)} +$$ +{{{\left( {s \sp 4}+{2 \ {a \sp 2} \ {s \sp 2}}+{a \sp 4} \right)} +\ {laplace \left({{e \sp {t \sp 2}}, t, s} \right)}}+ +{2\ {a \sp 3}}} \over {{s \sp 4}+{2 \ {a \sp 2} \ {s \sp 2}}+{a \sp 4}} +$$ +\returnType{Type: Expression Integer} + +\section{Integration} +\label{ugProblemIntegration} + +% +Integration is the reverse process of differentiation, that is, +\index{integration} an {\it integral} of a function $f$ with respect +to a variable $x$ is any function $g$ such that $D(g,x)$ is equal to +$f$. + +The package {\tt FunctionSpaceIntegration} provides the top-level +integration operation, \spadfunFrom{integrate}{FunctionSpaceIntegration}, +for integrating real-valued elementary functions. +\index{FunctionSpaceIntegration} + +\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)} +$$ +{{{\sinh \left({{a \ x}} \right)}\sp 2}+ +{{\cosh \left({{a \ x}} \right)}\sp 2}} +\over {4 \ a} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Unfortunately, antiderivatives of most functions cannot be expressed in +terms of elementary functions. + +\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)} +$$ +\int \sp{\displaystyle x} {{{\log +\left( +{{{\sqrt {{b+{ \%M \ a}}}}+1}} +\right)} +\over \%M} \ {d \%M}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Given an elementary function to integrate, Axiom returns a formal +integral as above only when it can prove that the integral is not +elementary and not when it cannot determine the integral. +In this rare case it prints a message that it cannot +determine if an elementary integral exists. + +Similar functions may have antiderivatives \index{antiderivative} +that look quite different because the form of the antiderivative +depends on the sign of a constant that appears in the function. + +\spadcommand{integrate(1/(x**2 - 2),x)} +$$ +{\log \left({{{{{\left( {x \sp 2}+2 \right)}\ {\sqrt {2}}} -{4 \ x}} +\over {{x \sp 2} -2}}} \right)} +\over {2 \ {\sqrt {2}}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +\spadcommand{integrate(1/(x**2 + 2),x)} +$$ +{\arctan \left({{{x \ {\sqrt {2}}} \over 2}} \right)}\over {\sqrt {2}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +If the integrand contains parameters, then there may be several possible +antiderivatives, depending on the signs of expressions of the parameters. + +In this case Axiom returns a list of answers that cover all the +possible cases. Here you use the answer involving the square root of +$a$ when $a > 0$ and \index{integration!result as list of real +functions} the answer involving the square root of $-a$ when $a < 0$. + +\spadcommand{integrate(x**2 / (x**4 - a**2), x)} +$$ +\begin{array}{@{}l} +\left[ +{{{\log +\left( +{{{{{\left( {x \sp 2}+a \right)} +\ {\sqrt {a}}} -{2 \ a \ x}} \over {{x \sp 2} -a}}} +\right)}+ +{2\ {\arctan \left({{{x \ {\sqrt {a}}} \over a}} \right)}}} +\over {4 \ {\sqrt {a}}}}, +\right. +\\ +\\ +\displaystyle +\left. +{{{\log \left({{{{{\left( {x \sp 2} -a \right)} +\ {\sqrt {-a}}}+{2 \ a \ x}} \over {{x \sp 2}+a}}} +\right)} +-{2 \ {\arctan \left({{{x \ {\sqrt {-a}}} \over a}} \right)}}} +\over {4 \ {\sqrt {-a}}}} +\right] +\end{array} +$$ +\returnType{Type: Union(List Expression Integer,...)} + +If the parameters and the variables of integration can be complex +numbers rather than real, then the notion of sign is not defined. In +this case all the possible answers can be expressed as one complex +function. To get that function, rather than a list of real functions, +use \spadfunFrom{complexIntegrate}{FunctionSpaceComplexIntegration}, +which is provided by the package \index{integration!result as a +complex functions} {\tt FunctionSpaceComplexIntegration}. +\index{FunctionSpaceComplexIntegration} + +This operation is used for integrating complex-valued elementary +functions. + +\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)} +$$ +\left( +\begin{array}{@{}l} +{{\sqrt {{4 \ a}}} \ {\log +\left( +{{{{x \ {\sqrt {-{4 \ a}}}}+{2 \ a}} \over {\sqrt {-{4 \ a}}}}} +\right)}} - +{{\sqrt {-{4 \ a}}} \ {\log +\left( +{{{{x \ {\sqrt {{4 \ a}}}}+{2 \ a}} \over {\sqrt {{4 \ a}}}}} +\right)}}+ +\\ +\\ +\displaystyle +{{\sqrt{-{4 \ a}}} \ {\log +\left( +{{{{x \ {\sqrt {{4 \ a}}}} -{2 \ a}} \over {\sqrt {{4 \ a}}}}} +\right)}} +-{{\sqrt {{4 \ a}}} \ {\log +\left( +{{{{x \ {\sqrt {-{4 \ a}}}} -{2 \ a}} \over {\sqrt {-{4 \ a}}}}} +\right)}} +\end{array} +\right) +\over {2 \ {\sqrt {-{4 \ a}}} \ {\sqrt {{4 \ a}}}} +$$ +\returnType{Type: Expression Integer} + +As with the real case, antiderivatives for most complex-valued +functions cannot be expressed in terms of elementary functions. + +\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} +$$ +\int \sp{\displaystyle x} +{{{\log \left({{{\sqrt {{b+{ \%M \ a}}}}+1}} \right)} +\over \%M} \ {d \%M}} +$$ +\returnType{Type: Expression Integer} + +Sometimes {\bf integrate} can involve symbolic algebraic numbers +such as those returned by \spadfunFrom{rootOf}{Expression}. +To see how to work with these strange generated symbols (such as +$\%\%a0$), see +\ref{ugxProblemSymRootAll} on page~\pageref{ugxProblemSymRootAll}. + +Definite integration is the process of computing the area between +\index{integration!definite} +the $x$-axis and the curve of a function $f(x)$. +The fundamental theorem of calculus states that if $f$ is +continuous on an interval $a..b$ and if there exists a function $g$ +that is differentiable on $a..b$ and such that $D(g, x)$ +is equal to $f$, then the definite integral of $f$ +for $x$ in the interval $a..b$ is equal to $g(b) - g(a)$. + +The package {\tt RationalFunctionDefiniteIntegration} provides +the top-level definite integration operation, +\spadfunFrom{integrate}{RationalFunctionDefiniteIntegration}, +for integrating real-valued rational functions. + +\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)} +$$ +{{2 \ {\arctan \left({8} \right)}}+ +{2\ {\arctan \left({5} \right)}}+ +{2\ {\arctan \left({2} \right)}}+ +{2\ {\arctan \left({{1 \over 2}} \right)}} +-\pi} \over 2 +$$ +\returnType{Type: Union(f1: OrderedCompletion Expression Integer,...)} + +Axiom checks beforehand that the function you are integrating is +defined on the interval $a..b$, and prints an error message if it +finds that this is not case, as in the following example: +\begin{verbatim} +integrate(1/(x**2-2), x = 1..2) + + >> Error detected within library code: + Pole in path of integration + You are being returned to the top level + of the interpreter. +\end{verbatim} +When parameters are present in the function, the function may or may not be +defined on the interval of integration. + +If this is the case, Axiom issues a warning that a pole might +lie in the path of integration, and does not compute the integral. + +\spadcommand{integrate(1/(x**2-a), x = 1..2)} +$$ +potentialPole +$$ +\returnType{Type: Union(pole: potentialPole,...)} + +If you know that you are using values of the parameter for which +the function has no pole in the interval of integration, use the +string {\tt ``noPole''} as a third argument to +\spadfunFrom{integrate}{RationalFunctionDefiniteIntegration}: + +The value here is, of course, incorrect if $sqrt(a)$ is between +$1$ and $2.$ + +\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")} +$$ +\begin{array}{@{}l} +\left[ +\left( +\begin{array}{@{}l} +-{\log \left({{{{{\left( -{4 \ {a \sp 2}} -{4 \ a} \right)} +\ {\sqrt {a}}}+{a \sp 3}+{6 \ {a \sp 2}}+a} \over {{a \sp 2} -{2 \ a}+1}}} +\right)}+ +\\ +\\ +\displaystyle +{\log\left({{{{{\left( -{8 \ {a \sp 2}} -{{32} \ a} \right)} +\ {\sqrt {a}}}+{a \sp 3}+{{24} \ {a \sp 2}}+{{16} \ a}} \over {{a \sp 2} +-{8 \ a}+{16}}}} +\right)} +\end{array} +\right) +\over {4 \ {\sqrt {a}}}, +\right. +\\ +\\ +\displaystyle +\left. +{{-{\arctan \left({{{2 \ {\sqrt {-a}}} \over a}} \right)}+ +{\arctan\left({{{\sqrt {-a}} \over a}} \right)}} +\over {\sqrt {-a}}} +\right] +\end{array} +$$ +\returnType{Type: Union(f2: List OrderedCompletion Expression Integer,...)} + +\section{Working with Power Series} +\label{ugProblemSeries} +% +Axiom has very sophisticated facilities for working with power +\index{series} +series. +\index{power series} + +Infinite series are represented by a list of the coefficients that +have already been determined, together with a function for computing +the additional coefficients if needed. + +The system command that determines how many terms of a series is +displayed is {\tt )set streams calculate}. For the purposes of this +book, we have used this system command to display fewer than ten +terms. \index{set streams calculate} Series can be created from +expressions, from functions for the series coefficients, and from +applications of operations on existing series. The most general +function for creating a series is called {\bf series}, although you +can also use {\bf taylor}, {\bf laurent} and {\bf puiseux} in +situations where you know what kind of exponents are involved. + +For information about solving differential equations in terms of +power series, see +\ref{ugxProblemDEQSeries} on page~\pageref{ugxProblemDEQSeries}. + +\subsection{Creation of Power Series} +\label{ugxProblemSeriesCreate} + +This is the easiest way to create a power series. This tells Axiom +that $x$ is to be treated as a power series, \index{series!creating} +so functions of $x$ are again power series. + +\spadcommand{x := series 'x } +$$ +x +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +We didn't say anything about the coefficients of the power series, so +the coefficients are general expressions over the integers. This +allows us to introduce denominators, symbolic constants, and other +variables as needed. + +Here the coefficients are integers (note that the coefficients are the +Fibonacci \index{Fibonacci numbers} numbers). + +\spadcommand{1/(1 - x - x**2) } +$$ +1+x+ +{2 \ {x \sp 2}}+ +{3 \ {x \sp 3}}+ +{5 \ {x \sp 4}}+ +{8 \ {x \sp 5}}+ +{{13} \ {x \sp 6}}+ +{{21} \ {x \sp 7}}+ +{{34} \ {x \sp 8}}+ +{{55} \ {x \sp 9}}+ +{{89} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +This series has coefficients that are rational numbers. + +\spadcommand{sin(x) } +$$ +x - +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {120}} \ {x \sp 5}} - +{{1 \over {5040}} \ {x \sp 7}}+ +{{1 \over {362880}} \ {x \sp 9}} - +{{1 \over {39916800}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +When you enter this expression you introduce the symbolic constants +$sin(1)$ and $cos(1).$ + +\spadcommand{sin(1 + x) } +$$ +\begin{array}{@{}l} +{\sin \left({1} \right)}+ +{{\cos\left({1} \right)}\ x} - +{{{\sin \left({1} \right)}\over 2} \ {x \sp 2}} - +{{{\cos \left({1} \right)}\over 6} \ {x \sp 3}}+ +{{{\sin \left({1} \right)}\over {24}} \ {x \sp 4}}+ +{{{\cos \left({1} \right)}\over {120}} \ {x \sp 5}} - +{{{\sin \left({1} \right)}\over {720}} \ {x \sp 6}} - +\\ +\\ +\displaystyle +{{{\cos \left({1} \right)}\over {5040}} \ {x \sp 7}}+ +{{{\sin \left({1} \right)}\over {40320}} \ {x \sp 8}}+ +{{{\cos \left({1} \right)}\over {362880}} \ {x \sp 9}} - +{{{\sin \left({1} \right)}\over {3628800}} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +When you enter the expression +the variable $a$ appears in the resulting series expansion. + +\spadcommand{sin(a * x) } +$$ +{a \ x} - +{{{a \sp 3} \over 6} \ {x \sp 3}}+ +{{{a \sp 5} \over {120}} \ {x \sp 5}} - +{{{a \sp 7} \over {5040}} \ {x \sp 7}}+ +{{{a \sp 9} \over {362880}} \ {x \sp 9}} - +{{{a \sp {11}} \over {39916800}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +You can also convert an expression into a series expansion. This +expression creates the series expansion of $1/log(y)$ about $y = 1$. +For details and more examples, see \ref{ugxProblemSeriesConversions} +on page~\pageref{ugxProblemSeriesConversions}. + +\spadcommand{series(1/log(y),y = 1)} +$$ +\begin{array}{@{}l} +{{\left( y -1 \right)}\sp {\left( -1 \right)}}+ +{1\over 2} -{{1 \over {12}} \ {\left( y -1 \right)}}+ +{{1\over {24}} \ {{\left( y -1 \right)}\sp 2}} - +{{{19} \over {720}} \ {{\left( y -1 \right)}\sp 3}}+ +{{3 \over {160}} \ {{\left( y -1 \right)}\sp 4}} - +\\ +\\ +\displaystyle +{{{863} \over {60480}} \ {{\left( y -1 \right)}\sp 5}}+ +{{{275} \over {24192}} \ {{\left( y -1 \right)}\sp 6}} - +{{{33953} \over {3628800}} \ {{\left( y -1 \right)}\sp 7}}+ +\\ +\\ +\displaystyle +{{{8183} \over {1036800}} \ {{\left( y -1 \right)}\sp 8}} - +{{{3250433} \over {479001600}} \ {{\left( y -1 \right)}\sp 9}}+ +{O \left({{{\left( y -1 \right)}\sp {10}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,y,1)} + +You can create power series with more general coefficients. You +normally accomplish this via a type declaration (see +\ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}). +See \ref{ugxProblemSeriesFunctions} on +page~\pageref{ugxProblemSeriesFunctions} for some warnings about working with +declared series. + +We declare that $y$ is a one-variable Taylor series +\index{series!Taylor} ({\tt UTS} is the abbreviation for +{\tt UnivariateTaylorSeries}) in the variable $z$ with {\tt FLOAT} +(that is, floating-point) coefficients, centered about $0.$ Then, by +assignment, we obtain the Taylor expansion of $exp(z)$ with +floating-point coefficients. \index{UnivariateTaylorSeries} + +\spadcommand{y : UTS(FLOAT,'z,0) := exp(z) } +$$ +\begin{array}{@{}l} +{1.0}+z+ +{{0.5} \ {z \sp 2}}+ +{{0.1666666666\ 6666666667} \ {z \sp 3}}+ +\\ +\\ +\displaystyle +{{0.0416666666\ 6666666666 7} \ {z \sp 4}}+ +{{0.0083333333\ 3333333333 34} \ {z \sp 5}}+ +\\ +\\ +\displaystyle +{{0.0013888888\ 8888888888 89} \ {z \sp 6}}+ +{{0.0001984126\ 9841269841 27} \ {z \sp 7}}+ +\\ +\\ +\displaystyle +{{0.0000248015\ 8730158730 1587} \ {z \sp 8}}+ +{{0.0000027557\ 3192239858 90653} \ {z \sp 9}}+ +\\ +\\ +\displaystyle +{{0.2755731922\ 3985890653 E -6} \ {z \sp {10}}}+ +{O \left({{z \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Float,z,0.0)} + +You can also create a power series by giving an explicit formula for +its $n$-th coefficient. For details and more examples, see +\ref{ugxProblemSeriesFormula} on page~\pageref{ugxProblemSeriesFormula}. + +To create a series about $w = 0$ whose $n$-th Taylor coefficient is +$1/n!$, you can evaluate this expression. This is the Taylor +expansion of $exp(w)$ at $w = 0$. + +\spadcommand{series(1/factorial(n),n,w = 0)} +$$ +\begin{array}{@{}l} +1+w+ +{{1 \over 2} \ {w \sp 2}}+ +{{1 \over 6} \ {w \sp 3}}+ +{{1 \over {24}} \ {w \sp 4}}+ +{{1 \over {120}} \ {w \sp 5}}+ +{{1 \over {720}} \ {w \sp 6}}+ +{{1 \over {5040}} \ {w \sp 7}}+ +\\ +\\ +\displaystyle +{{1 \over {40320}} \ {w \sp 8}}+ +{{1 \over {362880}} \ {w \sp 9}}+ +{{1 \over {3628800}} \ {w \sp {10}}}+ +{O \left({{w \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,w,0)} + +\subsection{Coefficients of Power Series} +\label{ugxProblemSeriesCoefficients} + +You can extract any coefficient from a power series---even one that +hasn't been computed yet. This is possible because in Axiom, infinite +series are represented by a list of the coefficients that have already +been determined, together with a function for computing the additional +coefficients. (This is known as {\it lazy evaluation}.) When you ask +for a \index{series!lazy evaluation} coefficient that hasn't yet been +computed, Axiom computes \index{lazy evaluation} whatever additional +coefficients it needs and then stores them in the representation of +the power series. + +Here's an example of how to extract the coefficients of a power series. +\index{series!extracting coefficients} + +\spadcommand{x := series(x) } +$$ +x +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{y := exp(x) * sin(x) } +$$ +\begin{array}{@{}l} +x+ +{x \sp 2}+ +{{1 \over 3} \ {x \sp 3}} - +{{1 \over {30}} \ {x \sp 5}} - +{{1 \over {90}} \ {x \sp 6}} - +{{1 \over {630}} \ {x \sp 7}}+ +{{1 \over {22680}} \ {x \sp 9}}+ +\\ +\\ +\displaystyle +{{1 \over {113400}} \ {x \sp {10}}}+ +{{1 \over {1247400}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +This coefficient is readily available. + +\spadcommand{coefficient(y,6) } +$$ +-{1 \over {90}} +$$ +\returnType{Type: Expression Integer} + +But let's get the fifteenth coefficient of $y$. + +\spadcommand{coefficient(y,15) } +$$ +-{1 \over {10216206000}} +$$ +\returnType{Type: Expression Integer} + +If you look at $y$ then you see that the coefficients up to order $15$ +have all been computed. + +\spadcommand{y } +$$ +\begin{array}{@{}l} +x+ +{x \sp 2}+ +{{1 \over 3} \ {x \sp 3}} - +{{1 \over {30}} \ {x \sp 5}} - +{{1 \over {90}} \ {x \sp 6}} - +{{1 \over {630}} \ {x \sp 7}}+ +{{1 \over {22680}} \ {x \sp 9}}+ +{{1 \over {113400}} \ {x \sp {10}}}+ +\\ +\\ +\displaystyle +{{1 \over {1247400}} \ {x \sp {11}}} - +{{1 \over {97297200}} \ {x \sp {13}}} - +{{1 \over {681080400}} \ {x \sp {14}}} - +{{1 \over {10216206000}} \ {x \sp {15}}}+ +{O \left({{x \sp {16}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\subsection{Power Series Arithmetic} +\label{ugxProblemSeriesArithmetic} + +You can manipulate power series using the usual arithmetic operations +\index{series!arithmetic} +$+$, $-$, $*$, and $/$ (from UnivariatePuiseuxSeries) + +The results of these operations are also power series. + +\spadcommand{x := series x } +$$ +x +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{(3 + x) / (1 + 7*x)} +$$ +\begin{array}{@{}l} +3 - +{{20} \ x}+ +{{140} \ {x \sp 2}} - +{{980} \ {x \sp 3}}+ +{{6860} \ {x \sp 4}} - +{{48020} \ {x \sp 5}}+ +{{336140} \ {x \sp 6}} - +{{2352980} \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{{16470860} \ {x \sp 8}} - +{{115296020} \ {x \sp 9}}+ +{{807072140} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +You can also compute $f(x) ** g(x)$, where $f(x)$ and $g(x)$ +are two power series. + +\spadcommand{base := 1 / (1 - x) } +$$ +1+x+ +{x \sp 2}+ +{x \sp 3}+ +{x \sp 4}+ +{x \sp 5}+ +{x \sp 6}+ +{x \sp 7}+ +{x \sp 8}+ +{x \sp 9}+ +{x \sp {10}}+ +{O \left({{x \sp {11}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{expon := x * base } +$$ +x+ +{x \sp 2}+ +{x \sp 3}+ +{x \sp 4}+ +{x \sp 5}+ +{x \sp 6}+ +{x \sp 7}+ +{x \sp 8}+ +{x \sp 9}+ +{x \sp {10}}+ +{x \sp {11}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{base ** expon } +$$ +\begin{array}{@{}l} +1+ +{x \sp 2}+ +{{3 \over 2} \ {x \sp 3}}+ +{{7 \over 3} \ {x \sp 4}}+ +{{{43} \over {12}} \ {x \sp 5}}+ +{{{649} \over {120}} \ {x \sp 6}}+ +{{{241} \over {30}} \ {x \sp 7}}+ +{{{3706} \over {315}} \ {x \sp 8}}+ +\\ +\\ +\displaystyle +{{{85763} \over {5040}} \ {x \sp 9}}+ +{{{245339} \over {10080}} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\subsection{Functions on Power Series} +\label{ugxProblemSeriesFunctions} + +Once you have created a power series, you can apply transcendental +functions +(for example, {\bf exp}, {\bf log}, {\bf sin}, {\bf tan}, +{\bf cosh}, etc.) to it. + +To demonstrate this, we first create the power series +expansion of the rational function + +$${\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}$$ + +about $x = 0$. + +\spadcommand{x := series 'x } +$$ +x +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\spadcommand{rat := x**2 / (1 - 6*x + x**2) } +$$ +\begin{array}{@{}l} +{x \sp 2}+ +{6 \ {x \sp 3}}+ +{{35} \ {x \sp 4}}+ +{{204} \ {x \sp 5}}+ +{{1189} \ {x \sp 6}}+ +{{6930} \ {x \sp 7}}+ +{{40391} \ {x \sp 8}}+ +{{235416} \ {x \sp 9}}+ +\\ +\\ +\displaystyle +{{1372105} \ {x \sp {10}}}+ +{{7997214} \ {x \sp {11}}}+ +{{46611179} \ {x \sp {12}}}+ +{O \left({{x \sp {13}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +If you want to compute the series expansion of + +$$\sin\left({\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}\right)$$ + +you simply compute the sine of $rat$. + +\spadcommand{sin(rat) } +$$ +\begin{array}{@{}l} +{x \sp 2}+ +{6 \ {x \sp 3}}+ +{{35} \ {x \sp 4}}+ +{{204} \ {x \sp 5}}+ +{{{7133} \over 6} \ {x \sp 6}}+ +{{6927} \ {x \sp 7}}+ +{{{80711} \over 2} \ {x \sp 8}}+ +{{235068} \ {x \sp 9}}+ +\\ +\\ +\displaystyle +{{{164285281} \over {120}} \ {x \sp {10}}}+ +{{{31888513} \over 4} \ {x \sp {11}}}+ +{{{371324777} \over 8} \ {x \sp {12}}}+ +{O \left({{x \sp {13}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\boxed{4.6in}{ +\vskip 0.1cm +\noindent {\bf Warning:} +the type of the coefficients of a power series may +affect the kind of computations that you can do with that series. +This can only happen when you have made a declaration to +specify a series domain with a certain type of coefficient.\\ +} + +If you evaluate then you have declared that $y$ is a one variable +Taylor series \index{series!Taylor} ({\tt UTS} is the abbreviation for +{\tt UnivariateTaylorSeries}) in the variable $y$ with {\tt FRAC INT} +(that is, fractions of integer) coefficients, centered about $0$. + +\spadcommand{y : UTS(FRAC INT,y,0) := y } +$$ +y +$$ +\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} + +You can now compute certain power series in $y$, {\it provided} that +these series have rational coefficients. + +\spadcommand{exp(y) } +$$ +\begin{array}{@{}l} +1+y+ +{{1 \over 2} \ {y \sp 2}}+ +{{1 \over 6} \ {y \sp 3}}+ +{{1 \over {24}} \ {y \sp 4}}+ +{{1 \over {120}} \ {y \sp 5}}+ +{{1 \over {720}} \ {y \sp 6}}+ +{{1 \over {5040}} \ {y \sp 7}}+ +{{1 \over {40320}} \ {y \sp 8}}+ +\\ +\\ +\displaystyle +{{1 \over {362880}} \ {y \sp 9}}+ +{{1 \over {3628800}} \ {y \sp {10}}}+ +{O \left({{y \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} + +You can get examples of such series by applying transcendental +functions to series in $y$ that have no constant terms. + +\spadcommand{tan(y**2) } +$$ +{y \sp 2}+ +{{1 \over 3} \ {y \sp 6}}+ +{{2 \over {15}} \ {y \sp {10}}}+ +{O \left({{y \sp {11}}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} + +\spadcommand{cos(y + y**5) } +$$ +1 - +{{1 \over 2} \ {y \sp 2}}+ +{{1 \over {24}} \ {y \sp 4}} - +{{{721} \over {720}} \ {y \sp 6}}+ +{{{6721} \over {40320}} \ {y \sp 8}} - +{{{1844641} \over {3628800}} \ {y \sp {10}}}+ +{O \left({{y \sp {11}}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} + +Similarly, you can compute the logarithm of a power series with rational +coefficients if the constant coefficient is $1.$ + +\spadcommand{log(1 + sin(y)) } +$$ +\begin{array}{@{}l} +y - +{{1 \over 2} \ {y \sp 2}}+ +{{1 \over 6} \ {y \sp 3}} - +{{1 \over {12}} \ {y \sp 4}}+ +{{1 \over {24}} \ {y \sp 5}} - +{{1 \over {45}} \ {y \sp 6}}+ +{{{61} \over {5040}} \ {y \sp 7}} - +{{{17} \over {2520}} \ {y \sp 8}}+ +{{{277} \over {72576}} \ {y \sp 9}} - +\\ +\\ +\displaystyle +{{{31} \over {14175}} \ {y \sp {10}}}+ +{O \left({{y \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} + +If you wanted to apply, say, the operation {\bf exp} to a power series +with a nonzero constant coefficient $a_0$, then the constant +coefficient of the result would be $e^{a_0}$, which is {\it not} a +rational number. Therefore, evaluating $exp(2 + tan(y))$ would +generate an error message. + +If you want to compute the Taylor expansion of $exp(2 + tan(y))$, you +must ensure that the coefficient domain has an operation {\bf exp} +defined for it. An example of such a domain is {\tt Expression +Integer}, the type of formal functional expressions over the integers. + +When working with coefficients of this type, + +\spadcommand{z : UTS(EXPR INT,z,0) := z } +$$ +z +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,z,0)} + +this presents no problems. + +\spadcommand{exp(2 + tan(z)) } +$$ +\begin{array}{@{}l} +{e \sp 2}+ +{{e \sp 2} \ z}+ +{{{e \sp 2} \over 2} \ {z \sp 2}}+ +{{{e \sp 2} \over 2} \ {z \sp 3}}+ +{{{3 \ {e \sp 2}} \over 8} \ {z \sp 4}}+ +{{{{37} \ {e \sp 2}} \over {120}} \ {z \sp 5}}+ +{{{{59} \ {e \sp 2}} \over {240}} \ {z \sp 6}}+ +{{{{137} \ {e \sp 2}} \over {720}} \ {z \sp 7}}+ +\\ +\\ +\displaystyle +{{{{871} \ {e \sp 2}} \over {5760}} \ {z \sp 8}}+ +{{{{41641} \ {e \sp 2}} \over {362880}} \ {z \sp 9}}+ +{{{{325249} \ {e \sp 2}} \over {3628800}} \ {z \sp {10}}}+ +{O \left({{z \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,z,0)} + +Another way to create Taylor series whose coefficients are expressions +over the integers is to use {\bf taylor} which works similarly to +\index{series!Taylor} {\bf series}. + +This is equivalent to the previous computation, except that now we +are using the variable $w$ instead of $z$. + +\spadcommand{w := taylor 'w } +$$ +w +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,w,0)} + +\spadcommand{exp(2 + tan(w)) } +$$ +\begin{array}{@{}l} +{e \sp 2}+ +{{e \sp 2} \ w}+ +{{{e \sp 2} \over 2} \ {w \sp 2}}+ +{{{e \sp 2} \over 2} \ {w \sp 3}}+ +{{{3 \ {e \sp 2}} \over 8} \ {w \sp 4}}+ +{{{{37} \ {e \sp 2}} \over {120}} \ {w \sp 5}}+ +{{{{59} \ {e \sp 2}} \over {240}} \ {w \sp 6}}+ +{{{{137} \ {e \sp 2}} \over {720}} \ {w \sp 7}}+ +\\ +\\ +\displaystyle +{{{{871} \ {e \sp 2}} \over {5760}} \ {w \sp 8}}+ +{{{{41641} \ {e \sp 2}} \over {362880}} \ {w \sp 9}}+ +{{{{325249} \ {e \sp 2}} \over {3628800}} \ {w \sp {10}}}+ +{O \left({{w \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,w,0)} + +\subsection{Converting to Power Series} +\label{ugxProblemSeriesConversions} + +The {\tt ExpressionToUnivariatePowerSeries} package provides +operations for computing series expansions of functions. +\index{ExpressionToUnivariatePowerSeries} + +Evaluate this to compute the Taylor expansion of $sin x$ about +\index{series!Taylor} $x = 0$. The first argument, $sin(x)$, +specifies the function whose series expansion is to be computed and +the second argument, $x = 0$, specifies that the series is to be +expanded in power of $(x - 0)$, that is, in power of $x$. + +\spadcommand{taylor(sin(x),x = 0)} +$$ +x - +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {120}} \ {x \sp 5}} - +{{1 \over {5040}} \ {x \sp 7}}+ +{{1 \over {362880}} \ {x \sp 9}}+ +{O \left({{x \sp {11}}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + +Here is the Taylor expansion of $sin x$ about $x = \frac{\pi}{6}$: + +\spadcommand{taylor(sin(x),x = \%pi/6)} +$$ +\begin{array}{@{}l} +{1 \over 2}+ +{{{\sqrt {3}} \over 2} \ {\left( x -{\pi \over 6} \right)}} +-{{1 \over 4} \ {{\left( x -{\pi \over 6} \right)}\sp 2}} - +{{{\sqrt {3}} \over {12}} \ {{\left( x -{\pi \over 6} \right)}\sp 3}}+ +{{1 \over {48}} \ {{\left( x -{\pi \over 6} \right)}\sp 4}}+ +\\ +\\ +\displaystyle +{{{\sqrt {3}} \over {240}} \ {{\left( x -{\pi \over 6} \right)}\sp 5}} - +{{1 \over {1440}} \ {{\left( x -{\pi \over 6} \right)}\sp 6}} - +{{{\sqrt {3}} \over {10080}} \ {{\left( x -{\pi \over 6} \right)}\sp 7}}+ +{{1 \over {80640}} \ {{\left( x -{\pi \over 6} \right)}\sp 8}}+ +\\ +\\ +\displaystyle +{{{\sqrt {3}} \over {725760}} \ {{\left( x -{\pi \over 6} \right)}\sp 9}} - +{{1 \over {7257600}} \ {{\left( x -{\pi \over 6} \right)}\sp {10}}}+ +{O \left({{{\left( x -{\pi \over 6} \right)}\sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,pi/6)} + + +The function to be expanded into a series may have variables other +than \index{series!multiple variables} the series variable. + +For example, we may expand $tan(x*y)$ as a Taylor series in $x$ + +\spadcommand{taylor(tan(x*y),x = 0)} +$$ +{y \ x}+ +{{{y \sp 3} \over 3} \ {x \sp 3}}+ +{{{2 \ {y \sp 5}} \over {15}} \ {x \sp 5}}+ +{{{{17} \ {y \sp 7}} \over {315}} \ {x \sp 7}}+ +{{{{62} \ {y \sp 9}} \over {2835}} \ {x \sp 9}}+ +{O \left({{x \sp {11}}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + +or as a Taylor series in $y$. + +\spadcommand{taylor(tan(x*y),y = 0)} +$$ +{x \ y}+ +{{{x \sp 3} \over 3} \ {y \sp 3}}+ +{{{2 \ {x \sp 5}} \over {15}} \ {y \sp 5}}+ +{{{{17} \ {x \sp 7}} \over {315}} \ {y \sp 7}}+ +{{{{62} \ {x \sp 9}} \over {2835}} \ {y \sp 9}}+ +{O \left({{y \sp {11}}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,y,0)} + +A more interesting function is +$${\displaystyle t e^{x t}} \over{\displaystyle e^t - 1}$$ +When we expand this function as a Taylor +series in $t$ the $n$-th order coefficient is the $n$-th Bernoulli +\index{Bernoulli!polynomial} polynomial \index{polynomial!Bernoulli} +divided by $n!$. + +\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0) } +$$ +\begin{array}{@{}l} +1+ +{{{{2 \ x} -1} \over 2} \ t}+ +{{{{6 \ {x \sp 2}} -{6 \ x}+1} \over {12}} \ {t \sp 2}}+ +{{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+x} \over {12}} \ {t \sp 3}}+ +\\ +\\ +\displaystyle +{{{{{30} \ {x \sp 4}} -{{60} \ {x \sp 3}}+{{30} \ {x \sp 2}} -1} \over +{720}} \ {t \sp 4}}+ +{{{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp +3}} -x} \over {720}} \ {t \sp 5}}+ +\\ +\\ +\displaystyle +{{{{{42} \ {x \sp 6}} -{{126} \ {x \sp 5}}+{{105} \ {x \sp 4}} -{{21} +\ {x \sp 2}}+1} \over {30240}} \ {t \sp 6}}+ +{{{{6 \ {x \sp 7}} -{{21} \ {x \sp 6}}+{{21} \ {x \sp 5}} -{7 \ {x +\sp 3}}+x} \over {30240}} \ {t \sp 7}}+ +\\ +\\ +\displaystyle +{{{{{30} \ {x \sp 8}} -{{120} \ {x \sp 7}}+{{140} \ {x \sp 6}} - +{{70} \ {x \sp 4}}+{{20} \ {x \sp 2}} -1} \over {1209600}} \ {t \sp 8}}+ +\\ +\\ +\displaystyle +{{{{{10} \ {x \sp 9}} -{{45} \ {x \sp 8}}+{{60} \ {x \sp 7}} - +{{42} \ {x \sp 5}}+{{20} \ {x \sp 3}} -{3 \ x}} +\over {3628800}} \ {t \sp 9}}+ +\\ +\\ +\displaystyle +{{{{{66} \ {x \sp {10}}} -{{330} \ {x \sp 9}}+{{495} \ {x \sp 8}} - +{{462} \ {x \sp 6}}+{{330} \ {x \sp 4}} -{{99} \ {x \sp 2}}+5} +\over {239500800}} \ {t \sp {10}}}+ +{O \left({{t \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,t,0)} + +Therefore, this and the next expression produce the same result. + +\spadcommand{factorial(6) * coefficient(bern,6) } +$$ +{{{42} \ {x \sp 6}} - +{{126} \ {x \sp 5}}+ +{{105} \ {x \sp 4}} - +{{21} \ {x \sp 2}}+1} +\over {42} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{bernoulliB(6,x)} +$$ +{x \sp 6} - +{3 \ {x \sp 5}}+ +{{5 \over 2} \ {x \sp 4}} - +{{1 \over 2} \ {x \sp 2}}+ +{1 \over {42}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +Technically, a series with terms of negative degree is not considered +to be a Taylor series, but, rather, a \index{series!Laurent} +{\it Laurent series}. \index{Laurent series} If you try to compute a +Taylor series expansion of $\frac{x}{\log x}$ at $x = 1$ via +$taylor(x/log(x),x = 1)$ you get an error message. The reason is that +the function has a {\it pole} at $x = 1$, meaning that its series +expansion about this point has terms of negative degree. A series +with finitely many terms of negative degree is called a Laurent +series. + +You get the desired series expansion by issuing this. + +\spadcommand{laurent(x/log(x),x = 1)} +$$ +\begin{array}{@{}l} +{{\left( x -1 \right)}\sp {\left( -1\right)}}+ +{3\over 2}+ +{{5 \over {12}} \ {\left( x -1 \right)}} +-{{1 \over {24}} \ {{\left( x -1 \right)}\sp 2}}+ +{{{11} \over {720}} \ {{\left( x -1 \right)}\sp 3}} - +{{{11} \over {1440}} \ {{\left( x -1 \right)}\sp 4}}+ +\\ +\\ +\displaystyle +{{{271} \over {60480}} \ {{\left( x -1 \right)}\sp 5}} - +{{{13} \over {4480}} \ {{\left( x -1 \right)}\sp 6}}+ +{{{7297} \over {3628800}} \ {{\left( x -1 \right)}\sp 7}} - +{{{425} \over {290304}} \ {{\left( x -1 \right)}\sp 8}}+ +\\ +\\ +\displaystyle +{{{530113} \over {479001600}} \ {{\left( x -1 \right)}\sp 9}}+ +{O \left({{{\left( x -1 \right)}\sp {10}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateLaurentSeries(Expression Integer,x,1)} + +Similarly, a series with terms of fractional degree is neither a +Taylor series nor a Laurent series. Such a series is called a +\index{series!Puiseux} {\it Puiseux series}. \index{Puiseux series} +The expression $laurent(sqrt(sec(x)),x = 3 * \%pi/2)$ results in an +error message because the series expansion about this point has terms +of fractional degree. + +However, this command produces what you want. + +\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} +$$ +{{\left( x -{{3 \ \pi} \over 2} \right)}\sp {\left( -{1 \over 2} \right)}}+ +{{1\over {12}} \ {{\left( x -{{3 \ \pi} \over 2} \right)}\sp {3 \over 2}}}+ +{{1 \over {160}} \ {{\left( x -{{3 \ \pi} \over 2} \right)}\sp {7 \over 2}}}+ +{O \left({{{\left( x -{{3 \ \pi} \over 2} \right)}\sp 5}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2)} + +Finally, consider the case of functions that do not have Puiseux +expansions about certain points. An example of this is $x^x$ about $x += 0$. $puiseux(x**x,x=0)$ produces an error message because of the +type of singularity of the function at $x = 0$. + +The general function {\bf series} can be used in this case. +Notice that the series returned is not, strictly speaking, a power series +because of the $log(x)$ in the expansion. + +\spadcommand{series(x**x,x=0)} +$$ +\begin{array}{@{}l} +1+ +{{\log \left({x} \right)}\ x}+ +{{{{\log \left({x} \right)}\sp 2} \over 2} \ {x \sp 2}}+ +{{{{\log \left({x} \right)}\sp 3} \over 6} \ {x \sp 3}}+ +{{{{\log \left({x} \right)}\sp 4} \over {24}} \ {x \sp 4}}+ +{{{{\log \left({x} \right)}\sp 5} \over {120}} \ {x \sp 5}}+ +{{{{\log \left({x} \right)}\sp 6} \over {720}} \ {x \sp 6}}+ +\\ +\\ +\displaystyle +{{{{\log \left({x} \right)}\sp 7} \over {5040}} \ {x \sp 7}}+ +{{{{\log \left({x} \right)}\sp 8} \over {40320}} \ {x \sp 8}}+ +{{{{\log \left({x} \right)}\sp 9} \over {362880}} \ {x \sp 9}}+ +{{{{\log \left({x} \right)}\sp {10}} \over {3628800}} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: GeneralUnivariatePowerSeries(Expression Integer,x,0)} + +\boxed{4.6in}{ +\vskip 0.1cm +The operation {\bf series} returns the most general type of +infinite series. +The user who is not interested in distinguishing +between various types of infinite series may wish to use this operation +exclusively.\\ +} + +\subsection{Power Series from Formulas} +\label{ugxProblemSeriesFormula} + +The {\tt GenerateUnivariatePowerSeries} package enables you to +\index{series!giving formula for coefficients} create power series +from explicit formulas for their $n$-th coefficients. In what +follows, we construct series expansions for certain transcendental +functions by giving formulas for their coefficients. You can also +compute such series expansions directly simply by specifying the +function and the point about which the series is to be expanded. +\index{GenerateUnivariatePowerSeries} See +\ref{ugxProblemSeriesConversions} on +page~\pageref{ugxProblemSeriesConversions} for more information. + +Consider the Taylor expansion of $e^x$ \index{series!Taylor} +about $x = 0$: + +$$ +\begin{array}{ccl} +e^x &=& \displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \\ \\ + &=& \displaystyle\sum_{n=0}^\infty \frac{x^n}{n!} +\end{array} +$$ + +The $n$-th Taylor coefficient is $1/n!$. + +This is how you create this series in Axiom. + +\spadcommand{series(n +-> 1/factorial(n),x = 0)} +$$ +\begin{array}{@{}l} +1+x+ +{{1 \over 2} \ {x \sp 2}}+ +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {24}} \ {x \sp 4}}+ +{{1 \over {120}} \ {x \sp 5}}+ +{{1 \over {720}} \ {x \sp 6}}+ +{{1 \over {5040}} \ {x \sp 7}}+ +{{1 \over {40320}} \ {x \sp 8}}+ +\\ +\\ +\displaystyle +{{1 \over {362880}} \ {x \sp 9}}+ +{{1 \over {3628800}} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +The first argument specifies a formula for the $n$-th coefficient by +giving a function that maps $n$ to $1/n!$. The second argument +specifies that the series is to be expanded in powers of $(x - 0)$, +that is, in powers of $x$. Since we did not specify an initial +degree, the first term in the series was the term of degree 0 (the +constant term). Note that the formula was given as an anonymous +function. These are discussed in \ref{ugUserAnon} on +page~\pageref{ugUserAnon}. + +Consider the Taylor expansion of $log x$ about $x = 1$: + +$$ +\begin{array}{ccl} +\log(x) &=& \displaystyle (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \cdots \\ \\ + &=& \displaystyle\sum_{n = 1}^\infty (-1)^{n-1} \frac{(x - 1)^n}{n} +\end{array}$$ + +If you were to evaluate the expression +$series(n +-> (-1)**(n-1) / n, x = 1)$ +you would get an error message because Axiom would try to +calculate a term of degree $0$ and therefore divide by $0.$ + +Instead, evaluate this. +The third argument, $1..$, indicates that only terms of degree +$n = 1, ...$ are to be computed. + +\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)} +$$ +\begin{array}{@{}l} +{\left( x -1 \right)} +-{{1 \over 2} \ {{\left( x -1 \right)}\sp 2}}+ +{{1 \over 3} \ {{\left( x -1 \right)}\sp 3}} - +{{1 \over 4} \ {{\left( x -1 \right)}\sp 4}}+ +{{1 \over 5} \ {{\left( x -1 \right)}\sp 5}} - +{{1 \over 6} \ {{\left( x -1 \right)}\sp 6}}+ +\\ +\\ +\displaystyle +{{1 \over 7} \ {{\left( x -1 \right)}\sp 7}} - +{{1 \over 8} \ {{\left( x -1 \right)}\sp 8}}+ +{{1 \over 9} \ {{\left( x -1 \right)}\sp 9}} - +{{1 \over {10}} \ {{\left( x -1 \right)}\sp {10}}}+ +{{1 \over {11}} \ {{\left( x -1 \right)}\sp {11}}}+ +\\ +\\ +\displaystyle +{O \left({{{\left( x -1 \right)}\sp {12}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,1)} + +Next consider the Taylor expansion of an odd function, say, $sin(x)$: + +$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ + +Here every other coefficient is zero and we would like to give an +explicit formula only for the odd Taylor coefficients. + +This is one way to do it. The third argument, $1..$, specifies that +the first term to be computed is the term of degree 1. The fourth +argument, $2$, specifies that we increment by $2$ to find the degrees +of subsequent terms, that is, the next term is of degree $1 + 2$, the +next of degree $1 + 2 + 2$, etc. + +\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)} +$$ +x - +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {120}} \ {x \sp 5}} - +{{1 \over {5040}} \ {x \sp 7}}+ +{{1 \over {362880}} \ {x \sp 9}} - +{{1 \over {39916800}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +The initial degree and the increment do not have to be integers. +For example, this expression produces a series expansion of +$\sin(x^{\frac{1}{3}})$. + +\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} +$$ +{x \sp {1 \over 3}} - +{{1 \over 6} \ x}+ +{{1 \over {120}} \ {x \sp {5 \over 3}}} - +{{1 \over {5040}} \ {x \sp {7 \over 3}}}+ +{{1 \over {362880}} \ {x \sp 3}} - +{{1 \over {39916800}} \ {x \sp {{11} \over 3}}}+ +{O \left({{x \sp 4}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +While the increment must be positive, the initial degree may be +negative. This yields the Laurent expansion of $csc(x)$ at $x = 0$. +(bernoulli(numer(n+1)) is necessary because bernoulli takes integer +arguments.) + +\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2) } +$$ +{x \sp {\left( -1 \right)}}+ +{{1\over 6} \ x}+ +{{7 \over {360}} \ {x \sp 3}}+ +{{{31} \over {15120}} \ {x \sp 5}}+ +{{{127} \over {604800}} \ {x \sp 7}}+ +{{{73} \over {3421440}} \ {x \sp 9}}+ +{O \left({{x \sp {10}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +Of course, the reciprocal of this power series is the Taylor expansion +of $sin(x)$. + +\spadcommand{1/cscx } +$$ +x - +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {120}} \ {x \sp 5}} - +{{1 \over {5040}} \ {x \sp 7}}+ +{{1 \over {362880}} \ {x \sp 9}} - +{{1 \over {39916800}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +As a final example,here is the Taylor expansion of $asin(x)$ about $x = 0$. + +\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2) } +$$ +x+ +{{1 \over 6} \ {x \sp 3}}+ +{{3 \over {40}} \ {x \sp 5}}+ +{{5 \over {112}} \ {x \sp 7}}+ +{{{35} \over {1152}} \ {x \sp 9}}+ +{{{63} \over {2816}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +When we compute the $sin$ of this series, we get $x$ +(in the sense that all higher terms computed so far are zero). + +\spadcommand{sin(asinx) } +$$ +x+{O \left({{x \sp {12}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\boxed{4.6in}{ +\vskip 0.1cm +Axiom isn't sufficiently ``symbolic'' in the sense we might wish. It +is an open problem to decide that ``x'' is the only surviving +term. Two attacks on the problem might be: + +(1) Notice that all of the higher terms are identically zero but +Axiom can't decide that from the information it knows. Presumably +we could attack this problem by looking at the sin function as +a taylor series around x=0 and seeing the term cancellation occur. +This uses a term-difference mechanism. + +(2) Notice that there is no way to decide that the stream for asinx +is actually the definition of asin(x). But we could recognize that +the stream for asin(x) has a generator term and so will a taylor +series expansion of sin(x). From these two generators it may be +possible in certain cases to decide that the application of one +generator to the other will yield only ``x''. This trick involves +finding the correct inverse for the stream functions. If we can +find an inverse for the ``remaining tail'' of the stream we could +conclude cancellation and thus turn an infinite stream into a +finite object. + +In general this is the zero-equivalence problem and is undecidable.\\ +} + +As we discussed in \ref{ugxProblemSeriesConversions} on +page~\pageref{ugxProblemSeriesConversions}, you can also use +the operations {\bf taylor}, {\bf laurent} and {\bf puiseux} instead +of {\bf series} if you know ahead of time what kind of exponents a +series has. You can't go wrong using {\bf series}, though. + +\subsection{Substituting Numerical Values in Power Series} +\label{ugxProblemSeriesSubstitute} + +Use \spadfunFrom{eval}{UnivariatePowerSeriesCategory} +\index{approximation} to substitute a numerical value for a variable +in \index{series!numerical approximation} a power series. For +example, here's a way to obtain numerical approximations of $\%e$ from +the Taylor series expansion of $exp(x)$. + +First you create the desired Taylor expansion. + +\spadcommand{f := taylor(exp(x)) } +$$ +\begin{array}{@{}l} +1+x+ +{{1 \over 2} \ {x \sp 2}}+ +{{1 \over 6} \ {x \sp 3}}+ +{{1 \over {24}} \ {x \sp 4}}+ +{{1 \over {120}} \ {x \sp 5}}+ +{{1 \over {720}} \ {x \sp 6}}+ +{{1 \over {5040}} \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{{1 \over {40320}} \ {x \sp 8}}+ +{{1 \over {362880}} \ {x \sp 9}}+ +{{1 \over {3628800}} \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + + +Then you evaluate the series at the value $1.0$. +The result is a sequence of the partial sums. + +\spadcommand{eval(f,1.0)} +$$ +\begin{array}{@{}l} +\left[ +{1.0}, +{2.0}, +{2.5}, +{2.6666666666\ 666666667}, +{2.7083333333\ 333333333}, +\right. +\\ +\\ +\displaystyle +{2.7166666666\ 666666667}, +{2.7180555555\ 555555556}, +{2.7182539682\ 53968254}, +\\ +\\ +\displaystyle +\left. +{2.7182787698\ 412698413}, +{2.7182815255\ 731922399}, +\ldots +\right] +\end{array} +$$ +\returnType{Type: Stream Expression Float} + +\subsection{Example: Bernoulli Polynomials and Sums of Powers} +\label{ugxProblemSeriesBernoulli} + +Axiom provides operations for computing definite and +\index{summation!definite} indefinite sums. +\index{summation!indefinite} + +You can compute the sum of the first ten fourth powers by evaluating +this. This creates a list whose entries are $m^4$ as $m$ ranges from +1 to 10, and then computes the sum of the entries of that list. + +\spadcommand{reduce(+,[m**4 for m in 1..10])} +$$ +25333 +$$ +\returnType{Type: PositiveInteger} + +You can also compute a formula for the sum of the first $k$ fourth +powers, where $k$ is an unspecified positive integer. + +\spadcommand{sum4 := sum(m**4, m = 1..k) } +$$ +{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} +$$ +\returnType{Type: Fraction Polynomial Integer} + +This formula is valid for any positive integer $k$. For instance, if +we replace $k$ by 10, \index{summation!definite} we obtain the number +we computed earlier. + +\spadcommand{eval(sum4, k = 10) } +$$ +25333 +$$ +\returnType{Type: Fraction Polynomial Integer} + +You can compute a formula for the sum of the first $k$ $n$-th powers +in a similar fashion. Just replace the $4$ in the definition of +{\bf sum4} by any expression not involving $k$. Axiom computes these +formulas using Bernoulli polynomials; \index{Bernoulli!polynomial} we +\index{polynomial!Bernoulli} use the rest of this section to describe +this method. + +First consider this function of $t$ and $x$. + +\spadcommand{f := t*exp(x*t) / (exp(t) - 1) } +$$ +{t \ {e \sp {\left( t \ x \right)}}}\over {{e \sp t} -1} +$$ +\returnType{Type: Expression Integer} + +Since the expressions involved get quite large, we tell +Axiom to show us only terms of degree up to $5.$ + +\spadcommand{)set streams calculate 5 } + +\index{set streams calculate} + +If we look at the Taylor expansion of $f(x, t)$ about $t = 0,$ +we see that the coefficients of the powers of $t$ are polynomials +in $x$. + +\spadcommand{ff := taylor(f,t = 0) } +$$ +\begin{array}{@{}l} +1+ +{{{{2 \ x} -1} \over 2} \ t}+ +{{{{6 \ {x \sp 2}} -{6 \ x}+1} \over {12}} \ {t \sp 2}}+ +{{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+x} \over {12}} \ {t \sp 3}}+ +\\ +\\ +\displaystyle +{{{{{30} \ {x \sp 4}} -{{60} \ {x \sp 3}}+{{30} \ {x \sp 2}} -1} \over +{720}} \ {t \sp 4}}+ +{{{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp +3}} -x} \over {720}} \ {t \sp 5}}+ +{O \left({{t \sp 6}} \right)} +\end{array} +$$ + + Type: UnivariateTaylorSeries(Expression Integer,t,0) + + +In fact, the $n$-th coefficient in this series is essentially the +$n$-th Bernoulli polynomial: the $n$-th coefficient of the series is +${1 \over {n!}} B_n(x)$, where $B_n(x)$ is the $n$-th Bernoulli +polynomial. Thus, to obtain the $n$-th Bernoulli polynomial, we +multiply the $n$-th coefficient of the series $ff$ by $n!$. + +For example, the sixth Bernoulli polynomial is this. + +\spadcommand{factorial(6) * coefficient(ff,6) } +$$ +{{{42} \ {x \sp 6}} -{{126} \ {x \sp 5}}+{{105} \ {x \sp 4}} - +{{21} \ {x \sp 2}}+1} \over {42} +$$ +\returnType{Type: Expression Integer} + +We derive some properties of the function $f(x,t)$. +First we compute $f(x + 1,t) - f(x,t)$. + +\spadcommand{g := eval(f, x = x + 1) - f } +$$ +{{t \ {e \sp {\left( {t \ x}+t \right)}}} +-{t \ {e \sp {\left( t \ x \right)}}}} +\over {{e \sp t} -1} +$$ +\returnType{Type: Expression Integer} + +If we normalize $g$, we see that it has a particularly simple form. + +\spadcommand{normalize(g) } +$$ +t \ {e \sp {\left( t \ x \right)}} +$$ +\returnType{Type: Expression Integer} + +From this it follows that the $n$-th coefficient in the Taylor +expansion of $g(x,t)$ at $t = 0$ is $${1\over{(n-1)!}}x^{n-1}$$. + +If you want to check this, evaluate the next expression. + +\spadcommand{taylor(g,t = 0) } +$$ +t+ +{x \ {t \sp 2}}+ +{{{x \sp 2} \over 2} \ {t \sp 3}}+ +{{{x \sp 3} \over 6} \ {t \sp 4}}+ +{{{x \sp 4} \over {24}} \ {t \sp 5}}+ +{O \left({{t \sp 6}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,t,0)} + +However, since +$$g(x,t) = f(x+1,t)-f(x,t)$$ +it follows that the $n$-th coefficient is +$${1 \over {n!}}(B_n(x+1)-B_n(x))$$ Equating +coefficients, we see that +$${1\over{(n-1)!}}x^{n-1} = {1\over{n!}}(B_n(x + 1) - B_n(x))$$ +and, therefore, +$$x^{n-1} = {1\over{n}}(B_n(x + 1) - B_n(x))$$ + +Let's apply this formula repeatedly, letting $x$ vary between two +integers $a$ and $b$, with $a < b$: + +$$ +\begin{array}{lcl} + a^{n-1} & = & {1 \over n} (B_n(a + 1) - B_n(a)) \\ + (a + 1)^{n-1} & = & {1 \over n} (B_n(a + 2) - B_n(a + 1)) \\ + (a + 2)^{n-1} & = & {1 \over n} (B_n(a + 3) - B_n(a + 2)) \\ + & \vdots & \\ + (b - 1)^{n-1} & = & {1 \over n} (B_n(b) - B_n(b - 1)) \\ + b^{n-1} & = & {1 \over n} (B_n(b + 1) - B_n(b)) +\end{array} +$$ + +When we add these equations we find that the sum of the left-hand +sides is +$$\sum_{m=a}^{b} m^{n-1},$$ +the sum of the +$$(n-1)^{\hbox{\small\rm st}}$$ +powers from $a$ to $b$. The sum of the right-hand sides is a +``telescoping series.'' After cancellation, the sum is simply +$${1\over{n}}(B_n(b + 1) - B_n(a))$$ + +Replacing $n$ by $n + 1$, we have shown that +$$ +\sum_{m = a}^{b} m^n = {1 \over {\displaystyle n + 1}} +(B_{n+1}(b + 1) - B_{n+1}(a)) +$$ + +Let's use this to obtain the formula for the sum of fourth powers. + +First we obtain the Bernoulli polynomial $B_5$. + +\spadcommand{B5 := factorial(5) * coefficient(ff,5) } +$$ +{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp 3}} -x} \over 6 +$$ +\returnType{Type: Expression Integer} + +To find the sum of the first $k$ 4th powers, +we multiply $1/5$ by $B_5(k+1) - B_5(1)$. + +\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1)) } +$$ +{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} +$$ +\returnType{Type: Expression Integer} + +This is the same formula that we obtained via $sum(m**4, m = 1..k)$. + +\spadcommand{sum4 } +$$ +{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} +$$ +\returnType{Type: Fraction Polynomial Integer} + +At this point you may want to do the same computation, but with an +exponent other than $4.$ For example, you might try to find a formula +for the sum of the first $k$ 20th powers. + +\section{Solution of Differential Equations} +\label{ugProblemDEQ} + +In this section we discuss Axiom's facilities for +\index{equation!differential!solving} solving \index{differential +equation} differential equations in closed-form and in series. + +Axiom provides facilities for closed-form solution of +\index{equation!differential!solving in closed-form} single +differential equations of the following kinds: +\begin{itemize} +\item linear ordinary differential equations, and +\item non-linear first order ordinary differential equations +when integrating factors can be found just by integration. +\end{itemize} + +For a discussion of the solution of systems of linear and polynomial +equations, see \ref{ugProblemLinPolEqn} on page~\pageref{ugProblemLinPolEqn}. + +\subsection{Closed-Form Solutions of Linear Differential Equations} +\label{ugxProblemLDEQClosed} + +A {\it differential equation} is an equation involving an unknown +{\it function} and one or more of its derivatives. +\index{differential equation} The equation is called {\it ordinary} +if derivatives with respect to \index{equation!differential} only +one dependent variable appear in the equation (it is called +{\it partial} otherwise). The package {\tt ElementaryFunctionODESolver} +provides the top-level operation {\bf solve} for finding closed-form +solutions of ordinary differential equations. +\index{ElementaryFunctionODESolver} + +To solve a differential equation, you must first create an operator +for \index{operator} the unknown function. + +We let $y$ be the unknown function in terms of $x$. + +\spadcommand{y := operator 'y } +$$ +y +$$ +\returnType{Type: BasicOperator} + +You then type the equation using {\tt D} to create the +derivatives of the unknown function $y(x)$ where $x$ is any +symbol you choose (the so-called {\it dependent variable}). + +This is how you enter +the equation $y'' + y' + y = 0$. + +\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0} +$$ +{{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+ +{{y \sb {{\ }} \sp {,}} \left({x} \right)}+ +{y \left({x} \right)}}=0 +$$ +\returnType{Type: Equation Expression Integer} + +The simplest way to invoke the {\bf solve} command is with three +arguments. +\begin{itemize} +\item the differential equation, +\item the operator representing the unknown function, +\item the dependent variable. +\end{itemize} + +So, to solve the above equation, we enter this. + +\spadcommand{solve(deq, y, x) } +$$ +\left[ +{particular=0}, {basis={\left[ +{{\cos \left({{{x \ {\sqrt {3}}} \over 2}} \right)} +\ {e \sp {\left( -{x \over 2} \right)}}}, +{{e \sp {\left( -{x \over 2} \right)}} +\ {\sin \left({{{x \ {\sqrt {3}}} \over 2}} \right)}}\right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +Since linear ordinary differential equations have infinitely many +solutions, {\bf solve} returns a {\it particular solution} $f_p$ and a +basis $f_1,\dots,f_n$ for the solutions of the corresponding +homogenuous equation. Any expression of the form +$$f_p + c_1 f_1 + \dots c_n f_n$$ +where the $c_i$ do not involve the dependent variable +is also a solution. This is similar to what you get when you solve +systems of linear algebraic equations. + +A way to select a unique solution is to specify {\it initial +conditions}: choose a value $a$ for the dependent variable and specify +the values of the unknown function and its derivatives at $a$. If the +number of initial conditions is equal to the order of the equation, +then the solution is unique (if it exists in closed form!) and {\bf +solve} tries to find it. To specify initial conditions to {\bf +solve}, use an {\tt Equation} of the form $x = a$ for the third +parameter instead of the dependent variable, and add a fourth +parameter consisting of the list of values $y(a), y'(a), ...$. + +To find the solution of $y'' + y = 0$ satisfying $y(0) = y'(0) = 1$, +do this. + +\spadcommand{deq := D(y x, x, 2) + y x } +$$ +{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+{y\left({x} \right)} +$$ +\returnType{Type: Expression Integer} + +You can omit the $= 0$ when you enter the equation to be solved. + +\spadcommand{solve(deq, y, x = 0, [1, 1]) } +$$ +{\sin \left({x} \right)}+{\cos\left({x} \right)} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Axiom is not limited to linear differential equations with constant +coefficients. It can also find solutions when the coefficients are +rational or algebraic functions of the dependent variable. +Furthermore, Axiom is not limited by the order of the equation. + +Axiom can solve the following third order equations with +polynomial coefficients. + +\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4 } +$$ +{{{x \sp 3} \ {{y \sb {{\ }} \sp {,,,}} \left({x} \right)}}+ +{{x\sp 2} \ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}- +{2 \ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ +{2\ {y \left({x} \right)}}}= +{2\ {x \sp 4}} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x) } +$$ +\begin{array}{@{}l} +\left[ +{particular={{{x \sp 5} -{{10} \ {x \sp 3}}+{{20} \ {x \sp 2}}+4} \over +{{15} \ x}}}, +\right. +\\ +\\ +\displaystyle +\left. +{basis={\left[ +{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+1} \over x}, +{{{x \sp 3} -1} \over x}, +{{{x \sp 3} -{3 \ {x \sp 2}} -1} \over x} +\right]}} +\right] +\end{array} +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +Here we are solving a homogeneous equation. + +\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x } +$$ +{{\left( {x \sp 9}+{x \sp 3} \right)}\ {{y \sb {{\ }} \sp {,,,}} +\left({x} \right)}}+ +{{18}\ {x \sp 8} \ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}- +{{90} \ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ +{{\left(-{{330} \ {x \sp 6}}+{90} \right)}\ {y \left({x} \right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{solve(deq, y, x) } +$$ +\left[ +{particular=0}, +{basis={\left[ +{x \over {{x \sp 6}+1}}, +{{x \ {e \sp {\left( -{{\sqrt {{91}}} \ {\log \left({x} \right)}}\right)}}} +\over {{x \sp 6}+1}}, +{{x \ {e \sp {\left( {\sqrt {{91}}} \ {\log \left({x} \right)}\right)}}} +\over {{x \sp 6}+1}} +\right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +On the other hand, and in contrast with the operation {\bf integrate}, +it can happen that Axiom finds no solution and that some closed-form +solution still exists. While it is mathematically complicated to +describe exactly when the solutions are guaranteed to be found, the +following statements are correct and form good guidelines for linear +ordinary differential equations: +\begin{itemize} +\item If the coefficients are constants, Axiom finds a complete basis +of solutions (i,e, all solutions). +\item If the coefficients are rational functions in the dependent variable, +Axiom at least finds all solutions that do not involve algebraic +functions. +\end{itemize} + +Note that this last statement does not mean that Axiom does not find +the solutions that are algebraic functions. It means that it is not +guaranteed that the algebraic function solutions will be found. + +This is an example where all the algebraic solutions are found. + +\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0 } +$$ +{{{\left( {x \sp 2}+1 \right)}\ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}+ +{3\ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ +{y\left({x} \right)}}=0 +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x) } +$$ +\left[ +{particular=0}, +{basis={\left[ {1 \over {\sqrt {{{x \sp 2}+1}}}}, +{{\log \left({{{\sqrt {{{x \sp 2}+1}}} -x}} \right)} +\over {\sqrt {{{x \sp 2}+1}}}} \right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +\subsection{Closed-Form Solutions of Non-Linear Differential Equations} +\label{ugxProblemNLDEQClosed} + +This is an example that shows how to solve a non-linear first order +ordinary differential equation manually when an integrating factor can +be found just by integration. At the end, we show you how to solve it +directly. + +Let's solve the differential equation $y' = y / (x + y log y)$. + +Using the notation $m(x, y) + n(x, y) y' = 0$, we have $m = -y$ and +$n = x + y log y$. + +\spadcommand{m := -y } +$$ +-y +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{n := x + y * log y } +$$ +{y \ {\log \left({y} \right)}}+x +$$ +\returnType{Type: Expression Integer} + +We first check for exactness, that is, does $dm/dy = dn/dx$? + +\spadcommand{D(m, y) - D(n, x) } +$$ +-2 +$$ +\returnType{Type: Expression Integer} + +This is not zero, so the equation is not exact. Therefore we must +look for an integrating factor: a function $mu(x,y)$ such that +$d(mu m)/dy = d(mu n)/dx$. Normally, we first search for $mu(x,y)$ +depending only on $x$ or only on $y$. + +Let's search for such a $mu(x)$ first. + +\spadcommand{mu := operator 'mu } +$$ +mu +$$ +\returnType{Type: BasicOperator} + +\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x) } +$$ +{{\left( -{y \ {\log \left({y} \right)}}-x \right)} +\ {{mu \sb {{\ }} \sp {,}} \left({x} \right)}}- +{2 \ {mu \left({x} \right)}} +$$ +\returnType{Type: Expression Integer} + +If the above is zero for a function $mu$ that does {\it not} depend on +$y$, then $mu(x)$ is an integrating factor. + +\spadcommand{solve(a = 0, mu, x) } +$$ +\left[ +{particular=0}, +{basis={\left[ +{1 \over +{{{y \sp 2} \ {{\log \left({y} \right)}\sp 2}}+ +{2 \ x \ y \ {\log \left({y} \right)}}+ +{x\sp 2}}} \right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +The solution depends on $y$, so there is no integrating factor that +depends on $x$ only. + +Let's look for one that depends on $y$ only. + +\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x) } +$$ +-{y \ {{mu \sb {{\ }} \sp {,}} \left({y} \right)}}- +{2 \ {mu \left({y} \right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{sb := solve(b = 0, mu, y) } +$$ +\left[ +{particular=0}, +{basis={\left[ {1 \over {y \sp 2}} \right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} + +\noindent +We've found one! + +The above $mu(y)$ is an integrating factor. We must multiply our +initial equation (that is, $m$ and $n$) by the integrating factor. + +\spadcommand{intFactor := sb.basis.1 } +$$ +1 \over {y \sp 2} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{m := intFactor * m } +$$ +-{1 \over y} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{n := intFactor * n } +$$ +{{y \ {\log \left({y} \right)}}+x}\over {y \sp 2} +$$ +\returnType{Type: Expression Integer} + +Let's check for exactness. + +\spadcommand{D(m, y) - D(n, x) } +$$ +0 +$$ +\returnType{Type: Expression Integer} + +We must solve the exact equation, that is, find a function $s(x,y)$ +such that $ds/dx = m$ and $ds/dy = n$. + +We start by writing $s(x, y) = h(y) + integrate(m, x)$ where $h(y)$ is +an unknown function of $y$. This guarantees that $ds/dx = m$. + +\spadcommand{h := operator 'h } +$$ +h +$$ +\returnType{Type: BasicOperator} + +\spadcommand{sol := h y + integrate(m, x) } +$$ +{{y \ {h \left({y} \right)}}-x} \over y +$$ +\returnType{Type: Expression Integer} + +All we want is to find $h(y)$ such that $ds/dy = n$. + +\spadcommand{dsol := D(sol, y) } +$$ +{{{y \sp 2} \ {{h \sb {{\ }} \sp {,}} +\left({y} \right)}}+x}\over {y \sp 2} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{nsol := solve(dsol = n, h, y) } +$$ +\left[ +{particular={{{\log \left({y} \right)}\sp 2} \over 2}}, +{basis={\left[ 1 \right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: +List Expression Integer),...)} + +The above particular solution is the $h(y)$ we want, so we just replace +$h(y)$ by it in the implicit solution. + +\spadcommand{eval(sol, h y = nsol.particular) } +$$ +{{y \ {{\log \left({y} \right)}\sp 2}} -{2 \ x}} \over {2 \ y} +$$ +\returnType{Type: Expression Integer} + +A first integral of the initial equation is obtained by setting this +result equal to an arbitrary constant. + +Now that we've seen how to solve the equation ``by hand,'' we show you +how to do it with the {\bf solve} operation. + +First define $y$ to be an operator. + +\spadcommand{y := operator 'y } +$$ +y +$$ +\returnType{Type: BasicOperator} + +Next we create the differential equation. + +\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x) } +$$ +{{y \sb {{\ }} \sp {,}} \left({x} \right)}= +{{y\left({x} \right)}\over +{{{y \left({x} \right)}\ {\log \left({{y \left({x} \right)}}\right)}}+x}} +$$ +\returnType{Type: Equation Expression Integer} + +Finally, we solve it. + +\spadcommand{solve(deq, y, x) } +$$ +{{{y \left({x} \right)}\ {{\log \left({{y \left({x} \right)}}\right)}\sp 2}}- +{2 \ x}} \over {2 \ {y \left({x} \right)}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +\subsection{Power Series Solutions of Differential Equations} +\label{ugxProblemDEQSeries} + +The command to solve differential equations in power +\index{equation!differential!solving in power series} series +\index{power series} around \index{series!power} a particular initial +point with specific initial conditions is called {\bf seriesSolve}. +It can take a variety of parameters, so we illustrate its use with +some examples. + +Since the coefficients of some solutions are quite large, we reset the +default to compute only seven terms. + +\spadcommand{)set streams calculate 7 } + +You can solve a single nonlinear equation of any order. For example, +we solve +$$y''' = sin(y'') * exp(y) + cos(x)$$ subject to +$$y(0) = 1, y'(0) = 0, y''(0) = 0$$ + +We first tell Axiom that the symbol $'y$ denotes a new operator. + +\spadcommand{y := operator 'y } +$$ +y +$$ +\returnType{Type: BasicOperator} + +Enter the differential equation using $y$ like any system function. + +\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)} +$$ +{{{y \sb {{\ }} \sp {,,,}} \left({x} \right)}- +{{e \sp {y \left({x} \right)}}\ +{\sin \left({{{y \sb {{\ }} \sp {,,}} \left({x} \right)}}\right)}}}= +{\cos\left({x} \right)} +$$ +\returnType{Type: Equation Expression Integer} + +Solve it around $x = 0$ with the initial conditions +$y(0) = 1, y'(0) = y''(0) = 0$. + +\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])} +$$ +1+ +{{1 \over 6} \ {x \sp 3}}+ +{{e \over {24}} \ {x \sp 4}}+ +{{{{e \sp 2} -1} \over {120}} \ {x \sp 5}}+ +{{{{e \sp 3} -{2 \ e}} \over {720}} \ {x \sp 6}}+ +{{{{e \sp 4} -{8 \ {e \sp 2}}+{4 \ e}+1} \over {5040}} \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + +You can also solve a system of nonlinear first order equations. For +example, we solve a system that has $tan(t)$ and $sec(t)$ as +solutions. + +We tell Axiom that $x$ is also an operator. + +\spadcommand{x := operator 'x} +$$ +x +$$ +\returnType{Type: BasicOperator} + +Enter the two equations forming our system. + +\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2} +$$ +{{x \sb {{\ }} \sp {,}} \left({t} \right)}= +{{{x\left({t} \right)}\sp 2}+1} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)} +$$ +{{y \sb {{\ }} \sp {,}} \left({t} \right)}= +{{x\left({t} \right)}\ {y \left({t} \right)}} +$$ +\returnType{Type: Equation Expression Integer} + +Solve the system around $t = 0$ with the initial conditions $x(0) = 0$ +and $y(0) = 1$. Notice that since we give the unknowns in the order +$[x, y]$, the answer is a list of two series in the order +$$[{\rm series\ for\ } x(t), {\rm \ series\ for\ }y(t)]$$ + +\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])} +\begin{verbatim} + Compiling function %BZ with type List UnivariateTaylorSeries( + Expression Integer,t,0) -> UnivariateTaylorSeries(Expression + Integer,t,0) + Compiling function %CA with type List UnivariateTaylorSeries( + Expression Integer,t,0) -> UnivariateTaylorSeries(Expression + Integer,t,0) +\end{verbatim} +$$ +\left[ +{t+ +{{1 \over 3} \ {t \sp 3}}+ +{{2 \over {15}} \ {t \sp 5}}+ +{{{17} \over {315}} \ {t \sp 7}}+ +{O \left({{t \sp 8}} \right)}}, +{1+{{1 \over 2} \ {t \sp 2}}+ +{{5 \over {24}} \ {t \sp 4}}+ +{{{61} \over {720}} \ {t \sp 6}}+ +{O \left({{t \sp 8}} \right)}} +\right] +$$ +\returnType{Type: List UnivariateTaylorSeries(Expression Integer,t,0)} + +\noindent +The order in which we give the equations and the initial conditions +has no effect on the order of the solution. + +\section{Finite Fields} +\label{ugProblemFinite} + +A {\it finite field} (also called a {\it Galois field}) is a finite +algebraic structure where one can add, multiply and divide under the +same laws (for example, commutativity, associativity or +distributivity) as apply to the rational, real or complex numbers. +Unlike those three fields, for any finite field there exists a +positive prime integer $p$, called the {\bf characteristic}, such that +$p x = 0$ for any element $x$ in the finite field. In fact, the +number of elements in a finite field is a power of the characteristic +and for each prime $p$ and positive integer $n$ there exists exactly +one finite field with $p^n$ elements, up to isomorphism.\footnote{For +more information about the algebraic structure and properties of +finite fields, see, for example, S. Lang, {\it Algebra}, Second +Edition, New York: Addison-Wesley Publishing Company, Inc., 1984, ISBN +0 201 05487 6; or R. Lidl, H. Niederreiter, {\it Finite Fields}, +Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge: +Cambridge Univ. Press, 1983, ISBN 0 521 30240 4.} + +When $n = 1,$ the field has $p$ elements and is called a {\it prime +field}, discussed in the next section. There are several ways of +implementing extensions of finite fields, and Axiom provides quite a +bit of freedom to allow you to choose the one that is best for your +application. Moreover, we provide operations for converting among the +different representations of extensions and different extensions of a +single field. Finally, note that you usually need to package-call +operations from finite fields if the operations do not take as an +argument an object of the field. See +\ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall} for more +information on package-calling. + +\subsection{Modular Arithmetic and Prime Fields} +\label{ugxProblemFinitePrime} +\index{finite field} +\index{Galois!field} +\index{field!finite!prime} +\index{field!prime} +\index{field!Galois} +\index{prime field} +\index{modular arithmetic} +\index{arithmetic!modular} + +Let $n$ be a positive integer. It is well known that you can get the +same result if you perform addition, subtraction or multiplication of +integers and then take the remainder on dividing by $n$ as if you had +first done such remaindering on the operands, performed the arithmetic +and then (if necessary) done remaindering again. This allows us to +speak of arithmetic {\it modulo} $n$ or, more simply {\it mod} $n$. + +In Axiom, you use {\tt IntegerMod} to do such arithmetic. + +\spadcommand{(a,b) : IntegerMod 12 } +\returnType{Type: Void} + +\spadcommand{(a, b) := (16, 7) } +$$ +7 +$$ +\returnType{Type: IntegerMod 12} + +\spadcommand{[a - b, a * b] } +$$ +\left[ +9, 4 +\right] +$$ +\returnType{Type: List IntegerMod 12} + +If $n$ is not prime, there is only a limited notion of reciprocals and +division. + +\spadcommand{a / b } +\begin{verbatim} + There are 12 exposed and 13 unexposed library operations named / + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op / + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named / + with argument type(s) + IntegerMod 12 + IntegerMod 12 + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +\spadcommand{recip a } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +Here $7$ and $12$ are relatively prime, so $7$ has a multiplicative +inverse modulo $12$. + +\spadcommand{recip b } +$$ +7 +$$ +\returnType{Type: Union(IntegerMod 12,...)} + +If we take $n$ to be a prime number $p$, then taking inverses and, +therefore, division are generally defined. + +Use {\tt PrimeField} instead of {\tt IntegerMod} for $n$ prime. + +\spadcommand{c : PrimeField 11 := 8 } +$$ +8 +$$ +\returnType{Type: PrimeField 11} + +\spadcommand{inv c } +$$ +7 +$$ +\returnType{Type: PrimeField 11} + +You can also use $1/c$ and $c**(-1)$ for the inverse of $c$. + +\spadcommand{9/c } +$$ +8 +$$ +\returnType{Type: PrimeField 11} + +{\tt PrimeField} (abbreviation {\tt PF}) checks if its argument is +prime when you try to use an operation from it. If you know the +argument is prime (particularly if it is large), {\tt InnerPrimeField} +(abbreviation {\tt IPF}) assumes the argument has already been +verified to be prime. If you do use a number that is not prime, you +will eventually get an error message, most likely a division by zero +message. For computer science applications, the most important finite +fields are {\tt PrimeField 2} and its extensions. + +In the following examples, we work with the finite field with +$p = 101$ elements. + +\spadcommand{GF101 := PF 101 } +$$ +\mbox{\rm PrimeField 101} +$$ +\returnType{Type: Domain} + +Like many domains in Axiom, finite fields provide an operation for +returning a random element of the domain. + +\spadcommand{x := random()\$GF101 } +$$ +8 +$$ +\returnType{Type: PrimeField 101} + +\spadcommand{y : GF101 := 37 } +$$ +37 +$$ +\returnType{Type: PrimeField 101} + +\spadcommand{z := x/y } +$$ +63 +$$ +\returnType{Type: PrimeField 101} + +\spadcommand{z * y - x } +$$ +0 +$$ +\returnType{Type: PrimeField 101} + +The element $2$ is a {\it primitive element} of this field, +\index{primitive element} +\index{element!primitive} + +\spadcommand{pe := primitiveElement()\$GF101 } +$$ +2 +$$ +\returnType{Type: PrimeField 101} + +in the sense that its powers enumerate all nonzero elements. + +\spadcommand{[pe**i for i in 0..99] } +$$ +\begin{array}{@{}l} +\left[ +1, 2, 4, 8, {16}, {32}, {64}, {27}, {54}, 7, {14}, {28}, {56}, +{11}, {22}, {44}, {88}, {75}, {49}, {98}, +\right. +\\ +\displaystyle +{95}, {89}, {77}, {53}, 5, {10}, {20}, {40}, {80}, {59}, {17}, +{34}, {68}, {35}, {70}, {39}, {78}, {55}, 9, +\\ +\displaystyle +{18}, {36}, {72}, {43}, {86}, {71}, {41}, {82}, {63}, {25}, +{50}, {100}, {99}, {97}, {93}, {85}, {69}, {37}, +\\ +\displaystyle +{74}, {47}, {94}, {87}, {73}, {45}, {90}, {79}, {57}, {13}, +{26}, {52}, 3, 6, {12}, {24}, {48}, {96}, {91}, +\\ +\displaystyle +{81}, {61}, {21}, {42}, {84}, {67}, {33}, {66}, {31}, {62}, +{23}, {46}, {92}, {83}, {65}, {29}, {58}, {15}, {30}, +\\ +\displaystyle +\left. +{60}, {19}, {38}, {76}, {51} +\right] +\end{array} +$$ +\returnType{Type: List PrimeField 101} + +If every nonzero element is a power of a primitive element, how do you +determine what the exponent is? Use \index{discrete logarithm} +{\bf discreteLog}. \index{logarithm!discrete} + +\spadcommand{ex := discreteLog(y) } +$$ +56 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{pe ** ex } +$$ +37 +$$ +\returnType{Type: PrimeField 101} + +The {\bf order} of a nonzero element $x$ is the smallest positive +integer $t$ such $x^t = 1$. + +\spadcommand{order y } +$$ +25 +$$ +\returnType{Type: PositiveInteger} + +The order of a primitive element is the defining $p-1$. + +\spadcommand{order pe } +$$ +100 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Extensions of Finite Fields} +\label{ugxProblemFiniteExtensionFinite} +\index{finite field} +\index{field!finite!extension of} + +When you want to work with an extension of a finite field in Axiom, +you have three choices to make: +\begin{enumerate} +\item Do you want to generate an extension of the prime field +(for example, {\tt PrimeField 2}) or an extension of a given field? +\item Do you want to use a representation that is particularly +efficient for multiplication, exponentiation and addition but uses a +lot of computer memory (a representation that models the cyclic group +structure of the multiplicative group of the field extension and uses +a Zech logarithm table), one that \index{Zech logarithm} uses a normal +basis for the vector space structure of the field extension, or one +that performs arithmetic modulo an irreducible polynomial? The cyclic +group representation is only usable up to ``medium'' (relative to your +machine's performance) sized fields. If the field is large and the +normal basis is relatively simple, the normal basis representation is +more efficient for exponentiation than the irreducible polynomial +representation. +\item Do you want to provide a polynomial explicitly, a root of which +``generates'' the extension in one of the three senses in (2), or do +you wish to have the polynomial generated for you? +\end{enumerate} + +This illustrates one of the most important features of Axiom: you can +choose exactly the right data-type and representation to suit your +application best. + +We first tell you what domain constructors to use for each case above, +and then give some examples. + +\hangafter=1\hangindent=2pc +Constructors that automatically generate extensions of the prime field: +\newline +{\tt FiniteField} \newline +{\tt FiniteFieldCyclicGroup} \newline +{\tt FiniteFieldNormalBasis} + +\hangafter=1\hangindent=2pc +Constructors that generate extensions of an arbitrary field: +\newline +{\tt FiniteFieldExtension} \newline +{\tt FiniteFieldExtensionByPolynomial} \newline +{\tt FiniteFieldCyclicGroupExtension} \newline +{\tt FiniteFieldCyclicGroupExtensionByPolynomial} \newline +{\tt FiniteFieldNormalBasisExtension} \newline +{\tt FiniteFieldNormalBasisExtensionByPolynomial} + +\hangafter=1\hangindent=2pc +Constructors that use a cyclic group representation: +\newline +{\tt FiniteFieldCyclicGroup} \newline +{\tt FiniteFieldCyclicGroupExtension} \newline +{\tt FiniteFieldCyclicGroupExtensionByPolynomial} + +\hangafter=1\hangindent=2pc +Constructors that use a normal basis representation: +\newline +{\tt FiniteFieldNormalBasis} \newline +{\tt FiniteFieldNormalBasisExtension} \newline +{\tt FiniteFieldNormalBasisExtensionByPolynomial} + +\hangafter=1\hangindent=2pc +Constructors that use an irreducible modulus polynomial representation: +\newline +{\tt FiniteField} \newline +{\tt FiniteFieldExtension} \newline +{\tt FiniteFieldExtensionByPolynomial} + +\hangafter=1\hangindent=2pc +Constructors that generate a polynomial for you: +\newline +{\tt FiniteField} \newline +{\tt FiniteFieldExtension} \newline +{\tt FiniteFieldCyclicGroup} \newline +{\tt FiniteFieldCyclicGroupExtension} \newline +{\tt FiniteFieldNormalBasis} \newline +{\tt FiniteFieldNormalBasisExtension} + +\hangafter=1\hangindent=2pc +Constructors for which you provide a polynomial: +\newline +{\tt FiniteFieldExtensionByPolynomial} \newline +{\tt FiniteFieldCyclicGroupExtensionByPolynomial} \newline +{\tt FiniteFieldNormalBasisExtensionByPolynomial} + +These constructors are discussed in the following sections where we +collect together descriptions of extension fields that have the same +underlying representation.\footnote{For more information on the +implementation aspects of finite fields, see J. Grabmeier, +A. Scheerhorn, {\it Finite Fields in AXIOM,} Technical Report, IBM +Heidelberg Scientific Center, 1992.} + +If you don't really care about all this detail, just use {\tt +FiniteField}. As your knowledge of your application and its Axiom +implementation grows, you can come back and choose an alternative +constructor that may improve the efficiency of your code. Note that +the exported operations are almost the same for all constructors of +finite field extensions and include the operations exported by {\tt +PrimeField}. + +\subsection{Irreducible Modulus Polynomial Representations} +\label{ugxProblemFiniteModulus} + +All finite field extension constructors discussed in this +\index{finite field} section \index{field!finite!extension of} use a +representation that performs arithmetic with univariate (one-variable) +polynomials modulo an irreducible polynomial. This polynomial may be +given explicitly by you or automatically generated. The ground field +may be the prime field or one you specify. See +\ref{ugxProblemFiniteExtensionFinite} on +page~\pageref{ugxProblemFiniteExtensionFinite} for general information about +finite field extensions. + +For {\tt FiniteField} (abbreviation {\tt FF}) you provide a prime +number $p$ and an extension degree $n$. This degree can be 1. + +Axiom uses the prime field {\tt PrimeField(p)}, here {\tt PrimeField 2}, +and it chooses an irreducible polynomial of degree $n$, here 12, +over the ground field. + +\spadcommand{GF4096 := FF(2,12); } +\returnType{Type: Domain} + +The objects in the generated field extension are polynomials of degree +at most $n-1$ with coefficients in the prime field. The polynomial +indeterminate is automatically chosen by Axiom and is typically +something like $\%A$ or $\%D$. These (strange) variables are +{\it only} for output display; there are several ways to construct +elements of this field. + +The operation {\bf index} enumerates the elements of the field +extension and accepts as argument the integers from 1 to $p ^ n$. + +The expression $index(p)$ always gives the indeterminate. + +\spadcommand{a := index(2)\$GF4096 } +$$ +\%A +$$ +\returnType{Type: FiniteField(2,12)} + +You can build polynomials in $a$ and calculate in $GF4096$. + +\spadcommand{b := a**12 - a**5 + a } +$$ +{ \%A \sp 5}+{ \%A \sp 3}+ \%A+1 +$$ +\returnType{Type: FiniteField(2,12)} + +\spadcommand{b ** 1000 } +$$ +{ \%A \sp {10}}+ +{ \%A \sp 9}+ +{ \%A \sp 7}+ +{ \%A \sp 5}+ +{ \%A \sp 4}+ +{ \%A \sp 3}+ +\%A +$$ +\returnType{Type: FiniteField(2,12)} + +\spadcommand{c := a/b } +$$ +{ \%A \sp {11}}+ +{ \%A \sp 8}+ +{ \%A \sp 7}+ +{ \%A \sp 5}+ +{ \%A \sp 4}+ +{ \%A \sp 3}+ +{ \%A \sp 2} +$$ +\returnType{Type: FiniteField(2,12)} + +Among the available operations are {\bf norm} and {\bf trace}. + +\spadcommand{norm c } +$$ +1 +$$ +\returnType{Type: PrimeField 2} + +\spadcommand{trace c } +$$ +0 +$$ +\returnType{Type: PrimeField 2} + +Since any nonzero element is a power of a primitive element, how do we +discover what the exponent is? + +The operation {\bf discreteLog} calculates \index{discrete logarithm} +the exponent and, \index{logarithm!discrete} if it is called with only +one argument, always refers to the primitive element returned by {\bf +primitiveElement}. + +\spadcommand{dL := discreteLog a } +$$ +1729 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{g ** dL } +$$ +g \sp {1729} +$$ +\returnType{Type: Polynomial Integer} + +{\tt FiniteFieldExtension} (abbreviation {\tt FFX}) is similar to {\tt +FiniteField} except that the ground-field for {\tt FiniteFieldExtension} +is arbitrary and chosen by you. + +In case you select the prime field as ground field, there is +essentially no difference between the constructed two finite field +extensions. + +\spadcommand{GF16 := FF(2,4); } +\returnType{Type: Domain} + +\spadcommand{GF4096 := FFX(GF16,3); } +\returnType{Type: Domain} + +\spadcommand{r := (random()\$GF4096) ** 20 } +$$ +{{\left( { \%B \sp 2}+1 \right)}\ { \%C \sp 2}}+ +{{\left( { \%B \sp 3}+{ \%B \sp 2}+1 \right)}\ \%C}+ +{ \%B \sp 3}+ +{ \%B \sp 2}+ +\%B+1 +$$ +\returnType{Type: FiniteFieldExtension(FiniteField(2,4),3)} + +\spadcommand{norm(r) } +$$ +{ \%B \sp 2}+ \%B +$$ +\returnType{Type: FiniteField(2,4)} + +{\tt FiniteFieldExtensionByPolynomial} (abbreviation {\tt FFP}) +is similar to {\tt FiniteField} and {\tt FiniteFieldExtension} +but is more general. + +\spadcommand{GF4 := FF(2,2); } +\returnType{Type: Domain} + +\spadcommand{f := nextIrreduciblePoly(random(6)\$FFPOLY(GF4))\$FFPOLY(GF4) } +$$ +{? \sp 6}+ +{{\left( \%D+1 \right)}\ {? \sp 5}}+ +{{\left( \%D+1 \right)}\ {? \sp 4}}+ +{{\left( \%D+1 \right)}\ ?}+1 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...)} + +For {\tt FFP} you choose both the ground field and the irreducible +polynomial used in the representation. The degree of the extension is +the degree of the polynomial. + +\spadcommand{GF4096 := FFP(GF4,f); } +\returnType{Type: Domain} + +\spadcommand{discreteLog random()\$GF4096 } +$$ +582 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Cyclic Group Representations} +\label{ugxProblemFiniteCyclic} +\index{finite field} +\index{field!finite!extension of} + +In every finite field there exist elements whose powers are all the +nonzero elements of the field. Such an element is called a +{\it primitive element}. + +In {\tt FiniteFieldCyclicGroup} (abbreviation {\tt FFCG}) +\index{group!cyclic} the nonzero elements are represented by the +powers of a fixed primitive \index{element!primitive} element +\index{primitive element} of the field (that is, a generator of its +cyclic multiplicative group). Multiplication (and hence +exponentiation) using this representation is easy. To do addition, we +consider our primitive element as the root of a primitive polynomial +(an irreducible polynomial whose roots are all primitive). See +\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility} +for examples of how to compute such a polynomial. + +To use {\tt FiniteFieldCyclicGroup} you provide a prime number and an +extension degree. +\spadcommand{GF81 := FFCG(3,4); } +\returnType{Type: Domain} + +Axiom uses the prime field, here {\tt PrimeField 3}, as the ground +field and it chooses a primitive polynomial of degree $n$, here 4, +over the prime field. + +\spadcommand{a := primitiveElement()\$GF81 } +$$ + \%F \sp 1 +$$ +\returnType{Type: FiniteFieldCyclicGroup(3,4)} + +You can calculate in $GF81$. + +\spadcommand{b := a**12 - a**5 + a } +$$ + \%F \sp {72} +$$ +\returnType{Type: FiniteFieldCyclicGroup(3,4)} + +In this representation of finite fields the discrete logarithm of an +element can be seen directly in its output form. + +\spadcommand{b } +$$ + \%F \sp {72} +$$ +\returnType{Type: FiniteFieldCyclicGroup(3,4)} + +\spadcommand{discreteLog b } +$$ +72 +$$ +\returnType{Type: PositiveInteger} + +{\tt FiniteFieldCyclicGroupExtension} (abbreviation {\tt FFCGX}) is +similar to {\tt FiniteFieldCyclicGroup} except that the ground field +for {\tt FiniteFieldCyclicGroupExtension} is arbitrary and chosen by +you. In case you select the prime field as ground field, there is +essentially no difference between the constructed two finite field +extensions. + +\spadcommand{GF9 := FF(3,2); } +\returnType{Type: Domain} + +\spadcommand{GF729 := FFCGX(GF9,3); } +\returnType{Type: Domain} + +\spadcommand{r := (random()\$GF729) ** 20 } +$$ + \%H \sp {420} +$$ +\returnType{Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3)} + +\spadcommand{trace(r) } +$$ +0 +$$ +\returnType{Type: FiniteField(3,2)} + +{\tt FiniteFieldCyclicGroupExtensionByPolynomial} (abbreviation +{\tt FFCGP}) is similar to {\tt FiniteFieldCyclicGroup} and +{\tt FiniteFieldCyclicGroupExtension} but is more general. For +{\tt FiniteFieldCyclicGroupExtensionByPolynomial} you choose both the +ground field and the irreducible polynomial used in the +representation. The degree of the extension is the degree of the +polynomial. + +\spadcommand{GF3 := PrimeField 3; } +\returnType{Type: Domain} + +We use a utility operation to generate an irreducible primitive +polynomial (see +\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility}). +The polynomial has one variable that is ``anonymous'': +it displays as a question mark. + +\spadcommand{f := createPrimitivePoly(4)\$FFPOLY(GF3) } +$$ +{? \sp 4}+?+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 3} + +\spadcommand{GF81 := FFCGP(GF3,f); } +\returnType{Type: Domain} + +Let's look at a random element from this field. + +\spadcommand{random()\$GF81 } +$$ + \%K \sp {13} +$$ +\returnType{Type: +FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2)} + +\subsection{Normal Basis Representations} +\label{ugxProblemFiniteNormal} +\index{finite field} +\index{field!finite!extension of} +\index{basis!normal} +\index{normal basis} + +Let $K$ be a finite extension of degree $n$ of the finite field $F$ +and let $F$ have $q$ elements. An element $x$ of $K$ is said to be +{\it normal} over $F$ if the elements + +$$1, x^q, x^{q^2}, \ldots, x^{q^{n-1}}$$ + +form a basis of $K$ as a vector space over $F$. Such a basis is +called a {\it normal basis}.\footnote{This agrees with the general +definition of a normal basis because the $n$ distinct powers of the +automorphism $x \mapsto x^q$ constitute the Galois group of $K/F$.} + +If $x$ is normal over $F$, its minimal \index{polynomial!minimal} +polynomial is also said to be {\it normal} over $F$. +\index{minimal polynomial} +There exist normal bases for all finite extensions of arbitrary +finite fields. + +In {\tt FiniteFieldNormalBasis} (abbreviation {\tt FFNB}), the +elements of the finite field are represented by coordinate vectors +with respect to a normal basis. + +You provide a prime $p$ and an extension degree $n$. + +\spadcommand{K := FFNB(3,8) } +$$ +FiniteFieldNormalBasis(3,8) +$$ +\returnType{Type: Domain} + +Axiom uses the prime field {\tt PrimeField(p)}, here {\tt PrimeField +3}, and it chooses a normal polynomial of degree $n$, here 8, over the +ground field. The remainder class of the indeterminate is used as the +normal element. The polynomial indeterminate is automatically chosen +by Axiom and is typically something like $\%A$ or $\%D$. These +(strange) variables are only for output display; there are several +ways to construct elements of this field. The output of the basis +elements is something like $\%A^{q^i}.$ + +\spadcommand{a := normalElement()\$K } +$$ + \%I +$$ +\returnType{Type: FiniteFieldNormalBasis(3,8)} + +You can calculate in $K$ using $a$. + +\spadcommand{b := a**12 - a**5 + a } +$$ +{2 \ { \%I \sp {q \sp 7}}}+{ \%I \sp {q \sp 5}}+{ \%I \sp q} +$$ +\returnType{Type: FiniteFieldNormalBasis(3,8)} + +{\tt FiniteFieldNormalBasisExtension} (abbreviation {\tt FFNBX}) is +similar to {\tt FiniteFieldNormalBasis} except that the groundfield +for {\tt FiniteFieldNormalBasisExtension} is arbitrary and chosen by +you. In case you select the prime field as ground field, there is +essentially no difference between the constructed two finite field +extensions. + +\spadcommand{GF9 := FFNB(3,2); } +\returnType{Type: Domain} + +\spadcommand{GF729 := FFNBX(GF9,3); } +\returnType{Type: Domain} + +\spadcommand{r := random()\$GF729 } +$$ +2 \ \%K \ { \%L \sp q} +$$ +\returnType{Type: +FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)} + +\spadcommand{r + r**3 + r**9 + r**27 } +$$ +{2 \ \%K \ { \%L \sp {q \sp 2}}}+ +{{\left( {2 \ { \%K \sp q}}+{2 \ \%K} \right)}\ { \%L \sp q}}+ +{2 \ { \%K \sp q} \ \%L} +$$ +\returnType{Type: +FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)} + +{\tt FiniteFieldNormalBasisExtensionByPolynomial} (abbreviation +{\tt FFNBP}) is similar to {\tt FiniteFieldNormalBasis} and +{\tt FiniteFieldNormalBasisExtension} but is more general. For +{\tt FiniteFieldNormalBasisExtensionByPolynomial} you choose both the +ground field and the irreducible polynomial used in the representation. +The degree of the extension is the degree of the polynomial. + +\spadcommand{GF3 := PrimeField 3; } +\returnType{Type: Domain} + +We use a utility operation to generate an irreducible normal +polynomial (see +\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility}). +The polynomial has +one variable that is ``anonymous'': it displays as a question mark. + +\spadcommand{f := createNormalPoly(4)\$FFPOLY(GF3) } +$$ +{? \sp 4}+{2 \ {? \sp 3}}+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 3} + +\spadcommand{GF81 := FFNBP(GF3,f); } +\returnType{Type: Domain} + +Let's look at a random element from this field. + +\spadcommand{r := random()\$GF81 } +$$ +{ \%M \sp {q \sp 2}}+{2 \ { \%M \sp q}}+{2 \ \%M} +$$ +\returnType{Type: +FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)} + +\spadcommand{r * r**3 * r**9 * r**27 } +$$ +{2 \ { \%M \sp {q \sp 3}}}+ +{2 \ { \%M \sp {q \sp 2}}}+ +{2 \ { \%M \sp q}}+ +{2 \ \%M} +$$ +\returnType{Type: +FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)} + +\spadcommand{norm r } +$$ +2 +$$ +\returnType{Type: PrimeField 3} + +\subsection{Conversion Operations for Finite Fields} +\label{ugxProblemFiniteConversion} +\index{field!finite!conversions} + +Let $K$ be a finite field. + +\spadcommand{K := PrimeField 3 } +$$ +\mbox{\rm PrimeField 3} +$$ +\returnType{Type: Domain} + +An extension field $K_m$ of degree $m$ over $K$ is a subfield of an +extension field $K_n$ of degree $n$ over $K$ if and only if $m$ +divides $n$. + +\begin{center} +\begin{tabular}{ccc} +$K_n$ \\ +$|$ \\ +$K_m$ & $\Longleftrightarrow$ & $m | n$ \\ +$|$ \\ +K +\end{tabular} +\end{center} + +{\tt FiniteFieldHomomorphisms} provides conversion operations between +different extensions of one fixed finite ground field and between +different representations of these finite fields. + +Let's choose $m$ and $n$, + +\spadcommand{(m,n) := (4,8) } +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +build the field extensions, + +\spadcommand{Km := FiniteFieldExtension(K,m) } +$$ +\mbox{\rm FiniteFieldExtension(PrimeField 3,4)} +$$ +\returnType{Type: Domain} + +and pick two random elements from the smaller field. + +\spadcommand{Kn := FiniteFieldExtension(K,n) } +$$ +\mbox{\rm FiniteFieldExtension(PrimeField 3,8)} +$$ +\returnType{Type: Domain} + +\spadcommand{a1 := random()\$Km } +$$ +{2 \ { \%A \sp 3}}+{ \%A \sp 2} +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} + +\spadcommand{b1 := random()\$Km } +$$ +{ \%A \sp 3}+{ \%A \sp 2}+{2 \ \%A}+1 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} + +Since $m$ divides $n$, +$K_m$ is a subfield of $K_n$. + +\spadcommand{a2 := a1 :: Kn } +$$ + \%B \sp 4 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,8)} + +Therefore we can convert the elements of $K_m$ +into elements of $K_n$. + +\spadcommand{b2 := b1 :: Kn } +$$ +{2 \ { \%B \sp 6}}+{2 \ { \%B \sp 4}}+{ \%B \sp 2}+1 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,8)} + +To check this, let's do some arithmetic. + +\spadcommand{a1+b1 - ((a2+b2) :: Km) } +$$ +0 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} + +\spadcommand{a1*b1 - ((a2*b2) :: Km) } +$$ +0 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} + +There are also conversions available for the situation, when $K_m$ and +$K_n$ are represented in different ways (see +\ref{ugxProblemFiniteExtensionFinite} on +page~\pageref{ugxProblemFiniteExtensionFinite}). For example let's choose +$K_m$ where the representation is 0 plus the cyclic multiplicative +group and $K_n$ with a normal basis representation. + +\spadcommand{Km := FFCGX(K,m) } +$$ +\mbox{\rm FiniteFieldCyclicGroupExtension(PrimeField 3,4)} +$$ +\returnType{Type: Domain} + +\spadcommand{Kn := FFNBX(K,n) } +$$ +\mbox{\rm FiniteFieldNormalBasisExtension(PrimeField 3,8)} +$$ +\returnType{Type: Domain} + +\spadcommand{(a1,b1) := (random()\$Km,random()\$Km) } +$$ + \%C \sp {13} +$$ +\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} + +\spadcommand{a2 := a1 :: Kn } +$$ +{2 \ { \%D \sp {q \sp 6}}}+ +{2 \ { \%D \sp {q \sp 5}}}+ +{2 \ { \%D \sp {q \sp 4}}}+ +{2 \ { \%D \sp {q \sp 2}}}+ +{2 \ { \%D \sp q}}+ +{2 \ \%D} +$$ +\returnType{Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)} + +\spadcommand{b2 := b1 :: Kn } +$$ +{2 \ { \%D \sp {q \sp 7}}}+ +{ \%D \sp {q \sp 6}}+ +{ \%D \sp {q \sp 5}}+ +{ \%D \sp {q \sp 4}}+ +{2 \ { \%D \sp {q \sp 3}}}+ +{ \%D \sp {q \sp 2}}+ +{ \%D \sp q}+ +\%D +$$ +\returnType{Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)} + +Check the arithmetic again. + +\spadcommand{a1+b1 - ((a2+b2) :: Km) } +$$ +0 +$$ +\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} + +\spadcommand{a1*b1 - ((a2*b2) :: Km) } +$$ +0 +$$ +\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} + +\subsection{Utility Operations for Finite Fields} +\label{ugxProblemFiniteUtility} + +{\tt FiniteFieldPolynomialPackage} (abbreviation {\tt FFPOLY}) +provides operations for generating, counting and testing polynomials +over finite fields. Let's start with a couple of definitions: +\begin{itemize} +\item A polynomial is {\it primitive} if its roots are primitive +\index{polynomial!primitive} +elements in an extension of the coefficient field of degree equal +to the degree of the polynomial. +\item A polynomial is {\it normal} over its coefficient field +\index{polynomial!normal} +if its roots are linearly independent +elements in an extension of the coefficient field of degree equal +to the degree of the polynomial. +\end{itemize} + +In what follows, many of the generated polynomials have one +``anonymous'' variable. This indeterminate is displayed as a question +mark ({\tt ``?''}). + +To fix ideas, let's use the field with five elements for the first +few examples. + +\spadcommand{GF5 := PF 5; } +\returnType{Type: Domain} + +You can generate irreducible polynomials of any (positive) degree +\index{polynomial!irreducible} (within the storage capabilities of the +computer and your ability to wait) by using +\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}. + +\spadcommand{f := createIrreduciblePoly(8)\$FFPOLY(GF5) } +$$ +{? \sp 8}+{? \sp 4}+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +Does this polynomial have other important properties? Use +{\bf primitive?} to test whether it is a primitive polynomial. + +\spadcommand{primitive?(f)\$FFPOLY(GF5) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Use {\bf normal?} to test whether it is a normal polynomial. + +\spadcommand{normal?(f)\$FFPOLY(GF5) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\noindent +Note that this is actually a trivial case, because a normal polynomial +of degree $n$ must have a nonzero term of degree $n-1$. We will refer +back to this later. + +To get a primitive polynomial of degree 8 just issue this. + +\spadcommand{p := createPrimitivePoly(8)\$FFPOLY(GF5) } +$$ +{? \sp 8}+{? \sp 3}+{? \sp 2}+?+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +\spadcommand{primitive?(p)\$FFPOLY(GF5) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +This polynomial is not normal, + +\spadcommand{normal?(p)\$FFPOLY(GF5) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +but if you want a normal one simply write this. + +\spadcommand{n := createNormalPoly(8)\$FFPOLY(GF5) } +$$ +{? \sp 8}+{4 \ {? \sp 7}}+{? \sp 3}+1 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +This polynomial is not primitive! + +\spadcommand{primitive?(n)\$FFPOLY(GF5) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +This could have been seen directly, as the constant term is 1 here, +which is not a primitive element up to the factor ($-1$) raised to the +degree of the polynomial.\footnote{Cf. Lidl, R. \& Niederreiter, H., +{\it Finite Fields,} Encycl. of Math. 20, (Addison-Wesley, 1983), +p.90, Th. 3.18.} + +What about polynomials that are both primitive and normal? The +existence of such a polynomial is by no means obvious. +\footnote{The existence of such polynomials is proved in +Lenstra, H. W. \& Schoof, R. J., {\it Primitive +Normal Bases for Finite Fields,} Math. Comp. 48, 1987, pp. 217-231.} +% + +If you really need one use either +\spadfunFrom{createPrimitiveNormalPoly}{FiniteFieldPolynomialPackage} or +\spadfunFrom{createNormalPrimitivePoly}{FiniteFieldPolynomialPackage}. + +\spadcommand{createPrimitiveNormalPoly(8)\$FFPOLY(GF5) } +$$ +{? \sp 8}+{4 \ {? \sp 7}}+{2 \ {? \sp 5}}+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +If you want to obtain additional polynomials of the various types +above as given by the {\bf create...} operations above, you can use +the {\bf next...} operations. For instance, +\spadfunFrom{nextIrreduciblePoly}{FiniteFieldPolynomialPackage} yields +the next monic irreducible polynomial with the same degree as the +input polynomial. By ``next'' we mean ``next in a natural order using +the terms and coefficients.'' This will become more clear in the +following examples. + +This is the field with five elements. + +\spadcommand{GF5 := PF 5; } +\returnType{Type: Domain} + +Our first example irreducible polynomial, say of degree 3, must be +``greater'' than this. + +\spadcommand{h := monomial(1,8)\$SUP(GF5) } +$$ +? \sp 8 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +You can generate it by doing this. + +\spadcommand{nh := nextIrreduciblePoly(h)\$FFPOLY(GF5) } +$$ +{? \sp 8}+2 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +Notice that this polynomial is not the same as the one +\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}. + +\spadcommand{createIrreduciblePoly(3)\$FFPOLY(GF5) } +$$ +{? \sp 3}+?+1 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +You can step through all irreducible polynomials of degree 8 over +the field with 5 elements by repeatedly issuing this. + +\spadcommand{nh := nextIrreduciblePoly(nh)\$FFPOLY(GF5) } +$$ +{? \sp 8}+3 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +You could also ask for the total number of these. + +\spadcommand{numberOfIrreduciblePoly(5)\$FFPOLY(GF5) } +$$ +624 +$$ +\returnType{Type: PositiveInteger} + +We hope that ``natural order'' on polynomials is now clear: first we +compare the number of monomials of two polynomials (``more'' is +``greater''); then, if necessary, the degrees of these monomials +(lexicographically), and lastly their coefficients (also +lexicographically, and using the operation {\bf lookup} if our field +is not a prime field). Also note that we make both polynomials monic +before looking at the coefficients: multiplying either polynomial by a +nonzero constant produces the same result. + +The package {\tt FiniteFieldPolynomialPackage} also provides similar +operations for primitive and normal polynomials. With the exception of +the number of primitive normal polynomials; we're not aware of any +known formula for this. + +\spadcommand{numberOfPrimitivePoly(3)\$FFPOLY(GF5) } +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +Take these, + +\spadcommand{m := monomial(1,1)\$SUP(GF5) } +$$ +? +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +\spadcommand{f := m**3 + 4*m**2 + m + 2 } +$$ +{? \sp 3}+{4 \ {? \sp 2}}+?+2 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +and then we have: + +\spadcommand{f1 := nextPrimitivePoly(f)\$FFPOLY(GF5) } +$$ +{? \sp 3}+{4 \ {? \sp 2}}+{4 \ ?}+2 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +What happened? + +\spadcommand{nextPrimitivePoly(f1)\$FFPOLY(GF5) } +$$ +{? \sp 3}+{2 \ {? \sp 2}}+3 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +Well, for the ordering used in +\spadfunFrom{nextPrimitivePoly}{FiniteFieldPolynomialPackage} we use +as first criterion a comparison of the constant terms of the +polynomials. Analogously, in +\spadfunFrom{nextNormalPoly}{FiniteFieldPolynomialPackage} we first +compare the monomials of degree 1 less than the degree of the +polynomials (which is nonzero, by an earlier remark). + +\spadcommand{f := m**3 + m**2 + 4*m + 1 } +$$ +{? \sp 3}+{? \sp 2}+{4 \ ?}+1 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 5} + +\spadcommand{f1 := nextNormalPoly(f)\$FFPOLY(GF5) } +$$ +{? \sp 3}+{? \sp 2}+{4 \ ?}+3 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +\spadcommand{nextNormalPoly(f1)\$FFPOLY(GF5) } +$$ +{? \sp 3}+{2 \ {? \sp 2}}+1 +$$ +\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} + +\noindent +We don't have to restrict ourselves to prime fields. + +Let's consider, say, a field with 16 elements. + +\spadcommand{GF16 := FFX(FFX(PF 2,2),2); } +\returnType{Type: Domain} + +We can apply any of the operations described above. + +\spadcommand{createIrreduciblePoly(5)\$FFPOLY(GF16) } +$$ +{? \sp 5}+ \%G +$$ +\returnType{Type: SparseUnivariatePolynomial +FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} + +Axiom also provides operations for producing random polynomials of a +given degree + +\spadcommand{random(5)\$FFPOLY(GF16) } +$$ +\begin{array}{@{}l} +{? \sp 5}+ +{{\left( { \%F \ \%G}+1 \right)}\ {? \sp 4}}+ +{ \%F \ \%G \ {? \sp 3}}+ +{{\left( \%G+ \%F+1 \right)}\ {? \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( {{\left( \%F+1 \right)}\ \%G}+ \%F \right)}\ ?}+1 +\end{array} +$$ +\returnType{Type: SparseUnivariatePolynomial +FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} + +or with degree between two given bounds. + +\spadcommand{random(3,9)\$FFPOLY(GF16) } +$$ +{? \sp 3}+ +{{\left( { \%F \ \%G}+1 \right)}\ {? \sp 2}}+ +{{\left( \%G+ \%F+1 \right)}\ ?}+1 +$$ +\returnType{Type: SparseUnivariatePolynomial +FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} + +{\tt FiniteFieldPolynomialPackage2} (abbreviation {\tt FFPOLY2}) +exports an operation {\bf rootOfIrreduciblePoly} for finding one root +of an irreducible polynomial $f$ \index{polynomial!root of} in an +extension field of the coefficient field. The degree of the extension +has to be a multiple of the degree of $f$. It is not checked whether +$f$ actually is irreducible. + +To illustrate this operation, we fix a ground field $GF$ + +\spadcommand{GF2 := PrimeField 2; } +\returnType{Type: Domain} + +and then an extension field. + +\spadcommand{F := FFX(GF2,12) } +$$ +\mbox{\rm FiniteFieldExtension(PrimeField 2,12)} +$$ +\returnType{Type: Domain} + +We construct an irreducible polynomial over $GF2$. + +\spadcommand{f := createIrreduciblePoly(6)\$FFPOLY(GF2) } +$$ +{? \sp 6}+?+1 +$$ +\returnType{Type: SparseUnivariatePolynomial PrimeField 2} + +We compute a root of $f$. + +\spadcommand{root := rootOfIrreduciblePoly(f)\$FFPOLY2(F,GF2) } +$$ +{ \%H \sp {11}}+{ \%H \sp 8}+{ \%H \sp 7}+{ \%H \sp 5}+ \%H+1 +$$ +\returnType{Type: FiniteFieldExtension(PrimeField 2,12)} + +and check the result +\spadcommand{eval(f, monomial(1,1)\$SUP(F) = root) } +$$ +0 +$$ +\returnType{Type: SparseUnivariatePolynomial +FiniteFieldExtension(PrimeField 2,12)} + +\section{Primary Decomposition of Ideals} +\label{ugProblemIdeal} + +Axiom provides a facility for the primary decomposition +\index{ideal!primary decomposition} of \index{primary decomposition of +ideal} polynomial ideals over fields of characteristic zero. The +algorithm +%is discussed in \cite{gtz:gbpdpi} and +works in essentially two steps: +\begin{enumerate} +\item the problem is solved for 0-dimensional ideals by ``generic'' +projection on the last coordinate +\item a ``reduction process'' uses localization and ideal quotients +to reduce the general case to the 0-dimensional one. +\end{enumerate} +The Axiom constructor {\tt PolynomialIdeals} represents ideals with +coefficients in any field and supports the basic ideal operations, +including intersection, sum and quotient. {\tt IdealDecompositionPackage} +contains the specific operations for the +primary decomposition and the computation of the radical of an ideal +with polynomial coefficients in a field of characteristic 0 with an +effective algorithm for factoring polynomials. + +The following examples illustrate the capabilities of this facility. + +First consider the ideal generated by +$x^2 + y^2 - 1$ +(which defines a circle in the $(x,y)$-plane) and the ideal +generated by $x^2 - y^2$ (corresponding to the +straight lines $x = y$ and $x = -y$. + +\spadcommand{(n,m) : List DMP([x,y],FRAC INT) } +\returnType{Type: Void} + +\spadcommand{m := [x**2+y**2-1] } +$$ +\left[ +{{x \sp 2}+{y \sp 2} -1} +\right] +$$ +\returnType{Type: List +DistributedMultivariatePolynomial([x,y],Fraction Integer)} + +\spadcommand{n := [x**2-y**2] } +$$ +\left[ +{{x \sp 2} -{y \sp 2}} +\right] +$$ +\returnType{Type: List +DistributedMultivariatePolynomial([x,y],Fraction Integer)} + +We find the equations defining the intersection of the two loci. +This correspond to the sum of the associated ideals. + +\spadcommand{id := ideal m + ideal n } +$$ +\left[ +{{x \sp 2} -{1 \over 2}}, {{y \sp 2} -{1 \over 2}} +\right] +$$ +\returnType{Type: PolynomialIdeals(Fraction Integer, +DirectProduct(2,NonNegativeInteger),OrderedVariableList [x,y], +DistributedMultivariatePolynomial([x,y],Fraction Integer))} + +We can check if the locus contains only a finite number of points, +that is, if the ideal is zero-dimensional. + +\spadcommand{zeroDim? id } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{zeroDim?(ideal m) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{dimension ideal m } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +We can find polynomial relations among the generators ($f$ and $g$ are +the parametric equations of the knot). + +\spadcommand{(f,g):DMP([x,y],FRAC INT) } +\returnType{Type: Void} + +\spadcommand{f := x**2-1 } +$$ +{x \sp 2} -1 +$$ +\returnType{Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)} + +\spadcommand{g := x*(x**2-1) } +$$ +{x \sp 3} -x +$$ +\returnType{Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)} + +\spadcommand{relationsIdeal [f,g] } +$$ +{\left[ {-{ \%B \sp 2}+{ \%A \sp 3}+{ \%A \sp 2}} \right]} +\mid +{\left[ { \%A={{x \sp 2} -1}}, { \%B={{x \sp 3} -x}} \right]} +$$ +\returnType{Type: SuchThat(List Polynomial Fraction Integer, +List Equation Polynomial Fraction Integer)} + +We can compute the primary decomposition of an ideal. + +\spadcommand{l: List DMP([x,y,z],FRAC INT) } +\returnType{Type: Void} + +\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4] } +$$ +\left[ +{{x \sp 2}+{2 \ {y \sp 2}}}, {{x \ {z \sp 2}} -{y \ z}}, {{z \sp 2} -4} +\right] +$$ +\returnType{Type: List +DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} + +\spadcommand{ld:=primaryDecomp ideal l } +$$ +\left[ +{\left[ {x+{{1 \over 2} \ y}}, {y \sp 2}, {z+2} \right]}, +{\left[ {x -{{1 \over 2} \ y}}, {y \sp 2}, {z -2} \right]} +\right] +$$ +\returnType{Type: List PolynomialIdeals(Fraction Integer, +DirectProduct(3,NonNegativeInteger), +OrderedVariableList [x,y,z], +DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} + +We can intersect back. + +\spadcommand{reduce(intersect,ld) } +$$ +\left[ +{x -{{1 \over 4} \ y \ z}}, {y \sp 2}, {{z \sp 2} -4} +\right] +$$ +\returnType{Type: PolynomialIdeals(Fraction Integer, +DirectProduct(3,NonNegativeInteger), +OrderedVariableList [x,y,z], +DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} + +We can compute the radical of every primary component. + +\spadcommand{reduce(intersect,[radical ld.i for i in 1..2]) } +$$ +\left[ +x, y, {{z \sp 2} -4} +\right] +$$ +\returnType{Type: PolynomialIdeals(Fraction Integer, +DirectProduct(3,NonNegativeInteger), +OrderedVariableList [x,y,z], +DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} + +Their intersection is equal to the radical of the ideal of $l$. + +\spadcommand{radical ideal l } +$$ +\left[x, y, {{z \sp 2} -4} \right] +$$ +\returnType{Type: PolynomialIdeals(Fraction Integer, +DirectProduct(3,NonNegativeInteger), +OrderedVariableList [x,y,z], +DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} + +\section{Computation of Galois Groups} +\label{ugProblemGalois} + +As a sample use of Axiom's algebraic number facilities, +\index{group!Galois} +we compute +\index{Galois!group} +the Galois group of the polynomial +$p(x) = x^5 - 5 x + 12$. + +\spadcommand{p := x**5 - 5*x + 12 } +$$ +{x \sp 5} -{5 \ x}+{12} +$$ +\returnType{Type: Polynomial Integer} + +We would like to construct a polynomial $f(x)$ such that the splitting +\index{field!splitting} field \index{splitting field} of $p(x)$ is +generated by one root of $f(x)$. First we construct a polynomial +$r = r(x)$ such that one root of $r(x)$ generates the field generated by +two roots of the polynomial $p(x)$. (As it will turn out, the field +generated by two roots of $p(x)$ is, in fact, the splitting field of +$p(x)$.) + +From the proof of the primitive element theorem we know that if $a$ +and $b$ are algebraic numbers, then the field ${\bf Q}(a,b)$ is equal +to ${\bf Q}(a+kb)$ for an appropriately chosen integer $k$. In our +case, we construct the minimal polynomial of $a_i - a_j$, where $a_i$ +and $a_j$ are two roots of $p(x)$. We construct this polynomial using +{\bf resultant}. The main result we need is the following: If $f(x)$ +is a polynomial with roots $a_i \ldots a_m$ and $g(x)$ is a polynomial +with roots $b_i \ldots b_n$, then the polynomial $h(x) = +resultant(f(y), g(x-y), y)$ is a polynomial of degree $m*n$ with roots +$a_i + b_j, i = 1 \ldots m, j = 1 \ldots n$. + +For $f(x)$ we use the polynomial $p(x)$. For $g(x)$ we use the +polynomial $-p(-x)$. Thus, the polynomial we first construct is +$resultant(p(y), -p(y-x), y)$. + +\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y) } +$$ +\begin{array}{@{}l} +{x \sp {25}} - +{{50} \ {x \sp {21}}} - +{{2375} \ {x \sp {17}}}+ +{{90000} \ {x \sp {15}}} - +{{5000} \ {x \sp {13}}}+ +{{2700000} \ {x \sp {11}}}+ +{{250000} \ {x \sp 9}}+ +\\ +\\ +\displaystyle +{{18000000} \ {x \sp 7}}+ +{{64000000} \ {x \sp 5}} +\end{array} +$$ +\returnType{Type: Polynomial Integer} + +The roots of $q(x)$ are $a_i - a_j, i \leq 1, j \leq 5$. Of course, +there are five pairs $(i,j)$ with $i = j$, so $0$ is a 5-fold root of +$q(x)$. + +Let's get rid of this factor. + +\spadcommand{q1 := exquo(q, x**5) } +$$ +\begin{array}{@{}l} +{x \sp {20}} - +{{50} \ {x \sp {16}}} - +{{2375} \ {x \sp {12}}}+ +{{90000} \ {x \sp {10}}} - +{{5000} \ {x \sp 8}}+ +{{2700000} \ {x \sp 6}}+ +\\ +\\ +\displaystyle +{{250000} \ {x \sp 4}}+ +{{18000000} \ {x \sp 2}}+ +{64000000} +\end{array} +$$ +\returnType{Type: Union(Polynomial Integer,...)} + +Factor the polynomial $q1$. + +\spadcommand{factoredQ := factor q1 } +$$ +\begin{array}{@{}l} +{\left( +{x \sp {10}} - +{{10} \ {x \sp 8}} - +{{75} \ {x \sp 6}}+ +{{1500} \ {x \sp 4}} - +{{5500} \ {x \sp 2}}+ +{16000} +\right)} * +\\ +\\ +\displaystyle +{\left( +{x \sp {10}}+ +{{10} \ {x \sp 8}}+ +{{125} \ {x \sp 6}}+ +{{500} \ {x \sp 4}}+ +{{2500} \ {x \sp 2}}+ +{4000} +\right)} +\end{array} +$$ +\returnType{Type: Factored Polynomial Integer} + +We see that $q1$ has two irreducible factors, each of degree $10$. +(The fact that the polynomial $q1$ has two factors of degree $10$ is +enough to show that the Galois group of $p(x)$ is the dihedral group +of order $10$.\footnote{See McKay, Soicher, Computing Galois Groups +over the Rationals, Journal of Number Theory 20, 273-281 (1983). We +do not assume the results of this paper, however, and we continue with +the computation.} Note that the type of $factoredQ$ is {\tt FR POLY +INT}, that is, {\tt Factored Polynomial Integer}. \index{Factored} +This is a special data type for recording factorizations of +polynomials with integer coefficients. + +We can access the individual factors using the operation +\spadfunFrom{nthFactor}{Factored}. + +\spadcommand{r := nthFactor(factoredQ,1) } +$$ +{x \sp {10}} -{{10} \ {x \sp 8}} -{{75} \ {x \sp 6}}+{{1500} \ {x \sp 4}} +-{{5500} \ {x \sp 2}}+{16000} +$$ +\returnType{Type: Polynomial Integer} + +Consider the polynomial $r = r(x)$. This is the minimal polynomial of +the difference of two roots of $p(x)$. Thus, the splitting field of +$p(x)$ contains a subfield of degree $10$. We show that this subfield +is, in fact, the splitting field of $p(x)$ by showing that $p(x)$ +factors completely over this field. + +First we create a symbolic root of the polynomial $r(x)$. (We +replaced $x$ by $b$ in the polynomial $r$ so that our symbolic root +would be printed as $b$.) + +\spadcommand{beta:AN := rootOf(eval(r,x,b)) } +$$ +b +$$ +\returnType{Type: AlgebraicNumber} + +We next tell Axiom to view $p(x)$ as a univariate polynomial in $x$ +with algebraic number coefficients. This is accomplished with this +type declaration. + +\spadcommand{p := p::UP(x,INT)::UP(x,AN) } +$$ +{x \sp 5} -{5 \ x}+{12} +$$ +\returnType{Type: UnivariatePolynomial(x,AlgebraicNumber)} + +Factor $p(x)$ over the field ${\bf Q}(\beta)$. +(This computation will take some time!) + +\spadcommand{algFactors := factor(p,[beta]) } +$$ +\begin{array}{@{}l} +{\left( +x+ +{\left( +\begin{array}{@{}l} +-{{85} \ {b \sp 9}} - +{{116} \ {b \sp 8}}+ +{{780} \ {b \sp 7}}+ +{{2640} \ {b \sp 6}}+ +{{14895} \ {b \sp 5}} - +\\ +\\ +\displaystyle +{{8820} \ {b \sp 4}} - +{{127050} \ {b \sp 3}} - +{{327000} \ {b \sp 2}} - +{{405200} \ b}+ +{2062400} +\end{array} +\right) +\over {1339200}} +\right)} +\\ +\\ +\displaystyle +{\left( +x+ +{{-{{17} \ {b \sp 8}}+ +{{156} \ {b \sp 6}}+ +{{2979} \ {b \sp 4}} - +{{25410} \ {b \sp 2}} - +{14080}} \over {66960}} +\right)} +\\ +\\ +\displaystyle +\ {\left( +x+ +{{{{143} \ {b \sp 8}} - +{{2100} \ {b \sp 6}} - +{{10485} \ {b \sp 4}}+ +{{290550} \ {b \sp 2}} - +{{334800} \ b} - +{960800}} +\over {669600}} +\right)} +\\ +\\ +\displaystyle +\ {\left( +x+ +{{{{143} \ {b \sp 8}} - +{{2100} \ {b \sp 6}} - +{{10485} \ {b \sp 4}}+ +{{290550} \ {b \sp 2}}+ +{{334800} \ b} - +{960800}} +\over {669600}} +\right)} +\\ +\\ +\displaystyle +{\left( +x+ +{\left( +\begin{array}{@{}l} +{{85} \ {b \sp 9}} - +{{116} \ {b \sp 8}} - +{{780} \ {b \sp 7}}+ +{{2640} \ {b \sp 6}} - +{{14895} \ {b \sp 5}} - +\\ +\\ +\displaystyle +{{8820} \ {b \sp 4}}+ +{{127050} \ {b \sp 3}} - +{{327000} \ {b \sp 2}}+ +{{405200} \ b}+ +{2062400} +\end{array} +\right) +\over {1339200}} +\right)} +\end{array} +$$ +\returnType{Type: Factored UnivariatePolynomial(x,AlgebraicNumber)} + +When factoring over number fields, it is important to specify the +field over which the polynomial is to be factored, as polynomials have +different factorizations over different fields. When you use the +operation {\bf factor}, the field over which the polynomial is +factored is the field generated by +\begin{enumerate} +\item the algebraic numbers that appear +in the coefficients of the polynomial, and +\item the algebraic numbers that +appear in a list passed as an optional second argument of the operation. +\end{enumerate} +In our case, the coefficients of $p$ +are all rational integers and only $beta$ +appears in the list, so the field is simply +${\bf Q}(\beta)$. + +It was necessary to give the list $[beta]$ as a second argument of the +operation because otherwise the polynomial would have been factored +over the field generated by its coefficients, namely the rational +numbers. + +\spadcommand{factor(p) } +$$ +{x \sp 5} -{5 \ x}+{12} +$$ +\returnType{Type: Factored UnivariatePolynomial(x,AlgebraicNumber)} + +We have shown that the splitting field of $p(x)$ has degree $10$. +Since the symmetric group of degree 5 has only one transitive subgroup +of order $10$, we know that the Galois group of $p(x)$ must be this +group, the dihedral group \index{group!dihedral} of order $10$. +Rather than stop here, we explicitly compute the action of the Galois +group on the roots of $p(x)$. + +First we assign the roots of $p(x)$ as the values of five \index{root} +variables. + +We can obtain an individual root by negating the constant coefficient of +one of the factors of $p(x)$. + +\spadcommand{factor1 := nthFactor(algFactors,1) } +$$ +x+ +{ +\left( +\begin{array}{@{}l} +-{{85} \ {b \sp 9}} - +{{116} \ {b \sp 8}}+ +{{780} \ {b \sp 7}}+ +{{2640} \ {b \sp 6}}+ +{{14895} \ {b \sp 5}} - +\\ +\\ +\displaystyle +{{8820} \ {b \sp 4}} - +{{127050} \ {b \sp 3}} - +{{327000} \ {b \sp 2}} - +{{405200} \ b}+ +{2062400} +\end{array} +\right) +\over {1339200}} +$$ +\returnType{Type: UnivariatePolynomial(x,AlgebraicNumber)} + +\spadcommand{root1 := -coefficient(factor1,0) } +$$ +\left( +\begin{array}{@{}l} +{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}} - +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}} - +{{14895} \ {b \sp 5}}+ +\\ +\\ +\displaystyle +{{8820} \ {b \sp 4}}+ +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}}+ +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200} +$$ +\returnType{Type: AlgebraicNumber} + +We can obtain a list of all the roots in this way. + +\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5] } +$$ +\begin{array}{@{}l} +\left[ +\left( +\begin{array}{@{}l} +{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}} - +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}} - +{{14895} \ {b \sp 5}}+ +{{8820} \ {b \sp 4}}+ +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}}+ +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200}, +\right. +\\ +\\ +\displaystyle +{{{{17} \ {b \sp 8}} - +{{156} \ {b \sp 6}} - +{{2979} \ {b \sp 4}}+ +{{25410} \ {b \sp 2}}+ +{14080}} +\over {66960}}, +\\ +\\ +\displaystyle +{{-{{143} \ {b \sp 8}}+ +{{2100} \ {b \sp 6}}+ +{{10485} \ {b \sp 4}} - +{{290550} \ {b \sp 2}}+ +{{334800} \ b}+ +{960800}} +\over {669600}}, +\\ +\\ +\displaystyle +{{-{{143} \ {b \sp 8}}+ +{{2100} \ {b \sp 6}}+ +{{10485} \ {b \sp 4}} - +{{290550} \ {b \sp 2}} - +{{334800} \ b}+{960800}} +\over {669600}}, +\\ +\\ +\displaystyle +\left. +\left( +\begin{array}{@{}l} +-{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}}+ +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}}+ +{{14895} \ {b \sp 5}}+ +{{8820} \ {b \sp 4}} - +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}}- +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200} +\right] +\end{array} +$$ +\returnType{Type: List AlgebraicNumber} + +The expression +\begin{verbatim} +- coefficient(nthFactor(algFactors, i), 0)} +\end{verbatim} +is the $i $-th root of $p(x)$ and the elements of $roots$ are the +$i$-th roots of $p(x)$ as $i$ ranges from $1$ to $5$. + +Assign the roots as the values of the variables $a1,...,a5$. + +\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5) } +$$ +\left( +\begin{array}{@{}l} +-{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}}+ +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}}+ +{{14895} \ {b \sp 5}}+ +{{8820} \ {b \sp 4}} - +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}}- +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200} +$$ +\returnType{Type: AlgebraicNumber} + +Next we express the roots of $r(x)$ as polynomials in $beta$. We +could obtain these roots by calling the operation {\bf factor}: +$factor(r, [beta])$ factors $r(x)$ over ${\bf Q}(\beta)$. However, +this is a lengthy computation and we can obtain the roots of $r(x)$ as +differences of the roots $a1,...,a5$ of $p(x)$. Only ten of these +differences are roots of $r(x)$ and the other ten are roots of the +other irreducible factor of $q1$. We can determine if a given value +is a root of $r(x)$ by evaluating $r(x)$ at that particular value. +(Of course, the order in which factors are returned by the operation +{\bf factor} is unimportant and may change with different +implementations of the operation. Therefore, we cannot predict in +advance which differences are roots of $r(x)$ and which are not.) + +Let's look at four examples (two are roots of $r(x)$ and +two are not). + +\spadcommand{eval(r,x,a1 - a2) } +$$ +0 +$$ +\returnType{Type: Polynomial AlgebraicNumber} + +\spadcommand{eval(r,x,a1 - a3) } +$$ +\left( +\begin{array}{@{}l} +{{47905} \ {b \sp 9}}+ +{{66920} \ {b \sp 8}} - +{{536100} \ {b \sp 7}} - +{{980400} \ {b \sp 6}} - +{{3345075} \ {b \sp 5}} - +{{5787000} \ {b \sp 4}}+ +\\ +\\ +\displaystyle +{{75572250} \ {b \sp 3}}+ +{{161688000} \ {b \sp 2}} - +{{184600000} \ b} - +{710912000} +\end{array} +\right) +\over {4464} +$$ +\returnType{Type: Polynomial AlgebraicNumber} + +\spadcommand{eval(r,x,a1 - a4) } +$$ +0 +$$ +\returnType{Type: Polynomial AlgebraicNumber} + +\spadcommand{eval(r,x,a1 - a5) } +$$ +{{{405} \ {b \sp 8}}+ +{{3450} \ {b \sp 6}} - +{{19875} \ {b \sp 4}} - +{{198000} \ {b \sp 2}} - +{588000}} +\over {31} +$$ +\returnType{Type: Polynomial AlgebraicNumber} + +Take one of the differences that was a root of $r(x)$ and assign it to +the variable $bb$. + +For example, if $eval(r,x,a1 - a4)$ returned $0$, you would enter this. + +\spadcommand{bb := a1 - a4 } +$$ +\left( +\begin{array}{@{}l} +{{85} \ {b \sp 9}}+ +{{402} \ {b \sp 8}} - +{{780} \ {b \sp 7}} - +{{6840} \ {b \sp 6}} - +{{14895} \ {b \sp 5}} - +{{12150} \ {b \sp 4}}+ +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{908100} \ {b \sp 2}}+ +{{1074800} \ b} - +{3984000} +\end{array} +\right) +\over {1339200} +$$ +\returnType{Type: AlgebraicNumber} + +Of course, if the difference is, in fact, equal to the root $beta$, +you should choose another root of $r(x)$. + +Automorphisms of the splitting field are given by mapping a generator +of the field, namely $beta$, to other roots of its minimal polynomial. +Let's see what happens when $beta$ is mapped to $bb$. + +We compute the images of the roots $a1,...,a5$ under this automorphism: + +\spadcommand{aa1 := subst(a1,beta = bb) } +$$ +{-{{143} \ {b \sp 8}}+ +{{2100} \ {b \sp 6}}+ +{{10485} \ {b \sp 4}}- +{{290550} \ {b \sp 2}}+ +{{334800} \ b}+ +{960800}} +\over {669600} +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{aa2 := subst(a2,beta = bb) } +$$ +\left( +\begin{array}{@{}l} +-{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}}+ +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}}+ +{{14895} \ {b \sp 5}}+ +{{8820} \ {b \sp 4}} - +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}} - +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200} +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{aa3 := subst(a3,beta = bb) } +$$ +\left( +\begin{array}{@{}l} +{{85} \ {b \sp 9}}+ +{{116} \ {b \sp 8}} - +{{780} \ {b \sp 7}} - +{{2640} \ {b \sp 6}} - +{{14895} \ {b \sp 5}}+ +{{8820} \ {b \sp 4}}+ +\\ +\\ +\displaystyle +{{127050} \ {b \sp 3}}+ +{{327000} \ {b \sp 2}}+ +{{405200} \ b} - +{2062400} +\end{array} +\right) +\over {1339200} +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{aa4 := subst(a4,beta = bb) } +$$ +{-{{143} \ {b \sp 8}}+ +{{2100} \ {b \sp 6}}+ +{{10485} \ {b \sp 4}}- +{{290550} \ {b \sp 2}} - +{{334800} \ b}+ +{960800}} +\over {669600} +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{aa5 := subst(a5,beta = bb) } +$$ +{{{17} \ {b \sp 8}} - +{{156} \ {b \sp 6}} - +{{2979} \ {b \sp 4}}+ +{{25410} \ {b \sp 2}}+ +{14080}} +\over {66960} +$$ +\returnType{Type: AlgebraicNumber} + +Of course, the values $aa1,...,aa5$ are simply a permutation of the values +$a1,...,a5$. + +Let's find the value of $aa1$ (execute as many of the following five commands +as necessary). + +\spadcommand{(aa1 = a1) :: Boolean } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{(aa1 = a2) :: Boolean } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{(aa1 = a3) :: Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{(aa1 = a4) :: Boolean } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{(aa1 = a5) :: Boolean } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Proceeding in this fashion, you can find the values of +$aa2,...aa5$. You have represented the automorphism $beta -> bb$ as a +permutation of the roots $a1,...,a5$. If you wish, you can repeat +this computation for all the roots of $r(x)$ and represent the Galois +group of $p(x)$ as a subgroup of the symmetric group on five letters. + +Here are two other problems that you may attack in a similar fashion: +\begin{enumerate} +\item Show that the Galois group of +$p(x) = x^4 + 2 x^3 - 2 x^2 - 3 x + 1$ +is the dihedral group of order eight. \index{group!dihedral} +(The splitting field of this polynomial is the Hilbert class field +\index{Hilbert class field} of \index{field!Hilbert class} the quadratic field +${\bf Q}(\sqrt{145})$.) +\item Show that the Galois group of +$p(x) = x^6 + 108$ +has order 6 and is isomorphic to $S_3,$ the symmetric group on three letters. +\index{group!symmetric} (The splitting field of this polynomial is the +splitting field of $x^3 - 2$.) +\end{enumerate} + +\section{Non-Associative Algebras and Modelling Genetic Laws} +\label{ugProblemGenetic} + +Many algebraic structures of mathematics and Axiom have a +multiplication operation {\tt *} that satisfies the associativity law +\index{associativity law} $a*(b*c) = (a*b)*c$ for all $a$, $b$ and +$c$. The octonions are a well known exception. There are many other +interesting non-associative structures, such as the class of +\index{Lie algebra} Lie algebras.\footnote{Two Axiom implementations +of Lie algebras are {\tt LieSquareMatrix} and {\tt FreeNilpotentLie}.} +Lie algebras can be used, for example, to analyse Lie symmetry +algebras of \index{symmetry} partial differential \index{differential +equation!partial} equations. \index{partial differential equation} In +this section we show a different application of non-associative +algebras, \index{non-associative algebra} the modelling of genetic +laws. \index{algebra!non-associative} + +The Axiom library contains several constructors for creating +non-assoc\-i\-a\-tive structures, ranging from the categories +{\tt Monad}, {\tt NonAssociativeRng}, and {\tt FramedNonAssociativeAlgebra}, +to the domains {\tt AlgebraGivenByStructuralConstants} and +{\tt GenericNonAssociativeAlgebra}. Furthermore, the package +{\tt AlgebraPackage} provides operations for analysing the structure of +such algebras.\footnote{% The interested reader can learn more about +these aspects of the Axiom library from the paper ``Computations in +Algebras of Finite Rank,'' by Johannes Grabmeier and Robert Wisbauer, +Technical Report, IBM Heidelberg Scientific Center, 1992.} + +Mendel's genetic laws are often written in a form like + +$$Aa \times Aa = {1\over 4}AA + {1\over 2}Aa + {1\over 4}aa$$ + +The implementation of general algebras in Axiom allows us to +\index{Mendel's genetic laws} use this as the definition for +multiplication in an algebra. \index{genetics} Hence, it is possible +to study questions of genetic inheritance using Axiom. To demonstrate +this more precisely, we discuss one example from a monograph of +A. W\"orz-Busekros, where you can also find a general setting of this +theory.\footnote{% W\"{o}rz-Busekros, A., {\it Algebras in Genetics}, +Springer Lectures Notes in Biomathematics 36, Berlin e.a. (1980). In +particular, see example 1.3.} + +We assume that there is an infinitely large random mating population. +Random mating of two gametes $a_i$ and $a_j$ gives zygotes +\index{zygote} $a_ia_j$, which produce new gametes. \index{gamete} In +classical Mendelian segregation we have $a_ia_j = {1 \over 2}a_i+{1 +\over 2}a_j$. In general, we have + +$$a_ia_j = \sum_{k=1}^n \gamma_{i,j}^k\ a_k.$$ + +%{$ai aj = gammaij1 a1 + gammaij2 a2 + ... + gammaijn an$} + +The segregation rates $\gamma_{i,j}$ are the structural constants of +an $n$-dimensional algebra. This is provided in Axiom by the +constructor {\tt AlgebraGivenByStructuralConstants} (abbreviation +{\tt ALGSC}). + +Consider two coupled autosomal loci with alleles $A$, $a$, $B$, and +$b$, building four different gametes $a_1 = AB, a_2 = Ab, a_3 = aB,$ +and $a_4 = ab$ {$a1 := AB, a2 := Ab, a3 := aB,$ and $a4 := ab$}. The +zygotes $a_ia_j$ produce gametes $a_i$ and $a_j$ with classical +Mendelian segregation. Zygote $a_1a_4$ undergoes transition to +$a_2a_3$ and vice versa with probability +$0 \le \theta \le {1\over2}$. + +Define a list $[(\gamma_{i,j}^k) 1 \le k \le 4]$ of four four-by-four +matrices giving the segregation rates. We use the value $1/10$ for +$\theta$. + +\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ] } +$$ +\begin{array}{@{}l} +\left[ +{\left[ +\begin{array}{cccc} +1 & {1 \over 2} & {1 \over 2} & {9 \over {20}} \\ +{1 \over 2} & 0 & {1 \over {20}} & 0 \\ +{1 \over 2} & {1 \over {20}} & 0 & 0 \\ +{9 \over {20}} & 0 & 0 & 0 +\end{array} +\right]}, +{\left[ +\begin{array}{cccc} +0 & {1 \over 2} & 0 & {1 \over {20}} \\ +{1 \over 2} & 1 & {9 \over {20}} & {1 \over 2} \\ +0 & {9 \over {20}} & 0 & 0 \\ +{1 \over {20}} & {1 \over 2} & 0 & 0 +\end{array} +\right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ +\begin{array}{cccc} +0 & 0 & {1 \over 2} & {1 \over {20}} \\ +0 & 0 & {9 \over {20}} & 0 \\ +{1 \over 2} & {9 \over {20}} & 1 & {1 \over 2} \\ +{1 \over {20}} & 0 & {1 \over 2} & 0 +\end{array} +\right]}, +{\left[ +\begin{array}{cccc} +0 & 0 & 0 & {9 \over {20}} \\ +0 & 0 & {1 \over {20}} & {1 \over 2} \\ +0 & {1 \over {20}} & 0 & {1 \over 2} \\ +{9 \over {20}} & {1 \over 2} & {1 \over 2} & 1 +\end{array} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List SquareMatrix(4,Fraction Integer)} + +Choose the appropriate symbols for the basis of gametes, + +\spadcommand{gametes := ['AB,'Ab,'aB,'ab] } +$$ +\left[ +AB, Ab, aB, ab +\right] +$$ +\returnType{Type: List OrderedVariableList [AB,Ab,aB,ab]} + +Define the algebra. + +\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates)} +$$ +\begin{array}{@{}l} +{\rm AlgebraGivenByStructuralConstants(Fraction Integer, 4, } +\\ +\displaystyle +{\rm [AB,Ab,aB,ab], [MATRIX,MATRIX,MATRIX,MATRIX])} +\end{array} +$$ +\returnType{Type: Domain} + +What are the probabilities for zygote $a_1a_4$ to produce the +different gametes? + +\spadcommand{a := basis()\$A} +$$ +\left[ +AB, Ab, aB, ab +\right] +$$ +\returnType{Type: Vector +AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + +\spadcommand{a.1*a.4} +$$ +{{9 \over {20}} \ ab}+ +{{1 \over {20}} \ aB}+ +{{1 \over {20}} \ Ab}+ +{{9 \over {20}} \ AB} +$$ +\returnType{Type: +AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + +Elements in this algebra whose coefficients sum to one play a +distinguished role. They represent a population with the distribution +of gametes reflected by the coefficients with respect to the basis of +gametes. + +Random mating of different populations $x$ and $y$ is described by +their product $x*y$. + +This product is commutative only if the gametes are not sex-dependent, +as in our example. + +\spadcommand{commutative?()\$A } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +In general, it is not associative. + +\spadcommand{associative?()\$A } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Random mating within a population $x$ is described by $x*x$. The next +generation is $(x*x)*(x*x)$. + +Use decimal numbers to compare the distributions more easily. + +\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]} +$$ +{{0.4} \ ab}+{{0.1} \ aB}+{{0.2} \ Ab}+{{0.3} \ AB} +$$ +\returnType{Type: +AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + +To compute directly the gametic distribution in the fifth generation, +we use {\bf plenaryPower}. + +\spadcommand{plenaryPower(x,5) } +$$ +{{0.{36561}} \ ab}+{{0.{13439}} \ aB}+{{0.{23439}} \ Ab}+{{0.{26561}} \ +AB} +$$ +\returnType{Type: +AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + +We now ask two questions: Does this distribution converge to an +equilibrium state? What are the distributions that are stable? + +This is an invariant of the algebra and it is used to answer the first +question. The new indeterminates describe a symbolic distribution. + +\spadcommand{q := leftRankPolynomial()\$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)} +$$ +\begin{array}{@{}l} +{Y \sp 3}+ +{{\left( +-{{{29} \over {20}} \ \%x4} - +{{{29} \over {20}} \ \%x3} - +{{{29} \over {20}} \ \%x2} - +{{{29} \over {20}} \ \%x1} +\right)}\ {Y \sp 2}}+ +\\ +\\ +\displaystyle +{ +\left( +\begin{array}{@{}l} +\left( {{9 \over {20}} \ { \%x4 \sp 2}}+ +{{\left( +{{9 \over {10}} \ \%x3}+ +{{9 \over {10}} \ \%x2}+ +{{9 \over {10}} \ \%x1} +\right)}\ \%x4}+ +\right. +\\ +\\ +\displaystyle +{{9 \over {20}} \ { \%x3 \sp 2}}+ +{{\left( {{9 \over {10}} \ \%x2}+{{9 \over {10}} \ \%x1} \right)}\ \%x3}+ +{{9 \over {20}} \ { \%x2 \sp 2}}+ +\\ +\\ +\displaystyle +\left. +{{9 \over {10}} \ \%x1 \ \%x2}+ +{{9 \over {20}} \ { \%x1 \sp 2}} +\right) +\end{array} +\right) +\ Y} +\end{array} +$$ +\returnType{Type: UnivariatePolynomial(Y,Polynomial Fraction Integer)} + + +Because the coefficient ${9 \over 20}$ has absolute value less than 1, +all distributions do converge, by a theorem of this theory. + +\spadcommand{factor(q :: POLY FRAC INT) } +$$ +\begin{array}{@{}l} +{\left( Y - \%x4 - \%x3 - \%x2 - \%x1 \right)} * +\\ +\\ +\displaystyle +{\left( +Y - +{{9 \over {20}} \ \%x4} - +{{9 \over {20}} \ \%x3} - +{{9 \over {20}} \ \%x2} - +{{9 \over {20}} \ \%x1} +\right)} +\ Y +\end{array} +$$ +\returnType{Type: Factored Polynomial Fraction Integer} + +The second question is answered by searching for idempotents in the algebra. + +\spadcommand{cI := conditionsForIdempotents()\$GCNAALG(FRAC INT, 4, gametes, segregationRates) } +$$ +\begin{array}{@{}l} +\left[ +{{{9 \over {10}} \ \%x1 \ \%x4}+ +{{\left( {{1 \over {10}} \ \%x2}+ \%x1 \right)}\ \%x3}+ +{ \%x1 \ \%x2}+ +{ \%x1 \sp 2} - +\%x1}, +\right. +\\ +\\ +\displaystyle +{{{\left( \%x2+{{1 \over {10}} \ \%x1} \right)}\ \%x4}+ +{{9 \over {10}} \ \%x2 \ \%x3}+ +{ \%x2 \sp 2}+ +{{\left( \%x1 -1 \right)}\ \%x2}}, +\\ +\\ +\displaystyle +{{{\left( \%x3+{{1 \over {10}} \ \%x1} \right)}\ \%x4}+ +{ \%x3 \sp 2}+ +{{\left( {{9 \over {10}} \ \%x2}+ \%x1 -1 \right)}\ \%x3}}, +\\ +\\ +\displaystyle +\left. +{{ \%x4 \sp 2}+ +{{\left( \%x3+ \%x2+{{9 \over {10}} \ \%x1} -1 \right)}\ \%x4}+ +{{1 \over {10}} \ \%x2 \ \%x3}} +\right] +\end{array} +$$ +\returnType{Type: List Polynomial Fraction Integer} + +Solve these equations and look at the first solution. + +\spadcommand{gbs:= groebnerFactorize cI} +$$ +\begin{array}{@{}l} +\left[ +\begin{array}{@{}l} +\left[ { \%x4+ \%x3+ \%x2+ \%x1 -1}, +\right. +\\ +\displaystyle +\left. +\ \ {{{\left( \%x2+ \%x1 \right)}\ \%x3}+ +{ \%x1 \ \%x2}+{ \%x1 \sp 2} - \%x1} +\right], +\end{array} +\right. +\\ +\\ +\displaystyle +{\left[ 1 \right]}, +{\left[ { \%x4+ \%x3 -1}, \%x2, \%x1 \right]}, +\\ +\\ +\displaystyle +{\left[ { \%x4+ \%x2 -1}, \%x3, \%x1 \right]}, +{\left[ \%x4, \%x3, \%x2, \%x1 \right]}, +\\ +\\ +\displaystyle +\left. +{\left[ { \%x4 -1}, \%x3, \%x2, \%x1 \right]}, +{\left[ { \%x4 -{1 \over 2}}, { \%x3 -{1 \over 2}}, \%x2, \%x1 \right]} +\right] +\end{array} +$$ +\returnType{Type: List List Polynomial Fraction Integer} + +\spadcommand{gbs.1} +$$ +\begin{array}{@{}l} +\left[ +{ \%x4+ \%x3+ \%x2+ \%x1 -1}, +\right. +\\ +\displaystyle +\left. +{{{\left( \%x2+ \%x1 \right)}\ \%x3}+{ \%x1 \ \%x2}+{ \%x1 \sp 2} - \%x1} +\right] +\end{array} +$$ +\returnType{Type: List Polynomial Fraction Integer} + + +Further analysis using the package {\tt PolynomialIdeals} shows that +there is a two-dimensional variety of equilibrium states and all other +solutions are contained in it. + +Choose one equilibrium state by setting two indeterminates to concrete +values. + +\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10]) } +$$ +\left[ +{\left[ +{ \%x4={2 \over 5}}, +{ \%x3={2 \over 5}}, +{ \%x2={1 \over {10}}}, +{ \%x1={1 \over {10}}} +\right]} +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT) } +$$ +{{2 \over 5} \ ab}+ +{{2 \over 5} \ aB}+ +{{1 \over {10}} \ Ab}+ +{{1 \over {10}} \ AB} +$$ +\returnType{Type: +AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + +Verify the result. + +\spadcommand{e*e-e } +$$ +0 +$$ +\returnType{Type: +AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], +[MATRIX,MATRIX,MATRIX,MATRIX])} + + +%\setcounter{chapter}{9} % Chapter 10 +\chapter{Some Examples of Domains and Packages} +In this chapter we show examples of many of the most commonly used +AXIOM domains and packages. The sections are organized by constructor +names. + +\section{AssociationList} +\label{AssociationListXmpPage} + +The {\tt AssociationList} constructor provides a general structure for +associative storage. This type provides association lists in which +data objects can be saved according to keys of any type. For a given +association list, specific types must be chosen for the keys and +entries. You can think of the representation of an association list +as a list of records with key and entry fields. + +Association lists are a form of table and so most of the operations +available for {\tt Table} are also available for {\tt AssociationList}. +They can also be viewed as lists and can be manipulated accordingly. + +This is a {\tt Record} type with age and gender fields. + +\spadcommand{Data := Record(monthsOld : Integer, gender : String)} +$$ +\mbox{\rm Record(monthsOld: Integer,gender: String)} +$$ +\returnType{Type: Domain} + +In this expression, {\tt al} is declared to be an association +list whose keys are strings and whose entries are the above records. + +\spadcommand{al : AssociationList(String,Data)} +\returnType{Type: Void} + +The \spadfunFrom{table}{AssociationList} operation is used to create +an empty association list. + +\spadcommand{al := table()} +$$ +table() +$$ +\returnType{Type: +AssociationList(String,Record(monthsOld: Integer,gender: String))} + +You can use assignment syntax to add things to the association list. + +\spadcommand{al."bob" := [407,"male"]\$Data} +$$ +\left[ +{monthsOld={407}}, {gender= \mbox{\tt "male"} } +\right] +$$ +\returnType{Type: Record(monthsOld: Integer,gender: String)} + +\spadcommand{al."judith" := [366,"female"]\$Data} +$$ +\left[ +{monthsOld={366}}, {gender= \mbox{\tt "female"} } +\right] +$$ +\returnType{Type: Record(monthsOld: Integer,gender: String)} + +\spadcommand{al."katie" := [24,"female"]\$Data} +$$ +\left[ +{monthsOld={24}}, {gender= \mbox{\tt "female"} } +\right] +$$ +\returnType{Type: Record(monthsOld: Integer,gender: String)} + +Perhaps we should have included a species field. + +\spadcommand{al."smokie" := [200,"female"]\$Data} +$$ +\left[ +{monthsOld={200}}, {gender= \mbox{\tt "female"} } +\right] +$$ +\returnType{Type: Record(monthsOld: Integer,gender: String)} + +Now look at what is in the association list. Note that the last-added +(key, entry) pair is at the beginning of the list. + +\spadcommand{al} +$$ +\begin{array}{@{}l} +table +\left( +{ \mbox{\tt "smokie"} = +{\left[ {monthsOld={200}}, {gender= \mbox{\tt "female"} } \right]}}, +\right. +\\ +\\ +\displaystyle +\ \ \ \ \ \ \ \ { \mbox{\tt "katie"} = +{\left[ {monthsOld={24}}, {gender= \mbox{\tt "female"} } \right]}}, +\\ +\\ +\displaystyle +\ \ \ \ \ \ \ \ { \mbox{\tt "judith"} = +{\left[ {monthsOld={366}}, {gender= \mbox{\tt "female"} } \right]}}, +\\ +\\ +\displaystyle +\left. +\ \ \ \ \ \ \ \ { \mbox{\tt "bob"} = +{\left[ {monthsOld={407}}, {gender= \mbox{\tt "male"} } \right]}} +\right) +\end{array} +$$ +\returnType{Type: +AssociationList(String,Record(monthsOld: Integer,gender: String))} + +You can reset the entry for an existing key. + +\spadcommand{al."katie" := [23,"female"]\$Data} +$$ +\left[ +{monthsOld={23}}, {gender= \mbox{\tt "female"} } +\right] +$$ +\returnType{Type: Record(monthsOld: Integer,gender: String)} + +Use \spadfunFrom{delete!}{AssociationList} to destructively remove an +element of the association list. Use +\spadfunFrom{delete}{AssociationList} to return a copy of the +association list with the element deleted. The second argument is the +index of the element to delete. + +\spadcommand{delete!(al,1)} +$$ +\begin{array}{@{}l} +table +\left( +{ \mbox{\tt "katie"} = +{\left[ {monthsOld={23}}, {gender= \mbox{\tt "female"} } \right]}}, +\right. +\\ +\\ +\displaystyle +\ \ \ \ \ \ \ \ { \mbox{\tt "judith"} = +{\left[ {monthsOld={366}}, {gender= \mbox{\tt "female"} } \right]}}, +\\ +\\ +\displaystyle +\left. +\ \ \ \ \ \ \ \ { \mbox{\tt "bob"} = +{\left[ {monthsOld={407}}, {gender= \mbox{\tt "male"} } \right]}} +\right) +\end{array} +$$ +\returnType{Type: +AssociationList(String,Record(monthsOld: Integer,gender: String))} + +For more information about tables, +see \ref{TableXmpPage} on page~\pageref{TableXmpPage}. +For more information about lists, +see \ref{ListXmpPage} on page~\pageref{ListXmpPage}. + +\section{BalancedBinaryTree} +\label{BalancedBinaryTreeXmpPage} + +{\tt BalancedBinaryTrees(S)} is the domain of balanced binary trees +with elements of type {\tt S} at the nodes. A binary tree is either +{\tt empty} or else consists of a {\tt node} having a {\tt value} and +two branches, each branch a binary tree. A balanced binary tree is +one that is balanced with respect its leaves. One with $2^k$ leaves +is perfectly ``balanced'': the tree has minimum depth, and the {\tt +left} and {\tt right} branch of every interior node is identical in +shape. + +Balanced binary trees are useful in algebraic computation for +so-called ``divide-and-conquer'' algorithms. Conceptually, the data +for a problem is initially placed at the root of the tree. The +original data is then split into two subproblems, one for each +subtree. And so on. Eventually, the problem is solved at the leaves +of the tree. A solution to the original problem is obtained by some +mechanism that can reassemble the pieces. In fact, an implementation +of the Chinese Remainder Algorithm using balanced binary trees was +first proposed by David Y. Y. Yun at the IBM T. J. Watson Research +Center in Yorktown Heights, New York, in 1978. It served as the +prototype for polymorphic algorithms in Axiom. + +In what follows, rather than perform a series of computations with a +single expression, the expression is reduced modulo a number of +integer primes, a computation is done with modular arithmetic for each +prime, and the Chinese Remainder Algorithm is used to obtain the +answer to the original problem. We illustrate this principle with the +computation of $12^2 = 144$. + +A list of moduli. + +\spadcommand{lm := [3,5,7,11]} +$$ +\left[ +3, 5, 7, {11} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The expression {\tt modTree(n, lm)} creates a balanced binary tree +with leaf values {\tt n mod m} for each modulus {\tt m} in {\tt lm}. + +\spadcommand{modTree(12,lm)} +$$ +\left[ +0, 2, 5, 1 +\right] +$$ +\returnType{Type: List Integer} + +Operation {\tt modTree} does this using operations on balanced binary +trees. We trace its steps. Create a balanced binary tree {\tt t} of +zeros with four leaves. + +\spadcommand{t := balancedBinaryTree(\#lm, 0)} +$$ +\left[ +{\left[ 0, 0, 0 \right]}, 0, {\left[ 0, 0, 0\right]} +\right] +$$ +\returnType{Type: BalancedBinaryTree NonNegativeInteger} + +The leaves of the tree are set to the individual moduli. + +\spadcommand{setleaves!(t,lm)} +$$ +\left[ +{\left[ 3, 0, 5\right]}, 0, {\left[ 7, 0, {11} \right]} +\right] +$$ +\returnType{Type: BalancedBinaryTree NonNegativeInteger} + +Use {\tt mapUp!} to do a bottom-up traversal of {\tt t}, setting each +interior node to the product of the values at the nodes of its +children. + +\spadcommand{mapUp!(t,\_*)} +$$ +1155 +$$ +\returnType{Type: PositiveInteger} + +The value at the node of every subtree is the product of the moduli +of the leaves of the subtree. + +\spadcommand{t} +$$ +\left[ +{\left[ 3, {15}, 5\right]}, {1155}, {\left[ 7, {77}, {11} \right]} +\right] +$$ +\returnType{Type: BalancedBinaryTree NonNegativeInteger} + +Operation {\tt mapDown!}{\tt (t,a,fn)} replaces the value {\tt v} at +each node of {\tt t} by {\tt fn(a,v)}. + +\spadcommand{mapDown!(t,12,\_rem)} +$$ +\left[ +{\left[ 0, {12}, 2\right]}, {12}, {\left[ 5, {12}, 1 \right]} +\right] +$$ +\returnType{Type: BalancedBinaryTree NonNegativeInteger} + +The operation {\tt leaves} returns the leaves of the resulting tree. +In this case, it returns the list of {\tt 12 mod m} for each modulus +{\tt m}. + +\spadcommand{leaves \%} +$$ +\left[ +0, 2, 5, 1 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +Compute the square of the images of {\tt 12} modulo each {\tt m}. + +\spadcommand{squares := [x**2 rem m for x in \% for m in lm]} +$$ +\left[ +0, 4, 4, 1 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +Call the Chinese Remainder Algorithm to get the answer for $12^2$. + +\spadcommand{chineseRemainder(\%,lm)} +$$ +144 +$$ +\returnType{Type: PositiveInteger} + +\section{BasicOperator} +\label{BasicOperatorXmpPage} + +A basic operator is an object that can be symbolically applied to a +list of arguments from a set, the result being a kernel over that set +or an expression. In addition to this section, please see +\ref{ExpressionXmpPage} on page~\pageref{ExpressionXmpPage} and +\ref{KernelXmpPage} on page~\pageref{KernelXmpPage} for additional +information and examples. + +You create an object of type {\tt BasicOperator} by using the +\spadfunFrom{operator}{BasicOperator} operation. This first form of +this operation has one argument and it must be a symbol. The symbol +should be quoted in case the name has been used as an identifier to +which a value has been assigned. + +A frequent application of {\tt BasicOperator} is the creation of an +operator to represent the unknown function when solving a differential +equation. + +Let {\tt y} be the unknown function in terms of {\tt x}. + +\spadcommand{y := operator 'y} +$$ +y +$$ +\returnType{Type: BasicOperator} + +This is how you enter the equation {\tt y'' + y' + y = 0}. + +\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0} +$$ +{{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+ +{{y\sb {{\ }} \sp {,}} \left({x} \right)}+ +{y\left({x} \right)}}=0 +$$ +\returnType{Type: Equation Expression Integer} + +To solve the above equation, enter this. + +\spadcommand{solve(deq, y, x)} +$$ +\left[ +{particular=0}, +{basis={\left[ {{\cos \left({{{x \ {\sqrt {3}}} \over 2}} \right)} +\ {e \sp {\left( -{x \over 2} \right)}}}, +{{e \sp {\left( -{x \over 2} \right)}} +\ {\sin \left({{{x \ {\sqrt {3}}} \over 2}} \right)}} +\right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer, +basis: List Expression Integer),...)} + +See \ref{ugProblemDEQPage} on page~\pageref{ugProblemDEQPage} +in Section \ref{ugProblemDEQNumber} on page~\pageref{ugProblemDEQNumber} +for this kind of use of {\tt BasicOperator}. + +Use the single argument form of \spadfunFrom{operator}{BasicOperator} +(as above) when you intend to use the operator to create functional +expressions with an arbitrary number of arguments + +{\it Nary} means an arbitrary number of arguments can be used +in the functional expressions. + +\spadcommand{nary? y} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{unary? y} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Use the two-argument form when you want to restrict the number of +arguments in the functional expressions created with the operator. + +This operator can only be used to create functional expressions +with one argument. + +\spadcommand{opOne := operator('opOne, 1)} +$$ +opOne +$$ +\returnType{Type: BasicOperator} + +\spadcommand{nary? opOne} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{unary? opOne} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Use \spadfunFrom{arity}{BasicOperator} to learn the number of arguments +that can be used. It returns {\tt "false"} if the operator is nary. + +\spadcommand{arity opOne} +$$ +1 +$$ +\returnType{Type: Union(NonNegativeInteger,...)} + +Use \spadfunFrom{name}{BasicOperator} to learn the name of an operator. + +\spadcommand{name opOne} +$$ +opOne +$$ +\returnType{Type: Symbol} + +Use \spadfunFrom{is?}{BasicOperator} to learn if an operator has a +particular name. + +\spadcommand{is?(opOne, 'z2)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +You can also use a string as the name to be tested against. + +\spadcommand{is?(opOne, "opOne")} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +You can attached named properties to an operator. These are rarely +used at the top-level of the Axiom interactive environment but are +used with Axiom library source code. + +By default, an operator has no properties. + +\spadcommand{properties y} +$$ +table() +$$ +\returnType{Type: AssociationList(String,None)} + +The interface for setting and getting properties is somewhat awkward +because the property values are stored as values of type {\tt None}. + +Attach a property by using \spadfunFrom{setProperty}{BasicOperator}. + +\spadcommand{setProperty(y, "use", "unknown function" :: None )} +$$ +y +$$ +\returnType{Type: BasicOperator} + +\spadcommand{properties y} +$$ +table +\left( +{{ \mbox{\tt "use"} =NONE}} +\right) +$$ +\returnType{Type: AssociationList(String,None)} + +We {\it know} the property value has type {\tt String}. + +\spadcommand{property(y, "use") :: None pretend String} +$$ +\mbox{\tt "unknown function"} +$$ +\returnType{Type: String} + +Use \spadfunFrom{deleteProperty!}{BasicOperator} to destructively +remove a property. + +\spadcommand{deleteProperty!(y, "use")} +$$ +y +$$ +\returnType{Type: BasicOperator} + +\spadcommand{properties y} +$$ +table() +$$ +\returnType{Type: AssociationList(String,None)} + +\section{BinaryExpansion} +\label{BinaryExpansionXmpPage} + +All rational numbers have repeating binary expansions. Operations to +access the individual bits of a binary expansion can be obtained by +converting the value to {\tt RadixExpansion(2)}. More examples of +expansions are available in +\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, +\ref{HexadecimalExpansionXmpPage} on +page~\pageref{HexadecimalExpansionXmpPage}, and +\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. + +The expansion (of type {\tt BinaryExpansion}) of a rational number +is returned by the \spadfunFrom{binary}{BinaryExpansion} operation. + +\spadcommand{r := binary(22/7)} +$$ +{11}.{\overline {001}} +$$ +\returnType{Type: BinaryExpansion} + +Arithmetic is exact. + +\spadcommand{r + binary(6/7)} +$$ +100 +$$ +\returnType{Type: BinaryExpansion} + +The period of the expansion can be short or long \ldots + +\spadcommand{[binary(1/i) for i in 102..106] } +$$ +\begin{array}{@{}l} +\left[ +{0.0{\overline {00000101}}}, +{0.{\overline {000000100111110001000101100101111001110010010101001}}}, +\right. +\\ +\\ +\displaystyle +{0.{000}{\overline {000100111011}}}, +{0.{\overline {000000100111}}}, +\\ +\\ +\displaystyle +\left. +{0.0{\overline {0000010011010100100001110011111011001010110111100011}}} +\right] +\end{array} +$$ +\returnType{Type: List BinaryExpansion} + +or very long. + +\spadcommand{binary(1/1007) } +$$ +\begin{array}{@{}l} +0. +\overline +{000000000100000100010100100101111000001111110000101111110010110001111101} +\\ +\displaystyle +\ \ \overline +{000100111001001100110001100100101010111101101001100000000110000110011110} +\\ +\displaystyle +\ \ \overline +{111000110100010111101001000111101100001010111011100111010101110011001010} +\\ +\displaystyle +\ \ \overline +{010111000000011100011110010000001001001001101110010101001110100011011101} +\\ +\displaystyle +\ \ \overline +{101011100010010000011001011011000000101100101111100010100000101010101101} +\\ +\displaystyle +\ \ \overline +{011000001101101110100101011111110101110101001100100001010011011000100110} +\\ +\displaystyle +\ \ \overline +{001000100001000011000111010011110001} +\end{array} +$$ +\returnType{Type: BinaryExpansion} + +These numbers are bona fide algebraic objects. + +\spadcommand{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)} +$$ +{{0.{01}} \ {x \sp 2}}+{{0.{\overline {10}}} \ x}+{0.{\overline {011100}}} +$$ +\returnType{Type: Polynomial BinaryExpansion} + +\spadcommand{q := D(p, x)} +$$ +{{0.1} \ x}+{0.{\overline {10}}} +$$ +\returnType{Type: Polynomial BinaryExpansion} + +\spadcommand{g := gcd(p, q)} +$$ +x+{1.{\overline {01}}} +$$ +\returnType{Type: Polynomial BinaryExpansion} + +\section{BinarySearchTree} +\label{BinarySearchTreeXmpPage} + +{\tt BinarySearchTree(R)} is the domain of binary trees with elements +of type {\tt R}, ordered across the nodes of the tree. A non-empty +binary search tree has a value of type {\tt R}, and {\tt right} and +{\tt left} binary search subtrees. If a subtree is empty, it is +displayed as a period (``.''). + +Define a list of values to be placed across the tree. The resulting +tree has {\tt 8} at the root; all other elements are in the left +subtree. + +\spadcommand{lv := [8,3,5,4,6,2,1,5,7]} +$$ +\left[ +8, 3, 5, 4, 6, 2, 1, 5, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +A convenient way to create a binary search tree is to apply the +operation {\tt binarySearchTree} to a list of elements. + +\spadcommand{t := binarySearchTree lv} +$$ +\left[ +{\left[ {\left[ 1, 2, . \right]}, 3, +{\left[ 4, 5, {\left[ 5, 6, 7 \right]}\right]} +\right]}, +8, . +\right] +$$ +\returnType{Type: BinarySearchTree PositiveInteger} + +Another approach is to first create an empty binary search tree of integers. + +\spadcommand{emptybst := empty()\$BSTREE(INT)} +$$ +[\ ] +$$ +\returnType{Type: BinarySearchTree Integer} + +Insert the value {\tt 8}. This establishes {\tt 8} as the root of the +binary search tree. Values inserted later that are less than {\tt 8} +get stored in the {\tt left} subtree, others in the {\tt right} subtree. + +\spadcommand{t1 := insert!(8,emptybst)} +$$ +8 +$$ +\returnType{Type: BinarySearchTree Integer} + +Insert the value {\tt 3}. This number becomes the root of the {\tt +left} subtree of {\tt t1}. For optimal retrieval, it is thus +important to insert the middle elements first. + +\spadcommand{insert!(3,t1)} +$$ +\left[3, 8, . \right] +$$ +\returnType{Type: BinarySearchTree Integer} + +We go back to the original tree {\tt t}. The leaves of the binary +search tree are those which have empty {\tt left} and {\tt right} subtrees. + +\spadcommand{leaves t} +$$ +\left[ +1, 4, 5, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +The operation {\tt split}{\tt (k,t)} returns a \index{record} +containing the two subtrees: one with all elements ``less'' than +{\tt k}, another with elements ``greater'' than {\tt k}. + +\spadcommand{split(3,t)} +$$ +\left[ +{less={\left[ 1, 2, . \right]}}, +{greater={\left[ {\left[ ., 3, +{\left[ 4, 5, +{\left[ 5, 6, 7 \right]} +\right]} +\right]}, +8, . +\right]}} +\right] +$$ +\returnType{Type: +Record(less: BinarySearchTree PositiveInteger,greater: +BinarySearchTree PositiveInteger)} + +Define {\tt insertRoot} to insert new elements by creating a new node. + +\spadcommand{insertRoot: (INT,BSTREE INT) -> BSTREE INT} +\returnType{Type: Void} + +The new node puts the inserted value between its ``less'' tree and +``greater'' tree. + +\begin{verbatim} +insertRoot(x, t) == + a := split(x, t) + node(a.less, x, a.greater) +\end{verbatim} + +Function {\tt buildFromRoot} builds a binary search tree from a list +of elements {\tt ls} and the empty tree {\tt emptybst}. + +\spadcommand{buildFromRoot ls == reduce(insertRoot,ls,emptybst)} +\returnType{Type: Void} + +Apply this to the reverse of the list {\tt lv}. + +\spadcommand{rt := buildFromRoot reverse lv} +$$ +\left[ +{\left[ {\left[ 1, 2, . \right]}, 3, +{\left[ 4, 5, {\left[ 5, 6, 7\right]}\right]} +\right]}, +8, . +\right] +$$ +\returnType{Type: BinarySearchTree Integer} + +Have Axiom check that these are equal. + +\spadcommand{(t = rt)@Boolean} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\section{CardinalNumber} +\label{CardinalNumberXmpPage} + +The {\tt CardinalNumber} domain can be used for values indicating the +cardinality of sets, both finite and infinite. For example, the +\spadfunFrom{dimension}{VectorSpace} operation in the category +{\tt VectorSpace} returns a cardinal number. + +The non-negative integers have a natural construction as cardinals +\begin{verbatim} +0 = #{ }, 1 = {0}, 2 = {0, 1}, ..., n = {i | 0 <= i < n}. +\end{verbatim} + +The fact that {\tt 0} acts as a zero for the multiplication of cardinals is +equivalent to the axiom of choice. + +Cardinal numbers can be created by conversion from non-negative integers. + +\spadcommand{c0 := 0 :: CardinalNumber} +$$ +0 +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{c1 := 1 :: CardinalNumber} +$$ +1 +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{c2 := 2 :: CardinalNumber} +$$ +2 +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{c3 := 3 :: CardinalNumber} +$$ +3 +$$ +\returnType{Type: CardinalNumber} + +They can also be obtained as the named cardinal {\tt Aleph(n)}. + +\spadcommand{A0 := Aleph 0} +$$ +Aleph +\left( +{0} +\right) +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{A1 := Aleph 1} +$$ +Aleph +\left( +{1} +\right) +$$ +\returnType{Type: CardinalNumber} + +The \spadfunFrom{finite?}{CardinalNumber} operation tests whether a +value is a finite cardinal, that is, a non-negative integer. + +\spadcommand{finite? c2} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{finite? A0} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Similarly, the \spadfunFrom{countable?}{CardinalNumber} +operation determines whether a value is +a countable cardinal, that is, finite or {\tt Aleph(0)}. + +\spadcommand{countable? c2} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{countable? A0} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{countable? A1} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Arithmetic operations are defined on cardinal numbers as follows: +If {\tt x = \#X} and {\tt y = \#Y} then + +\noindent +$ +\begin{array}{lr} +{\tt x+y = \#(X+Y)} & cardinality of the disjoint union\\ +{\tt x-y = \#(X-Y)} & cardinality of the relative complement \\ +{\tt x*y = \#(X*Y)} & cardinality of the Cartesian product \\ +{\tt x**y = \#(X**Y)} & +cardinality of the set of maps from {\tt Y} to {\tt X} \\ +\end{array} +$ + +Here are some arithmetic examples. + +\spadcommand{[c2 + c2, c2 + A1]} +$$ +\left[4, {Aleph \left({1} \right)}\right] +$$ +\returnType{Type: List CardinalNumber} + +\spadcommand{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]} +$$ +\left[ +0, 2, 4, 0, {Aleph \left({1} \right)}, +{Aleph \left({1} \right)}, +{Aleph \left({1} \right)} +\right] +$$ +\returnType{Type: List CardinalNumber} + +\spadcommand{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]} +$$ +\left[ +1, 2, 4, 1, {Aleph \left({1} \right)}, +{Aleph \left({1} \right)} +\right] +$$ +\returnType{Type: List CardinalNumber} + +Subtraction is a partial operation: it is not defined +when subtracting a larger cardinal from a smaller one, nor +when subtracting two equal infinite cardinals. + +\spadcommand{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]} +$$ +\left[ +1, 0, \mbox{\tt "failed"} , {Aleph \left({1} \right)}, +{Aleph \left({1} \right)}, +\mbox{\tt "failed"} +\right] +$$ +\returnType{Type: List Union(CardinalNumber,"failed")} + +The generalized continuum hypothesis asserts that +\begin{verbatim} +2**Aleph i = Aleph(i+1) +\end{verbatim} +and is independent of the axioms of set theory.\footnote{Goedel, +{\it The consistency of the continuum hypothesis,} +Ann. Math. Studies, Princeton Univ. Press, 1940.} + +The {\tt CardinalNumber} domain provides an operation to assert +whether the hypothesis is to be assumed. + +\spadcommand{generalizedContinuumHypothesisAssumed true} + +When the generalized continuum hypothesis +is assumed, exponentiation to a transfinite power is allowed. + +\spadcommand{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]} +$$ +\left[ +0, 1, {Aleph \left({1} \right)}, +{Aleph \left({1} \right)}, +{Aleph \left({2} \right)}, +{Aleph \left({1} \right)}, +{Aleph \left({2} \right)} +\right] +$$ +\returnType{Type: List CardinalNumber} + +Three commonly encountered cardinal numbers are + +\noindent +$ +\begin{array}{lr} +{\tt a = \#}{\bf Z} & countable infinity \\ +{\tt c = \#}{\bf R} & the continuum \\ +{\tt f = \#\{g| g: [0,1] -> {\bf R}\}} \\ +\end{array} +$ + +In this domain, these values are obtained under the generalized +continuum hypothesis in this way. + +\spadcommand{a := Aleph 0} +$$ +Aleph \left({0} \right) +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{c := 2**a} +$$ +Aleph \left({1} \right) +$$ +\returnType{Type: CardinalNumber} + +\spadcommand{f := 2**c} +$$ +Aleph \left({2} \right) +$$ +\returnType{Type: CardinalNumber} + +\section{CartesianTensor} +\label{CartesianTensorXmpPage} + +{\tt CartesianTensor(i0,dim,R)} provides Cartesian tensors with +components belonging to a commutative ring {\tt R}. Tensors can be +described as a generalization of vectors and matrices. This gives a +concise {\it tensor algebra} for multilinear objects supported by the +{\tt CartesianTensor} domain. You can form the inner or outer product +of any two tensors and you can add or subtract tensors with the same +number of components. Additionally, various forms of traces and +transpositions are useful. + +The {\tt CartesianTensor} constructor allows you to specify the +minimum index for subscripting. In what follows we discuss in detail +how to manipulate tensors. + +Here we construct the domain of Cartesian tensors of dimension 2 over the +integers, with indices starting at 1. + +\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)} +$$ +CartesianTensor(1,2,Integer) +$$ +\returnType{Type: Domain} + +\subsubsection{Forming tensors} + +Scalars can be converted to tensors of rank zero. + +\spadcommand{t0: CT := 8} +$$ +8 +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{rank t0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +Vectors (mathematical direct products, rather than one dimensional array +structures) can be converted to tensors of rank one. + +\spadcommand{v: DirectProduct(2, Integer) := directProduct [3,4]} +$$ +\left[ +3, 4 +\right] +$$ +\returnType{Type: DirectProduct(2,Integer)} + +\spadcommand{Tv: CT := v} +$$ +\left[ +3, 4 +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +Matrices can be converted to tensors of rank two. + +\spadcommand{m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ]} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +4 & 5 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{Tm: CT := m} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +4 & 5 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ]} +$$ +\left[ +\begin{array}{cc} +2 & 3 \\ +0 & 1 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{Tn: CT := n} +$$ +\left[ +\begin{array}{cc} +2 & 3 \\ +0 & 1 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +In general, a tensor of rank {\tt k} can be formed by making a list of +rank {\tt k-1} tensors or, alternatively, a {\tt k}-deep nested list +of lists. + +\spadcommand{t1: CT := [2, 3]} +$$ +\left[ +2, 3 +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{rank t1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t2: CT := [t1, t1]} +$$ +\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{t3: CT := [t2, t2]} +$$ +\left[ +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]}, +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]} +$$ +\left[ +{\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +\end{array} +\right]}, +{\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +2 & 3 \\ +2 & 3 +\end{array} +\right]} +\end{array} +\right]} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{rank tt} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +\subsubsection{Multiplication} + +Given two tensors of rank {\tt k1} and {\tt k2}, the outer +\spadfunFrom{product}{CartesianTensor} forms a new tensor of rank +{\tt k1+k2}. Here + +$$T_{mn}(i,j,k,l) = T_m(i,j) \ T_n(k,l)$$ + +\spadcommand{Tmn := product(Tm, Tn)} +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 3 \\ +0 & 1 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +4 & 6 \\ +0 & 2 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +8 & {12} \\ +0 & 4 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +{10} & {15} \\ +0 & 5 +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +The inner product (\spadfunFrom{contract}{CartesianTensor}) forms a +tensor of rank {\tt k1+k2-2}. This product generalizes the vector dot +product and matrix-vector product by summing component products along +two indices. + +Here we sum along the second index of $T_m$ and the first index of +$T_v$. Here + +$$T_{mv} = \sum_{j=1}^{\hbox{\tiny\rm dim}} T_m(i,j) \ T_v(j)$$ + +\spadcommand{Tmv := contract(Tm,2,Tv,1)} +$$ +\left[ +{11}, {32} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +The multiplication operator \spadopFrom{*}{CartesianTensor} is scalar +multiplication or an inner product depending on the ranks of the arguments. + +If either argument is rank zero it is treated as scalar multiplication. +Otherwise, {\tt a*b} is the inner product summing the last index of +{\tt a} with the first index of {\tt b}. + +\spadcommand{Tm*Tv} +$$ +\left[ +{11}, {32} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +This definition is consistent with the inner product on matrices +and vectors. + +\spadcommand{Tmv = m * v} +$$ +{\left[ {11}, {32} \right]}= +{\left[{11}, {32} \right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\subsubsection{Selecting Components} + +For tensors of low rank (that is, four or less), components can be selected +by applying the tensor to its indices. + +\spadcommand{t0()} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t1(1+1)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t2(2,1)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t3(2,1,2)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{Tmn(2,1,2,1)} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +A general indexing mechanism is provided for a list of indices. + +\spadcommand{t0[]} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t1[2]} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{t2[2,1]} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +The general mechanism works for tensors of arbitrary rank, but is +somewhat less efficient since the intermediate index list must be created. + +\spadcommand{t3[2,1,2]} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{Tmn[2,1,2,1]} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\subsubsection{Contraction} + +A ``contraction'' between two tensors is an inner product, as we have +seen above. You can also contract a pair of indices of a single +tensor. This corresponds to a ``trace'' in linear algebra. The +expression {\tt contract(t,k1,k2)} forms a new tensor by summing the +diagonal given by indices in position {\tt k1} and {\tt k2}. + +This is the tensor given by +$$xT_{mn} = \sum_{k=1}^{\hbox{\tiny\rm dim}} T_{mn}(k,k,i,j)$$ + +\spadcommand{cTmn := contract(Tmn,1,2)} +$$ +\left[ +\begin{array}{cc} +{12} & {18} \\ +0 & 6 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +Since {\tt Tmn} is the outer product of matrix {\tt m} and matrix {\tt n}, +the above is equivalent to this. + +\spadcommand{trace(m) * n} +$$ +\left[ +\begin{array}{cc} +{12} & {18} \\ +0 & 6 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +In this and the next few examples, we show all possible contractions +of {\tt Tmn} and their matrix algebra equivalents. + +\spadcommand{contract(Tmn,1,2) = trace(m) * n} +$$ +{\left[ +\begin{array}{cc} +{12} & {18} \\ +0 & 6 +\end{array} +\right]}={\left[ +\begin{array}{cc} +{12} & {18} \\ +0 & 6 +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\spadcommand{contract(Tmn,1,3) = transpose(m) * n} +$$ +{\left[ +\begin{array}{cc} +2 & 7 \\ +4 & {11} +\end{array} +\right]}={\left[ +\begin{array}{cc} +2 & 7 \\ +4 & {11} +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)} +$$ +{\left[ +\begin{array}{cc} +{14} & 4 \\ +{19} & 5 +\end{array} +\right]}={\left[ +\begin{array}{cc} +{14} & 4 \\ +{19} & 5 +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\spadcommand{contract(Tmn,2,3) = m * n} +$$ +{\left[ +\begin{array}{cc} +2 & 5 \\ +8 & {17} +\end{array} +\right]}={\left[ +\begin{array}{cc} +2 & 5 \\ +8 & {17} +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\spadcommand{contract(Tmn,2,4) = m * transpose(n)} +$$ +{\left[ +\begin{array}{cc} +8 & 2 \\ +{23} & 5 +\end{array} +\right]}={\left[ +\begin{array}{cc} +8 & 2 \\ +{23} & 5 +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\spadcommand{contract(Tmn,3,4) = trace(n) * m} +$$ +{\left[ +\begin{array}{cc} +3 & 6 \\ +{12} & {15} +\end{array} +\right]}={\left[ +\begin{array}{cc} +3 & 6 \\ +{12} & {15} +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\subsubsection{Transpositions} + +You can exchange any desired pair of indices using the +\spadfunFrom{transpose}{CartesianTensor} operation. + +Here the indices in positions one and three are exchanged, that is, +$tT_{mn}(i,j,k,l) = T_{mn}(k,j,i,l).$ + +\spadcommand{tTmn := transpose(Tmn,1,3)} +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 3 \\ +8 & {12} +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +4 & 6 \\ +{10} & {15} +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 4 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +0 & 2 \\ +0 & 5 +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +If no indices are specified, the first and last index are exchanged. + +\spadcommand{transpose Tmn} +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 8 \\ +0 & 0 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +4 & {10} \\ +0 & 0 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +3 & {12} \\ +1 & 4 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +6 & {15} \\ +2 & 5 +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +This is consistent with the matrix transpose. + +\spadcommand{transpose Tm = transpose m} +$$ +{\left[ +\begin{array}{cc} +1 & 4 \\ +2 & 5 +\end{array} +\right]}={\left[ +\begin{array}{cc} +1 & 4 \\ +2 & 5 +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +If a more complicated reordering of the indices is required, then the +\spadfunFrom{reindex}{CartesianTensor} operation can be used. +This operation allows the indices to be arbitrarily permuted. + +This defines $rT_{mn}(i,j,k,l) = \allowbreak T_{mn}(i,l,j,k).$ + +\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])} +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 0 \\ +4 & 0 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +3 & 1 \\ +6 & 2 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +8 & 0 \\ +{10} & 0 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +{12} & 4 \\ +{15} & 5 +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\subsubsection{Arithmetic} + +Tensors of equal rank can be added or subtracted so arithmetic +expressions can be used to produce new tensors. + +\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)} +$$ +\left[ +\begin{array}{cc} +-6 & -{16} \\ +2 & 6 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{Tv*(tt+Tn)} +$$ +\left[ +-4, -{11} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)} +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +{46} & {84} \\ +{174} & {212} +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +{57} & {114} \\ +{228} & {285} +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +{18} & {24} \\ +{57} & {63} +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +{17} & {30} \\ +{63} & {76} +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +\subsubsection{Specific Tensors} + +Two specific tensors have properties which depend only on the +dimension. + +The Kronecker delta satisfies +\begin{verbatim} + +- + | 1 if i = j +delta(i,j) = | + | 0 if i ^= j + +- +\end{verbatim} + +\spadcommand{delta: CT := kroneckerDelta()} +$$ +\left[ +\begin{array}{cc} +1 & 0 \\ +0 & 1 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +This can be used to reindex via contraction. + +\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])} +$$ +{\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 4 \\ +3 & 6 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +0 & 0 \\ +1 & 2 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +8 & {10} \\ +{12} & {15} +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +0 & 0 \\ +4 & 5 +\end{array} +\right]} +\end{array} +\right]}={\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +2 & 4 \\ +3 & 6 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +0 & 0 \\ +1 & 2 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +8 & {10} \\ +{12} & {15} +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +0 & 0 \\ +4 & 5 +\end{array} +\right]} +\end{array} +\right]} +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +The Levi Civita symbol determines the sign of a permutation of indices. + +\spadcommand{epsilon:CT := leviCivitaSymbol()} +$$ +\left[ +\begin{array}{cc} +0 & 1 \\ +-1 & 0 +\end{array} +\right] +$$ +\returnType{Type: CartesianTensor(1,2,Integer)} + +Here we have: +\begin{verbatim} +epsilon(i1,...,idim) + = +1 if i1,...,idim is an even permutation of i0,...,i0+dim-1 + = -1 if i1,...,idim is an odd permutation of i0,...,i0+dim-1 + = 0 if i1,...,idim is not a permutation of i0,...,i0+dim-1 +\end{verbatim} + +This property can be used to form determinants. + +\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m} +$$ +-6=-6 +$$ +\returnType{Type: Equation CartesianTensor(1,2,Integer)} + +\subsubsection{Properties of the CartesianTensor domain} + +{\tt GradedModule(R,E)} denotes ``{\tt E}-graded {\tt R}-module'', +that is, a collection of {\tt R}-modules indexed by an abelian monoid +{\tt E.} An element {\tt g} of {\tt G[s]} for some specific {\tt s} +in {\tt E} is said to be an element of {\tt G} with +\spadfunFrom{degree}{GradedModule} {\tt s}. Sums are defined in each +module {\tt G[s]} so two elements of {\tt G} can be added if they have +the same degree. Morphisms can be defined and composed by degree to +give the mathematical category of graded modules. + +{\tt GradedAlgebra(R,E)} denotes ``{\tt E}-graded {\tt R}-algebra.'' +A graded algebra is a graded module together with a degree preserving +{\tt R}-bilinear map, called the \spadfunFrom{product}{GradedAlgebra}. + +\begin{verbatim} +degree(product(a,b)) = degree(a) + degree(b) + +product(r*a,b) = product(a,r*b) = r*product(a,b) +product(a1+a2,b) = product(a1,b) + product(a2,b) +product(a,b1+b2) = product(a,b1) + product(a,b2) +product(a,product(b,c)) = product(product(a,b),c) +\end{verbatim} + +The domain {\tt CartesianTensor(i0, dim, R)} belongs to the category +{\tt GradedAlgebra(R, NonNegativeInteger)}. The non-negative integer +\spadfunFrom{degree}{GradedAlgebra} is the tensor rank and the graded +algebra \spadfunFrom{product}{GradedAlgebra} is the tensor outer +product. The graded module addition captures the notion that only +tensors of equal rank can be added. + +If {\tt V} is a vector space of dimension {\tt dim} over {\tt R}, +then the tensor module {\tt T[k](V)} is defined as +\begin{verbatim} +T[0](V) = R +T[k](V) = T[k-1](V) * V +\end{verbatim} +where {\tt *} denotes the {\tt R}-module tensor +\spadfunFrom{product}{GradedAlgebra}. {\tt CartesianTensor(i0,dim,R)} +is the graded algebra in which the degree {\tt k} module is {\tt +T[k](V)}. + +\subsubsection{Tensor Calculus} + +It should be noted here that often tensors are used in the context of +tensor-valued manifold maps. This leads to the notion of covariant +and contravariant bases with tensor component functions transforming +in specific ways under a change of coordinates on the manifold. This +is no more directly supported by the {\tt CartesianTensor} domain than +it is by the {\tt Vector} domain. However, it is possible to have the +components implicitly represent component maps by choosing a +polynomial or expression type for the components. In this case, it is +up to the user to satisfy any constraints which arise on the basis of +this interpretation. + +\section{Character} +\label{CharacterXmpPage} +The members of the domain {\tt Character} are values +representing letters, numerals and other text elements. +For more information on related topics, see +\ref{CharacterClassXmpPage} on page~\pageref{CharacterClassXmpPage} and +\ref{StringXmpPage} on page~\pageref{StringXmpPage}. + +Characters can be obtained using {\tt String} notation. + +\spadcommand{chars := [char "a", char "A", char "X", char "8", char "+"]} +$$ +\left[ +a, A, X, 8, + +\right] +$$ +\returnType{Type: List Character} + +Certain characters are available by name. +This is the blank character. + +\spadcommand{space()} +$$ +\ +$$ +\returnType{Type: Character} + +This is the quote that is used in strings. + +\spadcommand{quote()} +$$ +\mbox{\tt "} +$$ +\returnType{Type: Character} + +This is the escape character that allows quotes and other characters +within strings. + +\spadcommand{escape()} +$$ +\_ +$$ +\returnType{Type: Character} + +Characters are represented as integers in a machine-dependent way. +The integer value can be obtained using the +\spadfunFrom{ord}{Character} operation. It is always true that {\tt +char(ord c) = c} and {\tt ord(char i) = i}, provided that {\tt i} is +in the range {\tt 0..size()\$Character-1}. + +\spadcommand{[ord c for c in chars]} +$$ +\left[ +{97}, {65}, {88}, {56}, {43} +\right] +$$ +\returnType{Type: List Integer} + +The \spadfunFrom{lowerCase}{Character} operation converts an upper +case letter to the corresponding lower case letter. If the argument +is not an upper case letter, then it is returned unchanged. + +\spadcommand{[upperCase c for c in chars]} +$$ +\left[ +A, A, X, 8, + +\right] +$$ +\returnType{Type: List Character} + +Likewise, the \spadfunFrom{upperCase}{Character} operation converts lower +case letters to upper case. + +\spadcommand{[lowerCase c for c in chars] } +$$ +\left[ +a, a, x, 8, + +\right] +$$ +\returnType{Type: List Character} + +A number of tests are available to determine whether characters +belong to certain families. + +\spadcommand{[alphabetic? c for c in chars] } +$$ +\left[ +{\tt true}, {\tt true}, {\tt true}, {\tt false}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\spadcommand{[upperCase? c for c in chars] } +$$ +\left[ +{\tt false}, {\tt true}, {\tt true}, {\tt false}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\spadcommand{[lowerCase? c for c in chars] } +$$ +\left[ +{\tt true}, {\tt false}, {\tt false}, {\tt false}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\spadcommand{[digit? c for c in chars] } +$$ +\left[ +{\tt false}, {\tt false}, {\tt false}, {\tt true}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\spadcommand{[hexDigit? c for c in chars] } +$$ +\left[ +{\tt true}, {\tt true}, {\tt false}, {\tt true}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\spadcommand{[alphanumeric? c for c in chars] } +$$ +\left[ +{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt false} +\right] +$$ +\returnType{Type: List Boolean} + +\section{CharacterClass} +\label{CharacterClassXmpPage} +The {\tt CharacterClass} domain allows classes of characters to be +defined and manipulated efficiently. + +Character classes can be created by giving either a string or a list +of characters. + +\spadcommand{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"] } +$$ +\mbox{\tt "aeiouy"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{cl2 := charClass "bcdfghjklmnpqrstvwxyz" } +$$ +\mbox{\tt "bcdfghjklmnpqrstvwxyz"} +$$ +\returnType{Type: CharacterClass} + +A number of character classes are predefined for convenience. + +\spadcommand{digit()} +$$ +\mbox{\tt "0123456789"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{hexDigit()} +$$ +\mbox{\tt "0123456789ABCDEFabcdef"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{upperCase()} +$$ +\mbox{\tt "ABCDEFGHIJKLMNOPQRSTUVWXYZ"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{lowerCase()} +$$ +\mbox{\tt "abcdefghijklmnopqrstuvwxyz"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{alphabetic()} +$$ +\mbox{\tt "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{alphanumeric()} +$$ +\mbox{\tt "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"} +$$ +\returnType{Type: CharacterClass} + +You can quickly test whether a character belongs to a class. + +\spadcommand{member?(char "a", cl1) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{member?(char "a", cl2) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Classes have the usual set operations because the {\tt CharacterClass} +domain belongs to the category {\tt FiniteSetAggregate(Character)}. + +\spadcommand{intersect(cl1, cl2) } +$$ +\mbox{\tt "y"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{union(cl1,cl2) } +$$ +\mbox{\tt "abcdefghijklmnopqrstuvwxyz"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{difference(cl1,cl2) } +$$ +\mbox{\tt "aeiou"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{intersect(complement(cl1),cl2) } +$$ +\mbox{\tt "bcdfghjklmnpqrstvwxz"} +$$ +\returnType{Type: CharacterClass} + +You can modify character classes by adding or removing characters. + +\spadcommand{insert!(char "a", cl2) } +$$ +\mbox{\tt "abcdfghjklmnpqrstvwxyz"} +$$ +\returnType{Type: CharacterClass} + +\spadcommand{remove!(char "b", cl2) } +$$ +\mbox{\tt "acdfghjklmnpqrstvwxyz"} +$$ +\returnType{Type: CharacterClass} + +For more information on related topics, see +\ref{CharacterXmpPage} on page~\pageref{CharacterXmpPage} and +\ref{StringXmpPage} on page~\pageref{StringXmpPage}. + +\section{CliffordAlgebra} +\label{CliffordAlgebraXmpPage} +\noindent + +{\tt CliffordAlgebra(n,K,Q)} defines a vector space of dimension $2^n$ +over the field $K$ with a given quadratic form {\tt Q}. If $\{e_1, +\ldots, e_n\}$ is a basis for $K^n$ then +\begin{verbatim} +{ 1, + e(i) 1 <= i <= n, + e(i1)*e(i2) 1 <= i1 < i2 <=n, + ..., + e(1)*e(2)*...*e(n) } +\end{verbatim} +is a basis for the Clifford algebra. The algebra is defined by the relations +\begin{verbatim} +e(i)*e(i) = Q(e(i)) +e(i)*e(j) = -e(j)*e(i), i ^= j +\end{verbatim} +Examples of Clifford Algebras are +gaussians (complex numbers), quaternions, +exterior algebras and spin algebras. + +\subsection{The Complex Numbers as a Clifford Algebra} + +This is the field over which we will work, rational functions with +integer coefficients. + +\spadcommand{K := Fraction Polynomial Integer } +$$ +\mbox{\rm Fraction Polynomial Integer} +$$ +\returnType{Type: Domain} + +We use this matrix for the quadratic form. + +\spadcommand{m := matrix [ [-1] ] } +$$ +\left[ +\begin{array}{c} +-1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +We get complex arithmetic by using this domain. + +\spadcommand{C := CliffordAlgebra(1, K, quadraticForm m) } +$$ +\mbox{\rm CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} +$$ +\returnType{Type: Domain} + +Here is {\tt i}, the usual square root of {\tt -1.} + +\spadcommand{i: C := e(1) } +$$ +e \sb {1} +$$ +\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} + +Here are some examples of the arithmetic. + +\spadcommand{x := a + b * i } +$$ +a+{b \ {e \sb {1}}} +$$ +\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{y := c + d * i } +$$ +c+{d \ {e \sb {1}}} +$$ +\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} + +See \ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} +for examples of Axiom's constructor implementing complex numbers. + +\spadcommand{x * y } +$$ +-{b \ d}+{a \ c}+{{\left( {a \ d}+{b \ c} +\right)} +\ {e \sb {1}}} +$$ +\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} + +\subsection{The Quaternion Numbers as a Clifford Algebra} + +This is the field over which we will work, rational functions with +integer coefficients. + +\spadcommand{K := Fraction Polynomial Integer } +$$ +\mbox{\rm Fraction Polynomial Integer} +$$ +\returnType{Type: Domain} + +We use this matrix for the quadratic form. + +\spadcommand{m := matrix [ [-1,0],[0,-1] ] } +$$ +\left[ +\begin{array}{cc} +-1 & 0 \\ +0 & -1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +The resulting domain is the quaternions. + +\spadcommand{H := CliffordAlgebra(2, K, quadraticForm m) } +$$ +\mbox{\rm CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} +$$ +\returnType{Type: Domain} + +We use Hamilton's notation for {\tt i},{\tt j},{\tt k}. + +\spadcommand{i: H := e(1) } +$$ +e \sb {1} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{j: H := e(2) } +$$ +e \sb {2} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{k: H := i * j } +$$ +{e \sb {1}} \ {e \sb {2}} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{x := a + b * i + c * j + d * k } +$$ +a+{b \ {e \sb {1}}}+{c \ {e \sb {2}}}+{d \ {e \sb {1}} \ {e \sb {2}}} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{y := e + f * i + g * j + h * k } +$$ +e+{f \ {e \sb {1}}}+{g \ {e \sb {2}}}+{h \ {e \sb {1}} \ {e \sb {2}}} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{x + y } +$$ +e+a+{{\left( f+b +\right)} +\ {e \sb {1}}}+{{\left( g+c +\right)} +\ {e \sb {2}}}+{{\left( h+d +\right)} +\ {e \sb {1}} \ {e \sb {2}}} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{x * y } +$$ +\begin{array}{@{}l} +-{d \ h} - +{c \ g} - +{b \ f}+ +{a \ e}+ +{{\left( {c \ h} -{d \ g}+{a \ f}+{b \ e} \right)}\ {e \sb {1}}}+ +\\ +\\ +\displaystyle +{{\left( -{b \ h}+{a \ g}+{d \ f}+{c \ e} \right)}\ {e \sb {2}}}+ +{{\left( {a \ h}+{b \ g} -{c \ f}+{d \ e} \right)} +\ {e \sb {1}} \ {e \sb {2}}} +\end{array} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +See \ref{QuaternionXmpPage} on page~\pageref{QuaternionXmpPage} +for examples of Axiom's constructor implementing quaternions. + +\spadcommand{y * x } +$$ +\begin{array}{@{}l} +-{d \ h} -{c \ g} -{b \ f}+{a \ e}+ +{{\left( -{c \ h}+{d \ g}+{a \ f}+{b \ e} \right)}\ {e \sb {1}}}+ +\\ +\\ +\displaystyle +{{\left( {b \ h}+{a \ g} -{d \ f}+{c \ e} \right)}\ {e \sb {2}}}+ +{{\left( {a \ h} - +{b \ g}+{c \ f}+{d \ e} \right)}\ {e \sb {1}} \ {e \sb {2}}} +\end{array} +$$ +\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} + +\subsection{The Exterior Algebra on a Three Space} + +This is the field over which we will work, rational functions with +integer coefficients. + +\spadcommand{K := Fraction Polynomial Integer } +$$ +\mbox{\rm Fraction Polynomial Integer} +$$ +\returnType{Type: Domain} + +If we chose the three by three zero quadratic form, we obtain +the exterior algebra on {\tt e(1),e(2),e(3)}. + +\spadcommand{Ext := CliffordAlgebra(3, K, quadraticForm 0) } +$$ +\mbox{\rm CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} +$$ +\returnType{Type: Domain} + +This is a three dimensional vector algebra. +We define {\tt i}, {\tt j}, {\tt k} as the unit vectors. + +\spadcommand{i: Ext := e(1) } +$$ +e \sb {1} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{j: Ext := e(2) } +$$ +e \sb {2} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{k: Ext := e(3) } +$$ +e \sb {3} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +Now it is possible to do arithmetic. + +\spadcommand{x := x1*i + x2*j + x3*k } +$$ +{x1 \ {e \sb {1}}}+{x2 \ {e \sb {2}}}+{x3 \ {e \sb {3}}} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{y := y1*i + y2*j + y3*k } +$$ +{y1 \ {e \sb {1}}}+{y2 \ {e \sb {2}}}+{y3 \ {e \sb {3}}} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{x + y } +$$ +{{\left( y1+x1 +\right)} +\ {e \sb {1}}}+{{\left( y2+x2 +\right)} +\ {e \sb {2}}}+{{\left( y3+x3 +\right)} +\ {e \sb {3}}} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\spadcommand{x * y + y * x } +$$ +0 +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +On an {\tt n} space, a grade {\tt p} form has a dual {\tt n-p} form. +In particular, in three space the dual of a grade two element identifies +{\tt e1*e2->e3, e2*e3->e1, e3*e1->e2}. + +\spadcommand{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k } +\returnType{Type: Void} + +The vector cross product is then given by this. + +\spadcommand{dual2(x*y) } +\begin{verbatim} + Compiling function dual2 with type CliffordAlgebra(3,Fraction + Polynomial Integer,MATRIX) -> CliffordAlgebra(3,Fraction + Polynomial Integer,MATRIX) +\end{verbatim} +$$ +{{\left( {x2 \ y3} -{x3 \ y2} +\right)} +\ {e \sb {1}}}+{{\left( -{x1 \ y3}+{x3 \ y1} +\right)} +\ {e \sb {2}}}+{{\left( {x1 \ y2} -{x2 \ y1} +\right)} +\ {e \sb {3}}} +$$ +\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} + +\subsection{The Dirac Spin Algebra} + +In this section we will work over the field of rational numbers. + +\spadcommand{K := Fraction Integer } +$$ +\mbox{\rm Fraction Integer} +$$ +\returnType{Type: Domain} + +We define the quadratic form to be the Minkowski space-time metric. + +\spadcommand{g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ] } +$$ +\left[ +\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & -1 & 0 & 0 \\ +0 & 0 & -1 & 0 \\ +0 & 0 & 0 & -1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +We obtain the Dirac spin algebra used in Relativistic Quantum Field Theory. + +\spadcommand{D := CliffordAlgebra(4,K, quadraticForm g) } +$$ +\mbox{\rm CliffordAlgebra(4,Fraction Integer,MATRIX)} +$$ +\returnType{Type: Domain} + +The usual notation for the basis is $\gamma$ with a superscript. For +Axiom input we will use {\tt gam(i)}: + +\spadcommand{gam := [e(i)\$D for i in 1..4] } +$$ +\left[ +{e \sb {1}}, {e \sb {2}}, {e \sb {3}}, {e \sb {4}} +\right] +$$ +\returnType{Type: List CliffordAlgebra(4,Fraction Integer,MATRIX)} + +\noindent +There are various contraction identities of the form +\begin{verbatim} +g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) = + 2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s)) +\end{verbatim} +where a sum over {\tt l} and {\tt t} is implied. + +Verify this identity for particular values of {\tt m,n,r,s}. + +\spadcommand{m := 1; n:= 2; r := 3; s := 4; } +\returnType{Type: PositiveInteger} + +\spadcommand{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) } +$$ +-{4 \ {e \sb {1}} \ {e \sb {2}} \ {e \sb {3}} \ {e \sb {4}}} +$$ +\returnType{Type: CliffordAlgebra(4,Fraction Integer,MATRIX)} + +\spadcommand{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) } +$$ +-{4 \ {e \sb {1}} \ {e \sb {2}} \ {e \sb {3}} \ {e \sb {4}}} +$$ +\returnType{Type: CliffordAlgebra(4,Fraction Integer,MATRIX)} + +\section{Complex} +\label{ComplexXmpPage} + +The {\tt Complex} constructor implements complex objects over a +commutative ring {\tt R}. Typically, the ring {\tt R} is {\tt Integer}, +{\tt Fraction Integer}, {\tt Float} or {\tt DoubleFloat}. +{\tt R} can also be a symbolic type, like {\tt Polynomial Integer}. +For more information about the numerical and graphical aspects of +complex numbers, see \ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}. + +Complex objects are created by the \spadfunFrom{complex}{Complex} operation. + +\spadcommand{a := complex(4/3,5/2) } +$$ +{4 \over 3}+{{5 \over 2} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +\spadcommand{b := complex(4/3,-5/2) } +$$ +{4 \over 3} -{{5 \over 2} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +The standard arithmetic operations are available. + +\spadcommand{a + b } +$$ +8 \over 3 +$$ +\returnType{Type: Complex Fraction Integer} + +\spadcommand{a - b } +$$ +5 \ i +$$ +\returnType{Type: Complex Fraction Integer} + +\spadcommand{a * b } +$$ +{289} \over {36} +$$ +\returnType{Type: Complex Fraction Integer} + +If {\tt R} is a field, you can also divide the complex objects. + +\spadcommand{a / b } +$$ +-{{161} \over {289}}+{{{240} \over {289}} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +Use a conversion (\ref{ugTypesConvertPage} on +page~\pageref{ugTypesConvertPage} in Section +\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}) +to view the last object as a fraction of complex integers. + +\spadcommand{\% :: Fraction Complex Integer } +$$ +{-{15}+{8 \ i}} \over {{15}+{8 \ i}} +$$ +\returnType{Type: Fraction Complex Integer} + +The predefined macro {\tt \%i} is defined to be {\tt complex(0,1)}. + +\spadcommand{3.4 + 6.7 * \%i} +$$ +{3.4}+{{6.7} \ i} +$$ +\returnType{Type: Complex Float} + +You can also compute the \spadfunFrom{conjugate}{Complex} and +\spadfunFrom{norm}{Complex} of a complex number. + +\spadcommand{conjugate a } +$$ +{4 \over 3} -{{5 \over 2} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +\spadcommand{norm a } +$$ +{289} \over {36} +$$ +\returnType{Type: Fraction Integer} + +The \spadfunFrom{real}{Complex} and \spadfunFrom{imag}{Complex} operations +are provided to extract the real and imaginary parts, respectively. + +\spadcommand{real a } +$$ +4 \over 3 +$$ +\returnType{Type: Fraction Integer} + +\spadcommand{imag a } +$$ +5 \over 2 +$$ +\returnType{Type: Fraction Integer} + +The domain {\tt Complex Integer} is also called the Gaussian integers. +If {\tt R} is the integers (or, more generally, a {\tt EuclideanDomain}), +you can compute greatest common divisors. + +\spadcommand{gcd(13 - 13*\%i,31 + 27*\%i)} +$$ +5+i +$$ +\returnType{Type: Complex Integer} + +You can also compute least common multiples. + +\spadcommand{lcm(13 - 13*\%i,31 + 27*\%i)} +$$ +{143} -{{39} \ i} +$$ +\returnType{Type: Complex Integer} + +You can \spadfunFrom{factor}{Complex} Gaussian integers. + +\spadcommand{factor(13 - 13*\%i)} +$$ +-{{\left( 1+i +\right)} +\ {\left( 2+{3 \ i} +\right)} +\ {\left( 3+{2 \ i} +\right)}} +$$ +\returnType{Type: Factored Complex Integer} + +\spadcommand{factor complex(2,0)} +$$ +-{i \ {{\left( 1+i +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +\section{ContinuedFraction} +\label{ContinuedFractionXmpPage} + +Continued fractions have been a fascinating and useful tool in +mathematics for well over three hundred years. Axiom implements +continued fractions for fractions of any Euclidean domain. In +practice, this usually means rational numbers. In this section we +demonstrate some of the operations available for manipulating both +finite and infinite continued fractions. It may be helpful if you +review \ref{StreamXmpPage} on page~\pageref{StreamXmpPage} to remind +yourself of some of the operations with streams. + +The {\tt ContinuedFraction} domain is a field and therefore you can +add, subtract, multiply and divide the fractions. + +The \spadfunFrom{continuedFraction}{ContinuedFraction} operation +converts its fractional argument to a continued fraction. + +\spadcommand{c := continuedFraction(314159/100000) } +$$ +3+ \zag{1}{7}+ \zag{1}{{15}}+ \zag{1}{1}+ \zag{1}{{25}}+ \zag{1}{1}+ +\zag{1}{7}+ \zag{1}{4} +$$ +\returnType{Type: ContinuedFraction Integer} + +This display is a compact form of the bulkier +\begin{verbatim} + 3 + 1 + ------------------------------- + 7 + 1 + --------------------------- + 15 + 1 + ---------------------- + 1 + 1 + ------------------ + 25 + 1 + ------------- + 1 + 1 + --------- + 7 + 1 + ----- + 4 +\end{verbatim} + +You can write any rational number in a similar form. The fraction +will be finite and you can always take the ``numerators'' to be {\tt 1}. +That is, any rational number can be written as a simple, finite +continued fraction of the form + +\begin{verbatim} + a(1) + 1 + ------------------------- + a(2) + 1 + -------------------- + a(3) + + . + . + . + 1 + ------------- + a(n-1) + 1 + ---- + a(n) +\end{verbatim} + +The $a_i$ are called partial quotients and the operation +\spadfunFrom{partialQuotients}{ContinuedFraction} creates a stream of them. + +\spadcommand{partialQuotients c } +$$ +\left[ +3, 7, {15}, 1, {25}, 1, 7, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +By considering more and more of the fraction, you get the +\spadfunFrom{convergents}{ContinuedFraction}. For example, the first +convergent is $a_1$, the second is $a_1 + 1/a_2$ and so on. + +\spadcommand{convergents c } +$$ +\left[ +3, {{22} \over 7}, {{333} \over {106}}, {{355} \over {113}}, +{{9208} \over {2931}}, {{9563} \over {3044}}, {{76149} \over {24239}}, +\ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +Since this is a finite continued fraction, the last convergent is the +original rational number, in reduced form. The result of +\spadfunFrom{approximants}{ContinuedFraction} is always an infinite +stream, though it may just repeat the ``last'' value. + +\spadcommand{approximants c } +$$ +\left[ +3, {{22} \over 7}, {{333} \over {106}}, {{355} \over {113}}, +{{9208} \over {2931}}, {{9563} \over {3044}}, {{76149} \over {24239}}, +\ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +Inverting {\tt c} only changes the partial quotients of its fraction +by inserting a {\tt 0} at the beginning of the list. + +\spadcommand{pq := partialQuotients(1/c) } +$$ +\left[ +0, 3, 7, {15}, 1, {25}, 1, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Do this to recover the original continued fraction from this list of +partial quotients. The three-argument form of the +\spadfunFrom{continuedFraction}{ContinuedFraction} operation takes an +element which is the whole part of the fraction, a stream of elements +which are the numerators of the fraction, and a stream of elements +which are the denominators of the fraction. + +\spadcommand{continuedFraction(first pq,repeating [1],rest pq) } +$$ +\zag{1}{3}+ \zag{1}{7}+ \zag{1}{{15}}+ \zag{1}{1}+ \zag{1}{{25}}+ \zag{1}{1}+ +\zag{1}{7}+\ldots +$$ +\returnType{Type: ContinuedFraction Integer} + +The streams need not be finite for +\spadfunFrom{continuedFraction}{ContinuedFraction}. Can you guess +which irrational number has the following continued fraction? See the +end of this section for the answer. + +\spadcommand{z:=continuedFraction(3,repeating [1],repeating [3,6]) } +$$ +3+ \zag{1}{3}+ \zag{1}{6}+ \zag{1}{3}+ \zag{1}{6}+ \zag{1}{3}+ \zag{1}{6}+ +\zag{1}{3}+\ldots +$$ +\returnType{Type: ContinuedFraction Integer} + +In 1737 Euler discovered the infinite continued fraction expansion +\begin{verbatim} + e - 1 1 + ----- = --------------------- + 2 1 + 1 + ----------------- + 6 + 1 + ------------- + 10 + 1 + -------- + 14 + ... +\end{verbatim} + +We use this expansion to compute rational and floating point +approximations of {\tt e}.\footnote{For this and other interesting +expansions, see C. D. Olds, {\it Continued Fractions,} New +Mathematical Library, (New York: Random House, 1963), pp. 134--139.} + +By looking at the above expansion, we see that the whole part is {\tt 0} +and the numerators are all equal to {\tt 1}. This constructs the +stream of denominators. + +\spadcommand{dens:Stream Integer := cons(1,generate((x+->x+4),6)) } +$$ +\left[ +1, 6, {10}, {14}, {18}, {22}, {26}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Therefore this is the continued fraction expansion for +$(e - 1) / 2$. + +\spadcommand{cf := continuedFraction(0,repeating [1],dens) } +$$ +\zag{1}{1}+ \zag{1}{6}+ \zag{1}{{10}}+ \zag{1}{{14}}+ \zag{1}{{18}}+ +\zag{1}{{22}}+ \zag{1}{{26}}+\ldots +$$ +\returnType{Type: ContinuedFraction Integer} + +These are the rational number convergents. + +\spadcommand{ccf := convergents cf } +$$ +\left[ +0, 1, {6 \over 7}, {{61} \over {71}}, {{860} \over {1001}}, +{{15541} \over {18089}}, {{342762} \over {398959}}, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +You can get rational convergents for {\tt e} by multiplying by {\tt 2} and +adding {\tt 1}. + +\spadcommand{eConvergents := [2*e + 1 for e in ccf] } +$$ +\left[ +1, 3, {{19} \over 7}, {{193} \over {71}}, {{2721} \over {1001}}, +{{49171} \over {18089}}, {{1084483} \over {398959}}, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +You can also compute the floating point approximations to these convergents. + +\spadcommand{eConvergents :: Stream Float } +$$ +\begin{array}{@{}l} +\left[ +{1.0}, {3.0}, {2.7142857142 857142857}, {2.7183098591 549295775}, +\right. +\\ +\\ +\displaystyle +{2.7182817182 817182817}, {2.7182818287 356957267}, +\\ +\\ +\displaystyle +\left. +{2.7182818284\ 585634113}, \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream Float} + +Compare this to the value of {\tt e} computed by the +\spadfunFrom{exp}{Float} operation in {\tt Float}. + +\spadcommand{exp 1.0} +$$ +2.7182818284\ 590452354 +$$ +\returnType{Type: Float} + +In about 1658, Lord Brouncker established the following expansion +for $4 / \pi$, +\begin{verbatim} + 1 + 1 + ----------------------- + 2 + 9 + ------------------- + 2 + 25 + --------------- + 2 + 49 + ----------- + 2 + 81 + ------- + 2 + ... +\end{verbatim} + +Let's use this expansion to compute rational and floating point +approximations for $\pi$. + +\spadcommand{cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])} +$$ +1+ \zag{1}{2}+ \zag{9}{2}+ \zag{{25}}{2}+ \zag{{49}}{2}+ \zag{{81}}{2}+ +\zag{{121}}{2}+ \zag{{169}}{2}+\ldots +$$ +\returnType{Type: ContinuedFraction Integer} + +\spadcommand{ccf := convergents cf } +$$ +\left[ +1, {3 \over 2}, {{15} \over {13}}, {{105} \over {76}}, {{315} +\over {263}}, {{3465} \over {2578}}, {{45045} \over {36979}}, \ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +\spadcommand{piConvergents := [4/p for p in ccf] } +$$ +\left[ +4, {8 \over 3}, {{52} \over {15}}, {{304} \over {105}}, {{1052} +\over {315}}, {{10312} \over {3465}}, {{147916} \over {45045}}, +\ldots +\right] +$$ +\returnType{Type: Stream Fraction Integer} + +As you can see, the values are converging to +$\pi = 3.14159265358979323846...$, +but not very quickly. + +\spadcommand{piConvergents :: Stream Float } +$$ +\begin{array}{@{}l} +\left[ +{4.0}, {2.6666666666\ 666666667}, {3.4666666666\ 666666667}, +\right. +\\ +\\ +\displaystyle +{2.8952380952\ 380952381}, {3.3396825396\ 825396825}, +\\ +\\ +\displaystyle +\left. +{2.9760461760\ 461760462}, {3.2837384837\ 384837385}, \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream Float} + +You need not restrict yourself to continued fractions of integers. +Here is an expansion for a quotient of Gaussian integers. + +\spadcommand{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))} +$$ +-{90}+{{59} \ i}+ \zag{1}{{1 -{2 \ i}}}+ \zag{1}{{-1+{2 \ i}}} +$$ +\returnType{Type: ContinuedFraction Complex Integer} + +This is an expansion for a quotient of polynomials in one variable +with rational number coefficients. + +\spadcommand{r : Fraction UnivariatePolynomial(x,Fraction Integer) } +\returnType{Type: Void} + +\spadcommand{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) } +$$ +{{x \sp 2} -{3 \ x}+2} \over {{x \sp 2} -{7 \ x}+{12}} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{continuedFraction r } +$$ +1+ \zag{1}{{{{1 \over 4} \ x} -{9 \over 8}}}+ \zag{1}{{{{{16} \over 3} \ x} +-{{40} \over 3}}} +$$ +\returnType{Type: ContinuedFraction UnivariatePolynomial(x,Fraction Integer)} + +To conclude this section, we give you evidence that +\begin{verbatim} + z = 3 + 1 + ----------------------- + 3 + 1 + ------------------- + 6 + 1 + --------------- + 3 + 1 + ----------- + 6 + 1 + ------- + 3 + ... +\end{verbatim} + +is the expansion of $\sqrt{11}$. + +\spadcommand{[i*i for i in convergents(z) :: Stream Float] } +$$ +\begin{array}{@{}l} +\left[ +{9.0}, {11.1111111111\ 11111111}, {10.9944598337\ 9501385}, +\right. +\\ +\\ +\displaystyle +{11.0002777777\ 77777778}, {10.9999860763\ 98799786}, +\\ +\\ +\displaystyle +\left. +{11.0000006979\ 29731039}, {10.9999999650\ 15834446}, \ldots +\right] +\end{array} +$$ +\returnType{Type: Stream Float} + +\section{CycleIndicators} +\label{CycleIndicatorsXmpPage} + +This section is based upon the paper J. H. Redfield, ``The Theory of +Group-Reduced Distributions'', American J. Math.,49 (1927) 433-455, +and is an application of group theory to enumeration problems. It is +a development of the work by P. A. MacMahon on the application of +symmetric functions and Hammond operators to combinatorial theory. + +The theory is based upon the power sum symmetric functions +$s_i$ which are the sum of the $i$-th powers of the +variables. The cycle index of a permutation is an expression that +specifies the sizes of the cycles of a permutation, and may be +represented as a partition. A partition of a non-negative integer +{\tt n} is a collection of positive integers called its parts whose +sum is {\tt n}. For example, the partition $(3^2 \ 2 \ 1^2)$ will be +used to represent $s^2_3 s_2 s^2_1$ and will indicate that the +permutation has two cycles of length 3, one of length 2 and two of +length 1. The cycle index of a permutation group is the sum of the +cycle indices of its permutations divided by the number of +permutations. The cycle indices of certain groups are provided. + +The operation {\tt complete} returns the cycle index of the +symmetric group of order {\tt n} for argument {\tt n}. +Alternatively, it is the $n$-th complete homogeneous symmetric +function expressed in terms of power sum symmetric functions. + +\spadcommand{complete 1} +$$ +\left( +1 +\right) +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +\spadcommand{complete 2} +$$ +{{1 \over 2} \ {\left( 2 +\right)}}+{{1 +\over 2} \ {\left( 1 \sp 2 +\right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +\spadcommand{complete 3} +$$ +{{1 \over 3} \ {\left( 3 +\right)}}+{{1 +\over 2} \ {\left( {2 \sp {\ }} \ 1 +\right)}}+{{1 +\over 6} \ {\left( 1 \sp 3 +\right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +\spadcommand{complete 7} +$$ +\begin{array}{@{}l} +{{1 \over 7} \ {\left( 7 \right)}}+ +{{1\over 6} \ {\left( {6 \sp {\ }} \ 1 \right)}}+ +{{1\over {10}} \ {\left( {5 \sp {\ }} \ 2 \right)}}+ +{{1\over {10}} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}}+ +{{1\over {12}} \ {\left( {4 \sp {\ }} \ 3 \right)}}+ +{{1\over 8} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}}+ +\\ +\\ +\displaystyle +{{1\over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 3} \right)}}+ +{{1\over {18}} \ {\left( {3 \sp 2} \ 1 \right)}}+ +{{1\over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}}+ +{{1\over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 2} \right)}}+ +{{1\over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}}+ +\\ +\\ +\displaystyle +{{1\over {48}} \ {\left( {2 \sp 3} \ 1 \right)}}+ +{{1\over {48}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}}+ +{{1\over {240}} \ {\left( {2 \sp {\ }} \ {1 \sp 5} \right)}}+ +{{1\over {5040}} \ {\left( 1 \sp 7 \right)}} +\end{array} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The operation {\tt elementary} computes the $n$-th +elementary symmetric function for argument {\tt n.} + +\spadcommand{elementary 7} +$$ +\begin{array}{@{}l} +{{1 \over 7} \ {\left( 7 \right)}} +-{{1 \over 6} \ {\left( {6 \sp {\ }} \ 1 \right)}} +-{{1 \over {10}} \ {\left( {5 \sp {\ }} \ 2 \right)}}+ +{{1\over {10}} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}} +-{{1 \over {12}} \ {\left( {4 \sp {\ }} \ 3 \right)}}+ +{{1\over 8} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}} +\\ +\\ +\displaystyle +-{{1 \over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 3} \right)}}+ +{{1\over {18}} \ {\left( {3 \sp 2} \ 1 \right)}}+ +{{1\over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}} +-{{1 \over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 2} \right)}} ++{{1\over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}} +\\ +\\ +\displaystyle +-{{1 \over {48}} \ {\left( {2 \sp 3} \ 1 \right)}}+ +{{1\over {48}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}} +-{{1 \over {240}} \ {\left( {2 \sp {\ }} \ {1 \sp 5} \right)}}+ +{{1\over {5040}} \ {\left( 1 \sp 7 \right)}} +\end{array} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The operation {\tt alternating} returns the cycle index of the alternating +group having an even number of even parts in each cycle partition. + +\spadcommand{alternating 7} +$$ +\begin{array}{@{}l} +{{2 \over 7} \ {\left( 7 \right)}}+ +{{1\over 5} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}}+ +{{1\over 4} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}}+ +{{1\over 9} \ {\left( {3 \sp 2} \ 1 \right)}}+ +{{1\over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}}+ +{{1\over {36}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}}+ +\\ +\\ +\displaystyle +{{1\over {24}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}}+ +{{1\over {2520}} \ {\left( 1 \sp 7 \right)}} +\end{array} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The operation {\tt cyclic} returns the cycle index of the cyclic group. + +\spadcommand{cyclic 7} +$$ +{{6 \over 7} \ {\left( 7 \right)}}+ +{{1\over 7} \ {\left( 1 \sp 7 \right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The operation {\tt dihedral} is the cycle index of the +dihedral group. + +\spadcommand{dihedral 7} +$$ +{{3 \over 7} \ {\left( 7 \right)}}+ +{{1\over 2} \ {\left( {2 \sp 3} \ 1 \right)}}+ +{{1\over {14}} \ {\left( 1 \sp 7 \right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The operation {\tt graphs} for argument {\tt n} returns the cycle +index of the group of permutations on the edges of the complete graph +with {\tt n} nodes induced by applying the symmetric group to the +nodes. + +\spadcommand{graphs 5} +$$ +\begin{array}{@{}l} +{{1 \over 6} \ {\left( {6 \sp {\ }} \ {3 \sp {\ }} \ 1 \right)}}+ +{{1\over 5} \ {\left( 5 \sp 2 \right)}}+ +{{1\over 4} \ {\left( {4 \sp 2} \ 2 \right)}}+ +{{1\over 6} \ {\left( {3 \sp 3} \ 1 \right)}}+ +{{1\over 8} \ {\left( {2 \sp 4} \ {1 \sp 2} \right)}}+ +\\ +\\ +\displaystyle +{{1\over {12}} \ {\left( {2 \sp 3} \ {1 \sp 4} \right)}}+ +{{1\over {120}} \ {\left( 1 \sp {10} \right)}} +\end{array} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The cycle index of a direct product of two groups is the product of +the cycle indices of the groups. Redfield provided two operations on +two cycle indices which will be called ``cup'' and ``cap'' here. The +{\tt cup} of two cycle indices is a kind of scalar product that +combines monomials for permutations with the same cycles. The {\tt +cap} operation provides the sum of the coefficients of the result of +the {\tt cup} operation which will be an integer that enumerates what +Redfield called group-reduced distributions. + +We can, for example, represent {\tt complete 2 * complete 2} as the +set of objects {\tt a a b b} and +{\tt complete 2 * complete 1 * complete 1} as {\tt c c d e.} + +This integer is the number of different sets of four pairs. + +\spadcommand{cap(complete 2**2, complete 2*complete 1**2)} +$$ +4 +$$ +\returnType{Type: Fraction Integer} + +For example, +\begin{verbatim} +a a b b a a b b a a b b a a b b +c c d e c d c e c e c d d e c c +\end{verbatim} + +This integer is the number of different sets of four pairs no two +pairs being equal. + +\spadcommand{cap(elementary 2**2, complete 2*complete 1**2)} +$$ +2 +$$ +\returnType{Type: Fraction Integer} + +For example, +\begin{verbatim} +a a b b a a b b +c d c e c e c d +\end{verbatim} +In this case the configurations enumerated are easily constructed, +however the theory merely enumerates them providing little help in +actually constructing them. + +Here are the number of 6-pairs, first from {\tt a a a b b c,} second +from {\tt d d e e f g.} + +\spadcommand{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)} +$$ +24 +$$ +\returnType{Type: Fraction Integer} + +Here it is again, but with no equal pairs. + +\spadcommand{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)} +$$ +8 +$$ +\returnType{Type: Fraction Integer} + +\spadcommand{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)} +$$ +8 +$$ +\returnType{Type: Fraction Integer} + +The number of 6-triples, first from {\tt a a a b b c,} second from +{\tt d d e e f g,} third from {\tt h h i i j j.} + +\spadcommand{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))} +$$ +1500 +$$ +\returnType{Type: Fraction Integer} + +The cycle index of vertices of a square is dihedral 4. + +\spadcommand{square:=dihedral 4} +$$ +{{1 \over 4} \ {\left( 4 \right)}}+ +{{3\over 8} \ {\left( 2 \sp 2 \right)}}+ +{{1\over 4} \ {\left( {2 \sp {\ }} \ {1 \sp 2} \right)}}+ +{{1\over 8} \ {\left( 1 \sp 4 \right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The number of different squares with 2 red vertices and 2 blue vertices. + +\spadcommand{cap(complete 2**2,square)} +$$ +2 +$$ +\returnType{Type: Fraction Integer} + +The number of necklaces with 3 red beads, 2 blue beads and 2 green beads. + +\spadcommand{cap(complete 3*complete 2**2,dihedral 7)} +$$ +18 +$$ +\returnType{Type: Fraction Integer} + +The number of graphs with 5 nodes and 7 edges. + +\spadcommand{cap(graphs 5,complete 7*complete 3)} +$$ +4 +$$ +\returnType{Type: Fraction Integer} + +The cycle index of rotations of vertices of a cube. + +\spadcommand{s(x) == powerSum(x)} +\returnType{Type: Void} + +\spadcommand{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)} +\begin{verbatim} + Compiling function s with type PositiveInteger -> + SymmetricPolynomial Fraction Integer +\end{verbatim} +$$ +{{1 \over 4} \ {\left( 4 \sp 2 \right)}}+ +{{1\over 3} \ {\left( {3 \sp 2} \ {1 \sp 2} \right)}}+ +{{3\over 8} \ {\left( 2 \sp 4 \right)}}+ +{{1\over {24}} \ {\left( 1 \sp 8 \right)}} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +The number of cubes with 4 red vertices and 4 blue vertices. + +\spadcommand{cap(complete 4**2,cube)} +$$ +7 +$$ +\returnType{Type: Fraction Integer} + +The number of labeled graphs with degree sequence {\tt 2 2 2 1 1} +with no loops or multiple edges. + +\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))} +$$ +7 +$$ +\returnType{Type: Fraction Integer} + +Again, but with loops allowed but not multiple edges. + +\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))} +$$ +17 +$$ +\returnType{Type: Fraction Integer} + +Again, but with multiple edges allowed, but not loops + +\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))} +$$ +10 +$$ +\returnType{Type: Fraction Integer} + +Again, but with both multiple edges and loops allowed + +\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))} +$$ +23 +$$ +\returnType{Type: Fraction Integer} + +Having constructed a cycle index for a configuration we are at liberty +to evaluate the $s_i$ components any way we please. For example we +can produce enumerating generating functions. This is done by +providing a function {\tt f} on an integer {\tt i} to the value +required of $s_i$, and then evaluating {\tt eval(f, cycleindex)}. + +\spadcommand{x: ULS(FRAC INT,'x,0) := 'x } +$$ +x +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +\spadcommand{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0) } +\returnType{Type: Void} + +\spadcommand{Integers: INT -> ULS(FRAC INT, 'x, 0) } +\returnType{Type: Void} + +For the integers 0 and 1, or two colors. + +\spadcommand{ZeroOrOne n == 1+x**n } +\returnType{Type: Void} + +\spadcommand{ZeroOrOne 5 } +\begin{verbatim} + Compiling function ZeroOrOne with type Integer -> + UnivariateLaurentSeries(Fraction Integer,x,0) +\end{verbatim} +$$ +1+{x \sp 5} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +For the integers {\tt 0, 1, 2, ...} we have this. + +\spadcommand{Integers n == 1/(1-x**n) } +\returnType{Type: Void} + +\spadcommand{Integers 5 } +\begin{verbatim} + Compiling function Integers with type Integer -> + UnivariateLaurentSeries(Fraction Integer,x,0) +\end{verbatim} +$$ +1+{x \sp 5}+{O +\left( +{{x \sp 8}} +\right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of graphs with 5 nodes and {\tt n} +edges. + +Note that there is an eval function that takes two arguments. It has the +signature: +\begin{verbatim} +((Integer -> D1),SymmetricPolynomial Fraction Integer) -> D1 + from EvaluateCycleIndicators D1 if D1 has ALGEBRA FRAC INT +\end{verbatim} +This function is not normally exposed (it will not normally be considered +in the list of eval functions) as it is only useful for this particular +domain. To use it we ask that it be considered thus: + +\spadcommand{)expose EVALCYC} + +and now we can use it: + +\spadcommand{eval(ZeroOrOne, graphs 5) } +$$ +1+x+ +{2 \ {x \sp 2}}+ +{4 \ {x \sp 3}}+ +{6 \ {x \sp 4}}+ +{6 \ {x \sp 5}}+ +{6 \ {x \sp 6}}+ +{4 \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of necklaces with +{\tt n} red beads and {\tt n-8} green beads. + +\spadcommand{eval(ZeroOrOne,dihedral 8) } +$$ +1+x+ +{4 \ {x \sp 2}}+ +{5 \ {x \sp 3}}+ +{8 \ {x \sp 4}}+ +{5 \ {x \sp 5}}+ +{4 \ {x \sp 6}}+ +{x \sp 7}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of partitions of {\tt n} into 4 +or fewer parts. + +\spadcommand{eval(Integers,complete 4) } +$$ +1+x+ +{2 \ {x \sp 2}}+ +{3 \ {x \sp 3}}+ +{5 \ {x \sp 4}}+ +{6 \ {x \sp 5}}+ +{9 \ {x \sp 6}}+ +{{11} \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of partitions of {\tt n} into 4 +boxes containing ordered distinct parts. + +\spadcommand{eval(Integers,elementary 4) } +$$ +{x \sp 6}+ +{x \sp 7}+ +{2 \ {x \sp 8}}+ +{3 \ {x \sp 9}}+ +{5 \ {x \sp {10}}}+ +{6 \ {x \sp {11}}}+ +{9 \ {x \sp {12}}}+ +{{11} \ {x \sp {13}}}+ +{O \left({{x \sp {14}}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of different cubes with {\tt n} +red vertices and {\tt 8-n} green ones. + +\spadcommand{eval(ZeroOrOne,cube) } +$$ +1+x+ +{3 \ {x \sp 2}}+ +{3 \ {x \sp 3}}+ +{7 \ {x \sp 4}}+ +{3 \ {x \sp 5}}+ +{3 \ {x \sp 6}}+ +{x \sp 7}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of different cubes with integers +on the vertices whose sum is {\tt n.} + +\spadcommand{eval(Integers,cube) } +$$ +1+x+ +{4 \ {x \sp 2}}+ +{7 \ {x \sp 3}}+ +{{21} \ {x \sp 4}}+ +{{37} \ {x \sp 5}}+ +{{85} \ {x \sp 6}}+ +{{151} \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The coefficient of $x^n$ is the number of graphs with 5 nodes and with +integers on the edges whose sum is {\tt n.} In other words, the +enumeration is of multigraphs with 5 nodes and {\tt n} edges. + +\spadcommand{eval(Integers,graphs 5) } +$$ +1+x+ +{3 \ {x \sp 2}}+ +{7 \ {x \sp 3}}+ +{{17} \ {x \sp 4}}+ +{{35} \ {x \sp 5}}+ +{{76} \ {x \sp 6}}+ +{{149} \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +Graphs with 15 nodes enumerated with respect to number of edges. + +\spadcommand{eval(ZeroOrOne ,graphs 15) } +$$ +1+x+ +{2 \ {x \sp 2}}+ +{5 \ {x \sp 3}}+ +{{11} \ {x \sp 4}}+ +{{26} \ {x \sp 5}}+ +{{68} \ {x \sp 6}}+ +{{177} \ {x \sp 7}}+ +{O \left({{x \sp 8}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +Necklaces with 7 green beads, 8 white beads, 5 yellow beads and 10 +red beads. + +\spadcommand{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)} +$$ +49958972383320 +$$ +\returnType{Type: Fraction Integer} + +The operation {\tt SFunction} is the S-function or Schur function of a +partition written as a descending list of integers expressed in terms +of power sum symmetric functions. + +In this case the argument partition represents a tableau shape. For +example {\tt 3,2,2,1} represents a tableau with three boxes in the +first row, two boxes in the second and third rows, and one box in the +fourth row. {\tt SFunction [3,2,2,1]} counts the number of different +tableaux of shape {\tt 3, 2, 2, 1} filled with objects with an +ascending order in the columns and a non-descending order in the rows. + +\spadcommand{sf3221:= SFunction [3,2,2,1] } +$$ +\begin{array}{@{}l} +{{1 \over {12}} \ {\left( {6 \sp {\ }} \ 2 \right)}} +-{{1 \over {12}} \ {\left( {6 \sp {\ }} \ {1 \sp 2} \right)}} +-{{1 \over {16}} \ {\left( 4 \sp 2 \right)}}+ +{{1\over {12}} \ {\left( {4 \sp {\ }} \ {3 \sp {\ }} \ 1 \right)}}+ +{{1\over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 4} \right)}} +-{{1 \over {36}} \ {\left( {3 \sp 2} \ 2 \right)}}+ +\\ +\\ +\displaystyle +{{1\over {36}} \ {\left( {3 \sp 2} \ {1 \sp 2} \right)}} +-{{1 \over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \ 1 \right)}} +-{{1 \over {36}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 3} \right)}} +-{{1 \over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 5} \right)}} +-{{1 \over {192}} \ {\left( 2 \sp 4 \right)}}+ +\\ +\\ +\displaystyle +{{1\over {48}} \ {\left( {2 \sp 3} \ {1 \sp 2} \right)}}+ +{{1\over {96}} \ {\left( {2 \sp 2} \ {1 \sp 4} \right)}} +-{{1 \over {144}} \ {\left( {2 \sp {\ }} \ {1 \sp 6} \right)}}+ +{{1\over {576}} \ {\left( 1 \sp 8 \right)}} +\end{array} +$$ +\returnType{Type: SymmetricPolynomial Fraction Integer} + +This is the number filled with {\tt a a b b c c d d.} + +\spadcommand{cap(sf3221,complete 2**4) } +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +The configurations enumerated above are: +\begin{verbatim} +a a b a a c a a d +b c b b b b +c d c d c c +d d d +\end{verbatim} + +This is the number of tableaux filled with {\tt 1..8.} + +\spadcommand{cap(sf3221, powerSum 1**8)} +$$ +70 +$$ +\returnType{Type: Fraction Integer} + +The coefficient of $x^n$ is the number of column strict reverse plane +partitions of {\tt n} of shape {\tt 3 2 2 1.} + +\spadcommand{eval(Integers, sf3221)} +$$ +{x \sp 9}+ +{3 \ {x \sp {10}}}+ +{7 \ {x \sp {11}}}+ +{{14} \ {x \sp {12}}}+ +{{27} \ {x \sp {13}}}+ +{{47} \ {x \sp {14}}}+ +{O \left({{x \sp {15}}} \right)} +$$ +\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} + +The smallest is +\begin{verbatim} +0 0 0 +1 1 +2 2 +3 +\end{verbatim} + +\section{DeRhamComplex} +\label{DeRhamComplexXmpPage} + +The domain constructor {\tt DeRhamComplex} creates the class of +differential forms of arbitrary degree over a coefficient ring. The +De Rham complex constructor takes two arguments: a ring, {\tt +coefRing,} and a list of coordinate variables. + +This is the ring of coefficients. + +\spadcommand{coefRing := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +These are the coordinate variables. + +\spadcommand{lv : List Symbol := [x,y,z] } +$$ +\left[ +x, y, z +\right] +$$ +\returnType{Type: List Symbol} + +This is the De Rham complex of Euclidean three-space using coordinates +{\tt x, y} and {\tt z.} + +\spadcommand{der := DERHAM(coefRing,lv) } +$$ +DeRhamComplex(Integer,[x,y,z]) +$$ +\returnType{Type: Domain} + +This complex allows us to describe differential forms having +expressions of integers as coefficients. These coefficients can +involve any number of variables, for example, {\tt f(x,t,r,y,u,z).} +As we've chosen to work with ordinary Euclidean three-space, +expressions involving these forms are treated as functions of +{\tt x, y} and {\tt z} with the additional arguments {\tt t, r} +and {\tt u} regarded as symbolic constants. + +Here are some examples of coefficients. + +\spadcommand{R := Expression coefRing } +$$ +\mbox{\rm Expression Integer} +$$ +\returnType{Type: Domain} + +\spadcommand{f : R := x**2*y*z-5*x**3*y**2*z**5 } +$$ +-{5 \ {x \sp 3} \ {y \sp 2} \ {z \sp 5}}+{{x \sp 2} \ y \ z} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 } +$$ +-{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ +{y\ {z \sp 2} \ {\cos \left({z} \right)}} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{h : R :=x*y*z-2*x**3*y*z**2 } +$$ +-{2 \ {x \sp 3} \ y \ {z \sp 2}}+{x \ y \ z} +$$ +\returnType{Type: Expression Integer} + +We now define the multiplicative basis elements for the exterior +algebra over {\tt R}. + +\spadcommand{dx : der := generator(1) } +$$ +dx +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +\spadcommand{dy : der := generator(2)} +$$ +dy +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +\spadcommand{dz : der := generator(3)} +$$ +dz +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +This is an alternative way to give the above assignments. + +\spadcommand{[dx,dy,dz] := [generator(i)\$der for i in 1..3] } +$$ +\left[ +dx, dy, dz +\right] +$$ +\returnType{Type: List DeRhamComplex(Integer,[x,y,z])} + +Now we define some one-forms. + +\spadcommand{alpha : der := f*dx + g*dy + h*dz } +$$ +\begin{array}{@{}l} +{{\left( -{2 \ {x \sp 3} \ y \ {z \sp 2}}+{x \ y \ z} \right)}\ dz}+ +\\ +\\ +\displaystyle +{{\left( +-{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ +{y\ {z \sp 2} \ {\cos \left({z} \right)}} +\right)}\ dy}+ +\\ +\\ +\displaystyle +{{\left( -{5 \ {x \sp 3} \ {y \sp 2} \ {z \sp 5}}+{{x \sp 2} \ y \ z} +\right)}\ dx} +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy } +$$ +{x \ dy}+ +{{\cos +\left({{{\tan \left({{x \ y \ z}} \right)}+{x\ y \ z}}} \right)} +\ dx} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +A well-known theorem states that the composition of +\spadfunFrom{exteriorDifferential}{DeRhamComplex} with itself is the +zero map for continuous forms. Let's verify this theorem for {\tt alpha}. + +\spadcommand{exteriorDifferential alpha } +$$ +\begin{array}{@{}l} +{{\left( +{y \ {z \sp 2} \ {\sin \left({z} \right)}}+ +{{14}\ z \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} +-{2 \ y \ z \ {\cos \left({z} \right)}} +-{2 \ {x \sp 3} \ {z \sp 2}}+{x \ z} \right)}\ dy \ dz}+ +\\ +\\ +\displaystyle +{{\left( +{{25} \ {x \sp 3} \ {y \sp 2} \ {z \sp 4}} - +{6 \ {x \sp 2} \ y \ {z \sp 2}}+ +{y \ z} - +{{x \sp 2} \ y} +\right)}\ dx \ dz}+ +\\ +\\ +\displaystyle +{{\left( +-{{21} \ {x \sp 2} \ {y \sp 2} \ {z \sp 2} \ +{\cos \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ +{{10}\ {x \sp 3} \ y \ {z \sp 5}} -{{x \sp 2} \ z} \right)}\ dx \ dy} +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +We see a lengthy output of the last expression, but nevertheless, the +composition is zero. + +\spadcommand{exteriorDifferential \% } +$$ +0 +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +Now we check that \spadfunFrom{exteriorDifferential}{DeRhamComplex} +is a ``graded derivation'' {\tt D,} that is, {\tt D} satisfies: +\begin{verbatim} +D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b) +\end{verbatim} + +\spadcommand{gamma := alpha * beta } +$$ +\begin{array}{@{}l} +{{\left( +{2 \ {x \sp 4} \ y \ {z \sp 2}} - +{{x \sp 2} \ y \ z} +\right)}\ dy \ dz}+ +\\ +\\ +\displaystyle +{{\left( +{2 \ {x \sp 3} \ y \ {z \sp 2}} -{x \ y \ z} +\right)} +\ {\cos +\left( +{{{\tan \left({{x \ y \ z}} \right)}+ +{x\ y \ z}}} +\right)}\ dx \ dz}+ +\\ +\\ +\displaystyle +\left( +{{\left( +{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} +-{y \ {z \sp 2} \ {\cos \left({z} \right)}} +\right)} +\ {\cos \left({{{\tan \left({{x \ y \ z}} \right)}+{x\ y \ z}}} \right)}}- +\right. +\\ +\\ +\left. +\displaystyle +{5 \ {x \sp 4} \ {y \sp 2} \ {z \sp 5}}+ +{{x \sp 3} \ y \ z} +\right)\ dx \ dy +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +We try this for the one-forms {\tt alpha} and {\tt beta}. + +\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) } +$$ +0 +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +Now we define some ``basic operators'' (see +\ref{OperatorXmpPage} on page~\pageref{OperatorXmpPage}). + +\spadcommand{a : BOP := operator('a) } +$$ +a +$$ +\returnType{Type: BasicOperator} + +\spadcommand{b : BOP := operator('b) } +$$ +b +$$ +\returnType{Type: BasicOperator} + +\spadcommand{c : BOP := operator('c) } +$$ +c +$$ +\returnType{Type: BasicOperator} + +We also define some indeterminate one- and two-forms using these +operators. + +\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz } +$$ +{{c \left({x, y, z} \right)}\ dz}+ +{{b \left({x, y, z} \right)}\ dy}+ +{{a \left({x, y, z} \right)}\ dx} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz } +$$ +{{c \left({x, y, z} \right)}\ dy \ dz}+ +{{b \left({x, y, z} \right)}\ dx \ dz}+ +{{a \left({x, y, z} \right)}\ dx \ dy} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +This allows us to get formal definitions for the ``gradient'' \ldots + +\spadcommand{totalDifferential(a(x,y,z))\$der } +$$ +{{{a \sb {{,3}}} \left({x, y, z} \right)}\ dz}+ +{{{a \sb {{,2}}} \left({x, y, z} \right)}\ dy}+ +{{{a \sb {{,1}}} \left({x, y, z} \right)}\ dx} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +the ``curl'' \ldots + +\spadcommand{exteriorDifferential sigma } +$$ +\begin{array}{@{}l} +{{\left( +{{c \sb {{,2}}} \left({x, y, z} \right)} +-{{b \sb {{,3}}} \left({x, y, z} \right)} +\right)}\ dy \ dz}+ +\\ +\\ +\displaystyle +{{\left( +{{c \sb {{,1}}} \left({x, y, z} \right)} +-{{a \sb {{,3}}} \left({x, y, z} \right)} +\right)}\ dx \ dz}+ +\\ +\\ +\displaystyle +{{\left( +{{b \sb {{,1}}} \left({x, y, z} \right)} +-{{a \sb {{,2}}} \left({x, y, z} \right)} +\right)}\ dx \ dy} +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +and the ``divergence.'' + +\spadcommand{exteriorDifferential theta } +$$ +{\left( +{{c \sb {{,1}}} \left({x, y, z} \right)} +-{{b \sb {{,2}}} \left({x, y, z} \right)}+ +{{a\sb {{,3}}} \left({x, y, z} \right)} +\right)}\ dx \ dy \ dz +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +Note that the De Rham complex is an algebra with unity. This element +{\tt 1} is the basis for elements for zero-forms, that is, functions +in our space. + +\spadcommand{one : der := 1 } +$$ +1 +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +To convert a function to a function lying in the De Rham complex, +multiply the function by ``one.'' + +\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one } +$$ +a +\left( +{x, t, y, u, v, z, e} +\right) +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +A current limitation of Axiom forces you to write functions with more +than four arguments using square brackets in this way. + +\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one } +$$ +a +\left( +{x, y, x, t, x, z, y, r, u, x} +\right) +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +Now note how the system keeps track of where your coordinate functions +are located in expressions. + +\spadcommand{exteriorDifferential g1 } +$$ +\begin{array}{@{}l} +{{{a \sb {{,6}}} \left({x, t, y, u, v, z, e} \right)}\ dz}+ +\\ +\\ +\displaystyle +{{{a \sb {{,3}}} \left({x, t, y, u, v, z, e} \right)}\ dy}+ +\\ +\\ +\displaystyle +{{{a \sb {{,1}}} \left({x, t, y, u, v, z, e} \right)}\ dx} +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +\spadcommand{exteriorDifferential h1 } +$$ +\begin{array}{@{}l} +{{{a \sb {{,6}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)}\ dz}+ +\\ +\\ +\displaystyle +\begin{array}{@{}l} +\left( {{a \sb {{,7}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)}+ +\right. +\\ +\\ +\displaystyle +\left. +{{a\sb {{,2}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)} +\right)\ dy+ +\end{array} +\\ +\\ +\displaystyle +\begin{array}{@{}l} +\left( {{a \sb {{,{10}}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)}+ +\right. +\\ +\\ +\displaystyle +{{a\sb {{,5}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)}+ +\\ +\\ +\displaystyle +{{a\sb {{,3}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)}+ +\\ +\\ +\displaystyle +\left. +{{a\sb {{,1}}} +\left({x, y, x, t, x, z, y, r, u, x} \right)} +\right)\ dx +\end{array} +\end{array} +$$ +\returnType{Type: DeRhamComplex(Integer,[x,y,z])} + +In this example of Euclidean three-space, the basis for the De Rham complex +consists of the eight forms: {\tt 1}, {\tt dx}, {\tt dy}, {\tt dz}, +{\tt dx*dy}, {\tt dx*dz}, {\tt dy*dz}, and {\tt dx*dy*dz}. + +\spadcommand{coefficient(gamma, dx*dy) } +$$ +\begin{array}{@{}l} +{{\left( {7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} +-{y \ {z \sp 2} \ {\cos \left({z} \right)}}\right)}\ {\cos +\left( +{{{\tan \left({{x \ y \ z}} \right)}+ +{x\ y \ z}}} +\right)}} +\\ +\\ +\displaystyle +-{5 \ {x \sp 4} \ {y \sp 2} \ {z \sp 5}}+ +{{x \sp 3} \ y \ z} +\end{array} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{coefficient(gamma, one) } +$$ +0 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{coefficient(g1,one) } +$$ +a +\left( +{x, t, y, u, v, z, e} +\right) +$$ +\returnType{Type: Expression Integer} + +\section{DecimalExpansion} +\label{DecimalExpansionXmpPage} + +All rationals have repeating decimal expansions. Operations to access +the individual digits of a decimal expansion can be obtained by +converting the value to {\tt RadixExpansion(10)}. More examples of +expansions are available in +\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, +\ref{HexadecimalExpansionXmpPage} on +page~\pageref{HexadecimalExpansionXmpPage}, and +\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. + +The operation \spadfunFrom{decimal}{DecimalExpansion} is used to create +this expansion of type {\tt DecimalExpansion}. + +\spadcommand{r := decimal(22/7) } +$$ +3.{\overline {142857}} +$$ +\returnType{Type: DecimalExpansion} + +Arithmetic is exact. + +\spadcommand{r + decimal(6/7) } +$$ +4 +$$ +\returnType{Type: DecimalExpansion} + +The period of the expansion can be short or long \ldots + +\spadcommand{[decimal(1/i) for i in 350..354] } +$$ +\begin{array}{@{}l} +\left[ +{0.{00}{\overline {285714}}}, +{0.{\overline {002849}}}, +{0.{00284}{\overline {09}}}, +{0.{\overline {00283286118980169971671388101983}}}, +\right. +\\ +\\ +\displaystyle +\left. +{0.0{\overline {0282485875706214689265536723163841807909604519774011299435}}} +\right] +\end{array} +$$ +\returnType{Type: List DecimalExpansion} + +or very long. + +\spadcommand{decimal(1/2049) } +$$ +\begin{array}{@{}l} +0.{\overline +{000488042947779404587603709126403123474865788189360663738408979990239}} +\\ +\\ +\displaystyle +\ \ {\overline{ +141044411908247925817471937530502684236212786725231820400195217179111}} +\\ +\\ +\displaystyle +\ \ {\overline{ +761835041483650561249389946315275744265495363591996095656417764763299}} +\\ +\\ +\displaystyle +\ \ {\overline{ +170326988775012201073694485114690092728160078086871644704734016593460}} +\\ +\\ +\displaystyle +\ \ {\overline{ +22449975597852611029770619814543679843826256710590531966813079551}} +\end{array} +$$ +\returnType{Type: DecimalExpansion} + +These numbers are bona fide algebraic objects. + +\spadcommand{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9) } +$$ +{{0.{25}} \ {x \sp 2}}+{{0.{\overline 6}} \ x}+{0.{\overline 4}} +$$ +\returnType{Type: Polynomial DecimalExpansion} + +\spadcommand{q := differentiate(p, x) } +$$ +{{0.5} \ x}+{0.{\overline 6}} +$$ +\returnType{Type: Polynomial DecimalExpansion} + +\spadcommand{g := gcd(p, q) } +$$ +x+{1.{\overline 3}} +$$ +\returnType{Type: Polynomial DecimalExpansion} + +\section{DistributedMultivariatePolynomial} +\label{DistributedMultivariatePolynomialXmpPage} + +\hyphenation{Homo-gen-eous-Dis-tributed-Multi-var-i-ate-Pol-y-nomial} + +{\tt DistributedMultivariatePolynomial} which is abbreviated as {\tt DMP} +and {\tt HomogeneousDistributedMultivariatePolynomial}, which is abbreviated +as {\tt HDMP}, are very similar to {\tt MultivariatePolynomial} except that +they are represented and displayed in a non-recursive manner. + +\spadcommand{(d1,d2,d3) : DMP([z,y,x],FRAC INT) } +\returnType{Type: Void} + +The constructor {\tt DMP} orders its monomials lexicographically while +{\tt HDMP} orders them by total order refined by reverse lexicographic +order. + +\spadcommand{d1 := -4*z + 4*y**2*x + 16*x**2 + 1 } +$$ +-{4 \ z}+{4 \ {y \sp 2} \ x}+{{16} \ {x \sp 2}}+1 +$$ +\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +\spadcommand{d2 := 2*z*y**2 + 4*x + 1 } +$$ +{2 \ z \ {y \sp 2}}+{4 \ x}+1 +$$ +\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +\spadcommand{d3 := 2*z*x**2 - 2*y**2 - x } +$$ +{2 \ z \ {x \sp 2}} -{2 \ {y \sp 2}} -x +$$ +\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +These constructors are mostly used in Gr\"{o}bner basis calculations. + +\spadcommand{groebner [d1,d2,d3] } +$$ +\begin{array}{@{}l} +\left[ +{z - +{{{1568} \over {2745}} \ {x \sp 6}} - +{{{1264} \over {305}} \ {x \sp 5}}+ +{{6 \over {305}} \ {x \sp 4}}+ +{{{182} \over {549}} \ {x \sp 3}} +-{{{2047} \over {610}} \ {x \sp 2}} - +{{{103} \over {2745}} \ x} - +{{2857} \over {10980}}}, +\right. +\\ +\\ +\displaystyle +{{y \sp 2}+ +{{{112} \over {2745}} \ {x \sp 6}} - +{{{84} \over {305}} \ {x \sp 5}} - +{{{1264} \over {305}} \ {x \sp 4}} - +{{{13} \over {549}} \ {x \sp 3}}+ +{{{84} \over {305}} \ {x \sp 2}}+ +{{{1772} \over {2745}} \ x}+ +{2 \over {2745}}}, +\\ +\\ +\displaystyle +\left. +{{x \sp 7}+ +{{{29} \over 4} \ {x \sp 6}} - +{{{17} \over {16}} \ {x \sp 4}} - +{{{11} \over 8} \ {x \sp 3}}+ +{{1 \over {32}} \ {x \sp 2}}+ +{{{15} \over {16}} \ x}+ +{1 \over 4}} +\right] +\end{array} +$$ +\returnType{Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +\spadcommand{(n1,n2,n3) : HDMP([z,y,x],FRAC INT) } +\returnType{Type: Void} + +\spadcommand{n1 := d1 } +$$ +{4 \ {y \sp 2} \ x}+{{16} \ {x \sp 2}} -{4 \ z}+1 +$$ +\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +\spadcommand{n2 := d2 } +$$ +{2 \ z \ {y \sp 2}}+{4 \ x}+1 +$$ +\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +\spadcommand{n3 := d3 } +$$ +{2 \ z \ {x \sp 2}} -{2 \ {y \sp 2}} -x +$$ +\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +Note that we get a different Gr\"{o}bner basis when we use the +{\tt HDMP} polynomials, as expected. + +\spadcommand{groebner [n1,n2,n3] } +$$ +\begin{array}{@{}l} +\left[ +{{y \sp 4}+ +{2 \ {x \sp 3}} - +{{3 \over 2} \ {x \sp 2}}+ +{{1 \over 2} \ z} - +{1 \over 8}}, +\right. +\\ +\\ +\displaystyle +{{x \sp 4}+ +{{{29} \over 4} \ {x \sp 3}} - +{{1 \over 8} \ {y \sp 2}} - +{{7 \over 4} \ z \ x} - +{{9 \over {16}} \ x} - +{1 \over 4}}, +\\ +\\ +\displaystyle +{{z \ {y \sp 2}}+ +{2 \ x}+ +{1 \over 2}}, +\\ +\\ +\displaystyle +{{{y \sp 2} \ x}+ +{4 \ {x \sp 2}} - +z+ +{1 \over 4}}, +\\ +\\ +\displaystyle +{{z \ {x \sp 2}} - +{y \sp 2} - +{{1 \over 2} \ x}}, +\\ +\\ +\displaystyle +\left. +{{z \sp 2} - +{4 \ {y \sp 2}}+ +{2 \ {x \sp 2}} - +{{1 \over 4} \ z} - +{{3 \over 2} \ x}} +\right] +\end{array} +$$ +\returnType{Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} + +{\tt GeneralDistributedMultivariatePolynomial} is somewhat +more flexible in the sense that as well as accepting a list of +variables to specify the variable ordering, it also takes a +predicate on exponent vectors to specify the term ordering. +With this polynomial type the user can experiment with the effect +of using completely arbitrary term orderings. +This flexibility is mostly important for algorithms such as +Gr\"{o}bner basis calculations which can be very +sensitive to term ordering. + +For more information on related topics, see +\ref{ugIntroVariablesPage} on page~\pageref{ugIntroVariablesPage} in Section +\ref{ugIntroVariablesNumber} on page~\pageref{ugIntroVariablesNumber}, +\ref{ugTypesConvertPage} on page~\pageref{ugTypesConvertPage} in Section +\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}, +\ref{PolynomialXmpPage} on page~\pageref{PolynomialXmpPage}, +\ref{UnivariatePolynomialXmpPage} on +page~\pageref{UnivariatePolynomialXmpPage}, and +\ref{MultivariatePolynomialXmpPage} on +page~\pageref{MultivariatePolynomialXmpPage}. + +\section{DoubleFloat} +\label{DoubleFloatXmpPage} + +Axiom provides two kinds of floating point numbers. The domain +{\tt Float} (abbreviation {\tt FLOAT}) implements a model of arbitrary +precision floating point numbers. The domain {\tt DoubleFloat} +(abbreviation {\tt DFLOAT}) is intended to make available hardware +floating point arithmetic in Axiom. The actual model of floating +point {\tt DoubleFloat} that provides is system-dependent. For +example, on the IBM system 370 Axiom uses IBM double precision which +has fourteen hexadecimal digits of precision or roughly sixteen +decimal digits. Arbitrary precision floats allow the user to specify +the precision at which arithmetic operations are computed. Although +this is an attractive facility, it comes at a cost. Arbitrary-precision +floating-point arithmetic typically takes twenty to two hundred times +more time than hardware floating point. + +The usual arithmetic and elementary functions are available for +{\tt DoubleFloat}. Use {\tt )show DoubleFloat} to get a list of operations +or the HyperDoc browse facility to get more extensive documentation +about {\tt DoubleFloat}. + +By default, floating point numbers that you enter into Axiom are of +type {\tt Float}. + +\spadcommand{2.71828} +$$ +2.71828 +$$ +\returnType{Type: Float} + +You must therefore tell Axiom that you want to use {\tt DoubleFloat} +values and operations. The following are some conservative guidelines +for getting Axiom to use {\tt DoubleFloat}. + +To get a value of type {\tt DoubleFloat}, use a target with {\tt @}, \ldots + +\spadcommand{2.71828@DoubleFloat} +$$ +2.71828 +$$ +\returnType{Type: DoubleFloat} + +a conversion, \ldots + +\spadcommand{2.71828 :: DoubleFloat} +$$ +2.71828 +$$ +\returnType{Type: DoubleFloat} + +or an assignment to a declared variable. It is more efficient if you +use a target rather than an explicit or implicit conversion. + +\spadcommand{eApprox : DoubleFloat := 2.71828 } +$$ +2.71828 +$$ +\returnType{Type: DoubleFloat} + +You also need to declare functions that work with {\tt DoubleFloat}. + +\spadcommand{avg : List DoubleFloat -> DoubleFloat } +\returnType{Type: Void} + +\begin{verbatim} +avg l == + empty? l => 0 :: DoubleFloat + reduce(_+,l) / #l +\end{verbatim} +\returnType{Type: Void} + +%\spadcommand{avg [] } +% this complains but succeeds + +\spadcommand{avg [3.4,9.7,-6.8] } +\begin{verbatim} + Compiling function avg with type List Float -> DoubleFloat + +\end{verbatim} +$$ +2.1 +$$ +\returnType{Type: DoubleFloat} + +Use package-calling for operations from {\tt DoubleFloat} unless +the arguments themselves are already of type {\tt DoubleFloat}. + +\spadcommand{cos(3.1415926)\$DoubleFloat} +$$ +-{0.999999999999999} +$$ +\returnType{Type: DoubleFloat} + +\spadcommand{cos(3.1415926 :: DoubleFloat)} +$$ +-{0.999999999999999} +$$ +\returnType{Type: DoubleFloat} + +By far, the most common usage of {\tt DoubleFloat} is for functions to +be graphed. For more information about Axiom's numerical and +graphical facilities, see Section +\ref{ugGraph} on page~\pageref{ugGraph}, +\ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}, and +\ref{FloatXmpPage} on page~\pageref{FloatXmpPage}. + +\section{EqTable} +\label{EqTableXmpPage} + +The {\tt EqTable} domain provides tables where the keys are compared +using \spadfunFrom{eq?}{EqTable}. Keys are considered equal only if +they are the same instance of a structure. This is useful if the keys +are themselves updatable structures. Otherwise, all operations are +the same as for type {\tt Table}. See +\ref{TableXmpPage} on page~\pageref{TableXmpPage} for general +information about tables. + +The operation \spadfunFrom{table}{EqTable} is here used to create a table +where the keys are lists of integers. + +\spadcommand{e: EqTable(List Integer, Integer) := table() } +$$ +table() +$$ +\returnType{Type: EqTable(List Integer,Integer)} + +These two lists are equal according to \spadopFrom{=}{List}, but not +according to \spadfunFrom{eq?}{List}. + +\spadcommand{l1 := [1,2,3] } +$$ +\left[ +1, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{l2 := [1,2,3] } +$$ +\left[ +1, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Because the two lists are not \spadfunFrom{eq?}{List}, separate values +can be stored under each. + +\spadcommand{e.l1 := 111 } +$$ +111 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{e.l2 := 222 } +$$ +222 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{e.l1} +$$ +111 +$$ +\returnType{Type: PositiveInteger} + +\section{Equation} +\label{EquationXmpPage} + +The {\tt Equation} domain provides equations as mathematical objects. +These are used, for example, as the input to various +\spadfunFrom{solve}{TransSolvePackage} operations. + +Equations are created using the equals symbol, \spadopFrom{=}{Equation}. + +\spadcommand{eq1 := 3*x + 4*y = 5 } +$$ +{{4 \ y}+{3 \ x}}=5 +$$ +\returnType{Type: Equation Polynomial Integer} + +\spadcommand{eq2 := 2*x + 2*y = 3 } +$$ +{{2 \ y}+{2 \ x}}=3 +$$ +\returnType{Type: Equation Polynomial Integer} + +The left- and right-hand sides of an equation are accessible using +the operations \spadfunFrom{lhs}{Equation} and \spadfunFrom{rhs}{Equation}. + +\spadcommand{lhs eq1 } +$$ +{4 \ y}+{3 \ x} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{rhs eq1 } +$$ +5 +$$ +\returnType{Type: Polynomial Integer} + +Arithmetic operations are supported and operate on both sides of the +equation. + +\spadcommand{eq1 + eq2 } +$$ +{{6 \ y}+{5 \ x}}=8 +$$ +\returnType{Type: Equation Polynomial Integer} + +\spadcommand{eq1 * eq2 } +$$ +{{8 \ {y \sp 2}}+{{14} \ x \ y}+{6 \ {x \sp 2}}}={15} +$$ +\returnType{Type: Equation Polynomial Integer} + +\spadcommand{2*eq2 - eq1 } +$$ +x=1 +$$ +\returnType{Type: Equation Polynomial Integer} + +Equations may be created for any type so the arithmetic operations +will be defined only when they make sense. For example, exponentiation +is not defined for equations involving non-square matrices. + +\spadcommand{eq1**2 } +$$ +{{{16} \ {y \sp 2}}+{{24} \ x \ y}+{9 \ {x \sp 2}}}={25} +$$ +\returnType{Type: Equation Polynomial Integer} + +Note that an equals symbol is also used to {\it test} for equality of +values in certain contexts. For example, {\tt x+1} and {\tt y} are +unequal as polynomials. + +\spadcommand{if x+1 = y then "equal" else "unequal"} +$$ +\mbox{\tt "unequal"} +$$ +\returnType{Type: String} + +\spadcommand{eqpol := x+1 = y } +$$ +{x+1}=y +$$ +\returnType{Type: Equation Polynomial Integer} + +If an equation is used where a {\tt Boolean} value is required, then +it is evaluated using the equality test from the operand type. + +\spadcommand{if eqpol then "equal" else "unequal" } +$$ +\mbox{\tt "unequal"} +$$ +\returnType{Type: String} + +If one wants a {\tt Boolean} value rather than an equation, all one +has to do is ask! + +\spadcommand{eqpol::Boolean } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\section{Exit} +\label{ExitXmpPage} + +A function that does not return directly to its caller has {\tt Exit} +as its return type. The operation {\tt error} is an example of one +which does not return to its caller. Instead, it causes a return to +top-level. + +\spadcommand{n := 0 } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +The function {\tt gasp} is given return type {\tt Exit} since it is +guaranteed never to return a value to its caller. + +\begin{verbatim} +gasp(): Exit == + free n + n := n + 1 + error "Oh no!" + +Function declaration gasp : () -> Exit has been added to workspace. + +\end{verbatim} +\returnType{Type: Void} + +The return type of {\tt half} is determined by resolving the types of +the two branches of the {\tt if}. + +\begin{verbatim} +half(k) == + if odd? k then gasp() + else k quo 2 +\end{verbatim} + +Because {\tt gasp} has the return type {\tt Exit}, the type of +{\tt if} in {\tt half} is resolved to be {\tt Integer}. + +\spadcommand{half 4 } +\begin{verbatim} + Compiling function gasp with type () -> Exit + Compiling function half with type PositiveInteger -> Integer +\end{verbatim} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{half 3 } +\begin{verbatim} + Error signalled from user code in function gasp: + Oh no! +\end{verbatim} + +\spadcommand{n } +$$ +1 +$$ +\returnType{Type: NonNegativeInteger} + +For functions which return no value at all, use {\tt Void}. See +\ref{ugUserPage} on page~\pageref{ugUserPage} in Section +\ref{ugUserNumber} on page~\pageref{ugUserNumber} and +\ref{VoidXmpPage} on page~\pageref{VoidXmpPage} for +more information. + +\section{Expression} +\label{ExpressionXmpPage} + +{\tt Expression} is a constructor that creates domains whose objects +can have very general symbolic forms. Here are some examples: + +This is an object of type {\tt Expression Integer}. + +\spadcommand{sin(x) + 3*cos(x)**2} +$$ +{\sin \left({x} \right)}+{3\ {{\cos \left({x} \right)}\sp 2}} +$$ +\returnType{Type: Expression Integer} + +This is an object of type {\tt Expression Float}. + +\spadcommand{tan(x) - 3.45*x} +$$ +{\tan \left({x} \right)}-{{3.45} \ x} +$$ +\returnType{Type: Expression Float} + +This object contains symbolic function applications, sums, +products, square roots, and a quotient. + +\spadcommand{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))} +$$ +{-{{\tan \left({{\sqrt {7}}} \right)}\sp 2}+ +{2 \ {\sin \left({{\sqrt {{11}}}} \right)} +\ {\tan \left({{\sqrt {7}}} \right)}} +-{{\sin \left({{\sqrt {{11}}}} \right)}\sp 2}} +\over {{\cos \left({{y -x}} \right)} +-4} +$$ +\returnType{Type: Expression Integer} + +As you can see, {\tt Expression} actually takes an argument domain. +The {\it coefficients} of the terms within the expression belong to +the argument domain. {\tt Integer} and {\tt Float}, along with +{\tt Complex Integer} and {\tt Complex Float} are the most common +coefficient domains. + +The choice of whether to use a {\tt Complex} coefficient domain or not +is important since Axiom can perform some simplifications on +real-valued objects + +\spadcommand{log(exp x)@Expression(Integer)} +$$ +x +$$ +\returnType{Type: Expression Integer} + +... which are not valid on complex ones. + +\spadcommand{log(exp x)@Expression(Complex Integer)} +$$ +\log \left({{e \sp x}} \right) +$$ +\returnType{Type: Expression Complex Integer} + +Many potential coefficient domains, such as {\tt AlgebraicNumber}, are +not usually used because {\tt Expression} can subsume them. + +\spadcommand{sqrt 3 + sqrt(2 + sqrt(-5)) } +$$ +{\sqrt {{{\sqrt {-5}}+2}}}+{\sqrt {3}} +$$ +\returnType{Type: AlgebraicNumber} + +\spadcommand{\% :: Expression Integer } +$$ +{\sqrt {{{\sqrt {-5}}+2}}}+{\sqrt {3}} +$$ +\returnType{Type: Expression Integer} + +Note that we sometimes talk about ``an object of type {\tt +Expression}.'' This is not really correct because we should say, for +example, ``an object of type {\tt Expression Integer}'' or ``an object +of type {\tt Expression Float}.'' By a similar abuse of language, +when we refer to an ``expression'' in this section we will mean an +object of type {\tt Expression R} for some domain {\tt R}. + +The Axiom documentation contains many examples of the use of {\tt +Expression}. For the rest of this section, we'll give you some +pointers to those examples plus give you some idea of how to +manipulate expressions. + +It is important for you to know that {\tt Expression} creates domains +that have category {\tt Field}. Thus you can invert any non-zero +expression and you shouldn't expect an operation like {\tt factor} to +give you much information. You can imagine expressions as being +represented as quotients of ``multivariate'' polynomials where the +``variables'' are kernels (see +\ref{KernelXmpPage} on page~\pageref{KernelXmpPage}). A kernel can +either be a symbol such as {\tt x} or a symbolic function application +like {\tt sin(x + 4)}. The second example is actually a nested kernel +since the argument to {\tt sin} contains the kernel {\tt x}. + +\spadcommand{height mainKernel sin(x + 4)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Actually, the argument to {\tt sin} is an expression, and so the +structure of {\tt Expression} is recursive. +\ref{KernelXmpPage} on page~\pageref{KernelXmpPage} +demonstrates how to extract the kernels in an expression. + +Use the HyperDoc Browse facility to see what operations are applicable +to expression. At the time of this writing, there were 262 operations +with 147 distinct name in {\tt Expression Integer}. For example, +\spadfunFrom{numer}{Expression} and \spadfunFrom{denom}{Expression} +extract the numerator and denominator of an expression. + +\spadcommand{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) } +$$ +{-{{\sin \left({x} \right)}\sp 2}+ +{8 \ {\sin \left({x} \right)}} +-{16}} +\over {{2 \ y \ {\sqrt {-y}}} -1} +$$ +\returnType{Type: Expression Integer} + +\spadcommand{numer e } +$$ +-{{\sin \left({x} \right)}\sp 2}+ +{8 \ {\sin \left({x} \right)}} +-{16} +$$ +\returnType{Type: +SparseMultivariatePolynomial(Integer,Kernel Expression Integer)} + +\spadcommand{denom e } +$$ +{2 \ y \ {\sqrt {-y}}} -1 +$$ +\returnType{Type: +SparseMultivariatePolynomial(Integer,Kernel Expression Integer)} + +Use \spadfunFrom{D}{Expression} to compute partial derivatives. + +\spadcommand{D(e, x) } +$$ +{{{\left( +{4 \ y \ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}- +{{16} \ y \ {\cos \left({x} \right)}}\right)}\ {\sqrt {-y}}} - +{2 \ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ +{8\ {\cos \left({x} \right)}}} +\over {{4 \ y \ {\sqrt {-y}}}+{4 \ {y \sp 3}} -1} +$$ +\returnType{Type: Expression Integer} + +See +\ref{ugIntroCalcDerivPage} on page~\pageref{ugIntroCalcDerivPage} in Section +\ref{ugIntroCalcDerivNumber} on page~\pageref{ugIntroCalcDerivNumber} +for more examples of expressions and derivatives. + +\spadcommand{D(e, [x, y], [1, 2]) } +$$ +\left( +\begin{array}{@{}l} +{{\left( {{\left( -{{2304} \ {y \sp 7}}+{{960} \ {y \sp 4}} \right)} +\ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ +{{\left({{9216} \ {y \sp 7}} -{{3840} \ {y \sp 4}} \right)} +\ {\cos \left({x} \right)}}\right)}\ {\sqrt {-y}}}+ +\\ +\\ +\displaystyle +{{\left( -{{960} \ {y \sp 9}}+{{2160} \ {y \sp 6}} -{{180} \ {y \sp 3}} -3 +\right)}\ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ +\\ +\\ +\displaystyle +{{\left( +{{3840} \ {y \sp 9}} -{{8640} \ {y \sp 6}}+{{720} \ {y \sp 3}}+{12} +\right)}\ {\cos \left({x} \right)}} +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +{{\left( {{256} \ {y \sp {12}}} -{{1792} \ {y \sp 9}}+{{1120} \ {y +\sp 6}} -{{112} \ {y \sp 3}}+1 \right)}\ {\sqrt {-y}}} - +\\ +\\ +\displaystyle +{{1024} \ {y \sp {11}}}+{{1792} \ {y \sp 8}} -{{448} \ +{y \sp 5}}+{{16} \ {y \sp 2}} +\end{array} +\right) +$$ +\returnType{Type: Expression Integer} + +See +\ref{ugIntroCalcLimitsPage} on page~\pageref{ugIntroCalcLimitsPage} in Section +\ref{ugIntroCalcLimitsNumber} on page~\pageref{ugIntroCalcLimitsNumber} and +\ref{ugIntroSeriesPage} on page~\pageref{ugIntroSeriesPage} in Section +\ref{ugIntroSeriesNumber} on page~\pageref{ugIntroSeriesNumber} +for more examples of expressions and +calculus. Differential equations involving expressions are discussed +in \ref{ugProblemDEQPage} on page~\pageref{ugProblemDEQPage} in Section +\ref{ugProblemDEQNumber} on page~\pageref{ugProblemDEQNumber}. +Chapter 8 has many advanced examples: see +\ref{ugProblemIntegrationPage} on page~\pageref{ugProblemIntegrationPage} +in Section +\ref{ugProblemIntegrationNumber} on page~\pageref{ugProblemIntegrationNumber} +for a discussion of Axiom's integration facilities. + +When an expression involves no ``symbol kernels'' (for example, +{\tt x}), it may be possible to numerically evaluate the expression. + +If you suspect the evaluation will create a complex number, use +{\tt complexNumeric}. + +\spadcommand{complexNumeric(cos(2 - 3*\%i))} +$$ +-{4.1896256909\ 688072301}+{{9.1092278937\ 55336598} \ i} +$$ +\returnType{Type: Complex Float} + +If you know it will be real, use {\tt numeric}. + +\spadcommand{numeric(tan 3.8)} +$$ +0.7735560905\ 0312607286 +$$ +\returnType{Type: Float} + +The {\tt numeric} operation will display an error message if the +evaluation yields a calue with an non-zero imaginary part. Both of +these operations have an optional second argument {\tt n} which +specifies that the accuracy of the approximation be up to {\tt n} +decimal places. + +When an expression involves no ``symbolic application'' kernels, it +may be possible to convert it a polynomial or rational function in the +variables that are present. + +\spadcommand{e2 := cos(x**2 - y + 3) } +$$ +\cos \left({{y -{x \sp 2} -3}} \right) +$$ +\returnType{Type: Expression Integer} + +\spadcommand{e3 := asin(e2) - \%pi/2 } +$$ +-y+{x \sp 2}+3 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{e3 :: Polynomial Integer } +$$ +-y+{x \sp 2}+3 +$$ +\returnType{Type: Polynomial Integer} + +This also works for the polynomial types where specific variables +and their ordering are given. + +\spadcommand{e3 :: DMP([x, y], Integer) } +$$ +{x \sp 2} -y+3 +$$ +\returnType{Type: DistributedMultivariatePolynomial([x,y],Integer)} + +Finally, a certain amount of simplication takes place as expressions +are constructed. + +\spadcommand{sin \%pi} +$$ +0 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{cos(\%pi / 4)} +$$ +{\sqrt {2}} \over 2 +$$ +\returnType{Type: Expression Integer} + +For simplications that involve multiple terms of the expression, use +{\tt simplify}. + +\spadcommand{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 } +$$ +{{\tan \left({x} \right)}\sp 6}+ +{3 \ {{\tan \left({x} \right)}\sp 4}}+ +{3 \ {{\tan \left({x} \right)}\sp 2}}+1 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{simplify \% } +$$ +1 \over {{\cos \left({x} \right)}\sp 6} +$$ +\returnType{Type: Expression Integer} + +See \ref{ugUserRulesPage} on page~\pageref{ugUserRulesPage} in Section +\ref{ugUserRulesNumber} on page~\pageref{ugUserRulesNumber} for +examples of how to write your own rewrite rules for expressions. + +\section{Factored} +\label{FactoredXmpPage} + +{\tt Factored} creates a domain whose objects are kept in factored +form as long as possible. Thus certain operations like +\spadopFrom{*}{Factored} (multiplication) and +\spadfunFrom{gcd}{Factored} are relatively easy to do. Others, such +as addition, require somewhat more work, and the result may not be +completely factored unless the argument domain {\tt R} provides a +\spadfunFrom{factor}{Factored} operation. Each object consists of a +unit and a list of factors, where each factor consists of a member of +{\tt R} (the {\em base}), an exponent, and a flag indicating what is +known about the base. A flag may be one of ``{\tt nil}'', ``{\tt sqfr}'', +``{\tt irred}'' or ``{\tt prime}'', which mean that nothing is known about +the base, it is square-free, it is irreducible, or it is prime, +respectively. The current restriction to factored objects of integral +domains allows simplification to be performed without worrying about +multiplication order. + +\subsection{Decomposing Factored Objects} + +In this section we will work with a factored integer. + +\spadcommand{g := factor(4312) } +$$ +{2 \sp 3} \ {7 \sp 2} \ {11} +$$ +\returnType{Type: Factored Integer} + +Let's begin by decomposing {\tt g} into pieces. The only possible +units for integers are {\tt 1} and {\tt -1}. + +\spadcommand{unit(g) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +There are three factors. + +\spadcommand{numberOfFactors(g) } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +We can make a list of the bases, \ldots + +\spadcommand{[nthFactor(g,i) for i in 1..numberOfFactors(g)] } +$$ +\left[ +2, 7, {11} +\right] +$$ +\returnType{Type: List Integer} + +and the exponents, \ldots + +\spadcommand{[nthExponent(g,i) for i in 1..numberOfFactors(g)] } +$$ +\left[ +3, 2, 1 +\right] +$$ +\returnType{Type: List Integer} + +and the flags. You can see that all the bases (factors) are prime. + +\spadcommand{[nthFlag(g,i) for i in 1..numberOfFactors(g)] } +$$ +\left[ +\mbox{\tt "prime"} , \mbox{\tt "prime"} , \mbox{\tt "prime"} +\right] +$$ +\returnType{Type: List Union("nil","sqfr","irred","prime")} + +A useful operation for pulling apart a factored object into a list +of records of the components is \spadfunFrom{factorList}{Factored}. + +\spadcommand{factorList(g) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {flg= \mbox{\tt "prime"} }, {fctr=2}, {xpnt=3} \right]}, +\right. +\\ +\\ +\displaystyle +{\left[ {flg= \mbox{\tt "prime"} }, {fctr=7}, {xpnt=2} \right]}, +\\ +\\ +\left. +\displaystyle +{\left[ {flg= \mbox{\tt "prime"} }, {fctr={11}}, {xpnt=1} \right]} +\right] +\end{array} +$$ +\returnType{Type: +List Record(flg: Union("nil","sqfr","irred","prime"), +fctr: Integer,xpnt: Integer)} + +If you don't care about the flags, use \spadfunFrom{factors}{Factored}. + +\spadcommand{factors(g) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {factor=2}, {exponent=3} \right]}, +\right. +\\ +\\ +\displaystyle +{\left[ {factor=7}, {exponent=2} \right]}, +\\ +\\ +\displaystyle +\left. +{\left[ {factor={11}}, {exponent=1} \right]} +\right] +\end{array} +$$ +\returnType{Type: List Record(factor: Integer,exponent: Integer)} + +Neither of these operations returns the unit. + +\spadcommand{first(\%).factor } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Expanding Factored Objects} + +Recall that we are working with this factored integer. + +\spadcommand{g := factor(4312) } +$$ +{2 \sp 3} \ {7 \sp 2} \ {11} +$$ +\returnType{Type: Factored Integer} + +To multiply out the factors with their multiplicities, use +\spadfunFrom{expand}{Factored}. + +\spadcommand{expand(g) } +$$ +4312 +$$ +\returnType{Type: PositiveInteger} + +If you would like, say, the distinct factors multiplied together but +with multiplicity one, you could do it this way. + +\spadcommand{reduce(*,[t.factor for t in factors(g)]) } +$$ +154 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Arithmetic with Factored Objects} + +We're still working with this factored integer. + +\spadcommand{g := factor(4312) } +$$ +{2 \sp 3} \ {7 \sp 2} \ {11} +$$ +\returnType{Type: Factored Integer} + +We'll also define this factored integer. + +\spadcommand{f := factor(246960) } +$$ +{2 \sp 4} \ {3 \sp 2} \ 5 \ {7 \sp 3} +$$ +\returnType{Type: Factored Integer} + +Operations involving multiplication and division are particularly +easy with factored objects. + +\spadcommand{f * g } +$$ +{2 \sp 7} \ {3 \sp 2} \ 5 \ {7 \sp 5} \ {11} +$$ +\returnType{Type: Factored Integer} + +\spadcommand{f**500 } +$$ +{2 \sp {2000}} \ {3 \sp {1000}} \ {5 \sp {500}} \ {7 \sp {1500}} +$$ +\returnType{Type: Factored Integer} + +\spadcommand{gcd(f,g) } +$$ +{2 \sp 3} \ {7 \sp 2} +$$ +\returnType{Type: Factored Integer} + +\spadcommand{lcm(f,g) } +$$ +{2 \sp 4} \ {3 \sp 2} \ 5 \ {7 \sp 3} \ {11} +$$ +\returnType{Type: Factored Integer} + +If we use addition and subtraction things can slow down because +we may need to compute greatest common divisors. + +\spadcommand{f + g } +$$ +{2 \sp 3} \ {7 \sp 2} \ {641} +$$ +\returnType{Type: Factored Integer} + +\spadcommand{f - g } +$$ +{2 \sp 3} \ {7 \sp 2} \ {619} +$$ +\returnType{Type: Factored Integer} + +Test for equality with {\tt 0} and {\tt 1} by using +\spadfunFrom{zero?}{Factored} and \spadfunFrom{one?}{Factored}, +respectively. + +\spadcommand{zero?(factor(0))} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{zero?(g) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{one?(factor(1))} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{one?(f) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Another way to get the zero and one factored objects is to use +package calling (see +\ref{ugTypesPkgCallPage} on page~\pageref{ugTypesPkgCallPage} in Section +\ref{ugTypesPkgCallNumber} on page~\pageref{ugTypesPkgCallNumber}). + +\spadcommand{0\$Factored(Integer)} +$$ +0 +$$ +\returnType{Type: Factored Integer} + +\spadcommand{1\$Factored(Integer)} +$$ +1 +$$ +\returnType{Type: Factored Integer} + +\subsection{Creating New Factored Objects} + +The \spadfunFrom{map}{Factored} operation is used to iterate across +the unit and bases of a factored object. See +\ref{FactoredFunctionsTwoXmpPage} on +page~\pageref{FactoredFunctionsTwoXmpPage} for a discussion of +\spadfunFrom{map}{Factored}. + +The following four operations take a base and an exponent and create a +factored object. They differ in handling the flag component. + +\spadcommand{nilFactor(24,2) } +$$ +{24} \sp 2 +$$ +\returnType{Type: Factored Integer} + +This factor has no associated information. + +\spadcommand{nthFlag(\%,1) } +$$ +\mbox{\tt "nil"} +$$ +\returnType{Type: Union("nil",...)} + +This factor is asserted to be square-free. + +\spadcommand{sqfrFactor(30,2) } +$$ +{30} \sp 2 +$$ +\returnType{Type: Factored Integer} + +This factor is asserted to be irreducible. + +\spadcommand{irreducibleFactor(13,10) } +$$ +{13} \sp {10} +$$ +\returnType{Type: Factored Integer} + +This factor is asserted to be prime. + +\spadcommand{primeFactor(11,5) } +$$ +{11} \sp 5 +$$ +\returnType{Type: Factored Integer} + +A partial inverse to \spadfunFrom{factorList}{Factored} is +\spadfunFrom{makeFR}{Factored}. + +\spadcommand{h := factor(-720) } +$$ +-{{2 \sp 4} \ {3 \sp 2} \ 5} +$$ +\returnType{Type: Factored Integer} + +The first argument is the unit and the second is a list of records as +returned by \spadfunFrom{factorList}{Factored}. + +\spadcommand{h - makeFR(unit(h),factorList(h)) } +$$ +0 +$$ +\returnType{Type: Factored Integer} + +\subsection{Factored Objects with Variables} + +Some of the operations available for polynomials are also available +for factored polynomials. + +\spadcommand{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 } +$$ +{{\left( {4 \ {x \sp 2}} -{{12} \ x}+9 +\right)} +\ {y \sp 2}}+{{\left( {4 \ {x \sp 2}} -{{12} \ x}+9 +\right)} +\ y}+{{28} \ {x \sp 2}} -{{84} \ x}+{63} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{fp := factor(p) } +$$ +{{\left( {2 \ x} -3 +\right)} +\sp 2} \ {\left( {y \sp 2}+y+7 +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +You can differentiate with respect to a variable. + +\spadcommand{D(p,x) } +$$ +{{\left( {8 \ x} -{12} +\right)} +\ {y \sp 2}}+{{\left( {8 \ x} -{12} +\right)} +\ y}+{{56} \ x} -{84} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{D(fp,x) } +$$ +4 \ {\left( {2 \ x} -3 +\right)} +\ {\left( {y \sp 2}+y+7 +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +\spadcommand{numberOfFactors(\%) } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\section{FactoredFunctions2} +\label{FactoredFunctions2XmpPage} + +The {\tt FactoredFunctions2} package implements one operation, +\spadfunFrom{map}{FactoredFunctions2}, for applying an operation to every +base in a factored object and to the unit. + +\spadcommand{double(x) == x + x } +\returnType{Type: Void} + +\spadcommand{f := factor(720) } +$$ +{2 \sp 4} \ {3 \sp 2} \ 5 +$$ +\returnType{Type: Factored Integer} + +Actually, the \spadfunFrom{map}{FactoredFunctions2} operation used +in this example comes from {\tt Factored} itself, since {\tt double} +takes an integer argument and returns an integer result. + +\spadcommand{map(double,f) } +$$ +2 \ {4 \sp 4} \ {6 \sp 2} \ {10} +$$ +\returnType{Type: Factored Integer} + +If we want to use an operation that returns an object that has a type +different from the operation's argument, +the \spadfunFrom{map}{FactoredFunctions2} in {\tt Factored} +cannot be used and we use the one in {\tt FactoredFunctions2}. + +\spadcommand{makePoly(b) == x + b } +\returnType{Type: Void} + +In fact, the ``2'' in the name of the package means that we might +be using factored objects of two different types. + +\spadcommand{g := map(makePoly,f) } +$$ +{\left( x+1 \right)} +\ {{\left( x+2 \right)}\sp 4} +\ {{\left( x+3 \right)}\sp 2} +\ {\left( x+5 \right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +It is important to note that both versions of +\spadfunFrom{map}{FactoredFunctions2} destroy any information known +about the bases (the fact that they are prime, for instance). + +The flags for each base are set to ``nil'' in the object returned +by \spadfunFrom{map}{FactoredFunctions2}. + +\spadcommand{nthFlag(g,1) } +$$ +\mbox{\tt "nil"} +$$ +\returnType{Type: Union("nil",...)} + +For more information about factored objects and their use, see +\ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage} and +\ref{ugProblemGaloisPage} on page~\pageref{ugProblemGaloisPage} in Section +\ref{ugProblemGaloisNumber} on page~\pageref{ugProblemGaloisNumber}. + +\section{File} +\label{FileXmpPage} + +The {\tt File(S)} domain provides a basic interface to read and +write values of type {\tt S} in files. + +Before working with a file, it must be made accessible to Axiom with +the \spadfunFrom{open}{File} operation. + +\spadcommand{ifile:File List Integer:=open("/tmp/jazz1","output") } +$$ +\mbox{\tt "/tmp/jazz1"} +$$ +\returnType{Type: File List Integer} + +The \spadfunFrom{open}{File} function arguments are a {\tt FileName} +and a {\tt String} specifying the mode. If a full pathname is not +specified, the current default directory is assumed. The mode must be +one of ``{\tt input}'' or ``{\tt output}''. If it is not specified, +``{\tt input}'' is assumed. Once the file has been opened, you can read or +write data. + +The operations \spadfunFrom{read}{File} and \spadfunFrom{write}{File} are +provided. + +\spadcommand{write!(ifile, [-1,2,3])} +$$ +\left[ +-1, 2, 3 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{write!(ifile, [10,-10,0,111])} +$$ +\left[ +{10}, -{10}, 0, {111} +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{write!(ifile, [7])} +$$ +\left[ +7 +\right] +$$ +\returnType{Type: List Integer} + +You can change from writing to reading (or vice versa) by reopening a file. + +\spadcommand{reopen!(ifile, "input")} +$$ +\mbox{\tt "/tmp/jazz1"} +$$ +\returnType{Type: File List Integer} + +\spadcommand{read! ifile} +$$ +\left[ +-1, 2, 3 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{read! ifile} +$$ +\left[ +{10}, -{10}, 0, {111} +\right] +$$ +\returnType{Type: List Integer} + +The \spadfunFrom{read}{File} operation can cause an error if one tries +to read more data than is in the file. To guard against this +possibility the \spadfunFrom{readIfCan}{File} operation should be +used. + +\spadcommand{readIfCan! ifile } +$$ +\left[ +7 +\right] +$$ +\returnType{Type: Union(List Integer,...)} + +\spadcommand{readIfCan! ifile } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +You can find the current mode of the file, and the file's name. + +\spadcommand{iomode ifile} +$$ +\mbox{\tt "input"} +$$ +\returnType{Type: String} + +\spadcommand{name ifile} +$$ +\mbox{\tt "/tmp/jazz1"} +$$ +\returnType{Type: FileName} + +When you are finished with a file, you should close it. + +\spadcommand{close! ifile} +$$ +\mbox{\tt "/tmp/jazz1"} +$$ +\returnType{Type: File List Integer} + +\spadcommand{)system rm /tmp/jazz1} + +A limitation of the underlying LISP system is that not all values can +be represented in a file. In particular, delayed values containing +compiled functions cannot be saved. + +For more information on related topics, see +\ref{TextFileXmpPage} on page~\pageref{TextFileXmpPage}, +\ref{KeyedAccessFileXmpPage} on page~\pageref{KeyedAccessFileXmpPage}, +\ref{LibraryXmpPage} on page~\pageref{LibraryXmpPage}, and +\ref{FileNameXmpPage} on page~\pageref{FileNameXmpPage}. + +\section{FileName} +\label{FileNameXmpPage} + +The {\tt FileName} domain provides an interface to the computer's file +system. Functions are provided to manipulate file names and to test +properties of files. + +The simplest way to use file names in the Axiom interpreter is to rely +on conversion to and from strings. The syntax of these strings +depends on the operating system. + +\spadcommand{fn: FileName } +\returnType{Type: Void} + +On Linux, this is a proper file syntax: + +\spadcommand{fn := "/tmp/fname.input" } +$$ +\mbox{\tt "/tmp/fname.input"} +$$ +\returnType{Type: FileName} + +Although it is very convenient to be able to use string notation +for file names in the interpreter, it is desirable to have a portable +way of creating and manipulating file names from within programs. + +A measure of portability is obtained by considering a file name +to consist of three parts: the {\it directory}, the {\it name}, +and the {\it extension}. + +\spadcommand{directory fn } +$$ +\mbox{\tt "/tmp"} +$$ +\returnType{Type: String} + +\spadcommand{name fn } +$$ +\mbox{\tt "fname"} +$$ +\returnType{Type: String} + +\spadcommand{extension fn } +$$ +\mbox{\tt "input"} +$$ +\returnType{Type: String} + +The meaning of these three parts depends on the operating system. +For example, on CMS the file ``{\tt SPADPROF INPUT M}'' +would have directory ``{\tt M}'', name ``{\tt SPADPROF}'' and +extension ``{\tt INPUT}''. + +It is possible to create a filename from its parts. + +\spadcommand{fn := filename("/u/smwatt/work", "fname", "input") } +$$ +\mbox{\tt "/u/smwatt/work/fname.input"} +$$ +\returnType{Type: FileName} + +When writing programs, it is helpful to refer to directories via +variables. + +\spadcommand{objdir := "/tmp" } +$$ +\mbox{\tt "/tmp"} +$$ +\returnType{Type: String} + +\spadcommand{fn := filename(objdir, "table", "spad") } +$$ +\mbox{\tt "/tmp/table.spad"} +$$ +\returnType{Type: FileName} + +If the directory or the extension is given as an empty string, then +a default is used. On AIX, the defaults are the current directory +and no extension. + +\spadcommand{fn := filename("", "letter", "") } +$$ +\mbox{\tt "letter"} +$$ +\returnType{Type: FileName} + +Three tests provide information about names in the file system. + +The \spadfunFrom{exists?}{FileName} operation tests whether the named +file exists. + +\spadcommand{exists? "/etc/passwd"} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The operation \spadfunFrom{readable?}{FileName} tells whether the named file +can be read. If the file does not exist, then it cannot be read. + +\spadcommand{readable? "/etc/passwd"} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{readable? "/etc/security/passwd"} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{readable? "/ect/passwd"} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Likewise, the operation \spadfunFrom{writable?}{FileName} tells +whether the named file can be written. If the file does not exist, +the test is determined by the properties of the directory. + +\spadcommand{writable? "/etc/passwd"} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{writable? "/dev/null"} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{writable? "/etc/DoesNotExist"} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{writable? "/tmp/DoesNotExist"} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The \spadfunFrom{new}{FileName} operation constructs the name of a new +writable file. The argument sequence is the same as for +\spadfunFrom{filename}{FileName}, except that the name part is +actually a prefix for a constructed unique name. + +The resulting file is in the specified directory +with the given extension, and the same defaults are used. + +\spadcommand{fn := new(objdir, "xxx", "yy") } +$$ +\mbox{\tt "/tmp/xxx82404.yy"} +$$ +\returnType{Type: FileName} + +\section{FlexibleArray} +\label{FlexibleArrayXmpPage} + +The {\tt FlexibleArray} domain constructor creates one-dimensional +arrays of elements of the same type. Flexible arrays are an attempt +to provide a data type that has the best features of both +one-dimensional arrays (fast, random access to elements) and lists +(flexibility). They are implemented by a fixed block of storage. +When necessary for expansion, a new, larger block of storage is +allocated and the elements from the old storage area are copied into +the new block. + +Flexible arrays have available most of the operations provided by +{\tt OneDimensionalArray} (see +\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage} +and \ref{VectorXmpPage} on page~\pageref{VectorXmpPage}). +Since flexible arrays are also of category +{\tt ExtensibleLinearAggregate}, they have operations {\tt concat!}, +{\tt delete!}, {\tt insert!}, {\tt merge!}, {\tt remove!}, +{\tt removeDuplicates!}, and {\tt select!}. In addition, the operations +{\tt physicalLength} and {\tt physicalLength!} provide user-control +over expansion and contraction. + +A convenient way to create a flexible array is to apply the operation +{\tt flexibleArray} to a list of values. + +\spadcommand{flexibleArray [i for i in 1..6]} +$$ +\left[ +1, 2, 3, 4, 5, 6 +\right] +$$ +\returnType{Type: FlexibleArray PositiveInteger} + +Create a flexible array of six zeroes. + +\spadcommand{f : FARRAY INT := new(6,0)} +$$ +\left[ +0, 0, 0, 0, 0, 0 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +For $i=1\ldots 6$ set the $i$-th element to $i$. Display {\tt f}. + +\spadcommand{for i in 1..6 repeat f.i := i; f} +$$ +\left[ +1, 2, 3, 4, 5, 6 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Initially, the physical length is the same as the number of elements. + +\spadcommand{physicalLength f} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +Add an element to the end of {\tt f}. + +\spadcommand{concat!(f,11)} +$$ +\left[ +1, 2, 3, 4, 5, 6, {11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +See that its physical length has grown. + +\spadcommand{physicalLength f} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +Make {\tt f} grow to have room for {\tt 15} elements. + +\spadcommand{physicalLength!(f,15)} +$$ +\left[ +1, 2, 3, 4, 5, 6, {11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Concatenate the elements of {\tt f} to itself. The physical length +allows room for three more values at the end. + +\spadcommand{concat!(f,f)} +$$ +\left[ +1, 2, 3, 4, 5, 6, {11}, 1, 2, 3, 4, 5, 6, +{11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Use {\tt insert!} to add an element to the front of a flexible array. + +\spadcommand{insert!(22,f,1)} +$$ +\left[ +{22}, 1, 2, 3, 4, 5, 6, {11}, 1, 2, 3, 4, +5, 6, {11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Create a second flexible array from {\tt f} consisting of the elements +from index 10 forward. + +\spadcommand{g := f(10..)} +$$ +\left[ +2, 3, 4, 5, 6, {11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Insert this array at the front of {\tt f}. + +\spadcommand{insert!(g,f,1)} +$$ +\left[ +2, 3, 4, 5, 6, {11}, {22}, 1, 2, 3, 4, 5, +6, {11}, 1, 2, 3, 4, 5, 6, {11} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Merge the flexible array {\tt f} into {\tt g} after sorting each in place. + +\spadcommand{merge!(sort! f, sort! g)} +$$ +\left[ +1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, +4, 5, 5, 5, 5, 6, 6, 6, 6, {11}, {11}, {11}, +{11}, {22} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Remove duplicates in place. + +\spadcommand{removeDuplicates! f} +$$ +\left[ +1, 2, 3, 4, 5, 6, {11}, {22} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Remove all odd integers. + +\spadcommand{select!(i +-> even? i,f)} +$$ +\left[ +2, 4, 6, {22} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +All these operations have shrunk the physical length of {\tt f}. + +\spadcommand{physicalLength f} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +To force Axiom not to shrink flexible arrays call the {\tt shrinkable} +operation with the argument {\tt false}. You must package call this +operation. The previous value is returned. + +\spadcommand{shrinkable(false)\$FlexibleArray(Integer)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\section{Float} +\label{FloatXmpPage} + +Axiom provides two kinds of floating point numbers. The domain +{\tt Float} (abbreviation {\tt FLOAT}) implements a model of arbitrary +precision floating point numbers. The domain {\tt DoubleFloat} +(abbreviation {\tt DFLOAT}) is intended to make available hardware +floating point arithmetic in Axiom. The actual model of floating +point that {\tt DoubleFloat} provides is system-dependent. For +example, on the IBM system 370 Axiom uses IBM double precision which +has fourteen hexadecimal digits of precision or roughly sixteen +decimal digits. Arbitrary precision floats allow the user to specify +the precision at which arithmetic operations are computed. Although +this is an attractive facility, it comes at a cost. +Arbitrary-precision floating-point arithmetic typically takes twenty +to two hundred times more time than hardware floating point. + +For more information about Axiom's numeric and graphic facilities, see +\ref{ugGraphPage} on page~\pageref{ugGraphPage} in Section +\ref{ugGraphNumber} on page~\pageref{ugGraphNumber}, +\ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}, and +\ref{DoubleFloatXmpPage} on page~\pageref{DoubleFloatXmpPage}. + +\subsection{Introduction to Float} + +Scientific notation is supported for input and output of floating +point numbers. A floating point number is written as a string of +digits containing a decimal point optionally followed by the letter +``{\tt E}'', and then the exponent. + +We begin by doing some calculations using arbitrary precision floats. +The default precision is twenty decimal digits. + +\spadcommand{1.234} +$$ +1.234 +$$ +\returnType{Type: Float} + +A decimal base for the exponent is assumed, so the number +{\tt 1.234E2} denotes $1.234 \cdot 10^2$. + +\spadcommand{1.234E2} +$$ +123.4 +$$ +\returnType{Type: Float} + +The normal arithmetic operations are available for floating point numbers. + +\spadcommand{sqrt(1.2 + 2.3 / 3.4 ** 4.5)} +$$ +1.0996972790\ 671286226 +$$ +\returnType{Type: Float} + +\subsection{Conversion Functions} + +You can use conversion (\ref{ugTypesConvertPage} on +page~\pageref{ugTypesConvertPage} in Section +\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}) to +go back and forth between {\tt Integer}, {\tt Fraction Integer} and +{\tt Float}, as appropriate. + +\spadcommand{i := 3 :: Float } +$$ +3.0 +$$ +\returnType{Type: Float} + +\spadcommand{i :: Integer } +$$ +3 +$$ +\returnType{Type: Integer} + +\spadcommand{i :: Fraction Integer } +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +Since you are explicitly asking for a conversion, you must take +responsibility for any loss of exactness. + +\spadcommand{r := 3/7 :: Float } +$$ +0.4285714285\ 7142857143 +$$ +\returnType{Type: Float} + +\spadcommand{r :: Fraction Integer } +$$ +3 \over 7 +$$ +\returnType{Type: Fraction Integer} + +This conversion cannot be performed: use \spadfunFrom{truncate}{Float} +or \spadfunFrom{round}{Float} if that is what you intend. + +\spadcommand{r :: Integer } +\begin{verbatim} + Cannot convert from type Float to Integer for value + 0.4285714285 7142857143 +\end{verbatim} + +The operations \spadfunFrom{truncate}{Float} and \spadfunFrom{round}{Float} +truncate \ldots + +\spadcommand{truncate 3.6} +$$ +3.0 +$$ +\returnType{Type: Float} + +and round to the nearest integral {\tt Float} respectively. + +\spadcommand{round 3.6} +$$ +4.0 +$$ +\returnType{Type: Float} + +\spadcommand{truncate(-3.6)} +$$ +-{3.0} +$$ +\returnType{Type: Float} + +\spadcommand{round(-3.6)} +$$ +-{4.0} +$$ +\returnType{Type: Float} + +The operation \spadfunFrom{fractionPart}{Float} computes the +fractional part of {\tt x}, that is, {\tt x - truncate x}. + +\spadcommand{fractionPart 3.6} +$$ +0.6 +$$ +\returnType{Type: Float} + +The operation \spadfunFrom{digits}{Float} allows the user to set the +precision. It returns the previous value it was using. + +\spadcommand{digits 40 } +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{sqrt 0.2} +$$ +0.4472135954\ 9995793928\ 1834733746\ 2552470881 +$$ +\returnType{Type: Float} + +\spadcommand{pi()\$Float } +$$ +3.1415926535\ 8979323846\ 2643383279\ 502884197 +$$ +\returnType{Type: Float} + +The precision is only limited by the computer memory available. +Calculations at 500 or more digits of precision are not difficult. + +\spadcommand{digits 500 } +$$ +40 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{pi()\$Float } +$$ +\begin{array}{@{}l} +3.1415926535\ 8979323846\ 2643383279\ 5028841971\ 6939937510\ 5820974944 +\\ +\displaystyle +\ \ 5923078164\ 0628620899\ 8628034825\ 3421170679\ 8214808651\ 3282306647 +\\ +\displaystyle +\ \ 0938446095\ 5058223172\ 5359408128\ 4811174502\ 8410270193\ 8521105559 +\\ +\displaystyle +\ \ 6446229489\ 5493038196\ 4428810975\ 6659334461\ 2847564823\ 3786783165 +\\ +\displaystyle +\ \ 2712019091\ 4564856692\ 3460348610\ 4543266482\ 1339360726\ 0249141273 +\\ +\displaystyle +\ \ 7245870066\ 0631558817\ 4881520920\ 9628292540\ 9171536436\ 7892590360 +\\ +\displaystyle +\ \ 0113305305\ 4882046652\ 1384146951\ 9415116094\ 3305727036\ 5759591953 +\\ +\displaystyle +\ \ 0921861173\ 8193261179\ 3105118548\ 0744623799\ 6274956735\ 1885752724 +\\ +\displaystyle +\ \ 8912279381\ 830119491 +\end{array} +$$ +\returnType{Type: Float} + +Reset \spadfunFrom{digits}{Float} to its default value. + +\spadcommand{digits 20} +$$ +500 +$$ +\returnType{Type: PositiveInteger} + +Numbers of type {\tt Float} are represented as a record of two +integers, namely, the mantissa and the exponent where the base of the +exponent is binary. That is, the floating point number {\tt (m,e)} +represents the number $m \cdot 2^e$. A consequence of using a binary +base is that decimal numbers can not, in general, be represented +exactly. + +\subsection{Output Functions} + +A number of operations exist for specifying how numbers of type +{\tt Float} are to be displayed. By default, spaces are inserted every ten +digits in the output for readability.\footnote{Note that you cannot +include spaces in the input form of a floating point number, though +you can use underscores.} + +Output spacing can be modified with the \spadfunFrom{outputSpacing}{Float} +operation. This inserts no spaces and then displays the value of {\tt x}. + +\spadcommand{outputSpacing 0; x := sqrt 0.2 } +$$ +0.44721359549995793928 +$$ +\returnType{Type: Float} + +Issue this to have the spaces inserted every {\tt 5} digits. + +\spadcommand{outputSpacing 5; x } +$$ +0.44721\ 35954\ 99957\ 93928 +$$ +\returnType{Type: Float} + +By default, the system displays floats in either fixed format +or scientific format, depending on the magnitude of the number. + +\spadcommand{y := x/10**10 } +$$ +0.44721\ 35954\ 99957\ 93928\ {\rm E\ }-10 +$$ +\returnType{Type: Float} + +A particular format may be requested with the operations +\spadfunFrom{outputFloating}{Float} and \spadfunFrom{outputFixed}{Float}. + +\spadcommand{outputFloating(); x } +$$ +0.44721\ 35954\ 99957\ 93928\ {\rm E\ }0 +$$ +\returnType{Type: Float} + +\spadcommand{outputFixed(); y } +$$ +0.00000\ 00000\ 44721\ 35954\ 99957\ 93928 +$$ +\returnType{Type: Float} + +Additionally, you can ask for {\tt n} digits to be displayed after the +decimal point. + +\spadcommand{outputFloating 2; y } +$$ +0.45\ {\rm E\ } -10 +$$ +\returnType{Type: Float} + +\spadcommand{outputFixed 2; x } +$$ +0.45 +$$ +\returnType{Type: Float} + +This resets the output printing to the default behavior. + +\spadcommand{outputGeneral()} +\returnType{Type: Void} + +\subsection{An Example: Determinant of a Hilbert Matrix} + +Consider the problem of computing the determinant of a {\tt 10} by +{\tt 10} Hilbert matrix. The $(i,j)$-th entry of a Hilbert +matrix is given by {\tt 1/(i+j+1)}. + +First do the computation using rational numbers to obtain the +exact result. + +\spadcommand{a: Matrix Fraction Integer := matrix [ [1/(i+j+1) for j in 0..9] for i in 0..9] } +$$ +\left[ +\begin{array}{cccccccccc} +1 & {1 \over 2} & {1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & +{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} \\ +{1 \over 2} & {1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & +{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} \\ +{1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & {1 \over 7} & +{1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & +{1 \over {12}} \\ +{1 \over 4} & {1 \over 5} & {1 \over 6} & {1 \over 7} & {1 \over 8} & +{1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} & +{1 \over {13}} \\ +{1 \over 5} & {1 \over 6} & {1 \over 7} & {1 \over 8} & {1 \over 9} & +{1 \over {10}} & {1 \over {11}} & {1 \over {12}} & {1 \over {13}} & +{1 \over {14}} \\ +{1 \over 6} & {1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & +{1 \over {11}} & {1 \over {12}} & {1 \over {13}} & {1 \over {14}} & +{1 \over {15}} \\ +{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & +{1 \over {12}} & {1 \over {13}} & {1 \over {14}} & {1 \over {15}} & +{1 \over {16}} \\ +{1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} +& {1 \over {13}} & {1 \over {14}} & {1 \over {15}} & {1 \over {16}} & +{1 \over {17}} \\ +{1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} & +{1 \over {13}} & {1 \over {14}} & {1 \over {15}} & {1 \over {16}} & +{1 \over {17}} & {1 \over {18}} \\ +{1 \over {10}} & {1 \over {11}} & {1 \over {12}} & {1 \over {13}} & +{1 \over {14}} & {1 \over {15}} & {1 \over {16}} & {1 \over {17}} & +{1 \over {18}} & {1 \over {19}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +This version of \spadfunFrom{determinant}{Matrix} uses Gaussian elimination. + +\spadcommand{d:= determinant a } +$$ +1 \over {46206893947914691316295628839036278726983680000000000} +$$ +\returnType{Type: Fraction Integer} + +\spadcommand{d :: Float } +$$ +0.21641\ 79226\ 43149\ 18691\ {\rm E\ } -52 +$$ +\returnType{Type: Float} + +Now use hardware floats. Note that a semicolon (;) is used to prevent +the display of the matrix. + +\spadcommand{b: Matrix DoubleFloat := matrix [ [1/(i+j+1\$DoubleFloat) for j in 0..9] for i in 0..9]; } +\returnType{Type: Matrix DoubleFloat} + +The result given by hardware floats is correct only to four +significant digits of precision. In the jargon of numerical analysis, +the Hilbert matrix is said to be ``ill-conditioned.'' + +\spadcommand{determinant b } +$$ +2.1643677945721411e-53 +$$ +\returnType{Type: DoubleFloat} + +Now repeat the computation at a higher precision using {\tt Float}. + +\spadcommand{digits 40 } +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{c: Matrix Float := matrix [ [1/(i+j+1\$Float) for j in 0..9] for i in 0..9]; } +\returnType{Type: Matrix Float} + +\spadcommand{determinant c } +$$ +0.21641\ 79226\ 43149\ 18690\ 60594\ 98362\ 26174\ 36159\ {\rm E\ } -52 +$$ +\returnType{Type: Float} + +Reset \spadfunFrom{digits}{Float} to its default value. + +\spadcommand{digits 20} +$$ +40 +$$ +\returnType{Type: PositiveInteger} + +\section{Fraction} +\label{FractionXmpPage} + +The {\tt Fraction} domain implements quotients. The elements must +belong to a domain of category {\tt IntegralDomain}: multiplication +must be commutative and the product of two non-zero elements must not +be zero. This allows you to make fractions of most things you would +think of, but don't expect to create a fraction of two matrices! The +abbreviation for {\tt Fraction} is {\tt FRAC}. + +Use \spadopFrom{/}{Fraction} to create a fraction. + +\spadcommand{a := 11/12 } +$$ +{11} \over {12} +$$ +\returnType{Type: Fraction Integer} + +\spadcommand{b := 23/24 } +$$ +{23} \over {24} +$$ +\returnType{Type: Fraction Integer} + +The standard arithmetic operations are available. + +\spadcommand{3 - a*b**2 + a + b/a } +$$ +{313271} \over {76032} +$$ +\returnType{Type: Fraction Integer} + +Extract the numerator and denominator by using +\spadfunFrom{numer}{Fraction} and \spadfunFrom{denom}{Fraction}, +respectively. + +\spadcommand{numer(a) } +$$ +11 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{denom(b) } +$$ +24 +$$ +\returnType{Type: PositiveInteger} + +Operations like \spadfunFrom{max}{Fraction}, +\spadfunFrom{min}{Fraction}, \spadfunFrom{negative?}{Fraction}, +\spadfunFrom{positive?}{Fraction} and \spadfunFrom{zero?}{Fraction} +are all available if they are provided for the numerators and +denominators. +See \ref{IntegerXmpPage} on page~\pageref{IntegerXmpPage} for examples. + +Don't expect a useful answer from \spadfunFrom{factor}{Fraction}, +\spadfunFrom{gcd}{Fraction} or \spadfunFrom{lcm}{Fraction} if you apply +them to fractions. + +\spadcommand{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) } +$$ +{{x \sp 2}+{2 \ x}+1} \over {{x \sp 2} -{2 \ x}+1} +$$ +\returnType{Type: Fraction Polynomial Integer} + +Since all non-zero fractions are invertible, these operations have trivial +definitions. + +\spadcommand{factor(r) } +$$ +{{x \sp 2}+{2 \ x}+1} \over {{x \sp 2} -{2 \ x}+1} +$$ +\returnType{Type: Factored Fraction Polynomial Integer} + +Use \spadfunFrom{map}{Fraction} to apply \spadfunFrom{factor}{Fraction} to +the numerator and denominator, which is probably what you mean. + +\spadcommand{map(factor,r) } +$$ +{{\left( x+1 +\right)} +\sp 2} \over {{\left( x -1 +\right)} +\sp 2} +$$ +\returnType{Type: Fraction Factored Polynomial Integer} + +Other forms of fractions are available. Use {\tt continuedFraction} +to create a continued fraction. + +\spadcommand{continuedFraction(7/12)} +$$ +\zag{1}{1}+ \zag{1}{1}+ \zag{1}{2}+ \zag{1}{2} +$$ +\returnType{Type: ContinuedFraction Integer} + +Use {\tt partialFraction} to create a partial fraction. +See +\ref{ContinuedFractionXmpPage} on page~\pageref{ContinuedFractionXmpPage} +and \ref{PartialFractionXmpPage} on page~\pageref{PartialFractionXmpPage} for +additional information and examples. + +\spadcommand{partialFraction(7,12)} +$$ +1 -{3 \over {2 \sp 2}}+{1 \over 3} +$$ +\returnType{Type: PartialFraction Integer} + +Use conversion to create alternative views of fractions with objects +moved in and out of the numerator and denominator. + +\spadcommand{g := 2/3 + 4/5*\%i } +$$ +{2 \over 3}+{{4 \over 5} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +Conversion is discussed in detail in +Section~\ref{ugTypesConvertPage} +on page~\pageref{ugTypesConvertPage}. + +\spadcommand{g :: FRAC COMPLEX INT } +$$ +{{10}+{{12} \ i}} \over {15} +$$ +\returnType{Type: Fraction Complex Integer} + +\section{FullPartialFractionExpansion} +\label{FullPartialFractionExpansionXmpPage} + +The domain {\tt FullPartialFractionExpansion} implements +factor-free conversion of quotients to full partial fractions. + +Our examples will all involve quotients of univariate polynomials +with rational number coefficients. + +\spadcommand{Fx := FRAC UP(x, FRAC INT) } +$$ +\mbox{\rm Fraction UnivariatePolynomial(x,Fraction Integer)} +$$ +\returnType{Type: Domain} + +Here is a simple-looking rational function. + +\spadcommand{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) } +$$ +{36} \over {{x \sp 5} -{2 \ {x \sp 4}} -{2 \ {x \sp 3}}+{4 \ {x \sp 2}}+x +-2} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +We use \spadfunFrom{fullPartialFraction}{FullPartialFractionExpansion} +to convert it to an object of type {\tt FullPartialFractionExpansion}. + +\spadcommand{g := fullPartialFraction f } +$$ +{4 \over {x -2}} -{4 \over {x+1}}+ +{\sum \sb{\displaystyle {{{ \%A \sp 2} -1}=0}} +{{-{3 \ \%A} -6} \over {{\left( x - \%A \right)}\sp 2}}} +$$ +\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} + +Use a coercion to change it back into a quotient. + +\spadcommand{g :: Fx } +$$ +{36} \over {{x \sp 5} -{2 \ {x \sp 4}} -{2 \ {x \sp 3}}+{4 \ {x \sp 2}}+x +-2} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +Full partial fractions differentiate faster than rational functions. + +\spadcommand{g5 := D(g, 5) } +$$ +-{{480} \over {{\left( x -2 \right)}\sp 6}}+ +{{480} \over {{\left( x+1 \right)}\sp 6}}+ +{\sum \sb{\displaystyle {{{ \%A \sp 2} -1}=0}} +{{{{2160} \ \%A}+{4320}} \over {{\left( x - \%A \right)}\sp 7}}} +$$ +\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} + +\spadcommand{f5 := D(f, 5) } +$$ +\left( +\begin{array}{@{}l} +-{{544320} \ {x \sp {10}}}+ +{{4354560} \ {x \sp 9}} - +{{14696640} \ {x \sp 8}}+ +{{28615680} \ {x \sp 7}} - +\\ +\\ +\displaystyle +{{40085280} \ {x \sp 6}}+ +{{46656000} \ {x \sp 5}} - +{{39411360} \ {x \sp 4}}+ +{{18247680} \ {x \sp 3}} - +\\ +\\ +\displaystyle +{{5870880} \ {x \sp 2}}+ +{{3317760} \ x}+{246240} +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +{x \sp {20}} - +{{12} \ {x \sp {19}}}+ +{{53} \ {x \sp {18}}} - +{{76} \ {x \sp {17}}} - +{{159} \ {x \sp {16}}}+ +{{676} \ {x \sp {15}}} - +{{391} \ {x \sp {14}}} - +\\ +\\ +\displaystyle +{{1596} \ {x \sp {13}}}+ +{{2527} \ {x \sp {12}}}+ +{{1148} \ {x \sp {11}}} - +{{4977} \ {x \sp {10}}}+ +{{1372} \ {x \sp 9}}+ +\\ +\\ +\displaystyle +{{4907} \ {x \sp 8}} - +{{3444} \ {x \sp 7}} +-{{2381} \ {x \sp 6}}+ +{{2924} \ {x \sp 5}}+ +{{276} \ {x \sp 4}} - +\\ +\\ +\displaystyle +{{1184} \ {x \sp 3}}+ +{{208} \ {x \sp 2}}+ +{{192} \ x} - +{64} +\end{array} +\right) +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +We can check that the two forms represent the same function. + +\spadcommand{g5::Fx - f5 } +$$ +0 +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +Here are some examples that are more complicated. + +\spadcommand{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) } +$$ +{{x \sp 6} - +{x \sp 5}} +\over +{{x \sp 7} - +{4 \ {x \sp 6}}+ +{3 \ {x \sp 5}}+ +{9 \ {x \sp 3}} - +{6 \ {x \sp 2}} - +{4 \ x} - +8} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{g := fullPartialFraction f } +$$ +\begin{array}{@{}l} +{{{1952} \over {2401}} \over {x -2}}+ +{{{464} \over {343}} \over {{\left( x -2 \right)}\sp 2}}+ +{{{32} \over {49}} \over {{\left( x -2 \right)}\sp 3}}+ +\\ +\\ +\displaystyle +{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} +{{-{{{179} \over {2401}} \ \%A}+{{135} \over {2401}}} \over {x - \%A}}}+ +\\ +\\ +\displaystyle +{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} +{{{{{37} \over {1029}} \ \%A}+ +{{20} \over {1029}}} \over {{\left( x - \%A \right)}\sp 2}}} +\end{array} +$$ +\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} + +\spadcommand{g :: Fx - f } +$$ +0 +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) } +$$ +{{2 \ {x \sp 7}} -{7 \ {x \sp 5}}+{{26} \ {x \sp 3}}+{8 \ x}} +\over +{{x \sp 8} -{5 \ {x \sp 6}}+{6 \ {x \sp 4}}+{4 \ {x \sp 2}} -8} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{g := fullPartialFraction f } +$$ +\begin{array}{@{}l} +{\sum \sb{\displaystyle {{{ \%A \sp 2} -2}=0}} +{{1 \over 2} \over {x - \%A}}}+ +\\ +\\ +\displaystyle +{\sum \sb{\displaystyle {{{ \%A \sp 2} -2}=0}} +{1 \over {{\left( x - \%A \right)}\sp 3}}}+ +\\ +\\ +\displaystyle +{\sum \sb{\displaystyle {{{ \%A \sp 2}+1}=0}} +{{1 \over 2} \over {x - \%A}}} +\end{array} +$$ +\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} + +\spadcommand{g :: Fx - f } +$$ +0 +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)} +$$ +{x \sp 3} +\over +\left( +\begin{array}{@{}l} +{x \sp {21}}+ +{2 \ {x \sp {20}}}+ +{4 \ {x \sp {19}}}+ +{7 \ {x \sp {18}}}+ +{{10} \ {x \sp {17}}}+ +{{22} \ {x \sp {15}}}+ +{{30} \ {x \sp {14}}}+ +\\ +\\ +\displaystyle +{{36} \ {x \sp {13}}}+ +{{40} \ {x \sp {12}}}+ +{{47} \ {x \sp {11}}}+ +{{46} \ {x \sp {10}}}+ +{{49} \ {x \sp 9}}+ +{{43} \ {x \sp 8}}+ +{{38} \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{{32} \ {x \sp 6}}+ +{{23} \ {x \sp 5}}+ +{{19} \ {x \sp 4}}+ +{{10} \ {x \sp 3}}+ +{7 \ {x \sp 2}}+ +{2 \ x}+ +1 +\end{array} +\right) +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{g := fullPartialFraction f } +$$ +\begin{array}{@{}l} +{\sum \sb{\displaystyle {{{ \%A \sp 2}+1}=0}} +{{{1 \over 2} \ \%A} \over {x - \%A}}}+ +{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} +{{{{1 \over 9} \ \%A} -{{19} \over {27}}} \over {x - \%A}}}+ +\\ +\\ +\displaystyle +{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} +{{{{1 \over {27}} \ \%A} -{1 \over {27}}} +\over {{\left( x - \%A \right)}\sp 2}}}+ +\\ +\\ +\displaystyle +\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} +\left( +\begin{array}{@{}l} +-{{{96556567040} \over {912390759099}} \ { \%A \sp 4}}+ +{{{420961732891} \over {912390759099}} \ { \%A \sp 3}} - +\\ +\\ +\displaystyle +{{{59101056149} \over {912390759099}} \ { \%A \sp 2}} - +{{{373545875923} \over {912390759099}} \ \%A}+ +\\ +\\ +\displaystyle +{{529673492498} \over {912390759099}} +\end{array} +\right) +\over {x - \%A}+ +\\ +\\ +\displaystyle +\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} +\left( +\begin{array}{@{}l} +-{{{5580868} \over {94070601}} \ { \%A \sp 4}} - +{{{2024443} \over {94070601}} \ { \%A \sp 3}}+ +{{{4321919} \over {94070601}} \ { \%A \sp 2}} - +\\ +\\ +\displaystyle +{{{84614} \over {1542141}} \ \%A} - +{{5070620} \over {94070601}} +\end{array} +\right) +\over {{\left( x - \%A \right)}\sp 2}+ +\\ +\\ +\displaystyle +\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} +\left( +\begin{array}{@{}l} +{{{1610957} \over {94070601}} \ { \%A \sp 4}}+ +{{{2763014} \over {94070601}} \ { \%A \sp 3}} - +{{{2016775} \over {94070601}} \ { \%A \sp 2}}+ +\\ +\\ +\displaystyle +{{{266953} \over {94070601}} \ \%A}+ +{{4529359} \over {94070601}} +\end{array} +\right) +\over {{\left( x - \%A \right)}\sp 3} +\end{array} +$$ +\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} + +This verification takes much longer than the conversion to +partial fractions. + +\spadcommand{g :: Fx - f } +$$ +0 +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} + +For more information, see the paper: Bronstein, M and Salvy, B. +``Full Partial Fraction Decomposition of Rational Functions,'' +{\it Proceedings of ISSAC'93, Kiev}, ACM Press. Also see +\ref{PartialFractionXmpPage} on page~\pageref{PartialFractionXmpPage} +for standard partial fraction decompositions. + +\section{GeneralSparseTable} +\label{GeneralSparseTableXmpPage} + +Sometimes when working with tables there is a natural value to use as +the entry in all but a few cases. The {\tt GeneralSparseTable} +constructor can be used to provide any table type with a default value +for entries. See \ref{TableXmpPage} on page~\pageref{TableXmpPage} +for general information about tables. + +Suppose we launched a fund-raising campaign to raise fifty thousand dollars. +To record the contributions, we want a table with strings as keys +(for the names) and integer entries (for the amount). +In a data base of cash contributions, unless someone +has been explicitly entered, it is reasonable to assume they have made +a zero dollar contribution. + +This creates a keyed access file with default entry {\tt 0}. + +\spadcommand{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ; } +\returnType{Type: GeneralSparseTable(String,Integer,KeyedAccessFile Integer,0)} + + +Now {\tt patrons} can be used just as any other table. +Here we record two gifts. + +\spadcommand{patrons."Smith" := 10500 } +$$ +10500 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{patrons."Jones" := 22000 } +$$ +22000 +$$ +\returnType{Type: PositiveInteger} + +Now let us look up the size of the contributions from Jones and Stingy. + +\spadcommand{patrons."Jones" } +$$ +22000 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{patrons."Stingy" } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +Have we met our seventy thousand dollar goal? + +\spadcommand{reduce(+, entries patrons) } +$$ +32500 +$$ +\returnType{Type: PositiveInteger} + +So the project is cancelled and we can delete the data base: + +\spadcommand{)system rm -r kaf*.sdata } + +\section{GroebnerFactorizationPackage} +\label{GroebnerFactorizationPackageXmpPage} + +Solving systems of polynomial equations with the Gr\"{o}bner basis +algorithm can often be very time consuming because, in general, the +algorithm has exponential run-time. These systems, which often come +from concrete applications, frequently have symmetries which are not +taken advantage of by the algorithm. However, it often happens in +this case that the polynomials which occur during the Gr\"{o}bner +calculations are reducible. Since Axiom has an excellent polynomial +factorization algorithm, it is very natural to combine the Gr\"{o}bner +and factorization algorithms. + +{\tt GroebnerFactorizationPackage} exports the +\spadfunFrom{groebnerFactorize}{GroebnerFactorizationPackage} +operation which implements a modified Gr\"{o}bner basis algorithm. In +this algorithm, each polynomial that is to be put into the partial +list of the basis is first factored. The remaining calculation is +split into as many parts as there are irreducible factors. Call these +factors $p_1, \ldots,p_n.$ In the branches corresponding to $p_2, +\ldots,p_n,$ the factor $p_1$ can be divided out, and so on. This +package also contains operations that allow you to specify the +polynomials that are not zero on the common roots of the final +Gr\"{o}bner basis. + +Here is an example from chemistry. In a theoretical model of the +cyclohexan ${\rm C}_6{\rm H}_{12}$, the six carbon atoms each sit in +the center of gravity of a tetrahedron that has two hydrogen atoms and +two carbon atoms at its corners. We first normalize and set the +length of each edge to 1. Hence, the distances of one fixed carbon +atom to each of its immediate neighbours is 1. We will denote the +distances to the other three carbon atoms by $x$, $y$ and $z$. + +A.~Dress developed a theory to decide whether a set of points +and distances between them can be realized in an $n$-dimensional space. +Here, of course, we have $n = 3$. + +\spadcommand{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [ [0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0] ] } +$$ +\left[ +\begin{array}{cccccc} +0 & 1 & 1 & 1 & 1 & 1 \\ +1 & 0 & 1 & {8 \over 3} & x & {8 \over 3} \\ +1 & 1 & 0 & 1 & {8 \over 3} & y \\ +1 & {8 \over 3} & 1 & 0 & 1 & {8 \over 3} \\ +1 & x & {8 \over 3} & 1 & 0 & 1 \\ +1 & {8 \over 3} & y & {8 \over 3} & 1 & 0 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(6,DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} + +For the cyclohexan, the distances have to satisfy this equation. + +\spadcommand{eq := determinant mfzn } +$$ +\begin{array}{@{}l} +-{{x \sp 2} \ {y \sp 2}}+ +{{{22} \over 3} \ {x \sp 2} \ y} - +{{{25} \over 9} \ {x \sp 2}}+ +{{{22} \over 3} \ x \ {y \sp 2}} - +{{{388} \over 9} \ x \ y} - +\\ +\\ +\displaystyle +{{{250} \over {27}} \ x} - +{{{25} \over 9} \ {y \sp 2}} - +{{{250} \over {27}} \ y}+ +{{14575} \over {81}} +\end{array} +$$ +\returnType{Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} + +They also must satisfy the equations +given by cyclic shifts of the indeterminates. + +\spadcommand{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])] } +$$ +\begin{array}{@{}l} +\left[ + +\begin{array}{@{}l} +\left[ +{x \ y}+ +{x \ z} - +{{{22} \over 3} \ x}+ +{y \ z} - +{{{22} \over 3} \ y} - +{{{22} \over 3} \ z}+ +{{121} \over 3}, +\right. +\\ +\\ +\displaystyle +{x \ {z \sp 2}} - +{{{22} \over 3} \ x \ z}+ +{{{25} \over 9} \ x}+ +{y \ {z \sp 2}} - +{{{22} \over 3} \ y \ z}+ +{{{25} \over 9} \ y} - +{{{22} \over 3} \ {z \sp 2}}+ +{{{388} \over 9} \ z}+ +{{250} \over {27}}, +\\ +\\ +\displaystyle +\left. +\begin{array}{@{}l} +{{y \sp 2} \ {z \sp 2}} - +{{{22} \over 3} \ {y \sp 2} \ z}+ +{{{25} \over 9} \ {y \sp 2}} - +{{{22} \over 3} \ y \ {z \sp 2}}+ +{{{388} \over 9} \ y \ z}+ +{{{250} \over {27}} \ y}+ +\\ +\\ +\displaystyle +{{{25} \over 9} \ {z \sp 2}}+ +{{{250} \over {27}} \ z} - +{{14575} \over {81}} +\end{array} +\right], +\end{array} +\right. +\\ +\\ +\displaystyle +{\left[ +{x+y -{{21994} \over {5625}}}, +{{y \sp 2} -{{{21994} \over {5625}} \ y}+{{4427} \over {675}}}, +{z -{{463} \over {87}}} +\right]}, +\\ +\\ +\displaystyle +{\left[ +{{x \sp 2} - +{{1 \over 2} \ x \ z} - +{{{11} \over 2} \ x} - +{{5 \over 6} \ z}+ +{{265} \over {18}}}, +{y -z}, +{{z \sp 2} -{{{38} \over 3} \ z}+{{265} \over 9}} +\right]}, +\\ +\\ +\displaystyle +{\left[ +{x -{{25} \over 9}}, +{y -{{11} \over 3}}, +{z -{{11} \over 3}} \right]}, +\\ +\\ +\displaystyle +{\left[ +{x -{{11} \over 3}}, +{y -{{11} \over 3}}, +{z -{{11} \over 3}} +\right]}, +\\ +\\ +\displaystyle +{\left[ +{x+{5 \over 3}}, +{y+{5 \over 3}}, +{z+{5 \over 3}} +\right]}, +\\ +\\ +\displaystyle +\left. +{\left[ +{x -{{19} \over 3}}, +{y+{5 \over 3}}, +{z+{5 \over 3}} +\right]} +\right] +\end{array} +$$ +\returnType{Type: List List +DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} + +The union of the solutions of this list is the solution of our +original problem. If we impose positivity conditions, we get two +relevant ideals. One ideal is zero-dimensional, namely $x = y = z = 11/3$, +and this determines the ``boat'' form of the cyclohexan. The +other ideal is one-dimensional, which means that we have a solution +space given by one parameter. This gives the ``chair'' form of the +cyclohexan. The parameter describes the angle of the ``back of the +chair.'' + +\spadfunFrom{groebnerFactorize}{GroebnerFactorizationPackage} has an +optional {\tt Boolean}-valued second argument. When it is {\tt true} +partial results are displayed, since it may happen that the +calculation does not terminate in a reasonable time. See the source +code for {\tt GroebnerFactorizationPackage} in {\bf groebf.input} +for more details about the algorithms used. + +\section{Heap} +\label{HeapXmpPage} + +The domain {\tt Heap(S)} implements a priority queue of objects of +type {\tt S} such that the operation {\tt extract!} removes and +returns the maximum element. The implementation represents heaps as +flexible arrays (see +\ref{FlexibleArrayXmpPage} on page~\pageref{FlexibleArrayXmpPage}). +The representation and algorithms give complexity of $O(\log(n))$ +for insertion and extractions, and $O(n)$ for construction. + +Create a heap of six elements. + +\spadcommand{h := heap [-4,9,11,2,7,-7]} +$$ +\left[ +{11}, 7, 9, -4, 2, -7 +\right] +$$ +\returnType{Type: Heap Integer} + +Use {\tt insert!} to add an element. + +\spadcommand{insert!(3,h)} +$$ +\left[ +{11}, 7, 9, -4, 2, -7, 3 +\right] +$$ +\returnType{Type: Heap Integer} + +The operation {\tt extract!} removes and returns the maximum element. + +\spadcommand{extract! h} +$$ +11 +$$ +\returnType{Type: PositiveInteger} + +The internal structure of {\tt h} has been appropriately adjusted. + +\spadcommand{h} +$$ +\left[ +9, 7, 3, -4, 2, -7 +\right] +$$ +\returnType{Type: Heap Integer} + +Now {\tt extract!} elements repeatedly until none are left, collecting +the elements in a list. + +\spadcommand{[extract!(h) while not empty?(h)]} +$$ +\left[ +9, 7, 3, 2, -4, -7 +\right] +$$ +\returnType{Type: List Integer} + +Another way to produce the same result is by defining a {\tt heapsort} +function. + +\spadcommand{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))} +\returnType{Void} + +Create another sample heap. + +\spadcommand{h1 := heap [17,-4,9,-11,2,7,-7]} +$$ +\left[ +{17}, 2, 9, -{11}, -4, 7, -7 +\right] +$$ +\returnType{Type: Heap Integer} + +Apply {\tt heapsort} to present elements in order. + +\spadcommand{heapsort h1} +$$ +\left[ +{17}, 9, 7, 2, -4, -7, -{11} +\right] +$$ +\returnType{Type: List Integer} + +\section{HexadecimalExpansion} +\label{HexadecimalExpansionXmpPage} + +All rationals have repeating hexadecimal expansions. The operation +\spadfunFrom{hex}{HexadecimalExpansion} returns these expansions of +type {\tt HexadecimalExpansion}. Operations to access the individual +numerals of a hexadecimal expansion can be obtained by converting the +value to {\tt RadixExpansion(16)}. More examples of expansions are +available in the +\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, +\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, and +\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. + +This is a hexadecimal expansion of a rational number. + +\spadcommand{r := hex(22/7) } +$$ +3.{\overline {249}} +$$ +\returnType{Type: HexadecimalExpansion} + +Arithmetic is exact. + +\spadcommand{r + hex(6/7) } +$$ +4 +$$ +\returnType{Type: HexadecimalExpansion} + +The period of the expansion can be short or long \ldots + +\spadcommand{[hex(1/i) for i in 350..354] } +$$ +\begin{array}{@{}l} +\left[ +{0.0{\overline {\rm 0BB3EE721A54D88}}}, +{0.{\overline {\rm 00BAB6561}}}, +{0.{00}{\overline {\rm BA2E8}}}, +\right. +\\ +\\ +\displaystyle +\left. +{0.{\overline {\rm 00B9A7862A0FF465879D5F}}}, +{0.0{\overline {\rm 0B92143FA36F5E02E4850FE8DBD78}}} +\right] +\end{array} +$$ +\returnType{Type: List HexadecimalExpansion} + +or very long! + +\spadcommand{hex(1/1007) } +$$ +\begin{array}{@{}l} +0.{\overline +{\rm 0041149783F0BF2C7D13933192AF6980619EE345E91EC2BB9D5CC}} +\\ +\displaystyle +\ \ {\overline +{\rm A5C071E40926E54E8DDAE24196C0B2F8A0AAD60DBA57F5D4C8}} +\\ +\displaystyle +\ \ {\overline +{\rm 536262210C74F1}} +\end{array} +$$ +\returnType{Type: HexadecimalExpansion} + +These numbers are bona fide algebraic objects. + +\spadcommand{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9) } +$$ +{{0.4} \ {x \sp 2}}+{{0.{\overline {\rm A}}} \ x}+{0.{\overline {\rm 71C}}} +$$ +\returnType{Type: Polynomial HexadecimalExpansion} + +\spadcommand{q := D(p, x) } +$$ +{{0.8} \ x}+{0.{\overline {\rm A}}} +$$ +\returnType{Type: Polynomial HexadecimalExpansion} + +\spadcommand{g := gcd(p, q)} +$$ +x+{1.{\overline 5}} +$$ +\returnType{Type: Polynomial HexadecimalExpansion} + +\section{Integer} +\label{IntegerXmpPage} + +Axiom provides many operations for manipulating arbitrary precision +integers. In this section we will show some of those that come from +{\tt Integer} itself plus some that are implemented in other packages. +More examples of using integers are in the following sections: +\ref{ugIntroNumbersPage} on page~\pageref{ugIntroNumbersPage} in section +\ref{ugIntroNumbersNumber} on page~\pageref{ugIntroNumbersNumber} +\ref{IntegerNumberTheoryFunctionsXmpPage} on +page~\pageref{IntegerNumberTheoryFunctionsXmpPage}, +\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, +\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, +\ref{HexadecimalExpansionXmpPage} on +page~\pageref{HexadecimalExpansionXmpPage}, and +\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. + +\subsection{Basic Functions} + +The size of an integer in Axiom is only limited by the amount of +computer storage you have available. The usual arithmetic operations +are available. + +\spadcommand{2**(5678 - 4856 + 2 * 17)} +$$ +\begin{array}{@{}l} +48048107704350081471815409251259243912395261398716822634738556100 +\\ +\displaystyle +88084200076308293086342527091412083743074572278211496076276922026 +\\ +\displaystyle +43343568752733498024953930242542523045817764949544214392905306388 +\\ +\displaystyle +478705146745768073877141698859815495632935288783334250628775936 +\end{array} +$$ +\returnType{Type: PositiveInteger} + +There are a number of ways of working with the sign of an integer. +Let's use this {\tt x} as an example. + +\spadcommand{x := -101 } +$$ +-{101} +$$ +\returnType{Type: Integer} + +First of all, there is the absolute value function. + +\spadcommand{abs(x) } +$$ +101 +$$ +\returnType{Type: PositiveInteger} + +The \spadfunFrom{sign}{Integer} operation returns {\tt -1} if its argument +is negative, {\tt 0} if zero and {\tt 1} if positive. + +\spadcommand{sign(x) } +$$ +-1 +$$ +\returnType{Type: Integer} + +You can determine if an integer is negative in several other ways. + +\spadcommand{x < 0 } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{x <= -1 } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{negative?(x) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Similarly, you can find out if it is positive. + +\spadcommand{x > 0 } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{x >= 1 } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{positive?(x) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +This is the recommended way of determining whether an integer is zero. + +\spadcommand{zero?(x) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\boxed{4.6in}{ +\vskip 0.1cm +Use the \spadfunFrom{zero?}{Integer} operation whenever you are +testing any mathematical object for equality with zero. This is +usually more efficient that using {\tt =} (think of matrices: it is +easier to tell if a matrix is zero by just checking term by term than +constructing another ``zero'' matrix and comparing the two matrices +term by term) and also avoids the problem that {\tt =} is usually used +for creating equations.\\ +} + +This is the recommended way of determining whether an integer is equal +to one. + +\spadcommand{one?(x) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +This syntax is used to test equality using \spadopFrom{=}{Integer}. +It says that you want a {\tt Boolean} ({\tt true} or {\tt false}) +answer rather than an equation. + +\spadcommand{(x = -101)@Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The operations \spadfunFrom{odd?}{Integer} and +\spadfunFrom{even?}{Integer} determine whether an integer is odd or +even, respectively. They each return a {\tt Boolean} object. + +\spadcommand{odd?(x) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{even?(x) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +The operation \spadfunFrom{gcd}{Integer} computes the greatest common +divisor of two integers. + +\spadcommand{gcd(56788,43688)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{lcm}{Integer} computes their least common multiple. + +\spadcommand{lcm(56788,43688)} +$$ +620238536 +$$ +\returnType{Type: PositiveInteger} + +To determine the maximum of two integers, use \spadfunFrom{max}{Integer}. + +\spadcommand{max(678,567)} +$$ +678 +$$ +\returnType{Type: PositiveInteger} + +To determine the minimum, use \spadfunFrom{min}{Integer}. + +\spadcommand{min(678,567)} +$$ +567 +$$ +\returnType{Type: PositiveInteger} + +The {\tt reduce} operation is used to extend binary operations to more +than two arguments. For example, you can use {\tt reduce} to find the +maximum integer in a list or compute the least common multiple of all +integers in the list. + +\spadcommand{reduce(max,[2,45,-89,78,100,-45])} +$$ +100 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{reduce(min,[2,45,-89,78,100,-45])} +$$ +-{89} +$$ +\returnType{Type: Integer} + +\spadcommand{reduce(gcd,[2,45,-89,78,100,-45])} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{reduce(lcm,[2,45,-89,78,100,-45])} +$$ +1041300 +$$ +\returnType{Type: PositiveInteger} + +The infix operator ``/'' is {\it not} used to compute the quotient of +integers. Rather, it is used to create rational numbers as described +in \ref{FractionXmpPage} on page~\pageref{FractionXmpPage}. + +\spadcommand{13 / 4} +$$ +{13} \over 4 +$$ +\returnType{Type: Fraction Integer} + +The infix operation \spadfunFrom{quo}{Integer} computes the integer +quotient. + +\spadcommand{13 quo 4} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +The infix operation \spadfunFrom{rem}{Integer} computes the integer +remainder. + +\spadcommand{13 rem 4} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +One integer is evenly divisible by another if the remainder is zero. +The operation \spadfunFrom{exquo}{Integer} can also be used. See +\ref{ugTypesUnionsPage} on page~\pageref{ugTypesUnionsPage} in Section +\ref{ugTypesUnionsNumber} on page~\pageref{ugTypesUnionsNumber} for an +example. + +\spadcommand{zero?(167604736446952 rem 2003644)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The operation \spadfunFrom{divide}{Integer} returns a record of the +quotient and remainder and thus is more efficient when both are needed. + +\spadcommand{d := divide(13,4) } +$$ +\left[ +{quotient=3}, {remainder=1} +\right] +$$ +\returnType{Type: Record(quotient: Integer,remainder: Integer)} + +\spadcommand{d.quotient } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +Records are discussed in detail in Section +\ref{ugTypesRecords} on page~\pageref{ugTypesRecords}. + +\spadcommand{d.remainder } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Primes and Factorization} + +Use the operation \spadfunFrom{factor}{Integer} to factor integers. +It returns an object of type {\tt Factored Integer}. +See \ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage} +for a discussion of the manipulation of factored objects. + +\spadcommand{factor 102400} +$$ +{2 \sp {12}} \ {5 \sp 2} +$$ +\returnType{Type: Factored Integer} + +The operation \spadfunFrom{prime?}{Integer} returns {\tt true} or +{\tt false} depending on whether its argument is a prime. + +\spadcommand{prime? 7} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{prime? 8} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +The operation \spadfunFrom{nextPrime}{IntegerPrimesPackage} returns the +least prime number greater than its argument. + +\spadcommand{nextPrime 100} +$$ +101 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{prevPrime}{IntegerPrimesPackage} returns +the greatest prime number less than its argument. + +\spadcommand{prevPrime 100} +$$ +97 +$$ +\returnType{Type: PositiveInteger} + +To compute all primes between two integers (inclusively), use the +operation \spadfunFrom{primes}{IntegerPrimesPackage}. + +\spadcommand{primes(100,175)} +$$ +\left[ +{173}, {167}, {163}, {157}, {151}, {149}, {139}, {137}, +{131}, {127}, {113}, {109}, {107}, {103}, {101} +\right] +$$ +\returnType{Type: List Integer} + +You might sometimes want to see the factorization of an integer +when it is considered a {\it Gaussian integer}. +See \ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} for more details. + +\spadcommand{factor(2 :: Complex Integer)} +$$ +-{i \ {{\left( 1+i +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +\subsection{Some Number Theoretic Functions} + +Axiom provides several number theoretic operations for integers. +More examples are in \ref{IntegerNumberTheoryFunctionsXmpPage} on +page~\pageref{IntegerNumberTheoryFunctionsXmpPage}. + +The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} +computes the Fibonacci numbers. The algorithm has running time +$O\,(\log^3(n))$ for argument {\tt n}. + +\spadcommand{[fibonacci(k) for k in 0..]} +$$ +\left[ +0, 1, 1, 2, 3, 5, 8, {13}, {21}, {34}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +The operation \spadfunFrom{legendre}{IntegerNumberTheoryFunctions} +computes the Legendre symbol for its two integer arguments where the +second one is prime. If you know the second argument to be prime, use +\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} instead where no +check is made. + +\spadcommand{[legendre(i,11) for i in 0..10]} +$$ +\left[ +0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1 +\right] +$$ +\returnType{Type: List Integer} + +The operation \spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} +computes the Jacobi symbol for its two integer arguments. By +convention, {\tt 0} is returned if the greatest common divisor of the +numerator and denominator is not {\tt 1}. + +\spadcommand{[jacobi(i,15) for i in 0..9]} +$$ +\left[ +0, 1, 1, 0, 1, 0, 0, -1, 1, 0 +\right] +$$ +\returnType{Type: List Integer} + +The operation \spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions} +computes the values of Euler's $\phi$-function where $\phi(n)$ equals +the number of positive integers less than or equal to {\tt n} that are +relatively prime to the positive integer {\tt n}. + +\spadcommand{[eulerPhi i for i in 1..]} +$$ +\left[ +1, 1, 2, 2, 4, 2, 6, 4, 6, 4, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +The operation \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions} +computes the M\"{o}bius $\mu$ function. + +\spadcommand{[moebiusMu i for i in 1..]} +$$ +\left[ +1, -1, -1, 0, -1, 1, -1, 0, 0, 1, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Although they have somewhat limited utility, Axiom provides Roman numerals. + +\spadcommand{a := roman(78) } +$$ +{\rm LXXVIII } +$$ +\returnType{Type: RomanNumeral} + +\spadcommand{b := roman(87) } +$$ +{\rm LXXXVII } +$$ +\returnType{Type: RomanNumeral} + +\spadcommand{a + b } +$$ +{\rm CLXV } +$$ +\returnType{Type: RomanNumeral} + +\spadcommand{a * b } +$$ +{\rm MMMMMMDCCLXXXVI } +$$ +\returnType{Type: RomanNumeral} + +\spadcommand{b rem a } +$$ +{\rm IX } +$$ +\returnType{Type: RomanNumeral} + +\section{IntegerLinearDependence} +\label{IntegerLinearDependenceXmpPage} + +The elements $v_1, \dots,v_n$ of a module {\tt M} over a ring {\tt R} +are said to be {\it linearly dependent over {\tt R}} if there exist +$c_1,\dots,c_n$ in {\tt R}, not all $0$, such that $c_1 v_1 + +\dots c_n v_n = 0$. If such $c_i$'s exist, they form what is called a +{\it linear dependence relation over {\tt R}} for the $v_i$'s. + +The package {\tt IntegerLinearDependence} provides functions +for testing whether some elements of a module over the integers are +linearly dependent over the integers, and to find the linear +dependence relations, if any. + +Consider the domain of two by two square matrices with integer entries. + +\spadcommand{M := SQMATRIX(2,INT) } +$$ +SquareMatrix(2,Integer) +$$ +\returnType{Type: Domain} + +Now create three such matrices. + +\spadcommand{m1: M := squareMatrix matrix [ [1, 2], [0, -1] ] } +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +0 & -1 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{m2: M := squareMatrix matrix [ [2, 3], [1, -2] ] } +$$ +\left[ +\begin{array}{cc} +2 & 3 \\ +1 & -2 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{m3: M := squareMatrix matrix [ [3, 4], [2, -3] ] } +$$ +\left[ +\begin{array}{cc} +3 & 4 \\ +2 & -3 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +This tells you whether {\tt m1}, {\tt m2} and {\tt m3} are linearly +dependent over the integers. + +\spadcommand{linearlyDependentOverZ? vector [m1, m2, m3] } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Since they are linearly dependent, you can ask for the dependence relation. + +\spadcommand{c := linearDependenceOverZ vector [m1, m2, m3] } +$$ +\left[ +1, -2, 1 +\right] +$$ +\returnType{Type: Union(Vector Integer,...)} + +This means that the following linear combination should be {\tt 0}. + +\spadcommand{c.1 * m1 + c.2 * m2 + c.3 * m3 } +$$ +\left[ +\begin{array}{cc} +0 & 0 \\ +0 & 0 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +When a given set of elements are linearly dependent over {\tt R}, this +also means that at least one of them can be rewritten as a linear +combination of the others with coefficients in the quotient field of +{\tt R}. + +To express a given element in terms of other elements, use the operation +\spadfunFrom{solveLinearlyOverQ}{IntegerLinearDependence}. + +\spadcommand{solveLinearlyOverQ(vector [m1, m3], m2) } +$$ +\left[ +{1 \over 2}, {1 \over 2} +\right] +$$ +\returnType{Type: Union(Vector Fraction Integer,...)} + +\section{IntegerNumberTheoryFunctions} +\label{IntegerNumberTheoryFunctionsXmpPage} + +The {\tt IntegerNumberTheoryFunctions} package contains a variety of +operations of interest to number theorists. Many of these operations +deal with divisibility properties of integers. (Recall that an +integer {\tt a} divides an integer {\tt b} if there is an integer +{\tt c} such that {\tt b = a * c}.) + +The operation \spadfunFrom{divisors}{IntegerNumberTheoryFunctions} +returns a list of the divisors of an integer. + +\spadcommand{div144 := divisors(144) } +$$ +\left[ +1, 2, 3, 4, 6, 8, 9, {12}, {16}, {18}, {24}, +{36}, {48}, {72}, {144} +\right] +$$ +\returnType{Type: List Integer} + +You can now compute the number of divisors of {\tt 144} and the sum of +the divisors of {\tt 144} by counting and summing the elements of the +list we just created. + +\spadcommand{\#(div144) } +$$ +15 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{reduce(+,div144) } +$$ +403 +$$ +\returnType{Type: PositiveInteger} + +Of course, you can compute the number of divisors of an integer +{\tt n}, usually denoted {\tt d(n)}, and the sum of the divisors of an +integer {\tt n}, usually denoted {\tt $\sigma$(n)}, without ever +listing the divisors of {\tt n}. + +In Axiom, you can simply call the operations +\spadfunFrom{numberOfDivisors}{IntegerNumberTheoryFunctions} and +\spadfunFrom{sumOfDivisors}{IntegerNumberTheoryFunctions}. + +\spadcommand{numberOfDivisors(144)} +$$ +15 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{sumOfDivisors(144)} +$$ +403 +$$ +\returnType{Type: PositiveInteger} + +The key is that {\tt d(n)} and {\tt $\sigma$(n)} are ``multiplicative +functions.'' This means that when {\tt n} and {\tt m} are relatively +prime, that is, when {\tt n} and {\tt m} have no prime factor in +common, then {\tt d(nm) = d(n) d(m)} and {\tt $\sigma$(nm) = +$\sigma$(n) $\sigma$(m)}. Note that these functions are trivial to +compute when {\tt n} is a prime power and are computed for general +{\tt n} from the prime factorization of {\tt n}. Other examples of +multiplicative functions are {\tt $\sigma_k$(n)}, the sum of the +$k$-th powers of the divisors of {\tt n} and $\varphi(n)$, the +number of integers between 1 and {\tt n} which are prime to {\tt n}. +The corresponding Axiom operations are called +\spadfunFrom{sumOfKthPowerDivisors}{IntegerNumberTheoryFunctions} and +\spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions}. + +An interesting function is {\tt $\mu$(n)}, the M\"{o}bius $\mu$ +function, defined as follows: {\tt $\mu$(1) = 1}, {\tt $\mu$(n) = 0}, +when {\tt n} is divisible by a square, and {\tt $\mu = {(-1)}^k$, when +{\tt n} is the product of {\tt k} distinct primes. The corresponding +Axiom operation is \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions}. +This function occurs in the following theorem: + +\noindent + +{\bf Theorem} (M\"{o}bius Inversion Formula): \newline Let {\tt f(n)} +be a function on the positive integers and let {\tt F(n)} be defined +by $${F(n) = \sum_{d \mid n} f(n)}$$ sum of {\tt f(n)} over +{\tt d | n}} where the sum is taken over the positive divisors of +{\tt n}. Then the values of {\tt f(n)} can be recovered from the values of +{\tt F(n)}: +$${f(n) = \sum_{d \mid n} \mu(n) F({{n}\over{d}})}$$ +where again the sum is taken over the positive divisors of {\tt n}. + +When {\tt f(n) = 1}, then {\tt F(n) = d(n)}. Thus, if you sum $\mu(d) +\cdot d(n/d)$ over the positive divisors {\tt d} of {\tt n}, you +should always get {\tt 1}. + +\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)]) } +\returnType{Void} + +\spadcommand{f1(200) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{f1(846) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Similarly, when {\tt f(n) = n}, then {\tt F(n) = $\sigma$(n)}. Thus, +if you sum {\tt $\mu$(d) $\cdot$ $\sigma$(n/d)} over the positive +divisors {\tt d} of {\tt n}, you should always get {\tt n}. + +\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)]) } +\returnType{Void} + +\spadcommand{f2(200) } +$$ +200 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{f2(846) } +$$ +846 +$$ +\returnType{Type: PositiveInteger} + +The Fibonacci numbers are defined by $F(1) = F(2) = 1$ and +$F(n) = F(n-1) + F(n-2)$ for $n = 3,4,\ldots$. + +The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} +computes the $n$-th Fibonacci number. + +\spadcommand{fibonacci(25)} +$$ +75025 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{[fibonacci(n) for n in 1..15]} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, {89}, +{144}, {233}, {377}, {610} +\right] +$$ +\returnType{Type: List Integer} + +Fibonacci numbers can also be expressed as sums of binomial coefficients. + +\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)]) } +\returnType{Void} + +\spadcommand{fib(25) } +$$ +75025 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{[fib(n) for n in 1..15] } +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, {89}, +{144}, {233}, {377}, {610} +\right] +$$ +\returnType{Type: List Integer} + +Quadratic symbols can be computed with the operations +\spadfunFrom{legendre}{IntegerNumberTheoryFunctions} and +\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions}. The Legendre +symbol $\left({a \over p}\right)$ is defined for integers $a$ and +$p$ with $p$ an odd prime number. By definition, +$\left({a\over p}\right)$ = +1, when $a$ is a square $({\rm mod\ }p)$, +$\left({a \over p}\right)$ = -1, when $a$ is not a square $({\rm mod\ }p)$, +and $\left({a \over p}\right)$ = 0, when $a$ is divisible by $p$. + +You compute $\left({a \over p}\right)$ via the command {\tt legendre(a,p)}. + +\spadcommand{legendre(3,5)} +$$ +-1 +$$ +\returnType{Type: Integer} + +\spadcommand{legendre(23,691)} +$$ +-1 +$$ +\returnType{Type: Integer} + +The Jacobi symbol $\left({a \over n}\right)$ is the usual extension of +the Legendre symbol, where {\tt n} is an arbitrary integer. The most +important property of the Jacobi symbol is the following: if {\tt K} +is a quadratic field with discriminant {\tt d} and quadratic character +$\chi$, then $\chi(n) = (d/n)$. Thus, you can use the Jacobi symbol +to compute, say, the class numbers of imaginary quadratic fields from +a standard class number formula. + +This function computes the class number of the imaginary quadratic +field with discriminant {\tt d}. + +\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2)) } +\returnType{Void} + +\spadcommand{h(-163) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{h(-499) } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{h(-1832) } +$$ +26 +$$ +\returnType{Type: PositiveInteger} + +\section{Kernel} +\label{KernelXmpPage} + +A {\it kernel} is a symbolic function application (such as {\tt sin(x+ y)}) +or a symbol (such as {\tt x}). More precisely, a non-symbol +kernel over a set {\it S} is an operator applied to a given list of +arguments from {\it S}. The operator has type {\tt BasicOperator} +(see \ref{BasicOperatorXmpPage} on page~\pageref{BasicOperatorXmpPage}) +and the kernel object is usually part of an expression object (see +\ref{ExpressionXmpPage} on page~\pageref{ExpressionXmpPage}). + +Kernels are created implicitly for you when you create expressions. + +\spadcommand{x :: Expression Integer} +$$ +x +$$ +\returnType{Type: Expression Integer} + +You can directly create a ``symbol'' kernel by using the +\spadfunFrom{kernel}{Kernel} operation. + +\spadcommand{kernel x} +$$ +x +$$ +\returnType{Type: Kernel Expression Integer} + +This expression has two different kernels. + +\spadcommand{sin(x) + cos(x) } +$$ +{\sin +\left( +{x} +\right)}+{\cos +\left( +{x} +\right)} +$$ +\returnType{Type: Expression Integer} + +The operator \spadfunFrom{kernels}{Expression} returns a list of the +kernels in an object of type {\tt Expression}. + +\spadcommand{kernels \% } +$$ +\left[ +{\sin +\left( +{x} +\right)}, +{\cos +\left( +{x} +\right)} +\right] +$$ +\returnType{Type: List Kernel Expression Integer} + +This expression also has two different kernels. + +\spadcommand{sin(x)**2 + sin(x) + cos(x) } +$$ +{{\sin +\left( +{x} +\right)} +\sp 2}+{\sin +\left( +{x} +\right)}+{\cos +\left( +{x} +\right)} +$$ +\returnType{Type: Expression Integer} + +The {\tt sin(x)} kernel is used twice. + +\spadcommand{kernels \% } +$$ +\left[ +{\sin +\left( +{x} +\right)}, +{\cos +\left( +{x} +\right)} +\right] +$$ +\returnType{Type: List Kernel Expression Integer} + +An expression need not contain any kernels. + +\spadcommand{kernels(1 :: Expression Integer)} +$$ +\left[\ +\right] +$$ +\returnType{Type: List Kernel Expression Integer} + +If one or more kernels are present, one of them is +designated the {\it main} kernel. + +\spadcommand{mainKernel(cos(x) + tan(x))} +$$ +\tan +\left( +{x} +\right) +$$ +\returnType{Type: Union(Kernel Expression Integer,...)} + +Kernels can be nested. Use \spadfunFrom{height}{Kernel} to determine +the nesting depth. + +\spadcommand{height kernel x} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +This has height 2 because the {\tt x} has height 1 and then we apply +an operator to that. + +\spadcommand{height mainKernel(sin x)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{height mainKernel(sin cos x)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{height mainKernel(sin cos (tan x + sin x))} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Use the \spadfunFrom{operator}{Kernel} operation to extract the +operator component of the kernel. The operator has type {\tt BasicOperator}. + +\spadcommand{operator mainKernel(sin cos (tan x + sin x))} +$$ +\sin +$$ +\returnType{Type: BasicOperator} + +Use the \spadfunFrom{name}{Kernel} operation to extract the name of +the operator component of the kernel. The name has type {\tt Symbol}. +This is really just a shortcut for a two-step process of extracting +the operator and then calling \spadfunFrom{name}{BasicOperator} on +the operator. + +\spadcommand{name mainKernel(sin cos (tan x + sin x))} +$$ +\sin +$$ +\returnType{Type: Symbol} + +Axiom knows about functions such as {\tt sin}, {\tt cos} and so on and +can make kernels and then expressions using them. To create a kernel +and expression using an arbitrary operator, use +\spadfunFrom{operator}{BasicOperator}. + +Now {\tt f} can be used to create symbolic function applications. + +\spadcommand{f := operator 'f } +$$ +f +$$ +\returnType{Type: BasicOperator} + +\spadcommand{e := f(x, y, 10) } +$$ +f +\left( +{x, y, {10}} +\right) +$$ +\returnType{Type: Expression Integer} + +Use the \spadfunFrom{is?}{Kernel} operation to learn if the +operator component of a kernel is equal to a given operator. + +\spadcommand{is?(e, f) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +You can also use a symbol or a string as the second argument to +\spadfunFrom{is?}{Kernel}. + +\spadcommand{is?(e, 'f) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Use the \spadfunFrom{argument}{Kernel} operation to get a list containing +the argument component of a kernel. + +\spadcommand{argument mainKernel e } +$$ +\left[ +x, y, {10} +\right] +$$ +\returnType{Type: List Expression Integer} + +Conceptually, an object of type {\tt Expression} can be thought of a +quotient of multivariate polynomials, where the ``variables'' are +kernels. The arguments of the kernels are again expressions and so +the structure recurses. See \ref{ExpressionXmpPage} on +page~\pageref{ExpressionXmpPage} for examples of +using kernels to take apart expression objects. + +\section{KeyedAccessFile} +\label{KeyedAccessFileXmpPage} + +The domain {\tt KeyedAccessFile(S)} provides files which can be used +as associative tables. Data values are stored in these files and can +be retrieved according to their keys. The keys must be strings so +this type behaves very much like the {\tt StringTable(S)} domain. The +difference is that keyed access files reside in secondary storage +while string tables are kept in memory. For more information on +table-oriented operations, see the description of {\tt Table}. + +Before a keyed access file can be used, it must first be opened. +A new file can be created by opening it for output. + +\spadcommand{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output") } +$$ +\mbox{\tt "/tmp/editor.year"} +$$ +\returnType{Type: KeyedAccessFile Integer} + +Just as for vectors, tables or lists, values are saved in a keyed access file +by setting elements. + +\spadcommand{ey."Char" := 1986 } +$$ +1986 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{ey."Caviness" := 1985 } +$$ +1985 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{ey."Fitch" := 1984 } +$$ +1984 +$$ +\returnType{Type: PositiveInteger} + +Values are retrieved using application, in any of its syntactic forms. + +\spadcommand{ey."Char"} +$$ +1986 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{ey("Char")} +$$ +1986 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{ey "Char"} +$$ +1986 +$$ +\returnType{Type: PositiveInteger} + +Attempting to retrieve a non-existent element in this way causes an error. +If it is not known whether a key exists, you should use the +\spadfunFrom{search}{KeyedAccessFile} operation. + +\spadcommand{search("Char", ey) } +$$ +1986 +$$ +\returnType{Type: Union(Integer,...)} + +\spadcommand{search("Smith", ey)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +When an entry is no longer needed, it can be removed from the file. + +\spadcommand{remove!("Char", ey) } +$$ +1986 +$$ +\returnType{Type: Union(Integer,...)} + +The \spadfunFrom{keys}{KeyedAccessFile} operation returns a list of all the +keys for a given file. + +\spadcommand{keys ey } +$$ +\left[ +\mbox{\tt "Fitch"} , \mbox{\tt "Caviness"} +\right] +$$ +\returnType{Type: List String} + +The \spadfunFrom{\#}{KeyedAccessFile} operation gives the +number of entries. + +\spadcommand{\#ey} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +The table view of keyed access files provides safe operations. That +is, if the Axiom program is terminated between file operations, the +file is left in a consistent, current state. This means, however, +that the operations are somewhat costly. For example, after each +update the file is closed. + +Here we add several more items to the file, then check its contents. + +\spadcommand{KE := Record(key: String, entry: Integer) } +$$ +\mbox{\rm Record(key: String,entry: Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{reopen!(ey, "output") } +$$ +\mbox{\tt "/tmp/editor.year"} +$$ +\returnType{Type: KeyedAccessFile Integer} + +If many items are to be added to a file at the same time, then +it is more efficient to use the \spadfunFrom{write}{KeyedAccessFile} operation. + +\spadcommand{write!(ey, ["van Hulzen", 1983]\$KE) } +$$ +\left[ +{key= \mbox{\tt "van Hulzen"} }, {entry={1983}} +\right] +$$ +\returnType{Type: Record(key: String,entry: Integer)} + +\spadcommand{write!(ey, ["Calmet", 1982]\$KE)} +$$ +\left[ +{key= \mbox{\tt "Calmet"} }, {entry={1982}} +\right] +$$ +\returnType{Type: Record(key: String,entry: Integer)} + +\spadcommand{write!(ey, ["Wang", 1981]\$KE)} +$$ +\left[ +{key= \mbox{\tt "Wang"} }, {entry={1981}} +\right] +$$ +\returnType{Type: Record(key: String,entry: Integer)} + +\spadcommand{close! ey} +$$ +\mbox{\tt "/tmp/editor.year"} +$$ +\returnType{Type: KeyedAccessFile Integer} + +The \spadfunFrom{read}{KeyedAccessFile} operation is also available +from the file view, but it returns elements in a random order. It is +generally clearer and more efficient to use the +\spadfunFrom{keys}{KeyedAccessFile} operation and to extract elements +by key. + +\spadcommand{keys ey} +$$ +\left[ +\mbox{\tt "Wang"} , \mbox{\tt "Calmet"} , \mbox{\tt "van Hulzen"} , +\mbox{\tt "Fitch"} , \mbox{\tt "Caviness"} +\right] +$$ +\returnType{Type: List String} + +\spadcommand{members ey} +$$ +\left[ +{1981}, {1982}, {1983}, {1984}, {1985} +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{)system rm -r /tmp/editor.year} + +For more information on related topics, see +\ref{FileXmpPage} on page~\pageref{FileXmpPage}, +\ref{TextFileXmpPage} on page~\pageref{TextFileXmpPage}, and +\ref{LibraryXmpPage} on page~\pageref{LibraryXmpPage}. + +\section{LexTriangularPackage} +\label{LexTriangularPackageXmpPage} + +The {\tt LexTriangularPackage} package constructor provides an +implementation of the {\em lexTriangular} algorithm (D. Lazard +``Solving Zero-dimensional Algebraic Systems'', J. of Symbol. Comput., +1992). This algorithm decomposes a zero-dimensional variety into +zero-sets of regular triangular sets. Thus the input system must have +a finite number of complex solutions. Moreover, this system needs to +be a lexicographical Groebner basis. + +This package takes two arguments: the coefficient-ring {\bf R} of the +polynomials, which must be a {\tt GcdDomain} and their set of +variables given by {\bf ls} a {\tt List Symbol}. The type of the +input polynomials must be {\tt NewSparseMultivariatePolynomial(R,V)} +where {\bf V} is {\tt OrderedVariableList(ls)}. The abbreviation for +{\tt LexTriangularPackage} is {\tt LEXTRIPK}. The main operations are +\spadfunFrom{lexTriangular}{LexTriangularPackage} and +\spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage}. The +later provide decompositions by means of square-free regular +triangular sets, built with the {\tt SREGSET} constructor, whereas the +former uses the {\tt REGSET} constructor. Note that these +constructors also implement another algorithm for solving algebraic +systems by means of regular triangular sets; in that case no +computations of Groebner bases are needed and the input system may +have any dimension (i.e. it may have an infinite number of solutions). + +The implementation of the {\em lexTriangular} algorithm provided in +the {\tt LexTriangularPackage} constructor differs from that reported +in ``Computations of gcd over algebraic towers of simple extensions'' by +M. Moreno Maza and R. Rioboo (in proceedings of AAECC11, Paris, 1995). +Indeed, the \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage} +operation removes all multiplicities of the solutions (i.e. the computed +solutions are pairwise different) and the +\spadfunFrom{lexTriangular}{LexTriangularPackage} operation may keep +some multiplicities; this later operation runs generally faster than +the former. + +The interest of the {\em lexTriangular} algorithm is due to the +following experimental remark. For some examples, a triangular +decomposition of a zero-dimensional variety can be computed faster via +a lexicographical Groebner basis computation than by using a direct +method (like that of {\tt SREGSET} and {\tt REGSET}). This happens +typically when the total degree of the system relies essentially on +its smallest variable (like in the {\em Katsura} systems). When this +is not the case, the direct method may give better timings (like in +the {\em Rose} system). + +Of course, the direct method can also be applied to a lexicographical +Groebner basis. However, the {\em lexTriangular} algorithm takes +advantage of the structure of this basis and avoids many unnecessary +computations which are performed by the direct method. + +For this purpose of solving algebraic systems with a finite number of +solutions, see also the {\tt ZeroDimensionalSolvePackage}. It allows +to use both strategies (the lexTriangular algorithm and the direct +method) for computing either the complex or real roots of a system. + +Note that the way of understanding triangular decompositions is +detailed in the example of the {\tt RegularTriangularSet} constructor. + +Since the {\tt LEXTRIPK} package constructor is limited to +zero-dimensional systems, it provides a +\spadfunFrom{zeroDimensional?}{LexTriangularPackage} operation to +check whether this requirement holds. There is also a +\spadfunFrom{groebner}{LexTriangularPackage} operation to compute the +lexicographical Groebner basis of a set of polynomials with type {\tt +NewSparseMultivariatePolynomial(R,V)}. The elimination ordering is +that given by {\bf ls} (the greatest variable being the first element +of {\bf ls}). This basis is computed by the {\em FLGM} algorithm +(Faugere et al. ``Efficient Computation of Zero-Dimensional Groebner +Bases by Change of Ordering'' , J. of Symbol. Comput., 1993) +implemented in the {\tt LinGroebnerPackage} package constructor. +Once a lexicographical Groebner basis is computed, +then one can call the operations +\spadfunFrom{lexTriangular}{LexTriangularPackage} +and \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage}. +Note that these operations admit an optional argument +to produce normalized triangular sets. +There is also a \spadfunFrom{zeroSetSplit}{LexTriangularPackage} operation +which does all the job from the input system; +an error is produced if this system is not zero-dimensional. + +Let us illustrate the facilities of the {\tt LEXTRIPK} constructor +by a famous example, the {\em cyclic-6 root} system. + +Define the coefficient ring. + +\spadcommand{R := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the list of variables, + +\spadcommand{ls : List Symbol := [a,b,c,d,e,f] } +$$ +\left[ +a, b, c, d, e, f +\right] +$$ +\returnType{Type: List Symbol} + +and make it an ordered set. + +\spadcommand{V := OVAR(ls) } +$$ +\mbox{\rm OrderedVariableList [a,b,c,d,e,f]} +$$ +\returnType{Type: Domain} + +Define the polynomial ring. + +\spadcommand{P := NSMP(R, V)} +$$ +\mbox{\rm NewSparseMultivariatePolynomial(Integer,OrderedVariableList +[a,b,c,d,e,f])} +$$ +\returnType{Type: Domain} + +Define the polynomials. + +\spadcommand{p1: P := a*b*c*d*e*f - 1 } +$$ +{f \ e \ d \ c \ b \ a} -1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f } +$$ +{{\left( {{\left( {{\left( {{\left( e+f +\right)} +\ d}+{f \ e} +\right)} +\ c}+{f \ e \ d} +\right)} +\ b}+{f \ e \ d \ c} +\right)} +\ a}+{f \ e \ d \ c \ b} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f } +$$ +{{\left( {{\left( {{\left( d+f +\right)} +\ c}+{f \ e} +\right)} +\ b}+{f \ e \ d} +\right)} +\ a}+{e \ d \ c \ b}+{f \ e \ d \ c} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f } +$$ +{{\left( {{\left( c+f +\right)} +\ b}+{f \ e} +\right)} +\ a}+{d \ c \ b}+{e \ d \ c}+{f \ e \ d} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f } +$$ +{{\left( b+f +\right)} +\ a}+{c \ b}+{d \ c}+{e \ d}+{f \ e} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{p6: P := a + b + c + d + e + f } +$$ +a+b+c+d+e+f +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +\spadcommand{lp := [p1, p2, p3, p4, p5, p6]} +$$ +\begin{array}{@{}l} +\left[ +{{f \ e \ d \ c \ b \ a} -1}, +\right. +\\ +\\ +\displaystyle +{{{\left( +{{\left( +{{\left( +{{\left( e+f \right)} +\ d}+{f \ e} +\right)} +\ c}+{f \ e \ d} +\right)} +\ b}+{f \ e \ d \ c} +\right)} +\ a}+{f \ e \ d \ c \ b}}, +\\ +\\ +\displaystyle +{{{\left( +{{\left( +{{\left( d+f +\right)} +\ c}+{f \ e} +\right)} +\ b}+{f \ e \ d} +\right)} +\ a}+{e \ d \ c \ b}+{f \ e \ d \ c}}, +\\ +\\ +\displaystyle +{{{\left( +{{\left( c+f +\right)} +\ b}+{f \ e} +\right)} +\ a}+{d \ c \ b}+{e \ d \ c}+{f \ e \ d}}, +\\ +\\ +\displaystyle +{{{\left( b+f +\right)} +\ a}+{c \ b}+{d \ c}+{e \ d}+{f \ e}}, +\\ +\\ +\displaystyle +\left. +{a+b+c+d+e+f} +\right] +\end{array} +$$ +\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +Now call {\tt LEXTRIPK} . + +\spadcommand{lextripack := LEXTRIPK(R,ls)} +$$ +LexTriangularPackage(Integer,[a,b,c,d,e,f]) +$$ +\returnType{Type: Domain} + +Compute the lexicographical Groebner basis of the system. +This may take between 5 minutes and one hour, depending on your machine. + +\spadcommand{lg := groebner(lp)\$lextripack} + +$$ +\left[ +{a+b+c+d+e+f}, +\right. +$$ +$$ +\begin{array}{@{}l} +{{3968379498283200} \ {b \sp 2}}+ +{{15873517993132800} \ f \ b}+ +\\ +\displaystyle +{{3968379498283200} \ {d \sp 2}}+ +{{15873517993132800} \ f \ d}+ +\\ +\displaystyle +{{3968379498283200} \ {f \sp 3} \ {e \sp 5}} - +{{15873517993132800} \ {f \sp 4} \ {e \sp 4}}+ +\\ +\displaystyle +{{23810276989699200} \ {f \sp 5} \ {e \sp 3}}+ +\left( +{{206355733910726400} \ {f \sp 6}}+ +\right. +\\ +\displaystyle +\left. +{230166010900425600} +\right)\ {e \sp 2}+ +\left( +-{{729705987316687} \ {f \sp {43}}}+ +\right. +\\ +\displaystyle +{{1863667496867205421} \ {f \sp {37}}}+ +{{291674853771731104461} \ {f \sp {31}}}+ +\\ +\displaystyle +{{365285994691106921745} \ {f \sp {25}}}+ +{{549961185828911895} \ {f \sp {19}}} - +\\ +\displaystyle +{{365048404038768439269} \ {f \sp {13}}} - +{{292382820431504027669} \ {f \sp 7}} - +\\ +\displaystyle +\left. +{{2271898467631865497} \ f} +\right)\ e - +{{3988812642545399} \ {f \sp {44}}}+ +\\ +\displaystyle +{{10187423878429609997} \ {f \sp {38}}}+ +{{1594377523424314053637} \ {f \sp {32}}}+ +\\ +\displaystyle +{{1994739308439916238065} \ {f \sp {26}}}+ +{{1596840088052642815} \ {f \sp {20}}} - +\\ +\displaystyle +{{1993494118301162145413} \ {f \sp {14}}} - +{{1596049742289689815053} \ {f \sp 8}} - +\\ +\displaystyle +{{11488171330159667449} \ {f \sp 2}}, \hbox{\hskip 8.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{23810276989699200} \ c} - +{{23810276989699200} \ f} +\right)\ b+ +\\ +\displaystyle +{{23810276989699200} \ {c \sp 2}}+ +{{71430830969097600} \ f \ c} - +\\ +\displaystyle +{{23810276989699200} \ {d \sp 2}} - +{{95241107958796800} \ f \ d} - +\\ +\displaystyle +{{55557312975964800} \ {f \sp 3} \ {e \sp 5}}+ +{{174608697924460800} \ {f \sp 4} \ {e \sp 4}} - +\\ +\displaystyle +{{174608697924460800} \ {f \sp 5} \ {e \sp 3}}+ +\left( +-{{2428648252949318400} \ {f \sp 6}} - +\right. +\\ +\displaystyle +\left. +{2611193709870345600} +\right)\ {e \sp 2}+ +\left( +{{8305444561289527} \ {f \sp {43}}} - +\right. +\\ +\displaystyle +{{21212087151945459641} \ {f \sp {37}}} - +{{3319815883093451385381} \ {f \sp {31}}} - +\\ +\displaystyle +{{4157691646261657136445} \ {f \sp {25}}} - +{{6072721607510764095} \ {f \sp {19}}}+ +\\ +\displaystyle +{{4154986709036460221649} \ {f \sp {13}}}+ +{{3327761311138587096749} \ {f \sp 7}}+ +\\ +\displaystyle +\left. +{{25885340608290841637} \ f} +\right)\ e+ +{{45815897629010329} \ {f \sp {44}}} - +\\ +\displaystyle +{{117013765582151891207} \ {f \sp {38}}} - +{{18313166848970865074187} \ {f \sp {32}}}- +\\ +\displaystyle +{{22909971239649297438915} \ {f \sp {26}}} - +{{16133250761305157265} \ {f \sp {20}}}+ +\\ +\displaystyle +{{22897305857636178256623} \ {f \sp {14}}}+ +{{18329944781867242497923} \ {f \sp 8}}+ +\\ +\displaystyle +{{130258531002020420699} \ {f \sp 2}}, \hbox{\hskip 8.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{7936758996566400} \ d} - +{{7936758996566400} \ f} +\right)\ b - +\\ +\displaystyle +{{7936758996566400} \ f \ d} - +{{7936758996566400} \ {f \sp 3} \ {e \sp 5}}+ +\\ +\displaystyle +{{23810276989699200} \ {f \sp 4} \ {e \sp 4}} - +{{23810276989699200} \ {f \sp 5} \ {e \sp 3}}+ +\\ +\displaystyle +\left( +-{{337312257354072000} \ {f \sp 6}} - +{369059293340337600} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( +{{1176345388640471} \ {f \sp {43}}} - +{{3004383582891473073} \ {f \sp {37}}} - +\right. +\\ +\displaystyle +{{470203502707246105653} \ {f \sp {31}}} - +{{588858183402644348085} \ {f \sp {25}}} - +\\ +\displaystyle +{{856939308623513535} \ {f \sp {19}}}+ +{{588472674242340526377} \ {f \sp {13}}}+ +\\ +\displaystyle +\left. +{{471313241958371103517} \ {f \sp 7}}+ +{{3659742549078552381} \ f} +\right)\ e+ +\\ +\displaystyle +{{6423170513956901} \ {f \sp {44}}} - +{{16404772137036480803} \ {f \sp {38}}} - +\\ +\displaystyle +{{2567419165227528774463} \ {f \sp {32}}} - +{{3211938090825682172335} \ {f \sp {26}}} - +\\ +\displaystyle +{{2330490332697587485} \ {f \sp {20}}}+ +{{3210100109444754864587} \ {f \sp {14}}}+ +\\ +\displaystyle +{{2569858315395162617847} \ {f \sp 8}}+ +{{18326089487427735751} \ {f \sp 2}}, \hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{11905138494849600} \ e} - +{{11905138494849600} \ f} +\right)\ b - +\\ +\displaystyle +{{3968379498283200} \ {f \sp 3} \ {e \sp 5}}+ +{{15873517993132800} \ {f \sp 4} \ {e \sp 4}} - +\\ +\displaystyle +{{27778656487982400} \ {f \sp 5} \ {e \sp 3}}+ +\left( +-{{208339923659868000} \ {f \sp 6}} - +\right. +\\ +\displaystyle +\left. +{240086959646133600} +\right)\ {e \sp 2}+ +\left( +{{786029984751110} \ {f \sp {43}}} - +\right. +\\ +\displaystyle +{{2007519008182245250} \ {f \sp {37}}} - +{{314188062908073807090} \ {f \sp {31}}} - +\\ +\displaystyle +{{393423667537929575250} \ {f \sp {25}}} - +{{550329120654394950} \ {f \sp {19}}}+ +\\ +\displaystyle +{{393196408728889612770} \ {f \sp {13}}}+ +{{314892372799176495730} \ {f \sp 7}}+ +\\ +\displaystyle +\left. +{{2409386515146668530} \ f} +\right)\ e+ +{{4177638546747827} \ {f \sp {44}}} - +\\ +\displaystyle +{{10669685294602576381} \ {f \sp {38}}} - +{{1669852980419949524601} \ {f \sp {32}}} - +\\ +\displaystyle +{{2089077057287904170745} \ {f \sp {26}}} - +{{1569899763580278795} \ {f \sp {20}}}+ +\\ +\displaystyle +{{2087864026859015573349} \ {f \sp {14}}}+ +{{1671496085945199577969} \ {f \sp 8}}+ +\\ +\displaystyle +{{11940257226216280177} \ {f \sp 2}}, \hbox{\hskip 8.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{11905138494849600} \ {f \sp 6}} -{11905138494849600} +\right)\ b - +\\ +\displaystyle +{{15873517993132800} \ {f \sp 2} \ {e \sp 5}}+ +{{39683794982832000} \ {f \sp 3} \ {e \sp 4}} - +\\ +\displaystyle +{{39683794982832000} \ {f \sp 4} \ {e \sp 3}}+ +\left( -{{686529653202993600} \ {f \sp {11}}} - +\right. +\\ +\displaystyle +\left. +{{607162063237329600} \ {f \sp 5}} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( {{65144531306704} \ {f \sp {42}}} - +{{166381280901088652} \ {f \sp {36}}} - +\right. +\\ +\displaystyle +{{26033434502470283472} \ {f \sp {30}}} - +{{31696259583860650140} \ {f \sp {24}}}+ +\\ +\displaystyle +{{971492093167581360} \ {f \sp {18}}}+ +{{32220085033691389548} \ {f \sp {12}}}+ +\\ +\displaystyle +\left. +{{25526177666070529808} \ {f \sp 6}}+ +{138603268355749244} +\right)\ e+ +\\ +\displaystyle +{{167620036074811} \ {f \sp {43}}} - +{{428102417974791473} \ {f \sp {37}}} - +\\ +\displaystyle +{{66997243801231679313} \ {f \sp {31}}} - +{{83426716722148750485} \ {f \sp {25}}}+ +\\ +\displaystyle +{{203673895369980765} \ {f \sp {19}}}+ +{{83523056326010432457} \ {f \sp {13}}}+ +\\ +\displaystyle +{{66995789640238066937} \ {f \sp 7}}+ +{{478592855549587901} \ f}, \hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{801692827936} \ {c \sp 3}}+ +{{2405078483808} \ f \ {c \sp 2}} - +\\ +\displaystyle +{{2405078483808} \ {f \sp 2} \ c} - +{{13752945467} \ {f \sp {45}}}+ +\\ +\displaystyle +{{35125117815561} \ {f \sp {39}}}+ +{{5496946957826433} \ {f \sp {33}}}+ +\\ +\displaystyle +{{6834659447749117} \ {f \sp {27}}} - +{{44484880462461} \ {f \sp {21}}} - +\\ +\displaystyle +{{6873406230093057} \ {f \sp {15}}} - +{{5450844938762633} \ {f \sp 9}}+ +\\ +\displaystyle +{{1216586044571} \ {f \sp 3}},\hbox{\hskip 9.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( {{23810276989699200} \ d} - +{{23810276989699200} \ f} +\right)\ c+ +\\ +\displaystyle +{{23810276989699200} \ {d \sp 2}}+ +{{71430830969097600} \ f \ d}+ +\\ +\displaystyle +{{7936758996566400} \ {f \sp 3} \ {e \sp 5}} - +{{31747035986265600} \ {f \sp 4} \ {e \sp 4}}+ +\\ +\displaystyle +{{31747035986265600} \ {f \sp 5} \ {e \sp 3}}+ +\left( {{404774708824886400} \ {f \sp 6}}+ +\right. +\\ +\displaystyle +\left. +{396837949828320000} +\right)\ {e \sp 2}+ +\left( +-{{1247372229446701} \ {f \sp {43}}}+ +\right. +\\ +\displaystyle +{{3185785654596621203} \ {f \sp {37}}}+ +{{498594866849974751463} \ {f \sp {31}}}+ +\\ +\displaystyle +{{624542545845791047935} \ {f \sp {25}}}+ +{{931085755769682885} \ {f \sp {19}}} - +\\ +\displaystyle +{{624150663582417063387} \ {f \sp {13}}} - +{{499881859388360475647} \ {f \sp 7}} - +\\ +\displaystyle +\left. +{{3926885313819527351} \ f} +\right)\ e - +{{7026011547118141} \ {f \sp {44}}}+ +\\ +\displaystyle +{{17944427051950691243} \ {f \sp {38}}}+ +{{2808383522593986603543} \ {f \sp {32}}}+ +\\ +\displaystyle +{{3513624142354807530135} \ {f \sp {26}}}+ +{{2860757006705537685} \ {f \sp {20}}} - +\\ +\displaystyle +{{3511356735642190737267} \ {f \sp {14}}} - +{{2811332494697103819887} \ {f \sp 8}} - +\\ +\displaystyle +{{20315011631522847311} \ {f \sp 2}}, \hbox{\hskip 8.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{7936758996566400} \ e} -{{7936758996566400} \ f} +\right)\ c+ +\\ +\displaystyle +\left( +-{{4418748183673} \ {f \sp {43}}}+ +\right. +\\ +\displaystyle +{{11285568707456559} \ {f \sp {37}}}+ +{{1765998617294451019} \ {f \sp {31}}}+ +\\ +\displaystyle +{{2173749283622606155} \ {f \sp {25}}} - +{{55788292195402895} \ {f \sp {19}}} - +\\ +\displaystyle +{{2215291421788292951} \ {f \sp {13}}} - +{{1718142665347430851} \ {f \sp 7}}+ +\\ +\displaystyle +\left. +{{30256569458230237} \ f} +\right)\ e+ +{{4418748183673} \ {f \sp {44}}} - +\\ +\displaystyle +{{11285568707456559} \ {f \sp {38}}} - +{{1765998617294451019} \ {f \sp {32}}} - +\\ +\displaystyle +{{2173749283622606155} \ {f \sp {26}}}+ +{{55788292195402895} \ {f \sp {20}}}+ +\\ +\displaystyle +{{2215291421788292951} \ {f \sp {14}}}+ +{{1718142665347430851} \ {f \sp 8}} - +\\ +\displaystyle +{{30256569458230237} \ {f \sp 2}}, \hbox{\hskip 9.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{72152354514240} \ {f \sp 6}} - +{72152354514240} +\right)\ c+ +\\ +\displaystyle +{{40950859449} \ {f \sp {43}}} - +{{104588980990367} \ {f \sp {37}}} - +\\ +\displaystyle +{{16367227395575307} \ {f \sp {31}}} - +{{20268523416527355} \ {f \sp {25}}}+ +\\ +\displaystyle +{{442205002259535} \ {f \sp {19}}}+ +{{20576059935789063} \ {f \sp {13}}}+ +\\ +\displaystyle +{{15997133796970563} \ {f \sp 7}} - +{{275099892785581} \ f}, \hbox{\hskip 5.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{1984189749141600} \ {d \sp 3}}+ +{{5952569247424800} \ f \ {d \sp 2}} - +\\ +\displaystyle +{{5952569247424800} \ {f \sp 2} \ d} - +{{3968379498283200} \ {f \sp 4} \ {e \sp 5}}+ +\\ +\displaystyle +{{15873517993132800} \ {f \sp 5} \ {e \sp 4}}+ +{{17857707742274400} \ {e \sp 3}}+ +\\ +\displaystyle +\left( +-{{148814231185620000} \ {f \sp 7}} - +{{162703559429611200} \ f} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( +-{{390000914678878} \ {f \sp {44}}}+ +{{996062704593756434} \ {f \sp {38}}}+ +\right. +\\ +\displaystyle +{{155886323972034823914} \ {f \sp {32}}}+ +{{194745956143985421330} \ {f \sp {26}}}+ +\\ +\displaystyle +{{6205077595574430} \ {f \sp {20}}} - +{{194596512653299068786} \ {f \sp {14}}} - +\\ +\displaystyle +\left. +{{155796897940756922666} \ {f \sp 8}} - +{{1036375759077320978} \ {f \sp 2}} +\right)\ e - +\\ +\displaystyle +{{374998630035991} \ {f \sp {45}}}+ +{{957747106595453993} \ {f \sp {39}}}+ +\\ +\displaystyle +{{149889155566764891693} \ {f \sp {33}}}+ +{{187154171443494641685} \ {f \sp {27}}} - +\\ +\displaystyle +{{127129015426348065} \ {f \sp {21}}} - +{{187241533243115040417} \ {f \sp {15}}} - +\\ +\displaystyle +{{149719983567976534037} \ {f \sp 9}} - +{{836654081239648061} \ {f \sp 3}}, \hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{5952569247424800} \ e} - +{{5952569247424800} \ f} +\right)\ d - +\\ +\displaystyle +{{3968379498283200} \ {f \sp 3} \ {e \sp 5}}+ +{{9920948745708000} \ {f \sp 4} \ {e \sp 4}} - +\\ +\displaystyle +{{3968379498283200} \ {f \sp 5} \ {e \sp 3}}+ +\left( +-{{148814231185620000} \ {f \sp 6}} - +\right. +\\ +\displaystyle +\left. +{150798420934761600} +\right)\ {e \sp 2}+ +\left( +{{492558110242553} \ {f \sp {43}}} - +\right. +\\ +\displaystyle +{{1257992359608074599} \ {f \sp {37}}} - +{{196883094539368513959} \ {f \sp {31}}} - +\\ +\displaystyle +{{246562115745735428055} \ {f \sp {25}}} - +{{325698701993885505} \ {f \sp {19}}}+ +\\ +\displaystyle +{{246417769883651808111} \ {f \sp {13}}}+ +{{197327352068200652911} \ {f \sp 7}}+ +\\ +\displaystyle +\left. +{{1523373796389332143} \ f} +\right)\ e+ +{{2679481081803026} \ {f \sp {44}}} - +\\ +\displaystyle +{{6843392695421906608} \ {f \sp {38}}} - +{{1071020459642646913578} \ {f \sp {32}}} - +\\ +\displaystyle +{{1339789169692041240060} \ {f \sp {26}}} - +{{852746750910750210} \ {f \sp {20}}}+ +\\ +\displaystyle +{{1339105101971878401312} \ {f \sp {14}}}+ +{{1071900289758712984762} \ {f \sp 8}}+ +\\ +\displaystyle +{{7555239072072727756} \ {f \sp 2}}, \hbox{\hskip 8.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{11905138494849600} \ {f \sp 6}} - +{11905138494849600} +\right)\ d - +\\ +\displaystyle +{{7936758996566400} \ {f \sp 2} \ {e \sp 5}}+ +{{31747035986265600} \ {f \sp 3} \ {e \sp 4}} - +\\ +\displaystyle +{{31747035986265600} \ {f \sp 4} \ {e \sp 3}}+ +\\ +\displaystyle +\left( -{{420648226818019200} \ {f \sp {11}}} - +{{404774708824886400} \ {f \sp 5}} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( +{{15336187600889} \ {f \sp {42}}} - +{{39169739565161107} \ {f \sp {36}}} - +\right. +\\ +\displaystyle +{{6127176127489690827} \ {f \sp {30}}} - +{{7217708742310509615} \ {f \sp {24}}}+ +\\ +\displaystyle +{{538628483890722735} \ {f \sp {18}}}+ +{{7506804353843507643} \ {f \sp {12}}}+ +\\ +\displaystyle +\left. +{{5886160769782607203} \ {f \sp 6}}+ +{63576108396535879} +\right)\ e+ +\\ +\displaystyle +{{71737781777066} \ {f \sp {43}}} - +{{183218856207557938} \ {f \sp {37}}} - +\\ +\displaystyle +{{28672874271132276078} \ {f \sp {31}}} - +{{35625223686939812010} \ {f \sp {25}}}+ +\\ +\displaystyle +{{164831339634084390} \ {f \sp {19}}}+ +{{35724160423073052642} \ {f \sp {13}}}+ +\\ +\displaystyle +{{28627022578664910622} \ {f \sp 7}}+ +{{187459987029680506} \ f}, \hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{1322793166094400} \ {e \sp 6}} - +{{3968379498283200} \ f \ {e \sp 5}}+ +\\ +\displaystyle +{{3968379498283200} \ {f \sp 2} \ {e \sp 4}} - +{{5291172664377600} \ {f \sp 3} \ {e \sp 3}}+ +\\ +\displaystyle +\left( -{{230166010900425600} \ {f \sp {10}}} - +{{226197631402142400} \ {f \sp 4}} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( +-{{152375364610443885} \ {f \sp {47}}}+ +{{389166626064854890415} \ {f \sp {41}}}+ +\right. +\\ +\displaystyle +{{60906097841360558987335} \ {f \sp {35}}}+ +{{76167367934608798697275} \ {f \sp {29}}}+ +\\ +\displaystyle +{{27855066785995181125} \ {f \sp {23}}} - +{{76144952817052723145495} \ {f \sp {17}}} - +\\ +\displaystyle +\left. +{{60933629892463517546975} \ {f \sp {11}}} - +{{411415071682002547795} \ {f \sp 5}} +\right)\ e - +\\ +\displaystyle +{{209493533143822} \ {f \sp {42}}}+ +{{535045979490560586} \ {f \sp {36}}}+ +\\ +\displaystyle +{{83737947964973553146} \ {f \sp {30}}}+ +{{104889507084213371570} \ {f \sp {24}}}+ +\\ +\displaystyle +{{167117997269207870} \ {f \sp {18}}} - +{{104793725781390615514} \ {f \sp {12}}} - +\\ +\displaystyle +{{83842685189903180394} \ {f \sp 6}} - +{569978796672974242}, \hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( {{25438330117200} \ {f \sp 6}}+ +{25438330117200} +\right)\ {e \sp 3}+ +\\ +\displaystyle +\left( {{76314990351600} \ {f \sp 7}}+ +{{76314990351600} \ f} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( -{{1594966552735} \ {f \sp {44}}}+ +{{4073543370415745} \ {f \sp {38}}}+ +\right. +\\ +\displaystyle +{{637527159231148925} \ {f \sp {32}}}+ +{{797521176113606525} \ {f \sp {26}}}+ +\\ +\displaystyle +{{530440941097175} \ {f \sp {20}}} - +{{797160527306433145} \ {f \sp {14}}} - +\\ +\displaystyle +\left. +{{638132320196044965} \ {f \sp 8}} - +{{4510507167940725} \ {f \sp 2}} +\right)\ e - +\\ +\displaystyle +{{6036376800443} \ {f \sp {45}}}+ +{{15416903421476909} \ {f \sp {39}}}+ +\\ +\displaystyle +{{2412807646192304449} \ {f \sp {33}}}+ +{{3017679923028013705} \ {f \sp {27}}}+ +\\ +\displaystyle +{{1422320037411955} \ {f \sp {21}}} - +{{3016560402417843941} \ {f \sp {15}}} - +\\ +\displaystyle +{{2414249368183033161} \ {f \sp 9}} - +{{16561862361763873} \ {f \sp 3}}, \hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{1387545279120} \ {f \sp {12}}} - +{1387545279120} +\right)\ {e \sp 2}+ +\\ +\displaystyle +\left( {{4321823003} \ {f \sp {43}}} - +{{11037922310209} \ {f \sp {37}}} - +\right. +\\ +\displaystyle +{{1727510711947989} \ {f \sp {31}}} - +{{2165150991154425} \ {f \sp {25}}} - +\\ +\displaystyle +{{5114342560755} \ {f \sp {19}}}+ +{{2162682824948601} \ {f \sp {13}}}+ +\\ +\displaystyle +\left. +{{1732620732685741} \ {f \sp 7}}+ +{{13506088516033} \ f} +\right)\ e+ +\\ +\displaystyle +{{24177661775} \ {f \sp {44}}} - +{{61749727185325} \ {f \sp {38}}} - +\\ +\displaystyle +{{9664106795754225} \ {f \sp {32}}} - +{{12090487758628245} \ {f \sp {26}}} - +\\ +\displaystyle +{{8787672733575} \ {f \sp {20}}}+ +{{12083693383005045} \ {f \sp {14}}}+ +\\ +\displaystyle +{{9672870290826025} \ {f \sp 8}}+ +{{68544102808525} \ {f \sp 2}}, \hbox{\hskip 5.0cm} +\end{array} +$$ +$$ +\left. +\begin{array}{@{}l} +{f \sp {48}} - +{{2554} \ {f \sp {42}}} - +{{399710} \ {f \sp {36}}}- +{{499722} \ {f \sp {30}}}+ +\\ +\displaystyle +{{499722} \ {f \sp {18}}}+ +{{399710} \ {f \sp {12}}}+ +{{2554} \ {f \sp 6}} - +1 \hbox{\hskip 6.0cm} +\end{array} +\right] +$$ +\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +Apply lexTriangular to compute a decomposition into regular triangular sets. +This should not take more than 5 seconds. + +\spadcommand{lexTriangular(lg,false)\$lextripack} +$$ +\begin{array}{@{}l} +\left[ +\begin{array}{@{}l} +\left\{ +{{f \sp 6}+1}, +{{e \sp 6} - +{3 \ f \ {e \sp 5}}+ +{3 \ {f \sp 2} \ {e \sp 4}} - +{4 \ {f \sp 3} \ {e \sp 3}}+ +{3 \ {f \sp 4} \ {e \sp 2}} - +{3 \ {f \sp 5} \ e} -1}, +\right. +\\ +\displaystyle +{{3 \ d}+{{f \sp 2} \ {e \sp 5}} - +{4 \ {f \sp 3} \ {e \sp 4}}+ +{4 \ {f \sp 4} \ {e \sp 3}} - +{2 \ {f \sp 5} \ {e \sp 2}} - +{2 \ e}+{2 \ f}}, +{c+f}, +\\ +\displaystyle +{{3 \ b}+ +{2 \ {f \sp 2} \ {e \sp 5}} - +{5 \ {f \sp 3} \ {e \sp 4}}+ +{5 \ {f \sp 4} \ {e \sp 3}} - +{{10} \ {f \sp 5} \ {e \sp 2}} - +{4 \ e}+{7 \ f}}, +\\ +\displaystyle +\left. +{a -{{f \sp 2} \ {e \sp 5}}+ +{3 \ {f \sp 3} \ {e \sp 4}} - +{3 \ {f \sp 4} \ {e \sp 3}}+ +{4 \ {f \sp 5} \ {e \sp 2}}+ +{3 \ e} -{3 \ f}} +\right\}, +\end{array} +\right. +\\ +\\ +\displaystyle +\left\{ +{{f \sp 6} -1}, +{e -f}, +{d -f}, +{{c \sp 2}+ +{4 \ f \ c}+ +{f \sp 2}}, +{{{\left( +c -f +\right)}\ b} - +{f \ c} - +{5 \ {f \sp 2}}}, +{a+b+c+{3 \ f}} +\right\}, +\\ +\\ +\displaystyle +\left\{ +{{f \sp 6} -1}, +{e -f}, +{d -f}, +{c -f}, +{{b \sp 2}+ +{4 \ f \ b}+ +{f \sp 2}}, +{a+b+{4 \ f}} +\right\}, +\\ +\\ +\displaystyle +\left\{ +{{f \sp 6} -1}, +{e -f}, +{{d \sp 2}+ +{4 \ f \ d}+ +{f \sp 2}}, +{{{\left( d -f \right)}\ c} - +{f \ d} - +{5 \ {f \sp 2}}}, +{b -f}, +{a+c+d+{3 \ f}} +\right\}, +\\ +\\ +\displaystyle +\begin{array}{@{}l} +\left\{ +{{f \sp {36}} - +{{2554} \ {f \sp {30}}} - +{{399709} \ {f \sp {24}}} - +{{502276} \ {f \sp {18}}} - +{{399709} \ {f \sp {12}}} - +{{2554} \ {f \sp 6}}+1}, +\right. +\\ +\displaystyle +\left( +{{161718564} \ {f \sp {12}}} - +{161718564} +\right)\ {e \sp 2}+ +\left( +-{{504205} \ {f \sp {31}}}+ +{{1287737951} \ {f \sp {25}}}+ +\right. +\\ +\displaystyle +\left. +{{201539391380} \ {f \sp {19}}}+ +{{253982817368} \ {f \sp {13}}}+ +{{201940704665} \ {f \sp 7}}+ +{{1574134601} \ f} +\right)\ e - +\\ +\displaystyle +{{2818405} \ {f \sp {32}}}+ +{{7198203911} \ {f \sp {26}}}+ +{{1126548149060} \ {f \sp {20}}}+ +\\ +\displaystyle +{{1416530563364} \ {f \sp {14}}}+ +{{1127377589345} \ {f \sp 8}}+ +{{7988820725} \ {f \sp 2}}, +\\ +\displaystyle +\left( +{{693772639560} \ {f \sp 6}} - +{693772639560} +\right)\ d - +{{462515093040} \ {f \sp 2} \ {e \sp 5}}+ +\\ +\displaystyle +{{1850060372160} \ {f \sp 3} \ {e \sp 4}} - +{{1850060372160} \ {f \sp 4} \ {e \sp 3}}+ +\left( +-{{24513299931120} \ {f \sp {11}}} - +\right. +\\ +\displaystyle +\left. +{{23588269745040} \ {f \sp 5}} +\right)\ {e \sp 2}+ +\left( +-{{890810428} \ {f \sp {30}}}+ +{{2275181044754} \ {f \sp {24}}}+ +\right. +\\ +\displaystyle +{{355937263869776} \ {f \sp {18}}}+ +{{413736880104344} \ {f \sp {12}}}+ +{{342849304487996} \ {f \sp 6}}+ +\\ +\displaystyle +\left. +{3704966481878} +\right)\ e - +{{4163798003} \ {f \sp {31}}}+ +{{10634395752169} \ {f \sp {25}}}+ +\\ +\displaystyle +{{1664161760192806} \ {f \sp {19}}}+ +{{2079424391370694} \ {f \sp {13}}}+ +{{1668153650635921} \ {f \sp 7}}+ +\\ +\displaystyle +{{10924274392693} \ f}, +\left( +{{12614047992} \ {f \sp 6}} - +{12614047992} +\right)\ c - +\\ +\displaystyle +{{7246825} \ {f \sp {31}}}+ +{{18508536599} \ {f \sp {25}}}+ +{{2896249516034} \ {f \sp {19}}}+ +\\ +\displaystyle +{{3581539649666} \ {f \sp {13}}}+ +{{2796477571739} \ {f \sp 7}} - +{{48094301893} \ f}, +\\ +\displaystyle +\left( +{{693772639560} \ {f \sp 6}} - +{693772639560} +\right)\ b - +{{925030186080} \ {f \sp 2} \ {e \sp 5}}+ +\\ +\displaystyle +{{2312575465200} \ {f \sp 3} \ {e \sp 4}} - +{{2312575465200} \ {f \sp 4} \ {e \sp 3}}+ +\left( +-{{40007555547960} \ {f \sp {11}}} - +\right. +\\ +\displaystyle +\left. +{{35382404617560} \ {f \sp 5}} +\right)\ {e \sp 2}+ +\left( +-{{3781280823} \ {f \sp {30}}}+ +{{9657492291789} \ {f \sp {24}}}+ +\right. +\\ +\displaystyle +{{1511158913397906} \ {f \sp {18}}}+ +{{1837290892286154} \ {f \sp {12}}}+ +{{1487216006594361} \ {f \sp 6}}+ +\\ +\displaystyle +\left. +{8077238712093} +\right)\ e - +{{9736390478} \ {f \sp {31}}}+ +{{24866827916734} \ {f \sp {25}}}+ +\\ +\displaystyle +{{3891495681905296} \ {f \sp {19}}}+ +{{4872556418871424} \ {f \sp {13}}}+ +{{3904047887269606} \ {f \sp 7}}+ +\\ +\displaystyle +\left. +{{27890075838538} \ f}, +{a+b+c+d+e+f} +\right\}, +\end{array} +\\ +\\ +\displaystyle +\left. +\left\{ +{{f \sp 6} -1}, +{{e \sp 2}+ +{4 \ f \ e}+ +{f \sp 2}}, +{{{\left( e -f \right)}\ d} - +{f \ e} - +{5 \ {f \sp 2}}}, +{c -f}, +{b -f}, +{a+d+e+{3 \ f}} +\right\} + +\right] +\end{array} +$$ +\returnType{Type: List RegularChain(Integer,[a,b,c,d,e,f])} + +Note that the first set of the decomposition is normalized (all +initials are integer numbers) but not the second one (normalized +triangular sets are defined in the description of the +{\tt NormalizedTriangularSetCategory} constructor). + +So apply now lexTriangular to produce normalized triangular sets. + +\spadcommand{lts := lexTriangular(lg,true)\$lextripack } +$$ +\begin{array}{@{}l} +\left[ +\begin{array}{@{}l} +\left\{ +{{f \sp 6}+1}, +{{e \sp 6} - +{3 \ f \ {e \sp 5}}+ +{3 \ {f \sp 2} \ {e \sp 4}} - +{4 \ {f \sp 3} \ {e \sp 3}}+ +{3 \ {f \sp 4} \ {e \sp 2}} - +{3 \ {f \sp 5} \ e} -1}, +\right. +\\ +\displaystyle +{{3 \ d}+ +{{f \sp 2} \ {e \sp 5}} - +{4 \ {f \sp 3} \ {e \sp 4}}+ +{4 \ {f \sp 4} \ {e \sp 3}} - +{2 \ {f \sp 5} \ {e \sp 2}} - +{2 \ e}+{2 \ f}}, +{c+f}, +\\ +\displaystyle +{{3 \ b}+ +{2 \ {f \sp 2} \ {e \sp 5}}- +{5 \ {f \sp 3} \ {e \sp 4}}+ +{5 \ {f \sp 4} \ {e \sp 3}} - +{{10} \ {f \sp 5} \ {e \sp 2}} - +{4 \ e}+{7 \ f}}, +\\ +\displaystyle +\left. +{a -{{f \sp 2} \ {e \sp 5}}+ +{3 \ {f \sp 3} \ {e \sp 4}} - +{3 \ {f \sp 4} \ {e \sp 3}}+ +{4 \ {f \sp 5} \ {e \sp 2}}+ +{3 \ e} - +{3 \ f}} +\right\}, +\end{array} +\right. +\\ +\\ +\displaystyle +{\left\{ {{f \sp 6} -1}, {e -f}, {d -f}, {{c \sp 2}+{4 \ f \ +c}+{f \sp 2}}, {b+c+{4 \ f}}, {a -f} +\right\}}, +\\ +\\ +\displaystyle +{\left\{ {{f \sp 6} -1}, {e -f}, {d -f}, {c -f}, {{b \sp 2}+{4 +\ f \ b}+{f \sp 2}}, {a+b+{4 \ f}} +\right\}}, +\\ +\\ +\displaystyle +{\left\{ {{f \sp 6} -1}, {e -f}, {{d \sp 2}+{4 \ f \ d}+{f \sp +2}}, {c+d+{4 \ f}}, {b -f}, {a -f} +\right\}}, +\\ +\\ +\displaystyle +\begin{array}{@{}l} +\left\{ +{{f \sp {36}} - +{{2554} \ {f \sp {30}}} - +{{399709} \ {f \sp {24}}} - +{{502276} \ {f \sp {18}}} - +{{399709} \ {f \sp {12}}} - +{{2554} \ {f \sp 6}}+ +1}, +\right. +\\ +\displaystyle +{{1387545279120} \ {e \sp 2}}+ +\left( +{{4321823003} \ {f \sp {31}}} - +{{11037922310209} \ {f \sp {25}}} - +\right. +\\ +\displaystyle +{{1727506390124986} \ {f \sp {19}}} - +{{2176188913464634} \ {f \sp {13}}} - +{{1732620732685741} \ {f \sp 7}} - +\\ +\displaystyle +\left. +{{13506088516033} \ f} +\right)\ e+ +{{24177661775} \ {f \sp {32}}} - +{{61749727185325} \ {f \sp {26}}} - +\\ +\displaystyle +{{9664082618092450} \ {f \sp {20}}} - +{{12152237485813570} \ {f \sp {14}}} - +{{9672870290826025} \ {f \sp 8}} - +\\ +\displaystyle +{{68544102808525} \ {f \sp 2}}, +\\ +\displaystyle +{{1387545279120} \ d}+ +\left( +-{{1128983050} \ {f \sp {30}}}+ +{{2883434331830} \ {f \sp {24}}}+ +\right. +\\ +\displaystyle +{{451234998755840} \ {f \sp {18}}}+ +{{562426491685760} \ {f \sp {12}}}+ +{{447129055314890} \ {f \sp 6}} - +\\ +\displaystyle +\left. +{165557857270} +\right)\ e - +{{1816935351} \ {f \sp {31}}}+ +{{4640452214013} \ {f \sp {25}}}+ +\\ +\displaystyle +{{726247129626942} \ {f \sp {19}}}+ +{{912871801716798} \ {f \sp {13}}}+ +{{726583262666877} \ {f \sp 7}}+ +\\ +\displaystyle +{{4909358645961} \ f}, +\\ +\displaystyle +{{1387545279120} \ c}+ +{{778171189} \ {f \sp {31}}} - +{{1987468196267} \ {f \sp {25}}} - +\\ +\displaystyle +{{310993556954378} \ {f \sp {19}}} - +{{383262822316802} \ {f \sp {13}}} - +{{300335488637543} \ {f \sp 7}}+ +\\ +\displaystyle +{{5289595037041} \ f}, +\\ +\displaystyle +{{1387545279120} \ b}+ +\left( +{{1128983050} \ {f \sp {30}}} - +{{2883434331830} \ {f \sp {24}}} - +\right. +\\ +\displaystyle +{{451234998755840} \ {f \sp {18}}}- +{{562426491685760} \ {f \sp {12}}} - +{{447129055314890} \ {f \sp 6}}+ +\\ +\displaystyle +\left. +{165557857270} +\right)\ e - +{{3283058841} \ {f \sp {31}}}+ +{{8384938292463} \ {f \sp {25}}}+ +\\ +\displaystyle +{{1312252817452422} \ {f \sp {19}}}+ +{{1646579934064638} \ {f \sp {13}}}+ +{{1306372958656407} \ {f \sp 7}}+ +\\ +\displaystyle +{{4694680112151} \ f}, +\\ +\displaystyle +{{1387545279120} \ a}+ +{{1387545279120} \ e}+ +{{4321823003} \ {f \sp {31}}} - +\\ +\displaystyle +{{11037922310209} \ {f \sp {25}}} - +{{1727506390124986} \ {f \sp {19}}} - +{{2176188913464634} \ {f \sp {13}}} - +\\ +\displaystyle +\left. +{{1732620732685741} \ {f \sp 7}} - +{{13506088516033} \ f} +\right\}, +\end{array} +\\ +\\ +\displaystyle +\left. +\left\{ +{{f \sp 6} -1}, +{{e \sp 2}+{4 \ f \ e}+{f \sp 2}}, +{d+e+{4 \ f}}, +{c -f}, +{b -f}, +{a -f} +\right\} +\right] +\end{array} +$$ +\returnType{Type: List RegularChain(Integer,[a,b,c,d,e,f])} + +We check that all initials are constant. + +\spadcommand{[ [init(p) for p in (ts :: List(P))] for ts in lts] } +$$ +\begin{array}{@{}l} +\left[ +{\left[ 1, 3, 1, 3, 1, 1 \right]}, +{\left[ 1, 1, 1, 1, 1, 1 \right]}, +{\left[ 1, 1, 1, 1, 1, 1 \right]}, +{\left[ 1, 1, 1, 1, 1, 1 \right]}, +\right. +\\ +\displaystyle +\left[ {1387545279120}, {1387545279120}, {1387545279120}, +\right. +\\ +\displaystyle +\left. +{1387545279120}, {1387545279120}, 1 \right], +\\ +\displaystyle +\left. +{\left[ 1, 1, 1, 1, 1, 1\right]} +\right] +\end{array} +$$ +\returnType{Type: List List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} + +Note that each triangular set in {\bf lts} is a lexicographical +Groebner basis. Recall that a point belongs to the variety associated +with {\bf lp} if and only if it belongs to that associated with one +triangular set {\bf ts} in {\bf lts}. + +By running the \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage} +operation, we retrieve the above decomposition. + +\spadcommand{squareFreeLexTriangular(lg,true)\$lextripack } +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{{f \sp 6}+1}, +{{e \sp 6} - +{3 \ f \ {e \sp 5}}+ +{3 \ {f \sp 2} \ {e \sp 4}} - +{4 \ {f \sp 3} \ {e \sp 3}}+ +{3 \ {f \sp 4} \ {e \sp 2}} - +{3 \ {f \sp 5} \ e} -1}, +\right. +\right. +\\ +\displaystyle +{{3 \ d}+ +{{f \sp 2} \ {e \sp 5}} - +{4 \ {f \sp 3} \ {e \sp 4}}+ +{4 \ {f \sp 4} \ {e \sp 3}} - +{2 \ {f \sp 5} \ {e \sp 2}} - +{2 \ e}+ +{2 \ f}}, +\\ +\displaystyle +{c+f}, +{{3 \ b}+ +{2 \ {f \sp 2} \ {e \sp 5}} - +{5 \ {f \sp 3} \ {e \sp 4}}+ +{5 \ {f \sp 4} \ {e \sp 3}} - +{{10} \ {f \sp 5} \ {e \sp 2}} - +{4 \ e}+{7 \ f}}, +\\ +\displaystyle +\left. +{a - +{{f \sp 2} \ {e \sp 5}}+ +{3 \ {f \sp 3} \ {e \sp 4}} - +{3 \ {f \sp 4} \ {e \sp 3}}+ +{4 \ {f \sp 5} \ {e \sp 2}}+ +{3 \ e} -{3 \ f}} +\right\},\hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\left\{ +{{f \sp 6} -1}, +{e -f}, +{d -f}, +{{c \sp 2}+ +{4 \ f \ c}+ +{f \sp 2}}, +{b+c+{4 \ f}}, +{a -f} +\right\},\hbox{\hskip 3.5cm} +$$ +$$ +\left\{ +{{f \sp 6} -1}, +{e -f}, +{d -f}, +{c -f}, +{{b \sp 2}+ +{4 \ f \ b}+ +{f \sp 2}}, +{a+b+{4 \ f}} +\right\},\hbox{\hskip 3.5cm} +$$ +$$ +\left\{ +{{f \sp 6} -1}, +{e -f}, +{{d \sp 2}+ +{4 \ f \ d}+ +{f \sp 2}}, +{c+d+{4 \ f}}, +{b -f}, +{a -f} +\right\},\hbox{\hskip 3.5cm} +$$ +$$ +\begin{array}{@{}l} +\left\{ +{{f \sp {36}} - +{{2554} \ {f \sp {30}}} - +{{399709} \ {f \sp {24}}} - +{{502276} \ {f \sp {18}}} - +{{399709} \ {f \sp {12}}} - +{{2554} \ {f \sp 6}}+1}, +\right. +\\ +\displaystyle +{{1387545279120} \ {e \sp 2}}+ +\left( {{4321823003} \ {f \sp {31}}} - +{{11037922310209} \ {f \sp {25}}} - +\right. +\\ +\displaystyle +{{1727506390124986} \ {f \sp {19}}} - +{{2176188913464634} \ {f \sp {13}}} - +{{1732620732685741} \ {f \sp 7}} - +\\ +\displaystyle +\left. +{{13506088516033} \ f} +\right)\ e+ +{{24177661775} \ {f \sp {32}}} - +{{61749727185325} \ {f \sp {26}}}- +\\ +\displaystyle +{{9664082618092450} \ {f \sp {20}}} - +{{12152237485813570} \ {f \sp {14}}} - +{{9672870290826025} \ {f \sp 8}} - +\\ +\displaystyle +{{68544102808525} \ {f \sp 2}}, +\\ +\displaystyle +{{1387545279120} \ d}+ +\left( -{{1128983050} \ {f \sp {30}}}+ +{{2883434331830} \ {f \sp {24}}}+ +\right. +\\ +\displaystyle +{{451234998755840} \ {f \sp {18}}}+ +{{562426491685760} \ {f \sp {12}}}+ +{{447129055314890} \ {f \sp 6}} - +\\ +\displaystyle +\left. +{165557857270} +\right)\ e - +{{1816935351} \ {f \sp {31}}}+ +{{4640452214013} \ {f \sp {25}}}+ +\\ +\displaystyle +{{726247129626942} \ {f \sp {19}}}+ +{{912871801716798} \ {f \sp {13}}}+ +{{726583262666877} \ {f \sp 7}}+ +\\ +\displaystyle +{{4909358645961} \ f}, +\\ +\displaystyle +{{1387545279120} \ c}+ +{{778171189} \ {f \sp {31}}} - +{{1987468196267} \ {f \sp {25}}} - +\\ +\displaystyle +{{310993556954378} \ {f \sp {19}}} - +{{383262822316802} \ {f \sp {13}}} - +{{300335488637543} \ {f \sp 7}}+ +\\ +\displaystyle +{{5289595037041} \ f}, +\\ +\displaystyle +{{1387545279120} \ b}+ +\left( +{{1128983050} \ {f \sp {30}}} - +{{2883434331830} \ {f \sp {24}}} - +\right. +\\ +\displaystyle +{{451234998755840} \ {f \sp {18}}} - +{{562426491685760} \ {f \sp {12}}} - +{{447129055314890} \ {f \sp 6}}+ +\\ +\displaystyle +\left. +{165557857270} +\right)\ e - +{{3283058841} \ {f \sp {31}}}+ +{{8384938292463} \ {f \sp {25}}}+ +\\ +\displaystyle +{{1312252817452422} \ {f \sp {19}}}+ +{{1646579934064638} \ {f \sp {13}}}+ +{{1306372958656407} \ {f \sp 7}}+ +\\ +\displaystyle +{{4694680112151} \ f}, +{{1387545279120} \ a}+ +{{1387545279120} \ e}+ +\\ +\displaystyle +{{4321823003} \ {f \sp {31}}} - +{{11037922310209} \ {f \sp {25}}} - +{{1727506390124986} \ {f \sp {19}}} - +\\ +\displaystyle +\left. +{{2176188913464634} \ {f \sp {13}}} - +{{1732620732685741} \ {f \sp 7}} - +{{13506088516033} \ f} +\right\},\hbox{\hskip 1.5cm} +\end{array} +$$ +$$ +\left. +\left\{ +{{f \sp 6} -1}, +{{e \sp 2}+ +{4 \ f \ e}+ +{f \sp 2}}, +{d+e+{4 \ f}}, +{c -f}, +{b -f}, +{a -f} +\right\} +\right]\hbox{\hskip 3.5cm} +$$ +\returnType{Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [a,b,c,d,e,f],OrderedVariableList [a,b,c,d,e,f],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]))} + +Thus the solutions given by {\bf lts} are pairwise different. + +We count them as follows. + +\spadcommand{reduce(+,[degree(ts) for ts in lts]) } +$$ +156 +$$ +\returnType{Type: PositiveInteger} + +We can investigate the triangular decomposition {\bf lts} by using the +{\tt ZeroDimensionalSolvePackage}. + +This requires to add an extra variable (smaller than the others) as follows. + +\spadcommand{ls2 : List Symbol := concat(ls,new()\$Symbol) } +$$ +\left[ +a, b, c, d, e, f, \%A +\right] +$$ +\returnType{Type: List Symbol} + +Then we call the package. + +\spadcommand{zdpack := ZDSOLVE(R,ls,ls2) } +$$ +ZeroDimensionalSolvePackage(Integer,[a,b,c,d,e,f],[a,b,c,d,e,f,%A]) +$$ +\returnType{Type: Domain} + +We compute a univariate representation of the variety associated with +the input system as follows. + +\spadcommand{concat [univariateSolve(ts)\$zdpack for ts in lts] } +$$ +\begin{array}{@{}l} +\left[ +\left[ +{complexRoots={{? \sp 4} -{{13} \ {? \sp 2}}+{49}}}, +\right. +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{7 \ a}+{ \%A \sp 3} -{6 \ \%A}}, +{{{21} \ b}+{ \%A \sp 3}+ \%A}, +\right. +\\ +\displaystyle +{{{21} \ c} -{2 \ { \%A \sp 3}}+{{19} \ \%A}}, +{{7 \ d} -{ \%A \sp 3}+{6 \ \%A}}, +{{{21} \ e} -{ \%A \sp 3} - \%A}, +\\ +\displaystyle +\left. +\left. +{{{21} \ f}+{2 \ { \%A \sp 3}} -{{19} \ \%A}} +\right] +\right],\hbox{\hskip 7.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4}+{{11} \ {? \sp 2}}+{49}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{35} \ a}+{3 \ { \%A \sp 3}}+{{19} \ \%A}}, +{{{35} \ b}+{ \%A \sp 3}+{{18} \ \%A}}, +{{{35} \ c} -{2 \ { \%A \sp 3}} - \%A}, +\right. +\\ +\displaystyle +\left. +\left. +{{{35} \ d} -{3 \ { \%A \sp 3}} -{{19} \ \%A}}, +{{{35} \ e} -{ \%A \sp 3} -{{18} \ \%A}}, +{{{35} \ f}+{2 \ { \%A \sp 3}}+ \%A} +\right] +\right],\hbox{\hskip 2.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{? \sp 8} -{{12} \ {? \sp 7}}+{{58} \ {? \sp 6}} -{{120} \ {? \sp 5}}+ +\right. +\\ +\displaystyle +{{207} \ {? \sp 4}} - +{{360} \ {? \sp 3}}+ +{{802} \ {? \sp 2}} - +{{1332} \ ?}+{1369}, +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{43054532} \ a}+{{33782} \ { \%A \sp 7}} - +{{546673} \ { \%A \sp 6}}+ +{{3127348} \ { \%A \sp 5}} -{{6927123} \ { \%A \sp 4}}+ +\right. +\\ +\displaystyle +{{4365212} \ { \%A \sp 3}} - +{{25086957} \ { \%A \sp 2}}+ +{{39582814} \ \%A} -{107313172}, +\\ +\displaystyle +{{43054532} \ b} -{{33782} \ { \%A \sp 7}}+ +{{546673} \ { \%A \sp 6}} - +{{3127348} \ { \%A \sp 5}}+ +\\ +\displaystyle +{{6927123} \ { \%A \sp 4}} - +{{4365212} \ { \%A \sp 3}}+ +{{25086957} \ { \%A \sp 2}} - +\\ +\displaystyle +{{39582814} \ \%A}+{107313172}, +\\ +\displaystyle +{{21527266} \ c} -{{22306} \ { \%A \sp 7}}+ +{{263139} \ { \%A \sp 6}} - +{{1166076} \ { \%A \sp 5}}+{{1821805} \ { \%A \sp 4}} - +\\ +\displaystyle +{{2892788} \ { \%A \sp 3}}+ +{{10322663} \ { \%A \sp 2}} - +{{9026596} \ \%A}+{12950740}, +\\ +\displaystyle +{{43054532} \ d}+ +{{22306} \ { \%A \sp 7}} - +{{263139} \ { \%A \sp 6}}+ +\\ +\displaystyle +{{1166076} \ { \%A \sp 5}} - +{{1821805} \ { \%A \sp 4}}+ +{{2892788} \ { \%A \sp 3}} - +\\ +\displaystyle +{{10322663} \ { \%A \sp 2}}+ +{{30553862} \ \%A} -{12950740}, +\\ +\displaystyle +{{43054532} \ e} - +{{22306} \ { \%A \sp 7}}+ +{{263139} \ { \%A \sp 6}} - +\\ +\displaystyle +{{1166076} \ { \%A \sp 5}}+ +{{1821805} \ { \%A \sp 4}} - +{{2892788} \ { \%A \sp 3}}+ +\\ +\displaystyle +{{10322663} \ { \%A \sp 2}} - +{{30553862} \ \%A}+{12950740}, +\\ +\displaystyle +{{21527266} \ f}+ +{{22306} \ { \%A \sp 7}} - +{{263139} \ { \%A \sp 6}}+ +\\ +\displaystyle +{{1166076} \ { \%A \sp 5}} - +{{1821805} \ { \%A \sp 4}}+ +{{2892788} \ { \%A \sp 3}} - +\\ +\displaystyle +\left. +\left. +{{10322663} \ { \%A \sp 2}}+ +{{9026596} \ \%A} -{12950740} +\right] +\right],\hbox{\hskip 5.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{? \sp 8}+ +{{12} \ {? \sp 7}}+ +{{58} \ {? \sp 6}}+ +{{120} \ {? \sp 5}}+ +\right. +\\ +\displaystyle +{{207} \ {? \sp 4}}+ +{{360} \ {? \sp 3}}+ +{{802} \ {? \sp 2}}+ +{{1332} \ ?}+{1369}, +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{43054532} \ a}+ +{{33782} \ { \%A \sp 7}}+ +{{546673} \ { \%A \sp 6}}+ +{{3127348} \ { \%A \sp 5}}+ +\right. +\\ +\displaystyle +{{6927123} \ { \%A \sp 4}}+ +{{4365212} \ { \%A \sp 3}}+ +{{25086957} \ { \%A \sp 2}}+ +{{39582814} \ \%A}+{107313172}, +\\ +\displaystyle +{{43054532} \ b} - +{{33782} \ { \%A \sp 7}} - +{{546673} \ { \%A \sp 6}} - +{{3127348} \ { \%A \sp 5}} - +\\ +\displaystyle +{{6927123} \ { \%A \sp 4}} - +{{4365212} \ { \%A \sp 3}} - +{{25086957} \ { \%A \sp 2}} - +{{39582814} \ \%A} - +{107313172}, +\\ +\displaystyle +{{21527266} \ c} - +{{22306} \ { \%A \sp 7}} - +{{263139} \ { \%A \sp 6}} - +{{1166076} \ { \%A \sp 5}} - +\\ +\displaystyle +{{1821805} \ { \%A \sp 4}} - +{{2892788} \ { \%A \sp 3}} - +{{10322663} \ { \%A \sp 2}} - +{{9026596} \ \%A} - +{12950740}, +\\ +\displaystyle +{{43054532} \ d}+ +{{22306} \ { \%A \sp 7}}+ +{{263139} \ { \%A \sp 6}}+ +{{1166076} \ { \%A \sp 5}}+ +\\ +\displaystyle +{{1821805} \ { \%A \sp 4}}+ +{{2892788} \ { \%A \sp 3}}+ +{{10322663} \ { \%A \sp 2}}+ +{{30553862} \ \%A}+ +{12950740}, +\\ +\displaystyle +{{43054532} \ e} - +{{22306} \ { \%A \sp 7}} - +{{263139} \ { \%A \sp 6}} - +{{1166076} \ { \%A \sp 5}} - +\\ +\displaystyle +{{1821805} \ { \%A \sp 4}} - +{{2892788} \ { \%A \sp 3}} - +{{10322663} \ { \%A \sp 2}} - +{{30553862} \ \%A} - +{12950740}, +\\ +\displaystyle +{{21527266} \ f}+ +{{22306} \ { \%A \sp 7}}+ +{{263139} \ { \%A \sp 6}}+ +{{1166076} \ { \%A \sp 5}}+ +\\ +\displaystyle +\left. +\left. +{{1821805} \ { \%A \sp 4}}+ +{{2892788} \ { \%A \sp 3}}+ +{{10322663} \ { \%A \sp 2}}+ +{{9026596} \ \%A}+ +{12950740} +\right] +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4} -{? \sp 2}+1}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ +{a - \%A}, +{b+{ \%A \sp 3} - \%A}, +{c+{ \%A \sp 3}}, +{d+ \%A}, +{e -{ \%A \sp 3}+ \%A}, +{f -{ \%A \sp 3}} +\right] +\right],\hbox{\hskip 1.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 8}+{4 \ {? \sp 6}}+{{12} \ {? \sp 4}}+{{16} \ {? \sp 2}}+4}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{4 \ a} - +{2 \ { \%A \sp 7}} - +{7 \ { \%A \sp 5}} - +{{20} \ { \%A \sp 3}} - +{{22} \ \%A}}, +\right. +\\ +\displaystyle +{{4 \ b}+ +{2 \ { \%A \sp 7}}+ +{7 \ { \%A \sp 5}}+ +{{20} \ { \%A \sp 3}}+ +{{22} \ \%A}}, +\\ +\displaystyle +{{4 \ c}+ +{ \%A \sp 7}+ +{3 \ { \%A \sp 5}}+ +{{10} \ { \%A \sp 3}}+ +{{10} \ \%A}}, +\\ +\displaystyle +{{4 \ d}+{ \%A \sp 7}+ +{3 \ { \%A \sp 5}}+{{10} \ { \%A \sp 3}}+ +{6 \ \%A}}, +\\ +\displaystyle +{{4 \ e} - +{ \%A \sp 7} - +{3 \ { \%A \sp 5}} - +{{10} \ { \%A \sp 3}} - +{6 \ \%A}}, +\\ +\displaystyle +\left. +\left. +{{4 \ f} - +{ \%A \sp 7} - +{3 \ { \%A \sp 5}} - +{{10} \ { \%A \sp 3}} - +{{10} \ \%A}} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4}+{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}+{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\right. +\\ +\displaystyle +{{6 \ b}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{24} \ \%A}+6}, +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4} -{6 \ {? \sp 3}}+{{30} \ {? \sp 2}} -{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\right. +\\ +\displaystyle +{{6 \ b}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{24} \ \%A} -6}, +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3}+{5 \ { \%A \sp 2}}+6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2}+{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ +{a+1}, {b - \%A -5}, {c+ \%A+1}, {d+1}, {e+1}, {f+1} +\right] +\right],\hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ +{a -1}, {b - \%A+5}, {c+ \%A -1}, {d -1}, {e -1}, {f -1} +\right] +\right],\hbox{\hskip 4.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4}+{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}+{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{6 \ a}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{24} \ \%A}+6}, +\right. +\\ +\displaystyle +{{{30} \ b} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -6}, +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4} -{6 \ {? \sp 3}}+{{30} \ {? \sp 2}} -{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{6 \ a}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{24} \ \%A} -6}, +\right. +\\ +\displaystyle +{{{30} \ b} -{ \%A \sp 3}+{5 \ { \%A \sp 2}}+6}, +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2}+{6 \ ?}+6}, +\right. +\\ +\displaystyle +\left. +coordinates= +\left[ +{a - \%A -5}, {b+ \%A+1}, {c+1}, {d+1}, {e+1}, {f+1} +\right] +\right],\hbox{\hskip 2.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{6 \ ?}+6}, +\right. +\\ +\displaystyle +\left. +coordinates= +\left[ +{a - \%A+5}, {b+ \%A -1}, {c -1}, {d -1}, {e -1}, {f -1} +\right] +\right],\hbox{\hskip 2.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4}+{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}+{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\right. +\\ +\displaystyle +{{{30} \ b} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +{{6 \ c}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{24} \ \%A}+6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4} -{6 \ {? \sp 3}}+{{30} \ {? \sp 2}} -{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\right. +\\ +\displaystyle +{{{30} \ b} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +{{6 \ c}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{24} \ \%A} -6}, +\\ +\displaystyle +{{{30} \ d} -{ \%A \sp 3}+{5 \ { \%A \sp 2}}+6}, +\\ +\displaystyle +{{{30} \ e} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ f} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6} +\right] +\right],\hbox{\hskip 6.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2}+{6 \ ?}+6}, +\right. +\\ +\displaystyle +\left. +coordinates= +\left[ +{a+1}, {b+1}, {c - \%A -5}, {d+ \%A+1}, {e+1}, {f+1} +\right] +\right],\hbox{\hskip 2.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{6 \ ?}+6}, +\right. +\\ +\displaystyle +\left. +coordinates= +\left[ +{a -1}, {b -1}, {c - \%A+5}, {d+ \%A -1}, {e -1}, {f -1} +\right] +\right],\hbox{\hskip 2.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{? \sp 8}+ +{6 \ {? \sp 7}}+ +{{16} \ {? \sp 6}}+ +{{24} \ {? \sp 5}}+ +{{18} \ {? \sp 4}} - +{8 \ {? \sp 2}}+ +4, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{2 \ a}+ +{2 \ { \%A \sp 7}}+ +{9 \ { \%A \sp 6}}+ +{{18} \ { \%A \sp 5}}+ +{{19} \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}} - +{{10} \ { \%A \sp 2}} - +{2 \ \%A}+ +4, +\right. +\\ +\displaystyle +{2 \ b}+ +{2 \ { \%A \sp 7}}+ +{9 \ { \%A \sp 6}}+ +{{18} \ { \%A \sp 5}}+ +{{19} \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}} - +{{10} \ { \%A \sp 2}} - +{4 \ \%A}+ +4, +\\ +\displaystyle +{2 \ c} - +{ \%A \sp 7} - +{4 \ { \%A \sp 6}} - +{8 \ { \%A \sp 5}} - +{9 \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}} - +{2 \ \%A} - +4, +\\ +\displaystyle +{2 \ d}+ +{ \%A \sp 7}+ +{4 \ { \%A \sp 6}}+ +{8 \ { \%A \sp 5}}+ +{9 \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}}+ +{2 \ \%A}+ +4, +\\ +\displaystyle +{2 \ e} - +{2 \ { \%A \sp 7}} - +{9 \ { \%A \sp 6}} - +{{18} \ { \%A \sp 5}} - +{{19} \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}}+ +{{10} \ { \%A \sp 2}}+ +{4 \ \%A} - +4, +\\ +\displaystyle +\left. +\left. +{2 \ f} - +{2 \ { \%A \sp 7}} - +{9 \ { \%A \sp 6}} - +{{18} \ { \%A \sp 5}} - +{{19} \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}}+ +{{10} \ { \%A \sp 2}}+ +{2 \ \%A} - +4 +\right] +\right],\hbox{\hskip 1.0cm} %mmmm +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +\right. +\\ +\displaystyle +{? \sp 8}+{{12} \ {? \sp 7}}+ +{{64} \ {? \sp 6}}+ +{{192} \ {? \sp 5}}+ +{{432} \ {? \sp 4}}+ +{{768} \ {? \sp 3}}+ +{{1024} \ {? \sp 2}}+ +{{768} \ ?}+ +{256}, +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{1408} \ a} - +{{19} \ { \%A \sp 7}} - +{{200} \ { \%A \sp 6}} - +{{912} \ { \%A \sp 5}} - +{{2216} \ { \%A \sp 4}} - +\right. +\\ +\displaystyle +{{4544} \ { \%A \sp 3}} - +{{6784} \ { \%A \sp 2}} - +{{6976} \ \%A} - +{1792}, +\\ +\displaystyle +{{1408} \ b} - +{{37} \ { \%A \sp 7}} - +{{408} \ { \%A \sp 6}} - +{{1952} \ { \%A \sp 5}} - +{{5024} \ { \%A \sp 4}} - +\\ +\displaystyle +{{10368} \ { \%A \sp 3}} - +{{16768} \ { \%A \sp 2}} - +{{17920} \ \%A} - +{5120}, +\\ +\displaystyle +{{1408} \ c}+ +{{37} \ { \%A \sp 7}}+ +{{408} \ { \%A \sp 6}}+ +{{1952} \ { \%A \sp 5}}+ +{{5024} \ { \%A \sp 4}}+ +\\ +\displaystyle +{{10368} \ { \%A \sp 3}}+ +{{16768} \ { \%A \sp 2}}+ +{{17920} \ \%A}+ +{5120}, +\\ +\displaystyle +{{1408} \ d}+ +{{19} \ { \%A \sp 7}}+ +{{200} \ { \%A \sp 6}}+ +{{912} \ { \%A \sp 5}}+ +{{2216} \ { \%A \sp 4}}+ +\\ +\displaystyle +{{4544} \ { \%A \sp 3}}+ +{{6784} \ { \%A \sp 2}}+ +{{6976} \ \%A}+ +{1792}, +\\ +\displaystyle +{{2 \ e}+ \%A}, +\\ +\displaystyle +\left. +\left. +{{2 \ f} - \%A} +\right] +\right],\hbox{\hskip 10.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 8}+ +{4 \ {? \sp 6}}+ +{{12} \ {? \sp 4}}+ +{{16} \ {? \sp 2}}+ +4}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{4 \ a} - +{ \%A \sp 7} - +{3 \ { \%A \sp 5}} - +{{10} \ { \%A \sp 3}} - +{6 \ \%A}}, +\right. +\\ +\displaystyle +{{4 \ b} - +{ \%A \sp 7} - +{3 \ { \%A \sp 5}} - +{{10} \ { \%A \sp 3}} - +{{10} \ \%A}}, +\\ +\displaystyle +{{4 \ c} - +{2 \ { \%A \sp 7}} - +{7 \ { \%A \sp 5}} - +{{20} \ { \%A \sp 3}} - +{{22} \ \%A}}, +\\ +\displaystyle +{{4 \ d}+ +{2 \ { \%A \sp 7}}+ +{7 \ { \%A \sp 5}}+ +{{20} \ { \%A \sp 3}}+ +{{22} \ \%A}}, +\\ +\displaystyle +{{4 \ e}+ +{ \%A \sp 7}+ +{3 \ { \%A \sp 5}}+ +{{10} \ { \%A \sp 3}}+ +{{10} \ \%A}}, +\\ +\displaystyle +\left. +\left. +{{4 \ f}+ +{ \%A \sp 7}+ +{3 \ { \%A \sp 5}}+ +{{10} \ { \%A \sp 3}}+ +{6 \ \%A}} +\right] +\right],\hbox{\hskip 5.9cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 8}+{{16} \ {? \sp 6}} -{{96} \ {? \sp 4}}+ +{{256} \ {? \sp 2}}+{256}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{512} \ a} - +{ \%A \sp 7} - +{{12} \ { \%A \sp 5}}+ +{{176} \ { \%A \sp 3}} - +{{448} \ \%A}}, +\right. +\\ +\displaystyle +{{{128} \ b} - +{ \%A \sp 7} - +{{16} \ { \%A \sp 5}}+ +{{96} \ { \%A \sp 3}} +-{{256} \ \%A}}, +\\ +\displaystyle +{{{128} \ c}+ +{ \%A \sp 7}+ +{{16} \ { \%A \sp 5}} - +{{96} \ { \%A \sp 3}}+ +{{256} \ \%A}}, +\\ +\displaystyle +{{{512} \ d}+ +{ \%A \sp 7}+ +{{12} \ { \%A \sp 5}} - +{{176} \ { \%A \sp 3}}+ +{{448} \ \%A}}, +\\ +\displaystyle +{{2 \ e}+ \%A}, +\\ +\displaystyle +\left. +\left. +{{2 \ f} - \%A} +\right] +\right],\hbox{\hskip 10.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +\right. +\\ +\displaystyle +{{? \sp 8} - +{{12} \ {? \sp 7}}+ +{{64} \ {? \sp 6}} - +{{192} \ {? \sp 5}}+ +{{432} \ {? \sp 4}} - +{{768} \ {? \sp 3}}+ +{{1024} \ {? \sp 2}} - +{{768} \ ?}+ +{256}}, +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{1408} \ a} - +{{19} \ { \%A \sp 7}}+ +{{200} \ { \%A \sp 6}} - +{{912} \ { \%A \sp 5}}+ +{{2216} \ { \%A \sp 4}} - +\right. +\\ +\displaystyle +{{4544} \ { \%A \sp 3}}+ +{{6784} \ { \%A \sp 2}} - +{{6976} \ \%A}+ +{1792}, +\\ +\displaystyle +{{1408} \ b} - +{{37} \ { \%A \sp 7}}+ +{{408} \ { \%A \sp 6}} - +{{1952} \ { \%A \sp 5}}+ +{{5024} \ { \%A \sp 4}} - +\\ +\displaystyle +{{10368} \ { \%A \sp 3}}+ +{{16768} \ { \%A \sp 2}} - +{{17920} \ \%A}+ +{5120}, +\\ +\displaystyle +{{1408} \ c}+ +{{37} \ { \%A \sp 7}} - +{{408} \ { \%A \sp 6}}+ +{{1952} \ { \%A \sp 5}} - +{{5024} \ { \%A \sp 4}}+ +\\ +\displaystyle +{{10368} \ { \%A \sp 3}} - +{{16768} \ { \%A \sp 2}}+ +{{17920} \ \%A} - +{5120}, +\\ +\displaystyle +{{1408} \ d}+ +{{19} \ { \%A \sp 7}} - +{{200} \ { \%A \sp 6}}+ +{{912} \ { \%A \sp 5}} - +{{2216} \ { \%A \sp 4}}+ +\\ +\displaystyle +{{4544} \ { \%A \sp 3}} - +{{6784} \ { \%A \sp 2}}+ +{{6976} \ \%A} - +{1792}, +\\ +\displaystyle +{{2 \ e}+ \%A}, +\\ +\displaystyle +\left. +\left. +{{2 \ f} - \%A} +\right] +\right],\hbox{\hskip 10.0cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 8} - +{6 \ {? \sp 7}}+ +{{16} \ {? \sp 6}} - +{{24} \ {? \sp 5}}+ +{{18} \ {? \sp 4}} - +{8 \ {? \sp 2}}+ +4}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{2 \ a}+ +{2 \ { \%A \sp 7}} - +{9 \ { \%A \sp 6}}+ +{{18} \ { \%A \sp 5}} - +{{19} \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}}+ +{{10} \ { \%A \sp 2}} - +{2 \ \%A} - +4}, +\right. +\\ +\displaystyle +{{2 \ b}+ +{2 \ { \%A \sp 7}} - +{9 \ { \%A \sp 6}}+ +{{18} \ { \%A \sp 5}} - +{{19} \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}}+ +{{10} \ { \%A \sp 2}} - +{4 \ \%A} - +4}, +\\ +\displaystyle +{{2 \ c} - +{ \%A \sp 7}+ +{4 \ { \%A \sp 6}} - +{8 \ { \%A \sp 5}}+ +{9 \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}} - +{2 \ \%A}+ +4}, +\\ +\displaystyle +{{2 \ d}+ +{ \%A \sp 7} - +{4 \ { \%A \sp 6}}+ +{8 \ { \%A \sp 5}} - +{9 \ { \%A \sp 4}}+ +{4 \ { \%A \sp 3}}+ +{2 \ \%A} - +4}, +\\ +\displaystyle +{{2 \ e} - +{2 \ { \%A \sp 7}}+ +{9 \ { \%A \sp 6}} - +{{18} \ { \%A \sp 5}}+ +{{19} \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}} - +{{10} \ { \%A \sp 2}}+ +{4 \ \%A}+ +4}, +\\ +\displaystyle +\left. +\left. +{{2 \ f} - +{2 \ { \%A \sp 7}}+ +{9 \ { \%A \sp 6}} - +{{18} \ { \%A \sp 5}}+ +{{19} \ { \%A \sp 4}} - +{4 \ { \%A \sp 3}} - +{{10} \ { \%A \sp 2}}+ +{2 \ \%A}+ +4} +\right] +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4}+{{12} \ {? \sp 2}}+{144}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{12} \ a} -{ \%A \sp 2} -{12}}, +{{{12} \ b} -{ \%A \sp 2} -{12}}, +{{{12} \ c} -{ \%A \sp 2} -{12}}, +\right. +\\ +\displaystyle +\left. +\left. +{{{12} \ d} -{ \%A \sp 2} -{12}}, +{{6 \ e}+{ \%A \sp 2}+{3 \ \%A}+{12}}, +{{6 \ f}+{ \%A \sp 2} -{3 \ \%A}+{12}} +\right] +\right],\hbox{\hskip 2.4cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4}+{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}+{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{6 \ a} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{24} \ \%A} -6}, +{{{30} \ b}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{30} \ \%A}+6}, +\right. +\\ +\displaystyle +{{{30} \ c}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{30} \ \%A}+6}, +{{{30} \ d}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{30} \ \%A}+6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ e}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{30} \ \%A}+6}, +{{{30} \ f}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+6} +\right] +\right],\hbox{\hskip 2.4cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots= +{{? \sp 4} -{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}-{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{6 \ a} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{24} \ \%A}+6}, +{{{30} \ b}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{30} \ \%A} -6}, +\right. +\\ +\displaystyle +{{{30} \ c}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{30} \ \%A} -6}, +{{{30} \ d}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{30} \ \%A} -6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ e}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{30} \ \%A} -6}, +{{{30} \ f}+{ \%A \sp 3} -{5 \ { \%A \sp 2}} -6} +\right] +\right],\hbox{\hskip 2.4cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4}+{{12} \ {? \sp 2}}+{144}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{12} \ a}+{ \%A \sp 2}+{12}}, {{{12} \ b}+{ \%A \sp 2}+{12}}, +{{{12} \ c}+{ \%A \sp 2}+{12}}, {{{12} \ d}+{ \%A \sp 2}+{12}}, +\right. +\\ +\displaystyle +\left. +\left. +{{6 \ e} -{ \%A \sp 2}+{3 \ \%A} -{12}}, +{{6 \ f} -{ \%A \sp 2} -{3 \ \%A} -{12}} +\right] +\right],\hbox{\hskip 3.7cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{12}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ {a -1}, {b -1}, {c -1}, {d -1}, {{2 \ e}+ \%A+4}, {{2 \ f} - \%A+4} +\right] +\right],\hbox{\hskip 3.7cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2}+{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ {a+ \%A+5}, {b -1}, {c -1}, {d -1}, {e -1}, {f - \%A -1} +\right] +\right],\hbox{\hskip 3.9cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ +{a+ \%A -5}, {b+1}, {c+1}, {d+1}, {e+1}, {f - \%A+1} +\right] +\right],\hbox{\hskip 3.9cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{12}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ {a+1}, {b+1}, {c+1}, {d+1}, {{2 \ e}+ \%A -4}, {{2 \ f} - \%A -4} +\right] +\right],\hbox{\hskip 3.3cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4}+{6 \ {? \sp 3}}+{{30} \ {? \sp 2}}+{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +{{{30} \ b} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +\right. +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6}, +{{6 \ d}+{ \%A \sp 3}+{5 \ { \%A \sp 2}}+{{24} \ \%A}+6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ e} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -6}, +{{{30} \ f} -{ \%A \sp 3} -{5 \ { \%A \sp 2}} -{{30} \ \%A} -6} +\right] +\right],\hbox{\hskip 1.7cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 4} -{6 \ {? \sp 3}}+{{30} \ {? \sp 2}} -{{36} \ ?}+{36}}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{{30} \ a} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +{{{30} \ b} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +\right. +\\ +\displaystyle +{{{30} \ c} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6}, +{{6 \ d}+{ \%A \sp 3} -{5 \ { \%A \sp 2}}+{{24} \ \%A} -6}, +\\ +\displaystyle +\left. +\left. +{{{30} \ e} -{ \%A \sp 3}+{5 \ { \%A \sp 2}}+6}, +{{{30} \ f} -{ \%A \sp 3}+{5 \ { \%A \sp 2}} -{{30} \ \%A}+6} +\right] +\right],\hbox{\hskip 1.7cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2}+{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ +{a+1}, {b+1}, {c+1}, {d - \%A -5}, {e+ \%A+1}, {f+1} +\right] +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\left. +\begin{array}{@{}l} +\left[ +complexRoots={{? \sp 2} -{6 \ ?}+6}, +\right. +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left. +\left[ {a -1}, {b -1}, {c -1}, {d - \%A+5}, {e+ \%A -1}, {f -1} +\right] +\right]\hbox{\hskip 3.5cm} +\end{array} +\right] +$$ +\returnType{Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)} + +Since the \spadfunFrom{univariateSolve}{ZeroDimensionalSolvePackage} +operation may split a regular set, it returns a list. This explains +the use of \spadfunFrom{concat}{List}. + +Look at the last item of the result. It consists of two parts. For +any complex root {\bf ?} of the univariate polynomial in the first +part, we get a tuple of univariate polynomials (in {\bf a}, ..., +{\bf f} respectively) by replacing {\bf \%A} by {\bf ?} in the second part. +Each of these tuples {\bf t} describes a point of the variety +associated with {\bf lp} by equaling to zero the polynomials in {\bf t}. + +Note that the way of reading these univariate representations is explained also +in the example illustrating the {\tt ZeroDimensionalSolvePackage} constructor. + +Now, we compute the points of the variety with real coordinates. + +\spadcommand{concat [realSolve(ts)\$zdpack for ts in lts] } +$$ +\left[ +{\left[ +{ \%B{23}}, { \%B{23}}, { \%B{23}}, { \%B{27}}, +{-{ \%B{27}} -{4 \ { \%B{23}}}}, { \%B{23}} +\right]}, +\right.\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ +{ \%B{23}}, { \%B{23}}, { \%B{23}}, { \%B{28}}, +{-{ \%B{28}} -{4 \ { \%B{23}}}}, { \%B{23}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ +{ \%B{24}}, { \%B{24}}, { \%B{24}}, { \%B{25}}, +{-{ \%B{25}} -{4 \ { \%B{24}}}}, { \%B{24}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ +{ \%B{24}}, { \%B{24}}, { \%B{24}}, { \%B{26}}, +{-{ \%B{26}} -{4 \ { \%B{24}}}}, { \%B{24}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{29}}, { \%B{29}}, { \%B{29}}, { \%B{29}}, +{ \%B{33}}, {-{ \%B{33}} -{4 \ { \%B{29}}}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ +{ \%B{29}}, { \%B{29}}, { \%B{29}}, { \%B{29}}, +{ \%B{34}}, {-{ \%B{34}} -{4 \ { \%B{29}}}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{30}}, { \%B{30}}, { \%B{30}}, { \%B{30}}, +{ \%B{31}}, {-{ \%B{31}} -{4 \ { \%B{30}}}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{30}}, { \%B{30}}, { \%B{30}}, { \%B{30}}, +{ \%B{32}}, {-{ \%B{32}} -{4 \ { \%B{30}}}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{35}}, { \%B{35}}, { \%B{39}}, +{-{ \%B{39}} -{4 \ { \%B{35}}}}, { \%B{35}}, { \%B{35}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{35}}, { \%B{35}}, { \%B{40}}, {-{ \%B{40}} -{4 \ { + \%B{35}}}}, { \%B{35}}, { \%B{35}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{36}}, { \%B{36}}, { \%B{37}}, {-{ \%B{37}} -{4 \ { + \%B{36}}}}, { \%B{36}}, { \%B{36}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{36}}, { \%B{36}}, { \%B{38}}, {-{ \%B{38}} -{4 \ { + \%B{36}}}}, { \%B{36}}, { \%B{36}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{41}}, +\right. +\\ +\displaystyle +{ \%B{51}}, +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{41}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{41}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{41}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{41}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{41}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{41}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{41}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{41}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{41}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{41}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{41}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{41}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{41}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{41}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{41}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +-{ \%B{51}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{41}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{41}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{41}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{41}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{41}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{41}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{41}}, { \%B{52}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{41}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{41}} \sp {25}}} - +{{{25769893181} \over {49235160}} \ {{ \%B{41}} \sp {19}}} - +\\ +\\ +\displaystyle +{{{1975912990729} \over {3003344760}} \ {{ \%B{41}} \sp {13}}} - +{{{1048460696489} \over {2002229840}} \ {{ \%B{41}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{41}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{41}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{41}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{41}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{41}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{41}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{41}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{41}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{41}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{41}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{41}}}, +\\ +\\ +\displaystyle +-{ \%B{52}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{41}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{41}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{41}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{41}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{41}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{41}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{42}}, { \%B{49}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{42}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{42}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{42}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{42}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{42}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{42}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{42}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{42}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{42}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{42}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{42}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{42}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{42}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{42}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{42}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +-{ \%B{49}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{42}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{42}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{42}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{42}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{42}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{42}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{42}}, { \%B{50}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{42}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{42}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{42}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{42}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{42}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{42}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{42}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{42}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{42}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{42}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{42}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{42}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{42}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{42}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{42}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{42}}}, +\\ +\\ +\displaystyle +-{ \%B{50}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{42}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{42}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{42}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{42}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{42}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{42}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{43}}, { \%B{47}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{43}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{43}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{43}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{43}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{43}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{43}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{43}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{43}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{43}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{43}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{43}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{43}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{43}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{43}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{43}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +-{ \%B{47}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{43}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{43}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{43}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{43}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{43}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{43}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{43}}, { \%B{48}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{43}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{43}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{43}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{43}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{43}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{43}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{43}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{43}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{43}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{43}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{43}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{43}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{43}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{43}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{43}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{43}}}, +\\ +\\ +\displaystyle +-{ \%B{48}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{43}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{43}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{43}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{43}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{43}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{43}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{44}}, { \%B{45}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{44}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{44}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{44}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{44}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{44}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{44}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{44}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{44}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{44}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{44}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{44}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{44}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{44}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{44}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{44}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +-{ \%B{45}} -{{{4321823003} \over {1387545279120}} \ {{ \%B{44}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{44}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{44}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{44}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{44}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{44}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{44}}, { \%B{46}}, +\right. +\\ +\\ +\displaystyle +{{{7865521} \over {6006689520}} \ {{ \%B{44}} \sp {31}}} - +{{{6696179241} \over {2002229840}} \ {{ \%B{44}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{25769893181} \over {49235160}} \ {{ \%B{44}} \sp {19}}} - +{{{1975912990729} \over {3003344760}} \ {{ \%B{44}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{1048460696489} \over {2002229840}} \ {{ \%B{44}} \sp 7}} - +{{{21252634831} \over {6006689520}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +-{{{778171189} \over {1387545279120}} \ {{ \%B{44}} \sp {31}}}+ +{{{1987468196267} \over {1387545279120}} \ {{ \%B{44}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{155496778477189} \over {693772639560}} \ {{ \%B{44}} \sp {19}}}+ +{{{191631411158401} \over {693772639560}} \ {{ \%B{44}} \sp {13}}}+ +\\ +\\ +\displaystyle +{{{300335488637543} \over {1387545279120}} \ {{ \%B{44}} \sp 7}} - +{{{755656433863} \over {198220754160}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +{{{1094352947} \over {462515093040}} \ {{ \%B{44}} \sp {31}}} - +{{{2794979430821} \over {462515093040}} \ {{ \%B{44}} \sp {25}}} - +\\ +\\ +\displaystyle +{{{218708802908737} \over {231257546520}} \ {{ \%B{44}} \sp {19}}} - +{{{91476663003591} \over {77085848840}} \ {{ \%B{44}} \sp {13}}} - +\\ +\\ +\displaystyle +{{{145152550961823} \over {154171697680}} \ {{ \%B{44}} \sp 7}} - +{{{1564893370717} \over {462515093040}} \ { \%B{44}}}, +\\ +\\ +\displaystyle +-{ \%B{46}} - +{{{4321823003} \over {1387545279120}} \ {{ \%B{44}} \sp {31}}}+ +{{{180949546069} \over {22746643920}} \ {{ \%B{44}} \sp {25}}}+ +\\ +\\ +\displaystyle +{{{863753195062493} \over {693772639560}} \ {{ \%B{44}} \sp {19}}}+ +{{{1088094456732317} \over {693772639560}} \ {{ \%B{44}} \sp {13}}}+ +\\ +\\ +\displaystyle +\left. +{{{1732620732685741} \over {1387545279120}} \ {{ \%B{44}} \sp 7}}+ +{{{13506088516033} \over {1387545279120}} \ { \%B{44}}} +\right],\hbox{\hskip 3.5cm} +\end{array} +$$ +$$ +{\left[ { \%B{53}}, { \%B{57}}, {-{ \%B{57}} -{4 \ { \%B{53}}}}, +{ \%B{53}}, { \%B{53}}, { \%B{53}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{53}}, { \%B{58}}, {-{ \%B{58}} -{4 \ { \%B{53}}}}, +{ \%B{53}}, { \%B{53}}, { \%B{53}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +{\left[ { \%B{54}}, { \%B{55}}, {-{ \%B{55}} -{4 \ { \%B{54}}}}, +{ \%B{54}}, { \%B{54}}, { \%B{54}} +\right]},\hbox{\hskip 3.5cm} +$$ +$$ +\left. +{\left[ { \%B{54}}, { \%B{56}}, {-{ \%B{56}} -{4 \ { \%B{54}}}}, +{ \%B{54}}, { \%B{54}}, { \%B{54}} +\right]}\hbox{\hskip 3.5cm} +\right] +$$ +\returnType{Type: List List RealClosure Fraction Integer} + +We obtain 24 points given by lists of elements in the {\tt RealClosure} +of {\tt Fraction} of {\bf R}. In each list, the first value corresponds +to the indeterminate {\bf f}, the second to {\bf e} and so on. See +{\tt ZeroDimensionalSolvePackage} to learn more about the +\spadfunFrom{realSolve}{ZeroDimensionalSolvePackage} operation. + +\section{LazardSetSolvingPackage} +\label{LazardSetSolvingPackageXmpPage} + +The {\tt LazardSetSolvingPackage} package constructor solves +polynomial systems by means of Lazard triangular sets. However one +condition is relaxed: Regular triangular sets whose saturated ideals +have positive dimension are not necessarily normalized. + +The decompositions are computed in two steps. First the algorithm of +Moreno Maza (implemented in the {\tt RegularTriangularSet} domain +constructor) is called. Then the resulting decompositions are +converted into lists of square-free regular triangular sets and the +redundant components are removed. Moreover, zero-dimensional regular +triangular sets are normalized. + +Note that the way of understanding triangular decompositions +is detailed in the example of the {\tt RegularTriangularSet} +constructor. + +The {\tt LazardSetSolvingPackage} constructor takes six arguments. +The first one, {\bf R}, is the coefficient ring of the polynomials; it +must belong to the category {\tt GcdDomain}. The second one, {\bf E}, +is the exponent monoid of the polynomials; it must belong to the +category {\tt OrderedAbelianMonoidSup}. the third one, {\bf V}, is +the ordered set of variables; it must belong to the category {\tt +OrderedSet}. The fourth one is the polynomial ring; it must belong to +the category {\tt RecursivePolynomialCategory(R,E,V)}. The fifth one +is a domain of the category {\tt RegularTriangularSetCategory(R,E,V,P)} +and the last one is a domain of +the category {\tt SquareFreeRegularTriangularSetCategory(R,E,V,P)}. +The abbreviation for {\tt LazardSetSolvingPackage} is {\tt LAZM3PK}. + +{\bf N.B.} For the purpose of solving zero-dimensional algebraic systems, +see also {\tt LexTriangularPackage} and {\tt ZeroDimensionalSolvePackage}. +These packages are easier to call than {\tt LAZM3PK}. +Moreover, the {\tt ZeroDimensionalSolvePackage} +package provides operations +to compute either the complex roots or the real roots. + +We illustrate now the use of the {\tt LazardSetSolvingPackage} package +constructor with two examples (Butcher and Vermeer). + +Define the coefficient ring. + +\spadcommand{R := Integer} +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the list of variables, + +\spadcommand{ls : List Symbol := [b1,x,y,z,t,v,u,w] } +$$ +\left[ +b1, x, y, z, t, v, u, w +\right] +$$ +\returnType{Type: List Symbol} + +and make it an ordered set: + +\spadcommand{V := OVAR(ls)} +$$ +\mbox{\rm OrderedVariableList [b1,x,y,z,t,v,u,w]} +$$ +\returnType{Type: Domain} + +then define the exponent monoid. + +\spadcommand{E := IndexedExponents V } +$$ +\mbox{\rm IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w]} +$$ +\returnType{Type: Domain} + +Define the polynomial ring. + +\spadcommand{P := NSMP(R, V)} +$$ +\begin{array}{@{}l} +{\rm NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w])} +\end{array} +$$ +\returnType{Type: Domain} + +Let the variables be polynomial. + +\spadcommand{b1: P := 'b1} +$$ +b1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{x: P := 'x } +$$ +x +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{y: P := 'y } +$$ +y +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{z: P := 'z} +$$ +z +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{t: P := 't} +$$ +t +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{u: P := 'u} +$$ +u +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{v: P := 'v} +$$ +v +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{w: P := 'w} +$$ +w +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +Now call the {\tt RegularTriangularSet} domain constructor. + +\spadcommand{T := REGSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm RegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList[b1,x,y,z,t,v,u,w]))} +\end{array} +$$ +\returnType{Type: Domain} + +Define a polynomial system (the Butcher example). + +\spadcommand{p0 := b1 + y + z - t - w} +$$ +b1+y+z -t -w +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1} +$$ +{2 \ v \ y}+{2 \ u \ z}+{2 \ w \ t} -{2 \ {w \sp 2}} -w -1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w} +$$ +{3 \ {v \sp 2} \ y}+{3 \ {u \sp 2} \ z}+{{\left( -{3 \ {w \sp 2}} -1 +\right)} +\ t}+{3 \ {w \sp 3}}+{3 \ {w \sp 2}}+{4 \ w} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w} +$$ +{6 \ v \ z \ x}+{{\left( -{6 \ {w \sp 2}} -{3 \ w} -1 +\right)} +\ t}+{6 \ {w \sp 3}}+{6 \ {w \sp 2}}+{4 \ w} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1} +$$ +{4 \ {v \sp 3} \ y}+{4 \ {u \sp 3} \ z}+{{\left( {4 \ {w \sp 3}}+{4 \ +w} +\right)} +\ t} -{4 \ {w \sp 4}} -{6 \ {w \sp 3}} -{{10} \ {w \sp 2}} -w -1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1} +$$ +{8 \ u \ v \ z \ x}+{{\left( {8 \ {w \sp 3}}+{4 \ {w \sp 2}}+{4 \ w} +\right)} +\ t} -{8 \ {w \sp 4}} -{{12} \ {w \sp 3}} -{{14} \ {w \sp 2}} -{3 \ w} +-1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1} +$$ +{{12} \ {v \sp 2} \ z \ x}+{{\left( {{12} \ {w \sp 3}}+{{12} \ {w \sp +2}}+{8 \ w} +\right)} +\ t} -{{12} \ {w \sp 4}} -{{18} \ {w \sp 3}} -{{14} \ {w \sp 2}} -w -1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1} +$$ +{{\left( -{{24} \ {w \sp 3}} -{{24} \ {w \sp 2}} -{8 \ w} +\right)} +\ t}+{{24} \ {w \sp 4}}+{{36} \ {w \sp 3}}+{{26} \ {w \sp 2}}+{7 \ w}+1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{lp := [p0, p1, p2, p3, p4, p5, p6, p7]} +$$ +\begin{array}{@{}l} +\left[ +{b1+y+z -t -w}, +\right. +\\ +\displaystyle +{{2 \ v \ y}+ +{2 \ u \ z}+ +{2 \ w \ t} - +{2 \ {w \sp 2}} - +w - +1}, +\\ +\displaystyle +{{3 \ {v \sp 2} \ y}+ +{3 \ {u \sp 2} \ z}+ +{{\left( -{3 \ {w \sp 2}} -1 \right)}\ t}+ +{3 \ {w \sp 3}}+ +{3 \ {w \sp 2}}+ +{4 \ w}}, +\\ +\displaystyle +{{6 \ v \ z \ x}+ +{{\left( -{6 \ {w \sp 2}} - +{3 \ w} -1 \right)}\ t}+ +{6 \ {w \sp 3}}+ +{6 \ {w \sp 2}}+ +{4 \ w}}, +\\ +\displaystyle +{{4 \ {v \sp 3} \ y}+ +{4 \ {u \sp 3} \ z}+ +{{\left( {4 \ {w \sp 3}}+{4 \ w} \right)}\ t} - +{4 \ {w \sp 4}} - +{6 \ {w \sp 3}} - +{{10} \ {w \sp 2}} - +w - +1}, +\\ +\displaystyle +{{8 \ u \ v \ z \ x}+ +{{\left( {8 \ {w \sp 3}}+{4 \ {w \sp 2}}+{4 \ w} \right)}\ t} - +{8 \ {w \sp 4}} - +{{12} \ {w \sp 3}} - +{{14} \ {w \sp 2}} - +{3 \ w} +-1}, +\\ +\displaystyle +{{{12} \ {v \sp 2} \ z \ x}+ +{{\left( {{12} \ {w \sp 3}}+{{12} \ {w \sp 2}}+{8 \ w} \right)}\ t} - +{{12} \ {w \sp 4}} - +{{18} \ {w \sp 3}} - +{{14} \ {w \sp 2}} - +w - +1}, +\\ +\displaystyle +\left. +{{{\left( -{{24} \ {w \sp 3}} -{{24} \ {w \sp 2}} -{8 \ w} \right)}\ t}+ +{{24} \ {w \sp 4}}+ +{{36} \ {w \sp 3}}+ +{{26} \ {w \sp 2}}+ +{7 \ w}+ +1} +\right] +\end{array} +$$ +\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +First of all, let us solve this system in the sense of Lazard by means +of the {\tt REGSET} constructor: + +\spadcommand{lts := zeroSetSplit(lp,false)\$T} +$$ +\begin{array}{@{}l} +\left[ +{\left\{ {w+1}, u, v, {t+1}, {b1+y+z+2} \right\}}, +{\left\{ {w+1}, v, {t+1}, z, {b1+y+2} \right\}}, +\right. +\\ +\displaystyle +{\left\{ {w+1}, {t+1}, z, y, {b1+2} \right\}}, +{\left\{ {w+1}, {v -u}, {t+1}, {y+z}, x, {b1+2} \right\}}, +\\ +\displaystyle +{\left\{ {w+1}, u, {t+1}, y, x, {b1+z+2} \right\}}, +\\ +\displaystyle +\left\{ +{{{144} \ {w \sp 5}}+{{216} \ {w \sp 4}}+{{96} \ {w \sp 3}}+{6 +\ {w \sp 2}} -{{11} \ w} -1}, +\right. +\\ +\displaystyle +{{{\left( {{12} \ {w \sp 2}}+{9 \ w}+1 \right)}\ u} - +{{72} \ {w \sp 5}} - +{{108} \ {w \sp 4}} - +{{42} \ {w \sp 3}} - +{9 \ {w \sp 2}} - +{3 \ w}}, +\\ +\displaystyle +{{{\left( {{12} \ {w \sp 2}}+{9 \ w}+1 \right)}\ v}+ +{{36} \ {w \sp 4}}+ +{{54} \ {w \sp 3}}+ +{{18} \ {w \sp 2}}}, +\\ +\displaystyle +{{{\left( {{24} \ {w \sp 3}}+{{24} \ {w \sp 2}}+{8 \ w} \right)}\ t} - +{{24} \ {w \sp 4}} - +{{36} \ {w \sp 3}} - +{{26} \ {w \sp 2}} - +{7 \ w} -1}, +\\ +\displaystyle +{{\left( {{12} \ u \ v} -{{12} \ {u \sp 2}} \right)}\ z}+ +{{\left( {{12} \ w \ v}+{{12} \ {w \sp 2}}+4 \right)}\ t}+ +{{\left( {3 \ w} -5 \right)}\ v}+ +\\ +\displaystyle +\ \ {{36} \ {w \sp 4}}+ +{{42} \ {w \sp 3}}+ +{6 \ {w \sp 2}} - +{{16} \ w}, +\\ +\displaystyle +{{2 \ v \ y}+ +{2 \ u \ z}+ +{2 \ w \ t} - +{2 \ {w \sp 2}} - +w -1}, +\\ +\displaystyle +\left. +\left. +{{6 \ v \ z \ x}+ +{{\left( -{6 \ {w \sp 2}} -{3 \ w} -1 \right)}\ t}+ +{6 \ {w \sp 3}}+ +{6 \ {w \sp 2}}+ +{4 \ w}}, +{b1+y+z -t -w} +\right\} +\right] +\end{array} +$$ +\returnType{Type: List +RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w], +OrderedVariableList [b1,x,y,z,t,v,u,w], +NewSparseMultivariatePolynomial( +Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))} + +We can get the dimensions of each component +of a decomposition as follows. + +\spadcommand{[coHeight(ts) for ts in lts] } +$$ +\left[ +3, 3, 3, 2, 2, 0 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +The first five sets have a simple shape. However, the last one, which +has dimension zero, can be simplified by using Lazard triangular sets. + +Thus we call the {\tt SquareFreeRegularTriangularSet} domain constructor, + +\spadcommand{ST := SREGSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm SquareFreeRegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList[b1,x,y,z,t,v,u,w]))} +\end{array} +$$ +\returnType{Type: Domain} + +and set the {\tt LAZM3PK} package constructor to our situation. + +\spadcommand{pack := LAZM3PK(R,E,V,P,T,ST)} +$$ +\begin{array}{@{}l} +{\rm LazardSetSolvingPackage(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList[b1,x,y,z,t,v,u,w]),} +\\ +\displaystyle +{\rm \ \ RegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList[b1,x,y,z,t,v,u,w])),} +\\ +\displaystyle +{\rm \ \ SquareFreeRegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [b1,x,y,z,t,v,u,w],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList[b1,x,y,z,t,v,u,w])))} +\end{array} +$$ +\returnType{Type: Domain} + +We are ready to solve the system by means of Lazard triangular sets: + +\spadcommand{zeroSetSplit(lp,false)\$pack} +$$ +\begin{array}{@{}l} +\left[ +{\left\{ {w+1}, {t+1}, z, y, {b1+2} \right\}}, +\right. +\\ +\displaystyle +{\left\{ {w+1}, v, {t+1}, z, {b1+y+2} \right\}}, +\\ +\displaystyle +{\left\{ {w+1}, u, v, {t+1}, {b1+y+z+2} \right\}}, +\\ +\displaystyle +{\left\{ {w+1}, {v -u}, {t+1}, {y+z}, x, {b1+2} \right\}}, +\\ +\displaystyle +{\left\{ {w+1}, u, {t+1}, y, x, {b1+z+2} \right\}}, +\\ +\displaystyle +\left\{ +{{{144} \ {w \sp 5}}+ +{{216} \ {w \sp 4}}+ +{{96} \ {w \sp 3}}+ +{6 \ {w \sp 2}} - +{{11} \ w} -1}, +\right. +\\ +\displaystyle +{u -{{24} \ {w \sp 4}} - +{{36} \ {w \sp 3}} - +{{14} \ {w \sp 2}}+w+1}, +\\ +\displaystyle +{{3 \ v} -{{48} \ {w \sp 4}} - +{{60} \ {w \sp 3}} - +{{10} \ {w \sp 2}}+ +{8 \ w}+2}, +\\ +\displaystyle +{t -{{24} \ {w \sp 4}} - +{{36} \ {w \sp 3}} - +{{14} \ {w \sp 2}} -w+1}, +{{486} \ z} - +{{2772} \ {w \sp 4}} - +\\ +\displaystyle +\ \ {{4662} \ {w \sp 3}} - +{{2055} \ {w \sp 2}}+ +{{30} \ w}+{127}, +\\ +\displaystyle +{{{2916} \ y} - +{{22752} \ {w \sp 4}} - +{{30312} \ {w \sp 3}} - +{{8220} \ {w \sp 2}}+ +{{2064} \ w}+{1561}}, +\\ +\displaystyle +{{{356} \ x} - +{{3696} \ {w \sp 4}} - +{{4536} \ {w \sp 3}} - +{{968} \ {w \sp 2}}+ +{{822} \ w}+{371}}, +\\ +\displaystyle +\left. +\left. +{{{2916} \ b1} - +{{30600} \ {w \sp 4}} - +{{46692} \ {w \sp 3}} - +{{20274} \ {w \sp 2}} - +{{8076} \ w}+{593}} +\right\} +\right] +\end{array} +$$ +\returnType{Type: List +SquareFreeRegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w], +OrderedVariableList [b1,x,y,z,t,v,u,w], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [b1,x,y,z,t,v,u,w]))} + +We see the sixth triangular set is {\em nicer} now: each one of its +polynomials has a constant initial. + +We follow with the Vermeer example. The ordering is the usual one +for this system. + +Define the polynomial system. + +\spadcommand{f0 := (w - v) ** 2 + (u - t) ** 2 - 1} +$$ +{t \sp 2} -{2 \ u \ t}+{v \sp 2} -{2 \ w \ v}+{u \sp 2}+{w \sp 2} -1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{f1 := t ** 2 - v ** 3} +$$ +{t \sp 2} -{v \sp 3} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t)} +$$ +{{\left( -{3 \ {v \sp 2}} -{2 \ v}+{2 \ w} +\right)} +\ t}+{3 \ u \ {v \sp 2}} +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1)} +$$ +{6 \ {v \sp 2} \ t \ {z \sp 2}}+{{\left( -{2 \ t} -{3 \ {v \sp 2}} +\right)} +\ z}+1 +$$ +\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +\spadcommand{lf := [f0, f1, f2, f3]} +$$ +\begin{array}{@{}l} +\left[ +{{t \sp 2} -{2 \ u \ t}+{v \sp 2} -{2 \ w \ v}+{u \sp 2}+{w \sp 2} -1}, +\right. +\\ +\displaystyle +{{t \sp 2} -{v \sp 3}}, +\\ +\displaystyle +{{{\left( -{3 \ {v \sp 2}} -{2 \ v}+{2 \ w} \right)}\ t}+ +{3 \ u \ {v \sp 2}}}, +\\ +\displaystyle +\left. +{{6 \ {v \sp 2} \ t \ {z \sp 2}}+ +{{\left( -{2 \ t} -{3 \ {v \sp 2}}\right)}\ z}+1} +\right] +\end{array} +$$ +\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])} + +First of all, let us solve this system in the sense of Kalkbrener by +means of the {\tt REGSET} constructor: + +\spadcommand{zeroSetSplit(lf,true)\$T} +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{{729} \ {u \sp 6}}+ +{{\left( +-{{1458} \ {w \sp 3}}+ +{{729} \ {w \sp 2}} - +{{4158} \ w} -{1685} +\right)}\ {u \sp 4}}+ +\right. +\right. +\\ +\displaystyle +{{\left( {{729} \ {w \sp 6}} - +{{1458} \ {w \sp 5}} -{{2619} \ {w \sp 4}} - +{{4892} \ {w \sp 3}} -{{297} \ {w \sp 2}}+ +{{5814} \ w}+{427} +\right)}\ {u \sp 2}}+ +\\ +\displaystyle +{{729} \ {w \sp 8}}+ +{{216} \ {w \sp 7}} - +{{2900} \ {w \sp 6}} - +{{2376} \ {w \sp 5}}+ +{{3870} \ {w \sp 4}}+ +\\ +\displaystyle +\ \ {{4072} \ {w \sp 3}} - +{{1188} \ {w \sp 2}} - +{{1656} \ w}+{529}, +\\ +\displaystyle +\left( +{{2187} \ {u \sp 4}}+ +\left( +-{{4374} \ {w \sp 3}} - +{{972} \ {w \sp 2}} - +{{12474} \ w} +-{2868} +\right)\ {u \sp 2}+ +\right. +\\ +\displaystyle +\left. +{{2187} \ {w \sp 6}} - +{{1944} \ {w \sp 5}} - +{{10125} \ {w \sp 4}} - +{{4800} \ {w \sp 3}}+ +{{2501} \ {w \sp 2}}+ +{{4968} \ w} -{1587} +\right)\ v+ +\\ +\displaystyle +\left( +{{1944} \ {w \sp 3}} - +{{108} \ {w \sp 2}} +\right)\ {u \sp 2}+ +\\ +\displaystyle +{{972} \ {w \sp 6}}+ +{{3024} \ {w \sp 5}} - +{{1080} \ {w \sp 4}}+ +{{496} \ {w \sp 3}}+ +{{1116} \ {w \sp 2}}, +\\ +\displaystyle +{{{\left( +{3 \ {v \sp 2}}+ +{2 \ v} - +{2 \ w} +\right)}\ t} - +{3 \ u \ {v \sp 2}}}, +\\ +\displaystyle +\left. +\left. +\left( +\left( +{4 \ v} -{4 \ w} +\right)\ t - +{6 \ u \ {v \sp 2}} +\right)\ {z \sp 2}+ +\left( +{2 \ t}+ +{3 \ {v \sp 2}} +\right) +\ z -1 +\right\} +\right] +\end{array} +$$ +\returnType{Type: List +RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w], +OrderedVariableList [b1,x,y,z,t,v,u,w], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [b1,x,y,z,t,v,u,w]))} + +We have obtained one regular chain (i.e. regular triangular set) with +dimension 1. This set is in fact a characterist set of the (radical +of) of the ideal generated by the input system {\bf lf}. Thus we have +only the {\em generic points} of the variety associated with {\bf lf} +(for the elimination ordering given by {\bf ls}). + +So let us get now a full description of this variety. + +Hence, we solve this system in the sense of Lazard by means of the {\tt REGSET} +constructor: + +\spadcommand{zeroSetSplit(lf,false)\$T} +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{{729} \ {u \sp 6}}+ +\left( +-{{1458} \ {w \sp 3}}+ +{{729} \ {w \sp 2}} - +{{4158} \ w} - +{1685} +\right)\ {u \sp 4}+ +\right. +\right. +\\ +\displaystyle +\left( +{{729} \ {w \sp 6}} - +{{1458} \ {w \sp 5}} - +{{2619} \ {w \sp 4}} - +{{4892} \ {w \sp 3}} - +{{297} \ {w \sp 2}}+ +{{5814} \ w}+ +{427} +\right)\ {u \sp 2}+ +\\ +\displaystyle +{{729} \ {w \sp 8}}+ +{{216} \ {w \sp 7}} - +{{2900} \ {w \sp 6}} - +{{2376} \ {w \sp 5}}+ +{{3870} \ {w \sp 4}}+ +{{4072} \ {w \sp 3}} - +\\ +\displaystyle +{{1188} \ {w \sp 2}} - +{{1656} \ w}+ +{529}, +\\ +\displaystyle +\left( +{{2187} \ {u \sp 4}}+ +\left( +-{{4374} \ {w \sp 3}} - +{{972} \ {w \sp 2}} - +{{12474} \ w} - +{2868} +\right)\ {u \sp 2}+ +\right. +\\ +\displaystyle +\left. +{{2187} \ {w \sp 6}} - +{{1944} \ {w \sp 5}} - +{{10125} \ {w \sp 4}} - +{{4800} \ {w \sp 3}}+ +{{2501} \ {w \sp 2}}+ +{{4968} \ w} - +{1587} +\right)\ v+ +\\ +\displaystyle +\left( +{{1944} \ {w \sp 3}} - +{{108} \ {w \sp 2}} +\right)\ {u \sp 2}+ +\\ +\displaystyle +{{972} \ {w \sp 6}}+ +{{3024} \ {w \sp 5}} - +{{1080} \ {w \sp 4}}+ +{{496} \ {w \sp 3}}+ +{{1116} \ {w \sp 2}}, +\\ +\displaystyle +\left( +{3 \ {v \sp 2}}+ +{2 \ v} - +{2 \ w} +\right)\ t - +{3 \ u \ {v \sp 2}}, +\\ +\displaystyle +\left. +\left( +\left( +{4 \ v} - +{4 \ w} +\right)\ t - +{6 \ u \ {v \sp 2}} +\right)\ {z \sp 2}+ +\left( +{2 \ t}+ +{3 \ {v \sp 2}} +\right) +\ z -1 +\right\}, +\\ +\displaystyle +\left\{ +{{27} \ {w \sp 4}}+ +{4 \ {w \sp 3}} - +{{54} \ {w \sp 2}} - +{{36} \ w}+ +{23}, +\right. +\\ +\displaystyle +u, +\\ +\displaystyle +\left( +{{12} \ w}+ +2 +\right)\ v - +{9 \ {w \sp 2}} - +{2 \ w}+ +9, +\\ +\displaystyle +{6 \ {t \sp 2}} - +{2 \ v} - +{3 \ {w \sp 2}}+ +{2 \ w}+3, +\\ +\displaystyle +\left. +{{2 \ t \ z} -1} +\right\}, +\\ +\displaystyle +\left\{ +{{{59049} \ {w \sp 6}}+ +{{91854} \ {w \sp 5}} - +{{45198} \ {w \sp 4}}+ +{{145152} \ {w \sp 3}}+ +{{63549} \ {w \sp 2}}+ +{{60922} \ w}+{21420}}, +\right. +\\ +\displaystyle +\left( +{{31484448266904} \ {w \sp 5}} - +{{18316865522574} \ {w \sp 4}}+ +{{23676995746098} \ {w \sp 3}}+ +{{6657857188965} \ {w \sp 2}}+ +\right. +\\ +\displaystyle +\left. +{{8904703998546} \ w}+ +{3890631403260} +\right)\ {u \sp 2}+ +{{94262810316408} \ {w \sp 5}} - +{{82887296576616} \ {w \sp 4}}+ +\\ +\displaystyle +{{89801831438784} \ {w \sp 3}}+ +{{28141734167208} \ {w \sp 2}}+ +{{38070359425432} \ w}+ +{16003865949120}, +\\ +\displaystyle +\left( +{{243} \ {w \sp 2}}+ +{{36} \ w}+ +{85} +\right)\ {v \sp 2}+ +\left( +-{{81} \ {u \sp 2}} - +{{162} \ {w \sp 3}}+ +{{36} \ {w \sp 2}}+ +{{154} \ w}+ +{72} +\right)\ v - +{{72} \ {w \sp 3}}+ +{4 \ {w \sp 2}}, +\\ +\displaystyle +\left( +{3 \ {v \sp 2}}+ +{2 \ v} - +{2 \ w} +\right)\ t - +{3 \ u \ {v \sp 2}}, +\\ +\displaystyle +\left. +\left( +\left( +{4 \ v} - +{4 \ w} +\right)\ t - +{6 \ u \ {v \sp 2}} +\right)\ {z \sp 2}+ +\left( {2 \ t}+ +{3 \ {v \sp 2}} +\right)\ z +-1 +\right\}, +\\ +\displaystyle +\left\{ +{{{27} \ {w \sp 4}}+ +{4 \ {w \sp 3}} - +{{54} \ {w \sp 2}} - +{{36} \ w}+ +{23}}, u, +\right. +\\ +\displaystyle +{{{\left( +{{12} \ w}+ +2 +\right)}\ v} - +{9 \ {w \sp 2}} - +{2 \ w}+9}, +\\ +\displaystyle +{{6 \ {t \sp 2}} - +{2 \ v} -{3 \ {w \sp 2}}+ +{2 \ w}+ +3}, +\\ +\displaystyle +\left. +\left. +{{3 \ {v \sp 2} \ z} +-1} +\right\} +\right] +\end{array} +$$ +\returnType{Type: List +RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w], +OrderedVariableList [b1,x,y,z,t,v,u,w], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [b1,x,y,z,t,v,u,w]))} + +We retrieve our regular chain of dimension 1 and we get three regular +chains of dimension 0 corresponding to the {\em degenerated cases}. +We want now to simplify these zero-dimensional regular chains by using +Lazard triangular sets. Moreover, this will allow us to prove that +the above decomposition has no redundant component. {\bf N.B.} +Generally, decompositions computed by the {\tt REGSET} constructor do +not have redundant components. However, to be sure that no redundant +component occurs one needs to use the {\tt SREGSET} or {\tt LAZM3PK} +constructors. + +So let us solve the input system in the sense of Lazard by means of +the {\tt LAZM3PK} constructor: + +\spadcommand{zeroSetSplit(lf,false)\$pack } +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{{729} \ {u \sp 6}}+ +\left( +-{{1458} \ {w \sp 3}}+ +{{729} \ {w \sp 2}} - +{{4158} \ w} - +{1685} +\right)\ {u \sp 4}+ +\right. +\right. +\\ +\displaystyle +\ \ \left( +{{729} \ {w \sp 6}} - +{{1458} \ {w \sp 5}} - +{{2619} \ {w \sp 4}} - +{{4892} \ {w \sp 3}} - +{{297} \ {w \sp 2}}+ +{{5814} \ w}+{427} +\right)\ {u \sp 2}+ +\\ +\displaystyle +\ \ {{729} \ {w \sp 8}}+ +{{216} \ {w \sp 7}} - +{{2900} \ {w \sp 6}} - +{{2376} \ {w \sp 5}}+ +{{3870} \ {w \sp 4}}+ +{{4072} \ {w \sp 3}} - +\\ +\displaystyle +\ \ {{1188} \ {w \sp 2}} - +{{1656} \ w}+{529}, +\\ +\displaystyle +\left( +{{2187} \ {u \sp 4}}+ +\left( +-{{4374} \ {w \sp 3}} - +{{972} \ {w \sp 2}} - +{{12474} \ w} +-{2868} +\right)\ {u \sp 2}+ +\right. +\\ +\displaystyle +\left. +\ \ {{2187} \ {w \sp 6}} - +{{1944} \ {w \sp 5}} - +{{10125} \ {w \sp 4}} - +{{4800} \ {w \sp 3}}+ +{{2501} \ {w \sp 2}}+ +{{4968} \ w} - +{1587} +\right)\ v+ +\\ +\displaystyle +\left( +{{1944} \ {w \sp 3}} - +{{108} \ {w \sp 2}} +\right)\ {u \sp 2}+ +{{972} \ {w \sp 6}}+ +{{3024} \ {w \sp 5}} - +{{1080} \ {w \sp 4}}+ +{{496} \ {w \sp 3}}+ +{{1116} \ {w \sp 2}}, +\\ +\displaystyle +\left( +{3 \ {v \sp 2}}+ +{2 \ v} - +{2 \ w} +\right)\ t - +{3 \ u \ {v \sp 2}}, +\\ +\displaystyle +\left. +\left( +\left( +{4 \ v} - +{4 \ w} +\right)\ t - +{6 \ u \ {v \sp 2}} +\right)\ {z \sp 2}+ +\left( +{2 \ t}+ +{3 \ {v \sp 2}} +\right)\ z -1 +\right\}, +\\ +\displaystyle +\left\{ +{{81} \ {w \sp 2}}+ +{{18} \ w}+ +{28}, +{{729} \ {u \sp 2}} - +{{1890} \ w} - +{533}, +{{81} \ {v \sp 2}}+ +\left( +-{{162} \ w}+ +{27} +\right)\ v - +\right. +\\ +\displaystyle +{{72} \ w} - +{112}, +\\ +\displaystyle +{{{11881} \ t}+ +\left( {{972} \ w}+{2997} +\right)\ u \ v+ +{{\left( -{{11448} \ w} -{11536} +\right)}\ u}}, +\\ +\displaystyle +{{641237934604288} \ {z \sp 2}}+ +\left( +\left( +\left( +{{78614584763904} \ w}+ +{26785578742272} +\right)\ u+ +\right. +\right. +\\ +\displaystyle +\left. +\ \ {{236143618655616} \ w}+ +{70221988585728} +\right)\ v+ +\left( +{{358520253138432} \ w}+ +\right. +\\ +\displaystyle +\left. +\left. +\ \ {101922133759488} +\right)\ u+ +{{142598803536000} \ w}+ +{54166419595008} +\right)\ z+ +\\ +\displaystyle +\ \ \left( +{{32655103844499} \ w} - +{44224572465882} +\right)\ u \ v+ +\\ +\displaystyle +\left. +\left( +{{43213900115457} \ w} - +{32432039102070} +\right)\ u +\right\}, +\\ +\displaystyle +\left\{ +{{27} \ {w \sp 4}}+ +{4 \ {w \sp 3}} - +{{54} \ {w \sp 2}} - +{{36} \ w}+ +{23}, +u, +{{218} \ v} - +{{162} \ {w \sp 3}}+ +{3 \ {w \sp 2}}+ +{{160} \ w}+ +{153}, +\right. +\\ +\displaystyle +\ \ {{109} \ {t \sp 2}} - +{{27} \ {w \sp 3}} - +{{54} \ {w \sp 2}}+ +{{63} \ w}+ +{80}, +\\ +\displaystyle +\left. +\ \ {{1744} \ z}+ +\left( +-{{1458} \ {w \sp 3}}+ +{{27} \ {w \sp 2}}+ +{{1440} \ w}+ +{505} +\right)\ t +\right\}, +\\ +\displaystyle +\left\{ +{{27} \ {w \sp 4}}+ +{4 \ {w \sp 3}} - +{{54} \ {w \sp 2}} - +{{36} \ w}+{23}, +u, +{{218} \ v} - +{{162} \ {w \sp 3}}+ +{3 \ {w \sp 2}}+ +{{160} \ w}+ +{153}, +\right. +\\ +\displaystyle +\left. +\ \ {{109} \ {t \sp 2}} - +{{27} \ {w \sp 3}} - +{{54} \ {w \sp 2}}+ +{{63} \ w}+{80}, +{{1308} \ z}+ +{{162} \ {w \sp 3}} - +{3 \ {w \sp 2}} - +{{814} \ w} - +{153} +\right\}, +\\ +\displaystyle +\left\{ +{{729} \ {w \sp 4}}+ +{{972} \ {w \sp 3}} - +{{1026} \ {w \sp 2}}+ +{{1684} \ w}+ +{765}, +{{81} \ {u \sp 2}}+ +{{72} \ {w \sp 2}}+ +{{16} \ w} -{72}, +\right. +\\ +\displaystyle +\ \ {{702} \ v} - +{{162} \ {w \sp 3}} - +{{225} \ {w \sp 2}}+ +{{40} \ w} - +{99}, +\\ +\displaystyle +\ \ {{11336} \ t}+ +\left( +{{324} \ {w \sp 3}} - +{{603} \ {w \sp 2}} - +{{1718} \ w} - +{1557} +\right)\ u, +\\ +\displaystyle +\ \ {{595003968} \ {z \sp 2}}+ +\left( +\left( +-{{963325386} \ {w \sp 3}} - +{{898607682} \ {w \sp 2}}+ +{{1516286466} \ w} - +\right. +\right. +\\ +\displaystyle +\left. +\ \ {3239166186} +\right)\ u - +{{1579048992} \ {w \sp 3}} - +{{1796454288} \ {w \sp 2}}+ +{{2428328160} \ w} - +\\ +\displaystyle +\left. +{4368495024} +\right)\ z+ +\left( +{{9713133306} \ {w \sp 3}}+ +{{9678670317} \ {w \sp 2}} - +{{16726834476} \ w}+ +\right. +\\ +\displaystyle +\left. +\left. +\left. +{28144233593} +\right)\ u +\right\} +\right] +\end{array} +$$ +\returnType{Type: List +SquareFreeRegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w], +OrderedVariableList [b1,x,y,z,t,v,u,w], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [b1,x,y,z,t,v,u,w]))} + +Due to square-free factorization, we obtained now four +zero-dimensional regular chains. Moreover, each of them is normalized +(the initials are constant). Note that these zero-dimensional +components may be investigated further with the +{\tt ZeroDimensionalSolvePackage} package constructor. + +\section{Library} +\label{LibraryXmpPage} + +The {\tt Library} domain provides a simple way to store Axiom values +in a file. This domain is similar to {\tt KeyedAccessFile} but fewer +declarations are needed and items of different types can be saved +together in the same file. + +To create a library, you supply a file name. + +\spadcommand{stuff := library "/tmp/Neat.stuff" } +$$ +\mbox{\tt "/tmp/Neat.stuff"} +$$ +\returnType{Type: Library} + +Now values can be saved by key in the file. +The keys should be mnemonic, just as the field names are for records. +They can be given either as strings or symbols. + +\spadcommand{stuff.int := 32**2} +$$ +1024 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{stuff."poly" := x**2 + 1} +$$ +{x \sp 2}+1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{stuff.str := "Hello"} +$$ +\mbox{\tt "Hello"} +$$ +\returnType{Type: String} + +You obtain the set of available keys using the +\spadfunFrom{keys}{Library} operation. + +\spadcommand{keys stuff} +$$ +\left[ +\mbox{\tt "str"} , \mbox{\tt "poly"} , \mbox{\tt "int"} +\right] +$$ +\returnType{Type: List String} + +You extract values by giving the desired key in this way. + +\spadcommand{stuff.poly} +$$ +{x \sp 2}+1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{stuff("poly")} +$$ +{x \sp 2}+1 +$$ +\returnType{Type: Polynomial Integer} + +When the file is no longer needed, you should remove it from the +file system. + +\spadcommand{)system rm -rf /tmp/Neat.stuff } + +For more information on related topics, see +\ref{FileXmpPage} on page~\pageref{FileXmpPage}, +\ref{TextFileXmpPage} on page~\pageref{TextFileXmpPage}, and +\ref{KeyedAccessFileXmpPage} on page~\pageref{KeyedAccessFileXmpPage}. + +\section{LieExponentials} +\label{LieExponentialsXmpPage} + +\spadcommand{ a: Symbol := 'a } +$$ +a +$$ +\returnType{Type: Symbol} + +\spadcommand{ b: Symbol := 'b } +$$ +b +$$ +\returnType{Type: Symbol} + +Declarations of domains + +\spadcommand{ coef := Fraction(Integer) } +$$ +\mbox{\rm Fraction Integer} +$$ +\returnType{Type: Domain} + +\spadcommand{ group := LieExponentials(Symbol, coef, 3) } +$$ +\mbox{\rm LieExponentials(Symbol,Fraction Integer,3)} +$$ +\returnType{Type: Domain} + +\spadcommand{ lpoly := LiePolynomial(Symbol, coef) } +$$ +\mbox{\rm LiePolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{ poly := XPBWPolynomial(Symbol, coef) } +$$ +\mbox{\rm XPBWPolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +Calculations + +\spadcommand{ ea := exp(a::lpoly)\$group} +$$ +e \sp {\left[ a +\right]} +$$ +\returnType{Type: LieExponentials(Symbol,Fraction Integer,3)} + +\spadcommand{ eb := exp(b::lpoly)\$group} +$$ +e \sp {\left[ b +\right]} +$$ +\returnType{Type: LieExponentials(Symbol,Fraction Integer,3)} + +\spadcommand{ g: group := ea*eb} +$$ +{e \sp {\left[ b +\right]}} +\ {e \sp {\left( {1 \over 2} \ {\left[ a \ {b \sp 2} +\right]} +\right)}} +\ {e \sp {\left[ a \ b +\right]}} +\ {e \sp {\left( {1 \over 2} \ {\left[ {a \sp 2} \ b +\right]} +\right)}} +\ {e \sp {\left[ a +\right]}} +$$ +\returnType{Type: LieExponentials(Symbol,Fraction Integer,3)} + +\spadcommand{ g :: poly } +$$ +\begin{array}{@{}l} +1+ +{\left[ a \right]}+ +{\left[b \right]}+ +{{1\over 2} \ {\left[ a \right]}\ {\left[ a \right]}}+ +{\left[a \ b \right]}+ +{{\left[b \right]}\ {\left[ a \right]}}+ +{{1\over 2} \ {\left[ b \right]}\ {\left[ b \right]}}+ +{{1\over 6} \ {\left[ a \right]}\ {\left[ a \right]}\ {\left[ a \right]}}+ +{{1\over 2} \ {\left[ {a \sp 2} \ b \right]}}+ +\\ +\\ +\displaystyle +{{\left[a \ b \right]}\ {\left[ a \right]}}+ +{{1\over 2} \ {\left[ a \ {b \sp 2} \right]}}+ +{{1\over 2} \ {\left[ b \right]}\ {\left[ a \right]}\ {\left[ a \right]}}+ +{{\left[b \right]}\ {\left[ a \ b \right]}}+ +{{1\over 2} \ {\left[ b \right]}\ {\left[ b \right]}\ {\left[ a \right]}}+ +{{1\over 6} \ {\left[ b \right]}\ {\left[ b \right]}\ {\left[ b \right]}} +\end{array} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{ log(g)\$group } +$$ +{\left[ a +\right]}+{\left[ +b +\right]}+{{1 +\over 2} \ {\left[ a \ b +\right]}}+{{1 +\over {12}} \ {\left[ {a \sp 2} \ b +\right]}}+{{1 +\over {12}} \ {\left[ a \ {b \sp 2} +\right]}} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{ g1: group := inv(g) } +$$ +{e \sp {\left( -{\left[ b +\right]} +\right)}} +\ {e \sp {\left( -{\left[ a +\right]} +\right)}} +$$ +\returnType{Type: LieExponentials(Symbol,Fraction Integer,3)} + +\spadcommand{ g*g1 } +$$ +1 +$$ +\returnType{Type: LieExponentials(Symbol,Fraction Integer,3)} + +\section{LiePolynomial} +\label{LiePolynomialXmpPage} +Declaration of domains + +\spadcommand{RN := Fraction Integer } +$$ +\mbox{\rm Fraction Integer} +$$ +\returnType{Type: Domain} + +\spadcommand{Lpoly := LiePolynomial(Symbol,RN) } +$$ +\mbox{\rm LiePolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{Dpoly := XDPOLY(Symbol,RN) } +$$ +\mbox{\rm XDistributedPolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{Lword := LyndonWord Symbol } +$$ +\mbox{\rm LyndonWord Symbol} +$$ +\returnType{Type: Domain} + +Initialisation + +\spadcommand{a:Symbol := 'a } +$$ +a +$$ +\returnType{Type: Symbol} + +\spadcommand{b:Symbol := 'b } +$$ +b +$$ +\returnType{Type: Symbol} + +\spadcommand{c:Symbol := 'c } +$$ +c +$$ +\returnType{Type: Symbol} + +\spadcommand{aa: Lpoly := a } +$$ +\left[ +a +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{bb: Lpoly := b } +$$ +\left[ +b +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{cc: Lpoly := c } +$$ +\left[ +c +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{p : Lpoly := [aa,bb]} +$$ +\left[ +a \ b +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{q : Lpoly := [p,bb]} +$$ +\left[ +a \ {b \sp 2} +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +All the Lyndon words of order 4 + +\spadcommand{liste : List Lword := LyndonWordsList([a,b], 4)} +$$ +\left[ +{\left[ a +\right]}, +{\left[ b +\right]}, +{\left[ a \ b +\right]}, +{\left[ {a \sp 2} \ b +\right]}, +{\left[ a \ {b \sp 2} +\right]}, +{\left[ {a \sp 3} \ b +\right]}, +{\left[ {a \sp 2} \ {b \sp 2} +\right]}, +{\left[ a \ {b \sp 3} +\right]} +\right] +$$ +\returnType{Type: List LyndonWord Symbol} + +\spadcommand{r: Lpoly := p + q + 3*LiePoly(liste.4)\$Lpoly} +$$ +{\left[ a \ b +\right]}+{3 +\ {\left[ {a \sp 2} \ b +\right]}}+{\left[ +a \ {b \sp 2} +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{s:Lpoly := [p,r]} +$$ +-{3 \ {\left[ {a \sp 2} \ b \ a \ b +\right]}}+{\left[ +a \ b \ a \ {b \sp 2} +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5)} +$$ +{2 \ {\left[ a \ b +\right]}} +-{5 \ {\left[ a \ {b \sp 2} +\right]}} +-{3 \ {\left[ {a \sp 2} \ b \ a \ b +\right]}}+{\left[ +a \ b \ a \ {b \sp 2} +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{degree t } +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{mirror t } +$$ +-{2 \ {\left[ a \ b +\right]}} +-{5 \ {\left[ a \ {b \sp 2} +\right]}} +-{3 \ {\left[ {a \sp 2} \ b \ a \ b +\right]}}+{\left[ +a \ b \ a \ {b \sp 2} +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +Jacobi Relation + +\spadcommand{Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [ [p,q]\$Lpoly, r] + [ [q,r]\$Lpoly, p] + [ [r,p]\$Lpoly, q] } +\begin{verbatim} +Function declaration Jacobi : ( + LiePolynomial(Symbol, Fraction Integer), + LiePolynomial(Symbol,Fraction Integer), + LiePolynomial(Symbol,Fraction Integer)) -> + LiePolynomial(Symbol,Fraction Integer) + has been added to workspace. +\end{verbatim} +\returnType{Void} + +Tests + +\spadcommand{test: Lpoly := Jacobi(a,b,b) } +$$ +0 +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{test: Lpoly := Jacobi(p,q,r) } +$$ +0 +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{test: Lpoly := Jacobi(r,s,t) } +$$ +0 +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +Evaluation + +\spadcommand{eval(p, a, p)\$Lpoly} +$$ +\left[ +a \ {b \sp 2} +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{eval(p, [a,b], [2*bb, 3*aa])\$Lpoly } +$$ +-{6 \ {\left[ a \ b +\right]}} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{r: Lpoly := [p,c] } +$$ +{\left[ a \ b \ c +\right]}+{\left[ +a \ c \ b +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])\$Lpoly } +$$ +-{\left[ a \ b \ c +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])\$Lpoly } +$$ +-{\left[ a \ c \ b +\right]} +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{r + r1 + r2 } +$$ +0 +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\section{LinearOrdinaryDifferentialOperator} +\label{LinearOrdinaryDifferentialOperatorXmpPage} + +{\tt LinearOrdinaryDifferentialOperator(A, diff)} is the domain of +linear ordinary differential operators with coefficients in a ring +{\tt A} with a given derivation. + +\subsection{Differential Operators with Series Coefficients} + +\noindent +{\bf Problem:} +Find the first few coefficients of {\tt exp(x)/x**i} of {\tt Dop phi} where +\begin{verbatim} +Dop := D**3 + G/x**2 * D + H/x**3 - 1 +phi := sum(s[i]*exp(x)/x**i, i = 0..) +\end{verbatim} + +\noindent +{\bf Solution:} + +Define the differential. + +\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x)) } +\returnType{Void} + +\spadcommand{Dx := D() } +$$ +D +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator(Expression Integer,theMap NIL)} + +Now define the differential operator {\tt Dop}. + +\spadcommand{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1 } +$$ +{D \sp 3}+{{G \over {x \sp 2}} \ D}+{{-{x \sp 3}+H} \over {x \sp 3}} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator(Expression Integer,theMap NIL)} + +\spadcommand{n == 3 } +\returnType{Void} + +\spadcommand{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n]) } +\returnType{Void} + +\spadcommand{phi1 == Dop(phi) / exp x } +\returnType{Void} + +\spadcommand{phi2 == phi1 *x**(n+3) } +\returnType{Void} + +\spadcommand{phi3 == retract(phi2)@(POLY INT) } +\returnType{Void} + +\spadcommand{pans == phi3 ::UP(x,POLY INT) } +\returnType{Void} + +\spadcommand{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] } +\returnType{Void} + +\spadcommand{leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) } +\returnType{Void} + +Evaluate this for several values of {\tt n}. + +\spadcommand{leq } +\begin{verbatim} + Compiling body of rule n to compute value of type PositiveInteger + Compiling body of rule phi to compute value of type Expression + Integer + Compiling body of rule phi1 to compute value of type Expression + Integer + Compiling body of rule phi2 to compute value of type Expression + Integer + Compiling body of rule phi3 to compute value of type Polynomial + Integer + Compiling body of rule pans to compute value of type + UnivariatePolynomial(x,Polynomial Integer) + Compiling body of rule pans1 to compute value of type List + Polynomial Integer + Compiling body of rule leq to compute value of type List List + Equation Fraction Polynomial Integer + Compiling function G83347 with type Integer -> Boolean +\end{verbatim} + +\spadcommand{n==4 } +$$ +\begin{array}{@{}l} +\left[ +\left[ +{{s \sb {1}}={{{s \sb {0}} \ G} \over 3}}, +{{s \sb {2}}= +{{{3 \ {s \sb {0}} \ H}+ +{{s \sb {0}} \ {G \sp 2}}+ +{6 \ {s \sb {0}} \ G}} \over {18}}}, +\right. +\right. +\\ +\\ +\displaystyle +\left. +\left. +{{s \sb {3}}= +{{{{\left( {9 \ {s \sb {0}} \ G}+ +{{54} \ {s \sb {0}}} \right)}\ H}+ +{{s \sb {0}} \ {G \sp 3}}+ +{{18} \ {s \sb {0}} \ {G \sp 2}}+ +{{72} \ {s \sb {0}} \ G}} \over {162}}} +\right] +\right] +\end{array} +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\spadcommand{leq } +$$ +\begin{array}{@{}l} +\left[ +\left[ +{{s \sb {1}}={{{s \sb {0}} \ G} \over 3}}, +{{s \sb {2}}= +{{{3 \ {s \sb {0}} \ H}+ +{{s \sb {0}} \ {G \sp 2}}+{6 \ {s \sb {0}} \ G}} \over {18}}}, +\right. +\right. +\\ +\\ +\displaystyle +\left. +\left. +{{s \sb {3}}= +{{{{\left( {9 \ {s \sb {0}} \ G}+ +{{54} \ {s \sb {0}}} \right)}\ H}+ +{{s \sb {0}} \ {G \sp 3}}+ +{{18} \ {s \sb {0}} \ {G \sp 2}}+ +{{72} \ {s \sb {0}} \ G}} \over {162}}} +\right] +\right] +\end{array} +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\spadcommand{n==7 } +\begin{verbatim} + Compiled code for n has been cleared. + Compiled code for leq has been cleared. + Compiled code for pans1 has been cleared. + Compiled code for phi2 has been cleared. + Compiled code for phi has been cleared. + Compiled code for phi3 has been cleared. + Compiled code for phi1 has been cleared. + Compiled code for pans has been cleared. + 1 old definition(s) deleted for function or rule n +\end{verbatim} +\returnType{Void} + +\spadcommand{leq } +\begin{verbatim} +Compiling body of rule n to compute value of type PositiveInteger + ++++ |*0;n;1;G82322| redefined +Compiling body of rule phi to compute value of type Expression + Integer + ++++ |*0;phi;1;G82322| redefined +Compiling body of rule phi1 to compute value of type Expression + Integer + ++++ |*0;phi1;1;G82322| redefined +Compiling body of rule phi2 to compute value of type Expression + Integer + ++++ |*0;phi2;1;G82322| redefined +Compiling body of rule phi3 to compute value of type Polynomial + Integer + ++++ |*0;phi3;1;G82322| redefined +Compiling body of rule pans to compute value of type + UnivariatePolynomial(x,Polynomial Integer) + ++++ |*0;pans;1;G82322| redefined +Compiling body of rule pans1 to compute value of type List + Polynomial Integer + ++++ |*0;pans1;1;G82322| redefined +Compiling body of rule leq to compute value of type List List + Equation Fraction Polynomial Integer + ++++ |*0;leq;1;G82322| redefined +\end{verbatim} +$$ +\left[ +\left[ +{{s \sb {1}}={{{s \sb {0}} \ G} \over 3}}, +\right. +\right.\hbox{\hskip 10.0cm} +$$ +$$ +{s \sb {2}}= +{{3 \ {s \sb {0}} \ H}+ +{{s \sb {0}} \ {G \sp 2}}+ +{6 \ {s \sb {0}} \ G} \over {18}}, \hbox{\hskip 8.0cm} +$$ +$$ +{s \sb {3}}= +{\left( +{9 \ {s \sb {0}} \ G}+ +{{54} \ {s \sb {0}}} +\right)\ H+ +{{s \sb {0}} \ {G \sp 3}}+ +{{18} \ {s \sb {0}} \ {G \sp 2}}+ +{{72} \ {s \sb {0}} \ G} \over {162}}, \hbox{\hskip 6.0cm} +$$ +$$ +{s \sb {4}}= +{\left( +\begin{array}{@{}l} +{{27} \ {s \sb {0}} \ {H \sp 2}}+ +\left( +{{18} \ {s \sb {0}} \ {G \sp 2}}+ +{{378} \ {s \sb {0}} \ G}+ +{{1296} \ {s \sb {0}}} +\right)\ H+ +\\ +\\ +\displaystyle +{{s \sb {0}} \ {G \sp 4}}+ +{{36} \ {s \sb {0}} \ {G \sp 3}}+ +{{396} \ {s \sb {0}} \ {G \sp 2}}+ +{{1296} \ {s \sb {0}} \ G} +\end{array} +\right) +\over {1944}}, \hbox{\hskip 4.0cm} +$$ +$$ +{s \sb {5}}= +{\left( +\begin{array}{@{}l} +\left( +{{135} \ {s \sb {0}} \ G}+ +{{2268} \ {s \sb {0}}} +\right)\ {H \sp 2}+ +\\ +\\ +\displaystyle +\left( +{{30} \ {s \sb {0}} \ {G \sp 3}}+ +{{1350} \ {s \sb {0}} \ {G \sp 2}}+ +{{16416} \ {s \sb {0}} \ G}+ +{{38880} \ {s \sb {0}}} +\right)\ H+ +\\ +\\ +\displaystyle +{{s \sb {0}} \ {G \sp 5}}+ +{{60} \ {s \sb {0}} \ {G \sp 4}}+ +{{1188} \ {s \sb {0}} \ {G \sp 3}}+ +{{9504} \ {s \sb {0}} \ {G \sp 2}}+ +{{25920} \ {s \sb {0}} \ G} +\end{array} +\right) +\over {29160}}, \hbox{\hskip 2.0cm} +$$ +$$ +{s \sb {6}}= +{\left( +\begin{array}{@{}l} +{{405} \ {s \sb {0}} \ {H \sp 3}}+ +\\ +\\ +\displaystyle +{{\left( {{405} \ {s \sb {0}} \ {G \sp 2}}+ +{{18468} \ {s \sb {0}} \ G}+ +{{174960} \ {s \sb {0}}} \right)}\ {H \sp 2}}+ +\\ +\\ +\displaystyle +\left( +{{45} \ {s \sb {0}} \ {G \sp 4}}+ +{{3510} \ {s \sb {0}} \ {G \sp 3}}+ +{{88776} \ {s \sb {0}} \ {G \sp 2}}+ +{{777600} \ {s \sb {0}} \ G}+ +\right. +\\ +\displaystyle +\left. +{{1166400} \ {s \sb {0}}} +\right)\ H+ +\\ +\\ +\displaystyle +{{s \sb {0}} \ {G \sp 6}}+ +{{90} \ {s \sb {0}} \ {G \sp 5}}+ +{{2628} \ {s \sb {0}} \ {G \sp 4}}+ +{{27864} \ {s \sb {0}} \ {G \sp 3}}+ +{{90720} \ {s \sb {0}} \ {G \sp 2}} +\end{array} +\right) +\over {524880}}, \hbox{\hskip 1.0cm} +$$ +$$ +\left. +\left. +{s \sb {7}}= +{\left( +\begin{array}{@{}l} +\left( +{{2835} \ {s \sb {0}} \ G}+ +{{91854} \ {s \sb {0}}} +\right)\ {H \sp 3}+ +\\ +\\ +\displaystyle +\left( +{{945} \ {s \sb {0}} \ {G \sp 3}}+ +{{81648} \ {s \sb {0}} \ {G \sp 2}}+ +{{2082996} \ {s \sb {0}} \ G}+ +{{14171760} \ {s \sb {0}}} +\right)\ {H \sp 2}+ +\\ +\\ +\displaystyle +\left( +{{63} \ {s \sb {0}} \ {G \sp 5}}+ +{{7560} \ {s \sb {0}} \ {G \sp 4}}+ +{{317520} \ {s \sb {0}} \ {G \sp 3}}+ +{{5554008} \ {s \sb {0}} \ {G \sp 2}}+ +\right. +\\ +\displaystyle +\left. +{{34058880} \ {s \sb {0}} \ G} +\right)\ H+ +\\ +\\ +\displaystyle +{{s \sb {0}} \ {G \sp 7}}+ +{{126} \ {s \sb {0}} \ {G \sp 6}}+ +{{4788} \ {s \sb {0}} \ {G \sp 5}}+ +{{25272} \ {s \sb {0}} \ {G \sp 4}} - +{{1744416} \ {s \sb {0}} \ {G \sp 3}} - +\\ +\displaystyle +{{26827200} \ {s \sb {0}} \ {G \sp 2}} - +{{97977600} \ {s \sb {0}} \ G} +\end{array} +\right) +\over {11022480}} +\right] +\right] +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\section{LinearOrdinaryDifferentialOperator1} +\label{LinearOrdinaryDifferentialOperator1XmpPage} + +{\tt LinearOrdinaryDifferentialOperator1(A)} is the domain of linear +ordinary differential operators with coefficients in the differential ring +{\tt A}. + +\subsection{Differential Operators with Rational Function Coefficients} + +This example shows differential operators with rational function +coefficients. In this case operator multiplication is non-commutative and, +since the coefficients form a field, an operator division algorithm exists. + +We begin by defining {\tt RFZ} to be the rational functions in +{\tt x} with integer coefficients and {\tt Dx} to be the differential +operator for {\tt d/dx}. + +\spadcommand{RFZ := Fraction UnivariatePolynomial('x, Integer) } +$$ +\mbox{\rm Fraction UnivariatePolynomial(x,Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{x : RFZ := 'x } +$$ +x +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Integer)} + +\spadcommand{Dx : LODO1 RFZ := D()} +$$ +D +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +Operators are created using the usual arithmetic operations. + +\spadcommand{b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x } +$$ +{3 \ {x \sp 2} \ {D \sp 2}}+{2 \ D}+{1 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +\spadcommand{a : LODO1 RFZ := b*(5*x*Dx + 7)} +$$ +{{15} \ {x \sp 3} \ {D \sp 3}}+{{\left( {{51} \ {x \sp 2}}+{{10} \ x} +\right)} +\ {D \sp 2}}+{{29} \ D}+{7 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +Operator multiplication corresponds to functional composition. + +\spadcommand{p := x**2 + 1/x**2 } +$$ +{{x \sp 4}+1} \over {x \sp 2} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Integer)} + +Since operator coefficients depend on {\tt x}, the multiplication is +not commutative. + +\spadcommand{(a*b - b*a) p } +$$ +{-{{75} \ {x \sp 4}}+{{540} \ x} -{75}} \over {x \sp 4} +$$ +\returnType{Type: Fraction UnivariatePolynomial(x,Integer)} + +When the coefficients of operator polynomials come from a field, as in +this case, it is possible to define operator division. Division on +the left and division on the right yield different results when the +multiplication is non-commutative. + +The results of +\spadfunFrom{leftDivide}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightDivide}{LinearOrdinaryDifferentialOperator1} are +quotient-remainder pairs satisfying: \newline + +{\tt leftDivide(a,b) = [q, r]} such that {\tt a = b*q + r} \newline +{\tt rightDivide(a,b) = [q, r]} such that {\tt a = q*b + r} \newline + +In both cases, the +\spadfunFrom{degree}{LinearOrdinaryDifferentialOperator1} of the +remainder, {\tt r}, is less than the degree of {\tt b}. + +\spadcommand{ld := leftDivide(a,b) } +$$ +\left[ +{quotient={{5 \ x \ D}+7}}, {remainder=0} +\right] +$$ +\returnType{Type: +Record(quotient: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer), +remainder: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer))} + +\spadcommand{a = b * ld.quotient + ld.remainder } +$$ +\begin{array}{@{}l} +{{{15} \ {x \sp 3} \ {D \sp 3}}+ +{{\left( +{{51} \ {x \sp 2}}+ +{{10} \ x} +\right)} +\ {D \sp 2}}+ +{{29} \ D}+ +{7 \over x}}= +\\ +\\ +\displaystyle +{{{15} \ {x \sp 3} \ {D \sp 3}}+ +{{\left( {{51} \ {x \sp 2}}+ +{{10} \ x} +\right)} +\ {D \sp 2}}+ +{{29} \ D}+{7 \over x}} +\end{array} +$$ +\returnType{Type: +Equation LinearOrdinaryDifferentialOperator1 +Fraction UnivariatePolynomial(x,Integer)} + +The operations of left and right division +are so-called because the quotient is obtained by dividing +{\tt a} on that side by {\tt b}. + +\spadcommand{rd := rightDivide(a,b) } +$$ +\left[ +{quotient={{5 \ x \ D}+7}}, {remainder={{{10} \ D}+{5 \over x}}} +\right] +$$ +\returnType{Type: +Record(quotient: +LinearOrdinaryDifferentialOperator1 Fraction +UnivariatePolynomial(x,Integer), +remainder: +LinearOrdinaryDifferentialOperator1 Fraction +UnivariatePolynomial(x,Integer))} + +\spadcommand{a = rd.quotient * b + rd.remainder } +$$ +\begin{array}{@{}l} +{{{15} \ {x \sp 3} \ {D \sp 3}}+ +{{\left( {{51} \ {x \sp 2}}+{{10} \ x} +\right)} +\ {D \sp 2}}+ +{{29} \ D}+ +{7 \over x}}= +\\ +\\ +\displaystyle +{{{15} \ {x \sp 3} \ {D \sp 3}}+ +{{\left( {{51} \ {x \sp 2}}+ +{{10} \ x} +\right)} +\ {D \sp 2}}+ +{{29} \ D}+ +{7 \over x}} +\end{array} +$$ +\returnType{Type: Equation +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +Operations +\spadfunFrom{rightQuotient}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightRemainder}{LinearOrdinaryDifferentialOperator1} are +available if only one of the quotient or remainder are of interest to +you. This is the quotient from right division. + +\spadcommand{rightQuotient(a,b) } +$$ +{5 \ x \ D}+7 +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +This is the remainder from right division. +The corresponding ``left'' functions +\spadfunFrom{leftQuotient}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{leftRemainder}{LinearOrdinaryDifferentialOperator1} +are also available. + +\spadcommand{rightRemainder(a,b) } +$$ +{{10} \ D}+{5 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +For exact division, the operations +\spadfunFrom{leftExactQuotient}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightExactQuotient}{LinearOrdinaryDifferentialOperator1} are supplied. +These return the quotient but only if the remainder is zero. +The call {\tt rightExactQuotient(a,b)} would yield an error. + +\spadcommand{leftExactQuotient(a,b) } +$$ +{5 \ x \ D}+7 +$$ +\returnType{Type: +Union(LinearOrdinaryDifferentialOperator1 +Fraction UnivariatePolynomial(x,Integer),...)} + +The division operations allow the computation of left and right greatest +common divisors (\spadfunFrom{leftGcd}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightGcd}{LinearOrdinaryDifferentialOperator1}) via remainder +sequences, and consequently the computation of left and right least common +multiples (\spadfunFrom{rightLcm}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{leftLcm}{LinearOrdinaryDifferentialOperator1}). + +\spadcommand{e := leftGcd(a,b) } +$$ +{3 \ {x \sp 2} \ {D \sp 2}}+{2 \ D}+{1 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +Note that a greatest common divisor doesn't necessarily divide {\tt a} +and {\tt b} on both sides. Here the left greatest common divisor does +not divide {\tt a} on the right. + +\spadcommand{leftRemainder(a, e) } +$$ +0 +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +\spadcommand{rightRemainder(a, e) } +$$ +{{10} \ D}+{5 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +Similarly, a least common multiple is not necessarily divisible from +both sides. + +% NOTE: the book has a different answer +\spadcommand{f := rightLcm(a,b) } +$$ +{{15} \ {x \sp 3} \ {D \sp 3}}+ +{{\left( +{{51} \ {x \sp 2}}+ +{{10} \ x} +\right)} +\ {D \sp 2}}+ +{{29} \ D}+{7 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +% NOTE: the book has a different answer +\spadcommand{rightRemainder(f, b) } +$$ +{{10} \ D}+{5 \over x} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +% NOTE: the book has a different answer +\spadcommand{leftRemainder(f, b) } +$$ +0 +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)} + +\section{LinearOrdinaryDifferentialOperator2} +\label{LinearOrdinaryDifferentialOperator2XmpPage} + +{\tt LinearOrdinaryDifferentialOperator2(A, M)} is the domain of +linear ordinary differential operators with coefficients in the +differential ring {\tt A} and operating on {\tt M}, an {\tt A}-module. +This includes the cases of operators which are polynomials in {\tt D} +acting upon scalar or vector expressions of a single variable. The +coefficients of the operator polynomials can be integers, rational +functions, matrices or elements of other domains. + +\subsection{Differential Operators with Constant Coefficients} + +This example shows differential operators with rational +number coefficients operating on univariate polynomials. + +We begin by making type assignments so we can conveniently refer +to univariate polynomials in {\tt x} over the rationals. + +\spadcommand{Q := Fraction Integer } +$$ +\mbox{\rm Fraction Integer} +$$ +\returnType{Type: Domain} + +\spadcommand{PQ := UnivariatePolynomial('x, Q) } +$$ +\mbox{\rm UnivariatePolynomial(x,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{x: PQ := 'x } +$$ +x +$$ +\returnType{Type: UnivariatePolynomial(x,Fraction Integer)} + +Now we assign {\tt Dx} to be the differential operator +\spadfunFrom{D}{LinearOrdinaryDifferentialOperator2} +corresponding to {\tt d/dx}. + +\spadcommand{Dx: LODO2(Q, PQ) := D() } +$$ +D +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +Fraction Integer, +UnivariatePolynomial(x,Fraction Integer))} + +New operators are created as polynomials in {\tt D()}. + +\spadcommand{a := Dx + 1 } +$$ +D+1 +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +Fraction Integer, +UnivariatePolynomial(x,Fraction Integer))} + +\spadcommand{b := a + 1/2*Dx**2 - 1/2 } +$$ +{{1 \over 2} \ {D \sp 2}}+D+{1 \over 2} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +Fraction Integer, +UnivariatePolynomial(x,Fraction Integer))} + +To apply the operator {\tt a} to the value {\tt p} the usual function +call syntax is used. + +\spadcommand{p := 4*x**2 + 2/3 } +$$ +{4 \ {x \sp 2}}+{2 \over 3} +$$ +\returnType{Type: UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{a p } +$$ +{4 \ {x \sp 2}}+{8 \ x}+{2 \over 3} +$$ +\returnType{Type: UnivariatePolynomial(x,Fraction Integer)} + +Operator multiplication is defined by the identity {\tt (a*b) p = a(b(p))} + +\spadcommand{(a * b) p = a b p } +$$ +{{2 \ {x \sp 2}}+{{12} \ x}+{{37} \over 3}}={{2 \ {x \sp 2}}+{{12} \ +x}+{{37} \over 3}} +$$ +\returnType{Type: Equation UnivariatePolynomial(x,Fraction Integer)} + +Exponentiation follows from multiplication. + +\spadcommand{c := (1/9)*b*(a + b)**2 } +$$ +{{1 \over {72}} \ {D \sp 6}}+{{5 \over {36}} \ {D \sp 5}}+{{{13} \over +{24}} \ {D \sp 4}}+{{{19} \over {18}} \ {D \sp 3}}+{{{79} \over {72}} \ {D +\sp 2}}+{{7 \over {12}} \ D}+{1 \over 8} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +Fraction Integer, +UnivariatePolynomial(x,Fraction Integer))} + +Finally, note that operator expressions may be applied directly. + +\spadcommand{(a**2 - 3/4*b + c) (p + 1) } +$$ +{3 \ {x \sp 2}}+{{{44} \over 3} \ x}+{{541} \over {36}} +$$ +\returnType{Type: UnivariatePolynomial(x,Fraction Integer)} + +\subsection{ +Differential Operators with Matrix Coefficients Operating on Vectors} + +This is another example of linear ordinary differential operators with +non-commutative multiplication. Unlike the rational function case, +the differential ring of square matrices (of a given dimension) with +univariate polynomial entries does not form a field. Thus the number +of operations available is more limited. + +In this section, the operators have three by three +matrix coefficients with polynomial entries. + +\spadcommand{PZ := UnivariatePolynomial(x,Integer)} +$$ +UnivariatePolynomial(x,Integer) +$$ +\returnType{Type: Domain} + +\spadcommand{x:PZ := 'x } +$$ +x +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +\spadcommand{Mat := SquareMatrix(3,PZ)} +$$ +SquareMatrix(3,UnivariatePolynomial(x,Integer)) +$$ +\returnType{Type: Domain} + +The operators act on the vectors considered as a {\tt Mat}-module. + +\spadcommand{Vect := DPMM(3, PZ, Mat, PZ)} +$$ +\begin{array}{@{}l} +{\rm DirectProductMatrixModule(3,} +\\ +\displaystyle +{\rm \ \ UnivariatePolynomial(x,Integer),} +\\ +\displaystyle +{\rm \ \ SquareMatrix(3,UnivariatePolynomial(x,Integer)),} +\\ +\displaystyle +{\rm \ \ UnivariatePolynomial(x,Integer))} +\end{array} +$$ +\returnType{Type: Domain} + +\spadcommand{Modo := LODO2(Mat, Vect)} +$$ +\begin{array}{@{}l} +{\rm LinearOrdinaryDifferentialOperator2(} +\\ +\displaystyle +{\rm \ \ SquareMatrix(3,UnivariatePolynomial(x,Integer)),} +\\ +\displaystyle +{\rm \ \ DirectProductMatrixModule(3,} +\\ +\displaystyle +{\rm \ \ UnivariatePolynomial(x,Integer),} +\\ +\displaystyle +{\rm \ \ SquareMatrix(3,UnivariatePolynomial(x,Integer)),} +\\ +\displaystyle +{\rm \ \ UnivariatePolynomial(x,Integer)))} +\end{array} +$$ +\returnType{Type: Domain} + +The matrix {\tt m} is used as a coefficient and the vectors {\tt p} +and {\tt q} are operated upon. + +\spadcommand{m:Mat := matrix [ [x**2,1,0],[1,x**4,0],[0,0,4*x**2] ]} +$$ +\left[ +\begin{array}{ccc} +{x \sp 2} & 1 & 0 \\ +1 & {x \sp 4} & 0 \\ +0 & 0 & {4 \ {x \sp 2}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(3,UnivariatePolynomial(x,Integer))} + +\spadcommand{p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]} +$$ +\left[ +{{3 \ {x \sp 2}}+1}, {2 \ x}, {{7 \ {x \sp 3}}+{2 \ x}} +\right] +$$ +\returnType{Type: +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3,UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer))} + +\spadcommand{q: Vect := m * p} +$$ +\left[ +{{3 \ {x \sp 4}}+{x \sp 2}+{2 \ x}}, {{2 \ {x \sp 5}}+{3 \ {x \sp +2}}+1}, {{{28} \ {x \sp 5}}+{8 \ {x \sp 3}}} +\right] +$$ +\returnType{Type: +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3,UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer))} + +Now form a few operators. + +\spadcommand{Dx : Modo := D()} +$$ +D +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +SquareMatrix(3,UnivariatePolynomial(x,Integer)), +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3,UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer)))} + +\spadcommand{a : Modo := Dx + m} +$$ +D+{\left[ +\begin{array}{ccc} +{x \sp 2} & 1 & 0 \\ +1 & {x \sp 4} & 0 \\ +0 & 0 & {4 \ {x \sp 2}} +\end{array} +\right]} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +SquareMatrix(3,UnivariatePolynomial(x,Integer)), +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer)))} + +\spadcommand{b : Modo := m*Dx + 1} +$$ +{{\left[ +\begin{array}{ccc} +{x \sp 2} & 1 & 0 \\ +1 & {x \sp 4} & 0 \\ +0 & 0 & {4 \ {x \sp 2}} +\end{array} +\right]} +\ D}+{\left[ +\begin{array}{ccc} +1 & 0 & 0 \\ +0 & 1 & 0 \\ +0 & 0 & 1 +\end{array} +\right]} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer)))} + +\spadcommand{c := a*b } +$$ +\begin{array}{@{}l} +{{\left[ +\begin{array}{ccc} +{x \sp 2} & 1 & 0 \\ +1 & {x \sp 4} & 0 \\ +0 & 0 & {4 \ {x \sp 2}} +\end{array} +\right]} +\ {D \sp 2}}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{ccc} +{{x \sp 4}+{2 \ x}+2} & {{x \sp 4}+{x \sp 2}} & 0 \\ +{{x \sp 4}+{x \sp 2}} & {{x \sp 8}+{4 \ {x \sp 3}}+2} & 0 \\ +0 & 0 & {{{16} \ {x \sp 4}}+{8 \ x}+1} +\end{array} +\right]} +\ D}+ +\\ +\\ +\displaystyle +{\left[ +\begin{array}{ccc} +{x \sp 2} & 1 & 0 \\ +1 & {x \sp 4} & 0 \\ +0 & 0 & {4 \ {x \sp 2}} +\end{array} +\right]} +\end{array} +$$ +\returnType{Type: +LinearOrdinaryDifferentialOperator2( +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer)))} + +These operators can be applied to vector values. + +\spadcommand{a p } +$$ +\left[ +{{3 \ {x \sp 4}}+{x \sp 2}+{8 \ x}}, {{2 \ {x \sp 5}}+{3 \ {x \sp +2}}+3}, {{{28} \ {x \sp 5}}+{8 \ {x \sp 3}}+{{21} \ {x \sp 2}}+2} +\right] +$$ +\returnType{Type: +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer))} + +\spadcommand{b p } +$$ +\left[ +{{6 \ {x \sp 3}}+{3 \ {x \sp 2}}+3}, {{2 \ {x \sp 4}}+{8 \ x}}, +{{{84} \ {x \sp 4}}+{7 \ {x \sp 3}}+{8 \ {x \sp 2}}+{2 \ x}} +\right] +$$ +\returnType{Type: +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer))} + +\spadcommand{(a + b + c) (p + q) } +$$ +\begin{array}{@{}l} +\left[ +{{{10} \ {x \sp 8}}+ +{{12} \ {x \sp 7}}+ +{{16} \ {x \sp 6}}+ +{{30} \ {x \sp 5}}+ +{{85} \ {x \sp 4}}+ +{{94} \ {x \sp 3}}+ +{{40} \ {x \sp 2}}+ +{{40} \ x}+ +{17}}, +\right. +\\ +\\ +\displaystyle +{{10} \ {x \sp {12}}}+ +{{10} \ {x \sp 9}}+ +{{12} \ {x \sp 8}}+ +{{92} \ {x \sp 7}}+ +{6 \ {x \sp 6}}+ +{{32} \ {x \sp 5}}+ +{{72} \ {x \sp 4}}+ +{{28} \ {x \sp 3}}+ +{{49} \ {x \sp 2}}+ +\\ +\displaystyle +{{32} \ x}+ +{19}, +\\ +\\ +\displaystyle +\left. +{{{2240} \ {x \sp 8}}+ +{{224} \ {x \sp 7}}+ +{{1280} \ {x \sp 6}}+ +{{3508} \ {x \sp 5}}+ +{{492} \ {x \sp 4}}+ +{{751} \ {x \sp 3}}+ +{{98} \ {x \sp 2}}+ +{{18} \ x}+ +4} +\right] +\end{array} +$$ +\returnType{Type: +DirectProductMatrixModule(3, +UnivariatePolynomial(x,Integer), +SquareMatrix(3, +UnivariatePolynomial(x,Integer)), +UnivariatePolynomial(x,Integer))} + +\section{List} +\label{ListXmpPage} + +A \index{list} is a finite collection of elements in a specified +order that can contain duplicates. A list is a convenient structure +to work with because it is easy to add or remove elements and the +length need not be constant. There are many different kinds of lists +in Axiom, but the default types (and those used most often) are +created by the {\tt List} constructor. For example, there are objects +of type {\tt List Integer}, {\tt List Float} and {\tt List Polynomial +Fraction Integer}. Indeed, you can even have {\tt List List List +Boolean} (that is, lists of lists of lists of Boolean values). You +can have lists of any type of Axiom object. + +\subsection{Creating Lists} + +The easiest way to create a list with, for example, the elements +{\tt 2, 4, 5, 6} is to enclose the elements with square +brackets and separate the elements with commas. + +The spaces after the commas are optional, but they do improve the +readability. + +\spadcommand{[2, 4, 5, 6]} +$$ +\left[ +2, 4, 5, 6 +\right] +$$ +\returnType{Type: List PositiveInteger} + +To create a list with the single element {\tt 1}, you can use +either {\tt [1]} or the operation \spadfunFrom{list}{List}. + +\spadcommand{[1]} +$$ +\left[ +1 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{list(1)} +$$ +\left[ +1 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Once created, two lists {\tt k} and {\tt m} can be concatenated by +issuing {\tt append(k,m)}. \spadfunFrom{append}{List} does {\it not} +physically join the lists, but rather produces a new list with the +elements coming from the two arguments. + +\spadcommand{append([1,2,3],[5,6,7])} +$$ +\left[ +1, 2, 3, 5, 6, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Use \spadfunFrom{cons}{List} to append an element onto the front of a +list. + +\spadcommand{cons(10,[9,8,7])} +$$ +\left[ +{10}, 9, 8, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\subsection{Accessing List Elements} + +To determine whether a list has any elements, use the operation +\spadfunFrom{empty?}{List}. + +\spadcommand{empty? [x+1]} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Alternatively, equality with the list constant \spadfunFrom{nil}{List} can +be tested. + +\spadcommand{([] = nil)@Boolean} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +We'll use this in some of the following examples. + +\spadcommand{k := [4,3,7,3,8,5,9,2] } +$$ +\left[ +4, 3, 7, 3, 8, 5, 9, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Each of the next four expressions extracts the \spadfunFrom{first}{List} +element of {\tt k}. + +\spadcommand{first k } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{k.first } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{k.1 } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{k(1) } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +The last two forms generalize to {\tt k.i} and {\tt k(i)}, +respectively, where $ 1 \leq i \leq n$ and {\tt n} equals the length +of {\tt k}. + +This length is calculated by \spadopFrom{\#}{List}. + +\spadcommand{n := \#k } +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +Performing an operation such as {\tt k.i} is sometimes referred to as +{\it indexing into k} or {\it elting into k}. The latter phrase comes +about because the name of the operation that extracts elements is +called \spadfunFrom{elt}{List}. That is, {\tt k.3} is just +alternative syntax for {\tt elt(k,3)}. It is important to remember +that list indices begin with 1. If we issue {\tt k := [1,3,2,9,5]} +then {\tt k.4} returns {\tt 9}. It is an error to use an index that +is not in the range from {\tt 1} to the length of the list. + +The last element of a list is extracted by any of the +following three expressions. + +\spadcommand{last k } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{k.last } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +This form computes the index of the last element and then extracts the +element from the list. + +\spadcommand{k.(\#k) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Changing List Elements} + +We'll use this in some of the following examples. + +\spadcommand{k := [4,3,7,3,8,5,9,2] } +$$ +\left[ +4, 3, 7, 3, 8, 5, 9, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +List elements are reset by using the {\tt k.i} form on the left-hand +side of an assignment. This expression resets the first element of +{\tt k} to {\tt 999}. + +\spadcommand{k.1 := 999 } +$$ +999 +$$ +\returnType{Type: PositiveInteger} + +As with indexing into a list, it is an error to use an index that is +not within the proper bounds. Here you see that {\tt k} was modified. + +\spadcommand{k } +$$ +\left[ +{999}, 3, 7, 3, 8, 5, 9, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +The operation that performs the assignment of an element to a +particular position in a list is called \spadfunFrom{setelt}{List}. +This operation is {\it destructive} in that it changes the list. In +the above example, the assignment returned the value {\tt 999} and +{\tt k} was modified. For this reason, lists are called +\index{mutable} objects: it is possible to change part of a list +(mutate it) rather than always returning a new list reflecting the +intended modifications. + +Moreover, since lists can share structure, changes to one list can +sometimes affect others. + +\spadcommand{k := [1,2] } +$$ +\left[ +1, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{m := cons(0,k) } +$$ +\left[ +0, 1, 2 +\right] +$$ +\returnType{Type: List Integer} + +Change the second element of {\tt m}. + +\spadcommand{m.2 := 99 } +$$ +99 +$$ +\returnType{Type: PositiveInteger} + +See, {\tt m} was altered. + +\spadcommand{m } +$$ +\left[ +0, {99}, 2 +\right] +$$ +\returnType{Type: List Integer} + +But what about {\tt k}? It changed too! + +\spadcommand{k } +$$ +\left[ +{99}, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\subsection{Other Functions} + +An operation that is used frequently in list processing is that +which returns all elements in a list after the first element. + +\spadcommand{k := [1,2,3] } +$$ +\left[ +1, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Use the \spadfunFrom{rest}{List} operation to do this. + +\spadcommand{rest k } +$$ +\left[ +2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +To remove duplicate elements in a list {\tt k}, use +\spadfunFrom{removeDuplicates}{List}. + +\spadcommand{removeDuplicates [4,3,4,3,5,3,4]} +$$ +\left[ +4, 3, 5 +\right] +$$ +\returnType{Type: List PositiveInteger} + +To get a list with elements in the order opposite to those in +a list {\tt k}, use \spadfunFrom{reverse}{List}. + +\spadcommand{reverse [1,2,3,4,5,6]} +$$ +\left[ +6, 5, 4, 3, 2, 1 +\right] +$$ +\returnType{Type: List PositiveInteger} + +To test whether an element is in a list, use +\spadfunFrom{member?}{List}: {\tt member?(a,k)} returns {\tt true} or +{\tt false} depending on whether {\tt a} is in {\tt k} or not. + +\spadcommand{member?(1/2,[3/4,5/6,1/2])} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{member?(1/12,[3/4,5/6,1/2])} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +As an exercise, the reader should determine how to get a list +containing all but the last of the elements in a given non-empty list +{\tt k}.\footnote{{\tt reverse(rest(reverse(k)))} works.} + +\subsection{Dot, Dot} + +Certain lists are used so often that Axiom provides an easy way of +constructing them. If {\tt n} and {\tt m} are integers, then +{\tt expand [n..m]} creates a list containing {\tt n, n+1, ... m}. If +{\tt n > m} then the list is empty. It is actually permissible to leave +off the {\tt m} in the dot-dot construction (see below). + +The dot-dot notation can be used more than once in a list construction +and with specific elements being given. Items separated by dots are +called {\it segments.} + +\spadcommand{[1..3,10,20..23]} +$$ +\left[ +{1..3}, {{10}..{10}}, {{20}..{23}} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +Segments can be expanded into the range of items between the +endpoints by using \spadfunFrom{expand}{Segment}. + +\spadcommand{expand [1..3,10,20..23]} +$$ +\left[ +1, 2, 3, {10}, {20}, {21}, {22}, {23} +\right] +$$ +\returnType{Type: List Integer} + +What happens if we leave off a number on the right-hand side of +\spadopFrom{..}{UniversalSegment}? + +\spadcommand{expand [1..]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +What is created in this case is a {\tt Stream} which is a +generalization of a list. See +\ref{StreamXmpPage} on page~\pageref{StreamXmpPage} for more +information. + +\section{LyndonWord} +\label{LyndonWordXmpPage} + +Initialisations + +\spadcommand{a:Symbol :='a } +$$ +a +$$ +\returnType{Type: Symbol} + +\spadcommand{b:Symbol :='b } +$$ +b +$$ +\returnType{Type: Symbol} + +\spadcommand{c:Symbol :='c } +$$ +c +$$ +\returnType{Type: Symbol} + +\spadcommand{lword:= LyndonWord(Symbol) } +$$ +\mbox{\rm LyndonWord Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{magma := Magma(Symbol) } +$$ +\mbox{\rm Magma Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{word := OrderedFreeMonoid(Symbol) } +$$ +\mbox{\rm OrderedFreeMonoid Symbol} +$$ +\returnType{Type: Domain} + +All Lyndon words of with a, b, c to order 3 + +\spadcommand{LyndonWordsList1([a,b,c],3)\$lword } +$$ +\begin{array}{@{}l} +\left[ +{\left[ +{\left[ a \right]}, +{\left[ b \right]}, +{\left[ c \right]} +\right]}, +{\left[ +{\left[ a \ b \right]}, {\left[ a \ c \right]},{\left[ b \ c \right]} +\right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ {\left[ {a \sp 2} \ b +\right]}, +{\left[ {a \sp 2} \ c +\right]}, +{\left[ a \ {b \sp 2} +\right]}, +{\left[ a \ b \ c +\right]}, +{\left[ a \ c \ b +\right]}, +{\left[ a \ {c \sp 2} +\right]}, +{\left[ {b \sp 2} \ c +\right]}, +{\left[ b \ {c \sp 2} +\right]} +\right]} +\right] +\end{array} +$$ +\returnType{Type: OneDimensionalArray List LyndonWord Symbol} + +All Lyndon words of with a, b, c to order 3 in flat list + +\spadcommand{LyndonWordsList([a,b,c],3)\$lword} +$$ +\begin{array}{@{}l} +\left[ +{\left[ a \right]}, +{\left[ b \right]}, +{\left[ c \right]}, +{\left[ a \ b \right]}, +{\left[ a \ c \right]}, +{\left[ b \ c \right]}, +{\left[ {a \sp 2} \ b \right]}, +{\left[ {a \sp 2} \ c \right]}, +{\left[ a \ {b \sp 2} \right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ a \ b \ c \right]}, +{\left[ a \ c \ b \right]}, +{\left[ a \ {c \sp 2} \right]}, +{\left[ {b \sp 2} \ c \right]}, +{\left[ b \ {c \sp 2} \right]} +\right] +\end{array} +$$ +\returnType{Type: List LyndonWord Symbol} + +All Lyndon words of with a, b to order 5 + +\spadcommand{lw := LyndonWordsList([a,b],5)\$lword } +$$ +\begin{array}{@{}l} +\left[ +{\left[ a \right]}, +{\left[ b \right]}, +{\left[ a \ b \right]}, +{\left[ {a \sp 2} \ b \right]}, +{\left[ a \ {b \sp 2} \right]}, +{\left[ {a \sp 3} \ b \right]}, +{\left[ {a \sp 2} \ {b \sp 2} \right]}, +{\left[ a \ {b \sp 3} \right]}, +{\left[ {a \sp 4} \ b \right]}, +\right. +\\ +\\ +\displaystyle +\left. +{\left[ {a \sp 3} \ {b \sp 2} \right]}, +{\left[ {a \sp 2} \ b \ a \ b \right]}, +{\left[ {a \sp 2} \ {b \sp 3} \right]}, +{\left[ a \ b \ a \ {b \sp 2} \right]}, +{\left[ a \ {b \sp 4} \right]} +\right] +\end{array} +$$ +\returnType{Type: List LyndonWord Symbol} + +\spadcommand{w1 : word := lw.4 :: word } +$$ +{a \sp 2} \ b +$$ +\returnType{Type: OrderedFreeMonoid Symbol} + +\spadcommand{w2 : word := lw.5 :: word } +$$ +a \ {b \sp 2} +$$ +\returnType{Type: OrderedFreeMonoid Symbol} + +Let's try factoring + +\spadcommand{factor(a::word)\$lword } +$$ +\left[ +{\left[ a +\right]} +\right] +$$ +\returnType{Type: List LyndonWord Symbol} + +\spadcommand{factor(w1*w2)\$lword } +$$ +\left[ +{\left[ {a \sp 2} \ b \ a \ {b \sp 2} +\right]} +\right] +$$ +\returnType{Type: List LyndonWord Symbol} + +\spadcommand{factor(w2*w2)\$lword } +$$ +\left[ +{\left[ a \ {b \sp 2} +\right]}, +{\left[ a \ {b \sp 2} +\right]} +\right] +$$ +\returnType{Type: List LyndonWord Symbol} + +\spadcommand{factor(w2*w1)\$lword } +$$ +\left[ +{\left[ a \ {b \sp 2} +\right]}, +{\left[ {a \sp 2} \ b +\right]} +\right] +$$ +\returnType{Type: List LyndonWord Symbol} + +Checks and coercions + +\spadcommand{lyndon?(w1)\$lword } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{lyndon?(w1*w2)\$lword } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{lyndon?(w2*w1)\$lword } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{lyndonIfCan(w1)\$lword } +$$ +\left[ +{a \sp 2} \ b +\right] +$$ +\returnType{Type: Union(LyndonWord Symbol,...)} + +\spadcommand{lyndonIfCan(w2*w1)\$lword } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\spadcommand{lyndon(w1)\$lword } +$$ +\left[ +{a \sp 2} \ b +\right] +$$ +\returnType{Type: LyndonWord Symbol} + +\spadcommand{lyndon(w1*w2)\$lword } +$$ +\left[ +{a \sp 2} \ b \ a \ {b \sp 2} +\right] +$$ +\returnType{Type: LyndonWord Symbol} + +\section{Magma} +\label{MagmaXmpPage} + +Initialisations + +\spadcommand{x:Symbol :='x } +$$ +x +$$ +\returnType{Type: Symbol} + +\spadcommand{y:Symbol :='y } +$$ +y +$$ +\returnType{Type: Symbol} + +\spadcommand{z:Symbol :='z } +$$ +z +$$ +\returnType{Type: Symbol} + +\spadcommand{word := OrderedFreeMonoid(Symbol) } +$$ +\mbox{\rm OrderedFreeMonoid Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{tree := Magma(Symbol) } +$$ +\mbox{\rm Magma Symbol} +$$ +\returnType{Type: Domain} + +Let's make some trees + +\spadcommand{a:tree := x*x } +$$ +\left[ +x, x +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{b:tree := y*y } +$$ +\left[ +y, y +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{c:tree := a*b } +$$ +\left[ +{\left[ x, x +\right]}, +{\left[ y, y +\right]} +\right] +$$ +\returnType{Type: Magma Symbol} + +Query the trees + +\spadcommand{left c } +$$ +\left[ +x, x +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{right c } +$$ +\left[ +y, y +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{length c } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Coerce to the monoid + +\spadcommand{c::word } +$$ +{x \sp 2} \ {y \sp 2} +$$ +\returnType{Type: OrderedFreeMonoid Symbol} + +Check ordering + +\spadcommand{a < b } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{a < c } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{b < c } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Navigate the tree + +\spadcommand{first c } +$$ +x +$$ +\returnType{Type: Symbol} + +\spadcommand{rest c } +$$ +\left[ +x, {\left[ y, y +\right]} +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{rest rest c } +$$ +\left[ +y, y +\right] +$$ +\returnType{Type: Magma Symbol} + +Check ordering + +\spadcommand{ax:tree := a*x } +$$ +\left[ +{\left[ x, x +\right]}, +x +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{xa:tree := x*a } +$$ +\left[ +x, {\left[ x, x +\right]} +\right] +$$ +\returnType{Type: Magma Symbol} + +\spadcommand{xa < ax } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{lexico(xa,ax) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\section{MakeFunction} +\label{MakeFunctionXmpPage} + +It is sometimes useful to be able to define a function given by +the result of a calculation. + +Suppose that you have obtained the following expression after several +computations and that you now want to tabulate the numerical values of +{\tt f} for {\tt x} between {\tt -1} and {\tt +1} with increment +{\tt 0.1}. + +\spadcommand{expr := (x - exp x + 1)**2 * (sin(x**2) * x + 1)**3 } +$$ +\begin{array}{@{}l} +{{\left( +{{x \sp 3} \ {{e \sp x} \sp 2}}+ +{{\left( -{2 \ {x \sp 4}} -{2 \ {x \sp 3}} +\right)}\ {e \sp x}}+ +{x \sp 5}+ +{2 \ {x \sp 4}}+ +{x \sp 3} +\right)}\ {{\sin +\left( +{{x \sp 2}} +\right)}\sp 3}}+ +\\ +\\ +\displaystyle +{{\left( +{3 \ {x \sp 2} \ {{e \sp x} \sp 2}}+ +{{\left( -{6 \ {x \sp 3}} - +{6 \ {x \sp 2}} +\right)}\ {e \sp x}}+ +{3 \ {x \sp 4}}+ +{6 \ {x \sp 3}}+ +{3 \ {x \sp 2}} +\right)} +\ {{\sin +\left( +{{x \sp 2}} +\right)}\sp 2}}+ +\\ +\\ +\displaystyle +{{\left( +{3 \ x \ {{e \sp x} \sp 2}}+ +{{\left( -{6 \ {x \sp 2}} -{6\ x} +\right)}\ {e \sp x}}+ +{3 \ {x \sp 3}}+ +{6 \ {x \sp 2}}+ +{3 \ x} +\right)}\ {\sin +\left( +{{x \sp 2}} +\right)}}+ +{{e\sp x} \sp 2}+ +\\ +\\ +\displaystyle +{{\left( +-{2 \ x} - +2 +\right)}\ {e \sp x}}+ +{x \sp 2}+{2 \ x}+ +1 +\end{array} +$$ +\returnType{Type: Expression Integer} + +You could, of course, use the function \spadfunFrom{eval}{Expression} +within a loop and evaluate {\tt expr} twenty-one times, but this would +be quite slow. A better way is to create a numerical function {\tt f} +such that {\tt f(x)} is defined by the expression {\tt expr} above, +but without retyping {\tt expr}! The package {\tt MakeFunction} +provides the operation \spadfunFrom{function}{MakeFunction} which does +exactly this. + +Issue this to create the function {\tt f(x)} given by {\tt expr}. + +\spadcommand{function(expr, f, x) } +$$ +f +$$ +\returnType{Type: Symbol} + +To tabulate {\tt expr}, we can now quickly evaluate {\tt f} 21 times. + +\spadcommand{tbl := [f(0.1 * i - 1) for i in 0..20]; } +$$ +\begin{array}{@{}l} +\left[ +{0.0005391844\ 0362701574}, +{0.0039657551\ 1844206653}, +\right. +\\ +\displaystyle +{0.0088545187\ 4833983689\ 2}, +{0.0116524883\ 0907069695}, +\\ +\displaystyle +{0.0108618220\ 9245751364\ 5}, +{0.0076366823\ 2120869965\ 06}, +\\ +\displaystyle +{0.0040584985\ 7597822062\ 55}, +{0.0015349542\ 8910500836\ 48}, +\\ +\displaystyle +{0.0003424903\ 1549879905\ 716}, +{0.0000233304\ 8276098819\ 6001}, +\\ +\displaystyle +{0.0}, +{0.0000268186\ 8782862599\ 4229}, +\\ +\displaystyle +{0.0004691571\ 3720051642\ 621}, +{0.0026924576\ 5968519586\ 08}, +\\ +\displaystyle +{0.0101486881\ 7369135148\ 8}, +{0.0313833725\ 8543810564\ 3}, +\\ +\displaystyle +{0.0876991144\ 5154615297\ 9}, +{0.2313019789\ 3439968362}, +\\ +\displaystyle +{0.5843743955\ 958098772}, +{1.4114930171\ 992819197}, +\\ +\displaystyle +\left. +{3.2216948276\ 75164252} +\right] +\end{array} +$$ +\returnType{Type: List Float} + +Use the list {\tt [x1,...,xn]} as the +third argument to \spadfunFrom{function}{MakeFunction} +to create a multivariate function {\tt f(x1,...,xn)}. + +\spadcommand{e := (x - y + 1)**2 * (x**2 * y + 1)**2 } +$$ +\begin{array}{@{}l} +{{x \sp 4} \ {y \sp 4}}+ +{{\left( +-{2 \ {x \sp 5}} - +{2 \ {x \sp 4}}+ +{2 \ {x \sp 2}} +\right)}\ {y \sp 3}}+ +{{\left( +{x \sp 6}+ +{2 \ {x \sp 5}}+ +{x \sp 4} - +{4 \ {x \sp 3}} - +{4 \ {x \sp 2}}+ +1 +\right)}\ {y \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( {2 \ {x \sp 4}}+ +{4 \ {x \sp 3}}+ +{2 \ {x \sp 2}} - +{2 \ x} - +2 +\right)}\ y}+ +{x \sp 2}+ +{2 \ x}+ +1 +\end{array} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{function(e, g, [x, y]) } +$$ +g +$$ +\returnType{Type: Symbol} + +In the case of just two variables, they can be given as arguments +without making them into a list. + +\spadcommand{function(e, h, x, y) } +$$ +h +$$ +\returnType{Type: Symbol} + +Note that the functions created by \spadfunFrom{function}{MakeFunction} +are not limited to floating point numbers, but can be applied to any type +for which they are defined. + +\spadcommand{m1 := squareMatrix [ [1, 2], [3, 4] ] } +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{m2 := squareMatrix [ [1, 0], [-1, 1] ] } +$$ +\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 1 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{h(m1, m2) } +$$ +\left[ +\begin{array}{cc} +-{7836} & {8960} \\ +-{17132} & {19588} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +For more information, see +\ref{ugUserMakePage} on page~\pageref{ugUserMakePage} in Section +\ref{ugUserMakeNumber} on page~\pageref{ugUserMakeNumber}. + +\section{MappingPackage1} +\label{MappingPackage1XmpPage} + +Function are objects of type {\tt Mapping}. In this section we +demonstrate some library operations from the packages +{\tt MappingPackage1}, {\tt MappingPackage2}, and {\tt MappingPackage3} +that manipulate and create functions. Some terminology: a +{\it nullary} function takes no arguments, a {\it unary} function takes +one argument, and a {\it binary} function takes two arguments. + +We begin by creating an example function that raises a +rational number to an integer exponent. + +\spadcommand{power(q: FRAC INT, n: INT): FRAC INT == q**n } +\begin{verbatim} +Function declaration power : (Fraction Integer,Integer) -> + Fraction Integer has been added to workspace. +\end{verbatim} +\returnType{Void} + +\spadcommand{power(2,3) } +\begin{verbatim} +Compiling function power with type (Fraction Integer,Integer) -> + Fraction Integer +\end{verbatim} +$$ +8 +$$ +\returnType{Type: Fraction Integer} + +The \spadfunFrom{twist}{MappingPackage3} operation transposes the +arguments of a binary function. Here {\tt rewop(a, b)} is +{\tt power(b, a)}. + +\spadcommand{rewop := twist power } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: ((Integer,Fraction Integer) -> Fraction Integer)} + +This is $2^3.$ + +\spadcommand{rewop(3, 2) } +$$ +8 +$$ +\returnType{Type: Fraction Integer} + +Now we define {\tt square} in terms of {\tt power}. + +\spadcommand{square: FRAC INT -> FRAC INT } +\returnType{Void} + +The \spadfunFrom{curryRight}{MappingPackage3} operation creates a +unary function from a binary one by providing a constant argument on +the right. + +\spadcommand{square:= curryRight(power, 2) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (Fraction Integer -> Fraction Integer)} + +Likewise, the \spadfunFrom{curryLeft}{MappingPackage3} operation +provides a constant argument on the left. + +\spadcommand{square 4 } +$$ +16 +$$ +\returnType{Type: Fraction Integer} + +The \spadfunFrom{constantRight}{MappingPackage3} operation creates +(in a trivial way) a binary function from a unary one: +{\tt constantRight(f)} is the function {\tt g} such that +{\tt g(a,b)= f(a).} + +\spadcommand{squirrel:= constantRight(square)\$MAPPKG3(FRAC INT,FRAC INT,FRAC INT) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: ((Fraction Integer,Fraction Integer) -> Fraction Integer)} + +Likewise, {\tt constantLeft(f)} is the function {\tt g} such that +{\tt g(a,b)= f(b).} + +\spadcommand{squirrel(1/2, 1/3) } +$$ +1 \over 4 +$$ +\returnType{Type: Fraction Integer} + +The \spadfunFrom{curry}{MappingPackage2} operation makes a unary +function nullary. + +\spadcommand{sixteen := curry(square, 4/1) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (() -> Fraction Integer)} + +\spadcommand{sixteen() } +$$ +16 +$$ +\returnType{Type: Fraction Integer} + +The \spadopFrom{*}{MappingPackage3} operation constructs composed +functions. + +\spadcommand{square2:=square*square } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (Fraction Integer -> Fraction Integer)} + +\spadcommand{square2 3 } +$$ +81 +$$ +\returnType{Type: Fraction Integer} + +Use the \spadopFrom{**}{MappingPackage1} operation to create functions +that are {\tt n}-fold iterations of other functions. + +\spadcommand{sc(x: FRAC INT): FRAC INT == x + 1 } +\begin{verbatim} +Function declaration sc : Fraction Integer -> + Fraction Integer has been added to workspace. +\end{verbatim} +\returnType{Void} + +This is a list of {\tt Mapping} objects. + +\spadcommand{incfns := [sc**i for i in 0..10] } +$$ +\begin{array}{@{}l} +\left[ +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\right. +\\ +\displaystyle +\left. +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)}, +\mbox{theMap(...)} +\right] +\end{array} +$$ +\returnType{Type: List (Fraction Integer -> Fraction Integer)} + +This is a list of applications of those functions. + +\spadcommand{[f 4 for f in incfns] } +$$ +\left[ +4, 5, 6, 7, 8, 9, {10}, {11}, {12}, {13}, {14} +\right] +$$ +\returnType{Type: List Fraction Integer} + +Use the \spadfunFrom{recur}{MappingPackage1} +operation for recursion: + +{\tt g := recur f} means {\tt g(n,x) == f(n,f(n-1,...f(1,x))).} + +\spadcommand{times(n:NNI, i:INT):INT == n*i } +\begin{verbatim} +Function declaration times : (NonNegativeInteger,Integer) -> + Integer has been added to workspace. +\end{verbatim} +\returnType{Void} + +\spadcommand{r := recur(times) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: ((NonNegativeInteger,Integer) -> Integer)} + +This is a factorial function. + +\spadcommand{fact := curryRight(r, 1) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (NonNegativeInteger -> Integer)} + +\spadcommand{fact 4 } +$$ +24 +$$ +\returnType{Type: PositiveInteger} + +Constructed functions can be used within other functions. + +\begin{verbatim} +mto2ton(m, n) == + raiser := square**n + raiser m +\end{verbatim} +\returnType{Void} + +This is $3^{2^3}.$ + +\spadcommand{mto2ton(3, 3) } +\begin{verbatim} + Compiling function mto2ton with type (PositiveInteger, + PositiveInteger) -> Fraction Integer +\end{verbatim} +$$ +6561 +$$ +\returnType{Type: Fraction Integer} + +Here {\tt shiftfib} is a unary function that modifies its argument. + +\begin{verbatim} +shiftfib(r: List INT) : INT == + t := r.1 + r.1 := r.2 + r.2 := r.2 + t + t + +Function declaration shiftfib : List Integer -> Integer + has been added to workspace. +\end{verbatim} +\returnType{Void} + +By currying over the argument we get a function with private state. + +\spadcommand{fibinit: List INT := [0, 1] } +$$ +\left[ +0, 1 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{fibs := curry(shiftfib, fibinit) } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: (() -> Integer)} + +\spadcommand{[fibs() for i in 0..30] } +$$ +\begin{array}{@{}l} +\left[ +0, 1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +{89}, {144}, {233}, {377}, {610}, {987}, {1597}, +\right. +\\ +\displaystyle +{2584}, {4181}, {6765}, {10946}, {17711}, {28657}, {46368}, +{75025}, {121393}, {196418}, +\\ +\displaystyle +\left. +{317811}, {514229}, {832040} +\right] +\end{array} +$$ +\returnType{Type: List Integer} + +\section{Matrix} +\label{MatrixXmpPage} + +The {\tt Matrix} domain provides arithmetic operations on matrices +and standard functions from linear algebra. +This domain is similar to the {\tt TwoDimensionalArray} domain, except +that the entries for {\tt Matrix} must belong to a {\tt Ring}. + +\subsection{Creating Matrices} + +There are many ways to create a matrix from a collection of values or +from existing matrices. + +If the matrix has almost all items equal to the same value, use +\spadfunFrom{new}{Matrix} to create a matrix filled with that value +and then reset the entries that are different. + +\spadcommand{m : Matrix(Integer) := new(3,3,0) } +$$ +\left[ +\begin{array}{ccc} +0 & 0 & 0 \\ +0 & 0 & 0 \\ +0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +To change the entry in the second row, third column to {\tt 5}, use +\spadfunFrom{setelt}{Matrix}. + +\spadcommand{setelt(m,2,3,5) } +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +An alternative syntax is to use assignment. + +\spadcommand{m(1,2) := 10 } +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +The matrix was {\it destructively modified}. + +\spadcommand{m } +$$ +\left[ +\begin{array}{ccc} +0 & {10} & 0 \\ +0 & 0 & 5 \\ +0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +If you already have the matrix entries as a list of lists, use +\spadfunFrom{matrix}{Matrix}. + +\spadcommand{matrix [ [1,2,3,4],[0,9,8,7] ]} +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +0 & 9 & 8 & 7 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +If the matrix is diagonal, use \spadfunFrom{diagonalMatrix}{Matrix}. + +\spadcommand{dm := diagonalMatrix [1,x**2,x**3,x**4,x**5] } +$$ +\left[ +\begin{array}{ccccc} +1 & 0 & 0 & 0 & 0 \\ +0 & {x \sp 2} & 0 & 0 & 0 \\ +0 & 0 & {x \sp 3} & 0 & 0 \\ +0 & 0 & 0 & {x \sp 4} & 0 \\ +0 & 0 & 0 & 0 & {x \sp 5} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +Use \spadfunFrom{setRow}{Matrix} and \spadfunFrom{setColumn}{Matrix} +to change a row or column of a matrix. + +\spadcommand{setRow!(dm,5,vector [1,1,1,1,1]) } +$$ +\left[ +\begin{array}{ccccc} +1 & 0 & 0 & 0 & 0 \\ +0 & {x \sp 2} & 0 & 0 & 0 \\ +0 & 0 & {x \sp 3} & 0 & 0 \\ +0 & 0 & 0 & {x \sp 4} & 0 \\ +1 & 1 & 1 & 1 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +\spadcommand{setColumn!(dm,2,vector [y,y,y,y,y]) } +$$ +\left[ +\begin{array}{ccccc} +1 & y & 0 & 0 & 0 \\ +0 & y & 0 & 0 & 0 \\ +0 & y & {x \sp 3} & 0 & 0 \\ +0 & y & 0 & {x \sp 4} & 0 \\ +1 & y & 1 & 1 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +Use \spadfunFrom{copy}{Matrix} to make a copy of a matrix. + +\spadcommand{cdm := copy(dm) } +$$ +\left[ +\begin{array}{ccccc} +1 & y & 0 & 0 & 0 \\ +0 & y & 0 & 0 & 0 \\ +0 & y & {x \sp 3} & 0 & 0 \\ +0 & y & 0 & {x \sp 4} & 0 \\ +1 & y & 1 & 1 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +This is useful if you intend to modify a matrix destructively but +want a copy of the original. + +\spadcommand{setelt(dm,4,1,1-x**7) } +$$ +-{x \sp 7}+1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{[dm,cdm] } +$$ +\left[ +{\left[ +\begin{array}{ccccc} +1 & y & 0 & 0 & 0 \\ +0 & y & 0 & 0 & 0 \\ +0 & y & {x \sp 3} & 0 & 0 \\ +{-{x \sp 7}+1} & y & 0 & {x \sp 4} & 0 \\ +1 & y & 1 & 1 & 1 +\end{array} +\right]}, +{\left[ +\begin{array}{ccccc} +1 & y & 0 & 0 & 0 \\ +0 & y & 0 & 0 & 0 \\ +0 & y & {x \sp 3} & 0 & 0 \\ +0 & y & 0 & {x \sp 4} & 0 \\ +1 & y & 1 & 1 & 1 +\end{array} +\right]} +\right] +$$ +\returnType{Type: List Matrix Polynomial Integer} + +Use \spadfunFrom{subMatrix}{Matrix} to extract part of an existing +matrix. The syntax is {\tt subMatrix({\it m, firstrow, lastrow, +firstcol, lastcol})}. + +\spadcommand{subMatrix(dm,2,3,2,4) } +$$ +\left[ +\begin{array}{ccc} +y & 0 & 0 \\ +y & {x \sp 3} & 0 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +To change a submatrix, use \spadfunFrom{setsubMatrix}{Matrix}. + +\spadcommand{d := diagonalMatrix [1.2,-1.3,1.4,-1.5] } +$$ +\left[ +\begin{array}{cccc} +{1.2} & {0.0} & {0.0} & {0.0} \\ +{0.0} & -{1.3} & {0.0} & {0.0} \\ +{0.0} & {0.0} & {1.4} & {0.0} \\ +{0.0} & {0.0} & {0.0} & -{1.5} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +If {\tt e} is too big to fit where you specify, an error message is +displayed. Use \spadfunFrom{subMatrix}{Matrix} to extract part of +{\tt e}, if necessary. + +\spadcommand{e := matrix [ [6.7,9.11],[-31.33,67.19] ] } +$$ +\left[ +\begin{array}{cc} +{6.7} & {9.11} \\ +-{31.33} & {67.19} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +This changes the submatrix of {\tt d} whose upper left corner is +at the first row and second column and whose size is that of {\tt e}. + +\spadcommand{setsubMatrix!(d,1,2,e) } +$$ +\left[ +\begin{array}{cccc} +{1.2} & {6.7} & {9.11} & {0.0} \\ +{0.0} & -{31.33} & {67.19} & {0.0} \\ +{0.0} & {0.0} & {1.4} & {0.0} \\ +{0.0} & {0.0} & {0.0} & -{1.5} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +\spadcommand{d } +$$ +\left[ +\begin{array}{cccc} +{1.2} & {6.7} & {9.11} & {0.0} \\ +{0.0} & -{31.33} & {67.19} & {0.0} \\ +{0.0} & {0.0} & {1.4} & {0.0} \\ +{0.0} & {0.0} & {0.0} & -{1.5} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +Matrices can be joined either horizontally or vertically to make +new matrices. + +\spadcommand{a := matrix [ [1/2,1/3,1/4],[1/5,1/6,1/7] ] } +$$ +\left[ +\begin{array}{ccc} +{1 \over 2} & {1 \over 3} & {1 \over 4} \\ +{1 \over 5} & {1 \over 6} & {1 \over 7} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +\spadcommand{b := matrix [ [3/5,3/7,3/11],[3/13,3/17,3/19] ] } +$$ +\left[ +\begin{array}{ccc} +{3 \over 5} & {3 \over 7} & {3 \over {11}} \\ +{3 \over {13}} & {3 \over {17}} & {3 \over {19}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +Use \spadfunFrom{horizConcat}{Matrix} to append them side to side. +The two matrices must have the same number of rows. + +\spadcommand{horizConcat(a,b) } +$$ +\left[ +\begin{array}{cccccc} +{1 \over 2} & {1 \over 3} & {1 \over 4} & {3 \over 5} & {3 \over 7} & {3 +\over {11}} \\ +{1 \over 5} & {1 \over 6} & {1 \over 7} & {3 \over {13}} & {3 \over {17}} & +{3 \over {19}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +Use \spadfunFrom{vertConcat}{Matrix} to stack one upon the other. +The two matrices must have the same number of columns. + +\spadcommand{vab := vertConcat(a,b) } +$$ +\left[ +\begin{array}{ccc} +{1 \over 2} & {1 \over 3} & {1 \over 4} \\ +{1 \over 5} & {1 \over 6} & {1 \over 7} \\ +{3 \over 5} & {3 \over 7} & {3 \over {11}} \\ +{3 \over {13}} & {3 \over {17}} & {3 \over {19}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +The operation \spadfunFrom{transpose}{Matrix} is used to create a new +matrix by reflection across the main diagonal. + +\spadcommand{transpose vab } +$$ +\left[ +\begin{array}{cccc} +{1 \over 2} & {1 \over 5} & {3 \over 5} & {3 \over {13}} \\ +{1 \over 3} & {1 \over 6} & {3 \over 7} & {3 \over {17}} \\ +{1 \over 4} & {1 \over 7} & {3 \over {11}} & {3 \over {19}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +\subsection{Operations on Matrices} + +Axiom provides both left and right scalar multiplication. + +\spadcommand{m := matrix [ [1,2],[3,4] ] } +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\spadcommand{4 * m * (-5)} +$$ +\left[ +\begin{array}{cc} +-{20} & -{40} \\ +-{60} & -{80} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +You can add, subtract, and multiply matrices provided, of course, that +the matrices have compatible dimensions. If not, an error message is +displayed. + +\spadcommand{n := matrix([ [1,0,-2],[-3,5,1] ]) } +$$ +\left[ +\begin{array}{ccc} +1 & 0 & -2 \\ +-3 & 5 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +This following product is defined but {\tt n * m} is not. + +\spadcommand{m * n } +$$ +\left[ +\begin{array}{ccc} +-5 & {10} & 0 \\ +-9 & {20} & -2 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +The operations \spadfunFrom{nrows}{Matrix} and +\spadfunFrom{ncols}{Matrix} return the number of rows and columns of a +matrix. You can extract a row or a column of a matrix using the +operations \spadfunFrom{row}{Matrix} and \spadfunFrom{column}{Matrix}. +The object returned is a {\tt Vector}. + +Here is the third column of the matrix {\tt n}. + +\spadcommand{vec := column(n,3) } +$$ +\left[ +-2, 1 +\right] +$$ +\returnType{Type: Vector Integer} + +You can multiply a matrix on the left by a ``row vector'' and on the right +by a ``column vector.'' + +\spadcommand{vec * m } +$$ +\left[ +1, 0 +\right] +$$ +\returnType{Type: Vector Integer} + +Of course, the dimensions of the vector and the matrix must be compatible +or an error message is returned. + +\spadcommand{m * vec } +$$ +\left[ +0, -2 +\right] +$$ +\returnType{Type: Vector Integer} + +The operation \spadfunFrom{inverse}{Matrix} computes the inverse of a +matrix if the matrix is invertible, and returns {\tt "failed"} if not. + +This Hilbert matrix is invertible. + +\spadcommand{hilb := matrix([ [1/(i + j) for i in 1..3] for j in 1..3]) } +$$ +\left[ +\begin{array}{ccc} +{1 \over 2} & {1 \over 3} & {1 \over 4} \\ +{1 \over 3} & {1 \over 4} & {1 \over 5} \\ +{1 \over 4} & {1 \over 5} & {1 \over 6} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +\spadcommand{inverse(hilb) } +$$ +\left[ +\begin{array}{ccc} +{72} & -{240} & {180} \\ +-{240} & {900} & -{720} \\ +{180} & -{720} & {600} +\end{array} +\right] +$$ +\returnType{Type: Union(Matrix Fraction Integer,...)} + +This matrix is not invertible. + +\spadcommand{mm := matrix([ [1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16] ]) } +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +5 & 6 & 7 & 8 \\ +9 & {10} & {11} & {12} \\ +{13} & {14} & {15} & {16} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\spadcommand{inverse(mm) } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +The operation \spadfunFrom{determinant}{Matrix} computes the +determinant of a matrix provided that the entries of the matrix belong +to a {\tt CommutativeRing}. + +The above matrix {\tt mm} is not invertible and, hence, must have +determinant {\tt 0}. + +\spadcommand{determinant(mm) } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +The operation \spadfunFrom{trace}{SquareMatrix} computes the trace of +a {\em square} matrix. + +\spadcommand{trace(mm) } +$$ +34 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{rank}{Matrix} computes the {\it rank} of a +matrix: the maximal number of linearly independent rows or columns. + +\spadcommand{rank(mm) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{nullity}{Matrix} computes the {\it nullity} of +a matrix: the dimension of its null space. + +\spadcommand{nullity(mm) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{nullSpace}{Matrix} returns a list +containing a basis for the null space of a matrix. Note that the +nullity is the number of elements in a basis for the null space. + +\spadcommand{nullSpace(mm) } +$$ +\left[ +{\left[ 1, -2, 1, 0 +\right]}, +{\left[ 2, -3, 0, 1 +\right]} +\right] +$$ +\returnType{Type: List Vector Integer} + +The operation \spadfunFrom{rowEchelon}{Matrix} returns the row echelon +form of a matrix. It is easy to see that the rank of this matrix is +two and that its nullity is also two. + +\spadcommand{rowEchelon(mm) } +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +0 & 4 & 8 & {12} \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +For more information on related topics, see +\ref{ugIntroTwoDimPage} on page~\pageref{ugIntroTwoDimPage} in Section +\ref{ugIntroTwoDimNumber} on page~\pageref{ugIntroTwoDimNumber}, +\ref{ugProblemEigenPage} on page~\pageref{ugProblemEigenPage} in Section +\ref{ugProblemEigenNumber} on page~\pageref{ugProblemEigenNumber}, +\ref{ugxFloatHilbertPage} on page~\pageref{ugxFloatHilbertPage} in Section +\ref{ugxFloatHilbertNumber} on page~\pageref{ugxFloatHilbertNumber}, +\ref{PermanentXmpPage} on page~\pageref{PermanentXmpPage}, +\ref{VectorXmpPage} on page~\pageref{VectorXmpPage}, +\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage}, +and +\ref{TwoDimensionalArrayXmpPage} on page~\pageref{TwoDimensionalArrayXmpPage}. + +\section{Multiset} +\label{MultisetXmpPage} + +The domain {\tt Multiset(R)} is similar to {\tt Set(R)} except that +multiplicities (counts of duplications) are maintained and displayed. +Use the operation \spadfunFrom{multiset}{Multiset} to create multisets +from lists. All the standard operations from sets are available for +multisets. An element with multiplicity greater than one has the +multiplicity displayed first, then a colon, and then the element. + +Create a multiset of integers. + +\spadcommand{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]} +$$ +\left\{ +7, {2 \mbox{\rm : } 5}, {3 \mbox{\rm : } 3}, 1, {10}, 6, {4 +\mbox{\rm : } 4}, {2 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset PositiveInteger} + +The operation {\tt insert!} adds an element to a multiset. + +\spadcommand{insert!(3,s)} +$$ +\left\{ +7, {2 \mbox{\rm : } 5}, {4 \mbox{\rm : } 3}, 1, {10}, 6, {4 +\mbox{\rm : } 4}, {2 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset PositiveInteger} + +Use {\tt remove!} to remove an element. If a third argument is +present, it specifies how many instances to remove. Otherwise all +instances of the element are removed. Display the resulting multiset. + +\spadcommand{remove!(3,s,1); s} +$$ +\left\{ +7, {2 \mbox{\rm : } 5}, {3 \mbox{\rm : } 3}, 1, {10}, 6, {4 +\mbox{\rm : } 4}, {2 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset PositiveInteger} + +\spadcommand{remove!(5,s); s} +$$ +\left\{ +7, {3 \mbox{\rm : } 3}, 1, {10}, 6, {4 \mbox{\rm : } 4}, {2 +\mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset PositiveInteger} + +The operation {\tt count} returns the number of copies of a given value. + +\spadcommand{count(5,s)} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +A second multiset. + +\spadcommand{t := multiset [2,2,2,-9]} +$$ +\left\{ +-9, {3 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset Integer} + +The {\tt union} of two multisets is additive. + +\spadcommand{U := union(s,t)} +$$ +\left\{ +7, {3 \mbox{\rm : } 3}, 1, -9, {10}, 6, {4 \mbox{\rm : } +4}, {5 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset Integer} + +The {\tt intersect} operation gives the elements that are in +common, with additive multiplicity. + +\spadcommand{I := intersect(s,t)} +$$ +\left\{ +{5 \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset Integer} + +The {\tt difference} of {\tt s} and {\tt t} consists of the elements +that {\tt s} has but {\tt t} does not. Elements are regarded as +indistinguishable, so that if {\tt s} and {\tt t} have any element in +common, the {\tt difference} does not contain that element. + +\spadcommand{difference(s,t)} +$$ +\left\{ +7, {3 \mbox{\rm : } 3}, 1, {10}, 6, {4 \mbox{\rm : } 4} +\right\} +$$ +\returnType{Type: Multiset Integer} + +The {\tt symmetricDifference} is the {\tt union} of {\tt difference(s, t)} +and {\tt difference(t, s)}. + +\spadcommand{S := symmetricDifference(s,t)} +$$ +\left\{ +7, {3 \mbox{\rm : } 3}, 1, -9, {10}, 6, {4 \mbox{\rm : } 4} +\right\} +$$ +\returnType{Type: Multiset Integer} + +Check that the {\tt union} of the {\tt symmetricDifference} and +the {\tt intersect} equals the {\tt union} of the elements. + +\spadcommand{(U = union(S,I))@Boolean} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Check some inclusion relations. + +\spadcommand{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]} +$$ +\left[ +{\tt false}, {\tt true}, {\tt false}, {\tt true} +\right] +$$ +\returnType{Type: List Boolean} + +\section{MultivariatePolynomial} +\label{MultivariatePolynomialXmpPage} + +The domain constructor {\tt MultivariatePolynomial} is similar to {\tt +Polynomial} except that it specifies the variables to be used. {\tt +Polynomial} are available for {\tt MultivariatePolynomial}. The +abbreviation for {\tt MultivariatePolynomial} is {\tt MPOLY}. The +type expressions \centerline{{{\tt MultivariatePolynomial([x,y],Integer)}}} +and \centerline{{{\tt MPOLY([x,y],INT)}}} refer to the domain of +multivariate polynomials in the variables {\tt x} and {\tt y} where the +coefficients are restricted to be integers. The first variable specified +is the main variable and the display of the polynomial reflects this. + +This polynomial appears with terms in descending powers of the +variable {\tt x}. + +\spadcommand{m : MPOLY([x,y],INT) := (x**2 - x*y**3 +3*y)**2 } +$$ +{x \sp 4} -{2 \ {y \sp 3} \ {x \sp 3}}+{{\left( {y \sp 6}+{6 \ y} +\right)} +\ {x \sp 2}} -{6 \ {y \sp 4} \ x}+{9 \ {y \sp 2}} +$$ +\returnType{Type: MultivariatePolynomial([x,y],Integer)} + +It is easy to see a different variable ordering by doing a conversion. + +\spadcommand{m :: MPOLY([y,x],INT) } +$$ +{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp +3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} +$$ +\returnType{Type: MultivariatePolynomial([y,x],Integer)} + +You can use other, unspecified variables, by using {\tt Polynomial} in +the coefficient type of {\tt MPOLY}. + +\spadcommand{p : MPOLY([x,y],POLY INT) } +\returnType{Void} + +\spadcommand{p := (a**2*x - b*y**2 + 1)**2 } +$$ +{{a \sp 4} \ {x \sp 2}}+ +{{\left( +-{2 \ {a \sp 2} \ b \ {y \sp 2}}+ +{2 \ {a \sp 2}} +\right)}\ x}+ +{{b \sp 2} \ {y \sp 4}} - +{2 \ b \ {y \sp 2}}+ +1 +$$ +\returnType{Type: MultivariatePolynomial([x,y],Polynomial Integer)} + +Conversions can be used to re-express such polynomials in terms of +the other variables. For example, you can first push all the +variables into a polynomial with integer coefficients. + +\spadcommand{p :: POLY INT } +$$ +{{b \sp 2} \ {y \sp 4}}+ +{{\left( +-{2 \ {a \sp 2} \ b \ x} - +{2 \ b} +\right)}\ {y \sp 2}}+ +{{a \sp 4} \ {x \sp 2}}+ +{2 \ {a \sp 2} \ x}+ +1 +$$ +\returnType{Type: Polynomial Integer} + +Now pull out the variables of interest. + +\spadcommand{\% :: MPOLY([a,b],POLY INT) } +$$ +{{x \sp 2} \ {a \sp 4}}+ +{{\left( +-{2 \ x \ {y \sp 2} \ b}+ +{2 \ x} +\right)}\ {a \sp 2}}+ +{{y \sp 4} \ {b \sp 2}} - +{2 \ {y \sp 2} \ b}+ +1 +$$ +\returnType{Type: MultivariatePolynomial([a,b],Polynomial Integer)} + +\boxed{4.6in}{ +\vskip 0.1cm +\noindent {\bf Restriction:} +\begin{quotation}\noindent +Axiom does not allow you to create types where +{\tt MultivariatePolynomial} is contained in the coefficient type of +{\tt Polynomial}. Therefore, +{\tt MPOLY([x,y],POLY INT)} is legal but +{\tt POLY MPOLY([x,y],INT)} is not. +\end{quotation} +. +} + +Multivariate polynomials may be combined with univariate polynomials +to create types with special structures. + +\spadcommand{q : UP(x, FRAC MPOLY([y,z],INT)) } +\returnType{Void} + +This is a polynomial in {\tt x} whose coefficients are quotients of +polynomials in {\tt y} and {\tt z}. + +\spadcommand{q := (x**2 - x*(z+1)/y +2)**2 } +$$ +{x \sp 4}+ +{{{-{2 \ z} -2} \over y} \ {x \sp 3}}+ +{{{{4 \ {y \sp 2}}+{z \sp 2}+{2 \ z}+ +1} \over {y \sp 2}} \ {x \sp 2}}+ +{{{-{4 \ z} -4} \over y} \ +x}+ +4 +$$ +\returnType{Type: +UnivariatePolynomial(x,Fraction MultivariatePolynomial([y,z],Integer))} + +Use conversions for structural rearrangements. {\tt z} does not +appear in a denominator and so it can be made the main variable. + +\spadcommand{q :: UP(z, FRAC MPOLY([x,y],INT)) } +$$ +{{{x \sp 2} \over {y \sp 2}} \ {z \sp 2}}+ +{{{-{2 \ y \ {x \sp 3}}+{2 \ {x \sp 2}} - +{4 \ y \ x}} \over {y \sp 2}} \ z}+ +{{{{y \sp 2} \ {x \sp 4}} - +{2 \ y \ {x \sp 3}}+ +{{\left( {4 \ {y \sp 2}}+ +1 +\right)}\ {x \sp 2}} - +{4 \ y \ x}+ +{4 \ {y \sp 2}}} \over {y \sp 2}} +$$ +\returnType{Type: +UnivariatePolynomial(z,Fraction MultivariatePolynomial([x,y],Integer))} + +Or you can make a multivariate polynomial in {\tt x} and {\tt z} +whose coefficients are fractions in polynomials in {\tt y}. + +\spadcommand{q :: MPOLY([x,z], FRAC UP(y,INT)) } +$$ +\begin{array}{@{}l} +{x \sp 4}+ +{{\left( - +{{2 \over y} \ z} - +{2 \over y} +\right)} +\ {x \sp 3}}+ +{{\left( +{{1 \over {y \sp 2}} \ {z \sp 2}}+ +{{2 \over {y \sp 2}} \ z}+ +{{{4 \ {y \sp 2}}+ +1} \over {y \sp 2}} +\right)} +\ {x \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( +-{{4 \over y} \ z} - +{4 \over y} +\right)} +\ x}+ +4 +\end{array} +$$ +\returnType{Type: +MultivariatePolynomial([x,z],Fraction UnivariatePolynomial(y,Integer))} + +A conversion like {\tt q :: MPOLY([x,y], FRAC UP(z,INT))} is not +possible in this example because {\tt y} appears in the denominator of +a fraction. As you can see, Axiom provides extraordinary flexibility +in the manipulation and display of expressions via its conversion +facility. + +For more information on related topics, see +\ref{PolynomialXmpPage} on page~\pageref{PolynomialXmpPage}, +\ref{UnivariatePolynomialXmpPage} on +page~\pageref{UnivariatePolynomialXmpPage}, and +\ref{DistributedMultivariatePolynomialXmpPage} on +page~\pageref{DistributedMultivariatePolynomialXmpPage}. + +\section{None} +\label{NoneXmpPage} + +The {\tt None} domain is not very useful for interactive work but it +is provided nevertheless for completeness of the Axiom type system. + +Probably the only place you will ever see it is if you enter an +empty list with no type information. + +\spadcommand{[ ]} +$$ +\left[\ +\right] +$$ +\returnType{Type: List None} +Such an empty list can be converted into an empty list +of any other type. + +\spadcommand{[ ] :: List Float} +$$ +\left[\ +\right] +$$ +\returnType{Type: List Float} + +If you wish to produce an empty list of a particular +type directly, such as {\tt List NonNegativeInteger}, do it this way. + +\spadcommand{[ ]\$List(NonNegativeInteger)} +$$ +\left[\ +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +\section{Octonion} +\label{OctonionXmpPage} + +The Octonions, also called the Cayley-Dixon algebra, defined over a +commutative ring are an eight-dimensional non-associative algebra. +Their construction from quaternions is similar to the construction +of quaternions from complex numbers (see +\ref{QuaternionXmpPage} on page~\pageref{QuaternionXmpPage}). + +As {\tt Octonion} creates an eight-dimensional algebra, you have to +give eight components to construct an octonion. + +\spadcommand{oci1 := octon(1,2,3,4,5,6,7,8) } +$$ +1+{2 \ i}+{3 \ j}+{4 \ k}+{5 \ E}+{6 \ I}+{7 \ J}+{8 \ K} +$$ +\returnType{Type: Octonion Integer} + +\spadcommand{oci2 := octon(7,2,3,-4,5,6,-7,0) } +$$ +7+{2 \ i}+{3 \ j} -{4 \ k}+{5 \ E}+{6 \ I} -{7 \ J} +$$ +\returnType{Type: Octonion Integer} + +Or you can use two quaternions to create an octonion. + +\spadcommand{oci3 := octon(quatern(-7,-12,3,-10), quatern(5,6,9,0)) } +$$ +-7 -{{12} \ i}+{3 \ j} -{{10} \ k}+{5 \ E}+{6 \ I}+{9 \ J} +$$ +\returnType{Type: Octonion Integer} + +You can easily demonstrate the non-associativity of multiplication. + +\spadcommand{(oci1 * oci2) * oci3 - oci1 * (oci2 * oci3) } +$$ +{{2696} \ i} -{{2928} \ j} -{{4072} \ k}+{{16} \ E} -{{1192} \ I}+{{832} +\ J}+{{2616} \ K} +$$ +\returnType{Type: Octonion Integer} + +As with the quaternions, we have a real part, the imaginary parts {\tt +i}, {\tt j}, {\tt k}, and four additional imaginary parts {\tt E}, +{\tt I}, {\tt J} and {\tt K}. These parts correspond to the canonical +basis {\tt (1,i,j,k,E,I,J,K)}. + +For each basis element there is a component operation to extract +the coefficient of the basis element for a given octonion. + +\spadcommand{[real oci1, imagi oci1, imagj oci1, imagk oci1, imagE oci1, imagI oci1, imagJ oci1, imagK oci1] } +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8 +\right] +$$ +\returnType{Type: List PositiveInteger} + +A basis with respect to the quaternions is given by {\tt (1,E)}. +However, you might ask, what then are the commuting rules? To answer +this, we create some generic elements. + +We do this in Axiom by simply changing the ground ring from {\tt +Integer} to {\tt Polynomial Integer}. + +\spadcommand{q : Quaternion Polynomial Integer := quatern(q1, qi, qj, qk) } +$$ +q1+{qi \ i}+{qj \ j}+{qk \ k} +$$ +\returnType{Type: Quaternion Polynomial Integer} + +\spadcommand{E : Octonion Polynomial Integer:= octon(0,0,0,0,1,0,0,0) } +$$ +E +$$ +\returnType{Type: Octonion Polynomial Integer} + +Note that quaternions are automatically converted to octonions in the +obvious way. + +\spadcommand{q * E } +$$ +{q1 \ E}+{qi \ I}+{qj \ J}+{qk \ K} +$$ +\returnType{Type: Octonion Polynomial Integer} + +\spadcommand{E * q } +$$ +{q1 \ E} -{qi \ I} -{qj \ J} -{qk \ K} +$$ +\returnType{Type: Octonion Polynomial Integer} + +\spadcommand{q * 1\$(Octonion Polynomial Integer) } +$$ +q1+{qi \ i}+{qj \ j}+{qk \ k} +$$ +\returnType{Type: Octonion Polynomial Integer} + +\spadcommand{1\$(Octonion Polynomial Integer) * q } +$$ +q1+{qi \ i}+{qj \ j}+{qk \ k} +$$ +\returnType{Type: Octonion Polynomial Integer} + +Finally, we check that the \spadfunFrom{norm}{Octonion}, defined as +the sum of the squares of the coefficients, is a multiplicative map. + +\spadcommand{o : Octonion Polynomial Integer := octon(o1, oi, oj, ok, oE, oI, oJ, oK) } +$$ +o1+{oi \ i}+{oj \ j}+{ok \ k}+{oE \ E}+{oI \ I}+{oJ \ J}+{oK \ K} +$$ +\returnType{Type: Octonion Polynomial Integer} + +\spadcommand{norm o } +$$ +{ok \sp 2}+{oj \sp 2}+{oi \sp 2}+{oK \sp 2}+{oJ \sp 2}+{oI \sp 2}+{oE \sp +2}+{o1 \sp 2} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{p : Octonion Polynomial Integer := octon(p1, pi, pj, pk, pE, pI, pJ, pK) } +$$ +p1+{pi \ i}+{pj \ j}+{pk \ k}+{pE \ E}+{pI \ I}+{pJ \ J}+{pK \ K} +$$ +\returnType{Type: Octonion Polynomial Integer} + +Since the result is {\tt 0}, the norm is multiplicative. + +\spadcommand{norm(o*p)-norm(p)*norm(o) } +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +\section{OneDimensionalArray} +\label{OneDimensionalArrayXmpPage} + +The {\tt OneDimensionalArray} domain is used for storing data in a +one-dimensional indexed data structure. Such an array is a +homogeneous data structure in that all the entries of the array must +belong to the same Axiom domain. Each array has a fixed length +specified by the user and arrays are not extensible. The indexing of +one-dimensional arrays is one-based. This means that the ``first'' +element of an array is given the index {\tt 1}. See also +\ref{VectorXmpPage} on page~\pageref{VectorXmpPage} and +\ref{FlexibleArrayXmpPage} on page~\pageref{FlexibleArrayXmpPage}. + +To create a one-dimensional array, apply the operation +{\tt oneDimensionalArray} to a list. + +\spadcommand{oneDimensionalArray [i**2 for i in 1..10]} +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +Another approach is to first create {\tt a}, a one-dimensional array +of 10 {\tt 0}'s. {\tt OneDimensionalArray} has the convenient +abbreviation {\tt ARRAY1}. + +\spadcommand{a : ARRAY1 INT := new(10,0)} +$$ +\left[ +0, 0, 0, 0, 0, 0, 0, 0, 0, 0 +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Set each {\tt i}th element to i, then display the result. + +\spadcommand{for i in 1..10 repeat a.i := i; a} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Square each element by mapping the function $i \mapsto i^2$ onto each +element. + +\spadcommand{map!(i +-> i ** 2,a); a} +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Reverse the elements in place. + +\spadcommand{reverse! a} +$$ +\left[ +{100}, {81}, {64}, {49}, {36}, {25}, {16}, 9, 4, 1 +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Swap the {\tt 4}th and {\tt 5}th element. + +\spadcommand{swap!(a,4,5); a} +$$ +\left[ +{100}, {81}, {64}, {36}, {49}, {25}, {16}, 9, 4, 1 +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Sort the elements in place. + +\spadcommand{sort! a } +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Create a new one-dimensional array {\tt b} containing the last 5 +elements of {\tt a}. + +\spadcommand{b := a(6..10)} +$$ +\left[ +{36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +Replace the first 5 elements of {\tt a} with those of {\tt b}. + +\spadcommand{copyInto!(a,b,1)} +$$ +\left[ +{36}, {49}, {64}, {81}, {100}, {36}, {49}, {64}, +{81}, {100} +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +\section{Operator} +\label{OperatorXmpPage} + +Given any ring {\tt R}, the ring of the {\tt Integer}-linear operators +over {\tt R} is called {\tt Operator(R)}. To create an operator over +{\tt R}, first create a basic operator using the operation +{\tt operator}, and then convert it to {\tt Operator(R)} for the {\tt R} +you want. + +We choose {\tt R} to be the two by two matrices over the integers. + +\spadcommand{R := SQMATRIX(2, INT)} +$$ +SquareMatrix(2,Integer) +$$ +\returnType{Type: Domain} + +Create the operator {\tt tilde} on {\tt R}. + +\spadcommand{t := operator("tilde") :: OP(R) } +$$ +tilde +$$ +\returnType{Type: Operator SquareMatrix(2,Integer)} + +Since {\tt Operator} is unexposed we must either package-call operations +from it, or expose it explicitly. For convenience we will do the latter. + +Expose {\tt Operator}. + +\spadcommand{)set expose add constructor Operator } +\begin{verbatim} + Operator is now explicitly exposed in frame G82322 +\end{verbatim} + +To attach an evaluation function (from {\tt R} to {\tt R}) to an +operator over {\tt R}, use {\tt evaluate(op, f)} where {\tt op} is an +operator over {\tt R} and {\tt f} is a function {\tt R -> R}. This +needs to be done only once when the operator is defined. Note that +{\tt f} must be {\tt Integer}-linear (that is, +{\tt f(ax+y) = a f(x) + f(y)} for any integer {\tt a}, and any {\tt x} +and {\tt y} in {\tt R}). + +We now attach the transpose map to the above operator {\tt t}. + +\spadcommand{evaluate(t, m +-> transpose m)} +$$ +tilde +$$ +\returnType{Type: Operator SquareMatrix(2,Integer)} + +Operators can be manipulated formally as in any ring: {\tt +} is +the pointwise addition and {\tt *} is composition. Any element +{\tt x} of {\tt R} can be converted to an operator +$op_x$ over {\tt R}, and the evaluation function of +$op_x$ is left-multiplication by {\tt x}. + +Multiplying on the left by this matrix swaps the two rows. + +\spadcommand{s : R := matrix [ [0, 1], [1, 0] ]} +$$ +\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +Can you guess what is the action of the following operator? + +\spadcommand{rho := t * s} +$$ +tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]} +$$ +\returnType{Type: Operator SquareMatrix(2,Integer)} + +Hint: applying {\tt rho} four times gives the identity, so +{\tt rho**4-1} should return 0 when applied to any two by two matrix. + +\spadcommand{z := rho**4 - 1} +$$ +-1+{tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]} +\ tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]} +\ tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]} +\ tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]}} +$$ +\returnType{Type: Operator SquareMatrix(2,Integer)} + +Now check with this matrix. + +\spadcommand{m:R := matrix [ [1, 2], [3, 4] ]} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{z m} +$$ +\left[ +\begin{array}{cc} +0 & 0 \\ +0 & 0 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +As you have probably guessed by now, {\tt rho} acts on matrices +by rotating the elements clockwise. + +\spadcommand{rho m} +$$ +\left[ +\begin{array}{cc} +3 & 1 \\ +4 & 2 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{rho rho m} +$$ +\left[ +\begin{array}{cc} +4 & 3 \\ +2 & 1 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +\spadcommand{(rho**3) m} +$$ +\left[ +\begin{array}{cc} +2 & 4 \\ +1 & 3 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +Do the swapping of rows and transposition commute? We can check by +computing their bracket. + +\spadcommand{b := t * s - s * t} +$$ +-{{\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]} +\ tilde}+{tilde \ {\left[ +\begin{array}{cc} +0 & 1 \\ +1 & 0 +\end{array} +\right]}} +$$ +\returnType{Type: Operator SquareMatrix(2,Integer)} + +Now apply it to {\tt m}. + +\spadcommand{b m } +$$ +\left[ +\begin{array}{cc} +1 & -3 \\ +3 & -1 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Integer)} + +Next we demonstrate how to define a differential operator on a +polynomial ring. + +This is the recursive definition of the {\tt n}-th Legendre polynomial. + +\begin{verbatim} +L n == + n = 0 => 1 + n = 1 => x + (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2) +\end{verbatim} +\returnType{Void} + +Create the differential operator $d \over {dx}$ on polynomials in {\tt x} +over the rational numbers. + +\spadcommand{dx := operator("D") :: OP(POLY FRAC INT) } +$$ +D +$$ +\returnType{Type: Operator Polynomial Fraction Integer} + +Now attach the map to it. + +\spadcommand{evaluate(dx, p +-> D(p, 'x)) } +$$ +D +$$ +\returnType{Type: Operator Polynomial Fraction Integer} + +This is the differential equation satisfied by the {\tt n}-th +Legendre polynomial. + +\spadcommand{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1) } +\returnType{Void} + +Now we verify this for {\tt n = 15}. Here is the polynomial. + +\spadcommand{L 15 } +$$ +\begin{array}{@{}l} +{{{9694845} \over {2048}} \ {x \sp {15}}} - +{{{35102025} \over {2048}} \ {x \sp {13}}}+ +{{{50702925} \over {2048}} \ {x \sp {11}}} - +{{{37182145} \over {2048}} \ {x \sp 9}}+ +{{{14549535} \over {2048}} \ {x \sp 7}} - +\\ +\\ +\displaystyle +{{{2909907} \over {2048}} \ {x \sp 5}}+ +{{{255255} \over {2048}} \ {x \sp 3}} - +{{{6435} \over {2048}} \ x} +\end{array} +$$ +\returnType{Type: Polynomial Fraction Integer} + +Here is the operator. + +\spadcommand{E 15 } +$$ +{240} -{2 \ x \ D} -{{\left( {x \sp 2} -1 +\right)} +\ {D \sp 2}} +$$ +\returnType{Type: Operator Polynomial Fraction Integer} + +Here is the evaluation. + +\spadcommand{(E 15)(L 15) } +$$ +0 +$$ +\returnType{Type: Polynomial Fraction Integer} + +\section{OrderedVariableList} +\label{OrderedVariableListXmpPage} + +The domain {\tt OrderedVariableList} provides symbols which are +restricted to a particular list and have a definite ordering. Those +two features are specified by a {\tt List Symbol} object that is the +argument to the domain. + +This is a sample ordering of three symbols. + +\spadcommand{ls:List Symbol:=['x,'a,'z] } +$$ +\left[ +x, a, z +\right] +$$ +\returnType{Type: List Symbol} + +Let's build the domain + +\spadcommand{Z:=OVAR ls } +$$ +\mbox{\rm OrderedVariableList [x,a,z]} +$$ +\returnType{Type: Domain} + +How many variables does it have? + +\spadcommand{size()\$Z } +$$ +3 +$$ +\returnType{Type: NonNegativeInteger} + +They are (in the imposed order) + +\spadcommand{lv:=[index(i::PI)\$Z for i in 1..size()\$Z] } +$$ +\left[ +x, a, z +\right] +$$ +\returnType{Type: List OrderedVariableList [x,a,z]} + +Check that the ordering is right + +\spadcommand{sorted?(>,lv) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\section{OrderlyDifferentialPolynomial} +\label{OrderlyDifferentialPolynomialXmpPage} + +Many systems of differential equations may be transformed to +equivalent systems of ordinary differential equations where the +equations are expressed polynomially in terms of the unknown +functions. In Axiom, the domain constructors +{\tt OrderlyDifferentialPolynomial} (abbreviated {\tt ODPOL}) and +{\tt SequentialDifferentialPolynomial} (abbreviation {\tt SDPOL}) implement +two domains of ordinary differential polynomials over any differential +ring. In the simplest case, this differential ring is usually either +the ring of integers, or the field of rational numbers. However, +Axiom can handle ordinary differential polynomials over a field of +rational functions in a single indeterminate. + +The two domains {\tt ODPOL} and {\tt SDPOL} are almost identical, the +only difference being the choice of a different ranking, which is an +ordering of the derivatives of the indeterminates. The first domain +uses an orderly ranking, that is, derivatives of higher order are +ranked higher, and derivatives of the same order are ranked +alphabetically. The second domain uses a sequential ranking, where +derivatives are ordered first alphabetically by the differential +indeterminates, and then by order. A more general domain constructor, +{\tt DifferentialSparseMultivariatePolynomial} (abbreviation +{\tt DSMP}) allows both a user-provided list of differential indeterminates +as well as a user-defined ranking. We shall illustrate +{\tt ODPOL(FRAC INT)}, which constructs a domain of ordinary differential +polynomials in an arbitrary number of differential indeterminates with +rational numbers as coefficients. + +\spadcommand{dpol:= ODPOL(FRAC INT) } +$$ +\mbox{\rm OrderlyDifferentialPolynomial Fraction Integer} +$$ +\returnType{Type: Domain} + +A differential indeterminate {\tt w} may be viewed as an infinite +sequence of algebraic indeterminates, which are the derivatives of +{\tt w}. To facilitate referencing these, Axiom provides the +operation \spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial} to +convert an element of type {\tt Symbol} to a map from the natural +numbers to the differential polynomial ring. + +\spadcommand{w := makeVariable('w)\$dpol } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: +(NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)} + +\spadcommand{z := makeVariable('z)\$dpol } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: +(NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)} + +The fifth derivative of {\tt w} can be obtained by applying the map +{\tt w} to the number {\tt 5.} Note that the order of differentiation +is given as a subscript (except when the order is 0). + +\spadcommand{w.5 } +$$ +w \sb {5} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +\spadcommand{w 0 } +$$ +w +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The first five derivatives of {\tt z} can be generated by a list. + +\spadcommand{[z.i for i in 1..5] } +$$ +\left[ +{z \sb {1}}, {z \sb {2}}, {z \sb {3}}, {z \sb {4}}, {z \sb {5}} +\right] +$$ +\returnType{Type: List OrderlyDifferentialPolynomial Fraction Integer} + +The usual arithmetic can be used to form a differential polynomial from +the derivatives. + +\spadcommand{f:= w.4 - w.1 * w.1 * z.3 } +$$ +{w \sb {4}} -{{{w \sb {1}} \sp 2} \ {z \sb {3}}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +\spadcommand{g:=(z.1)**3 * (z.2)**2 - w.2 } +$$ +{{{z \sb {1}} \sp 3} \ {{z \sb {2}} \sp 2}} -{w \sb {2}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The operation \spadfunFrom{D}{OrderlyDifferentialPolynomial} +computes the derivative of any differential polynomial. + +\spadcommand{D(f) } +$$ +{w \sb {5}} -{{{w \sb {1}} \sp 2} \ {z \sb {4}}} -{2 \ {w \sb {1}} \ {w +\sb {2}} \ {z \sb {3}}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The same operation can compute higher derivatives, like the +fourth derivative. + +\spadcommand{D(f,4) } +$$ +\begin{array}{@{}l} +{w \sb {8}} - +{{{w \sb {1}} \sp 2} \ {z \sb {7}}} - +{8 \ {w \sb {1}} \ {w \sb {2}} \ {z \sb {6}}}+ +{{\left( +-{{12} \ {w \sb {1}} \ {w \sb {3}}} - +{{12} \ {{w \sb {2}} \sp 2}} +\right)}\ {z \sb {5}}} - +{2 \ {w \sb {1}} \ {z \sb {3}} \ {w \sb {5}}}+ +\\ +\\ +\displaystyle +{{\left( +-{8 \ {w \sb {1}} \ {w \sb {4}}} - +{{24} \ {w \sb {2}} \ {w \sb {3}}} +\right)} +\ {z \sb {4}}} - +{8 \ {w \sb {2}} \ {z \sb {3}} \ {w \sb {4}}} - +{6 \ {{w \sb {3}} \sp 2} \ {z \sb {3}}} +\end{array} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The operation \spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial} +creates a map to facilitate referencing the derivatives of {\tt f}, +similar to the map {\tt w}. + +\spadcommand{df:=makeVariable(f)\$dpol } +$$ +\mbox{theMap(...)} +$$ +\returnType{Type: +(NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)} + +The fourth derivative of f may be referenced easily. + +\spadcommand{df.4 } +$$ +\begin{array}{@{}l} +{w \sb {8}} - +{{{w \sb {1}} \sp 2} \ {z \sb {7}}} - +{8 \ {w \sb {1}} \ {w \sb {2}} \ {z \sb {6}}}+ +{{\left( +-{{12} \ {w \sb {1}} \ {w \sb {3}}} - +{{12} \ {{w \sb {2}} \sp 2}} +\right)} +\ {z \sb {5}}} - +{2 \ {w \sb {1}} \ {z \sb {3}} \ {w \sb {5}}}+ +\\ +\\ +\displaystyle +{{\left( +-{8 \ {w \sb {1}} \ {w \sb {4}}} - +{{24} \ {w \sb {2}} \ {w \sb {3}}} +\right)}\ {z \sb {4}}} - +{8 \ {w \sb {2}} \ {z \sb {3}} \ {w \sb {4}}} - +{6 \ {{w \sb {3}} \sp 2} \ {z \sb {3}}} +\end{array} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The operation \spadfunFrom{order}{OrderlyDifferentialPolynomial} +returns the order of a differential polynomial, or the order +in a specified differential indeterminate. + +\spadcommand{order(g) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{order(g, 'w) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +The operation +\spadfunFrom{differentialVariables}{OrderlyDifferentialPolynomial} returns +a list of differential indeterminates occurring in a differential polynomial. + +\spadcommand{differentialVariables(g) } +$$ +\left[ +z, w +\right] +$$ +\returnType{Type: List Symbol} + +The operation \spadfunFrom{degree}{OrderlyDifferentialPolynomial} returns +the degree, or the degree in the differential indeterminate specified. + +\spadcommand{degree(g) } +$$ +{{z \sb {2}} \sp 2} \ {{z \sb {1}} \sp 3} +$$ +\returnType{Type: IndexedExponents OrderlyDifferentialVariable Symbol} + +\spadcommand{degree(g, 'w) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{weights}{OrderlyDifferentialPolynomial} returns +a list of weights of differential monomials appearing in differential +polynomial, or a list of weights in a specified differential +indeterminate. + +\spadcommand{weights(g) } +$$ +\left[ +7, 2 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +\spadcommand{weights(g,'w) } +$$ +\left[ +2 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +The operation \spadfunFrom{weight}{OrderlyDifferentialPolynomial} returns +the maximum weight of all differential monomials appearing in the +differential polynomial. + +\spadcommand{weight(g) } +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +A differential polynomial is {\em isobaric} if the weights of all +differential monomials appearing in it are equal. + +\spadcommand{isobaric?(g) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +To substitute {\em differentially}, use +\spadfunFrom{eval}{OrderlyDifferentialPolynomial}. Note that we must +coerce {\tt 'w} to {\tt Symbol}, since in {\tt ODPOL}, differential +indeterminates belong to the domain {\tt Symbol}. Compare this result +to the next, which substitutes {\em algebraically} (no substitution is +done since {\tt w.0} does not appear in {\tt g}). + +\spadcommand{eval(g,['w::Symbol],[f]) } +$$ +-{w \sb {6}}+ +{{{w \sb {1}} \sp 2} \ {z \sb {5}}}+ +{4 \ {w \sb {1}} \ {w \sb {2}} \ {z \sb {4}}}+ +{{\left( {2 \ {w \sb {1}} \ {w \sb {3}}}+ +{2 \ {{w \sb {2}} \sp 2}} +\right)} +\ {z \sb {3}}}+ +{{{z \sb {1}} \sp 3} \ {{z \sb {2}} \sp 2}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +\spadcommand{eval(g,variables(w.0),[f]) } +$$ +{{{z \sb {1}} \sp 3} \ {{z \sb {2}} \sp 2}} -{w \sb {2}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Since {\tt OrderlyDifferentialPolynomial} belongs to +{\tt PolynomialCategory}, all the operations defined in the latter +category, or in packages for the latter category, are available. + +\spadcommand{monomials(g) } +$$ +\left[ +{{{z \sb {1}} \sp 3} \ {{z \sb {2}} \sp 2}}, -{w \sb {2}} +\right] +$$ +\returnType{Type: List OrderlyDifferentialPolynomial Fraction Integer} + +\spadcommand{variables(g) } +$$ +\left[ +{z \sb {2}}, {w \sb {2}}, {z \sb {1}} +\right] +$$ +\returnType{Type: List OrderlyDifferentialVariable Symbol} + +\spadcommand{gcd(f,g) } +$$ +1 +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +\spadcommand{groebner([f,g]) } +$$ +\left[ +{{w \sb {4}} - +{{{w \sb {1}} \sp 2} \ {z \sb {3}}}}, +{{{{z \sb {1}} \sp 3} \ {{z \sb {2}} \sp 2}} - +{w \sb {2}}} +\right] +$$ +\returnType{Type: List OrderlyDifferentialPolynomial Fraction Integer} + +The next three operations are essential for elimination procedures in +differential polynomial rings. The operation +\spadfunFrom{leader}{OrderlyDifferentialPolynomial} returns the leader +of a differential polynomial, which is the highest ranked derivative +of the differential indeterminates that occurs. + +\spadcommand{lg:=leader(g) } +$$ +z \sb {2} +$$ +\returnType{Type: OrderlyDifferentialVariable Symbol} + +The operation \spadfunFrom{separant}{OrderlyDifferentialPolynomial} returns +the separant of a differential polynomial, which is the partial derivative +with respect to the leader. + +\spadcommand{sg:=separant(g) } +$$ +2 \ {{z \sb {1}} \sp 3} \ {z \sb {2}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +The operation \spadfunFrom{initial}{OrderlyDifferentialPolynomial} returns +the initial, which is the leading coefficient when the given differential +polynomial is expressed as a polynomial in the leader. + +\spadcommand{ig:=initial(g) } +$$ +{z \sb {1}} \sp 3 +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Using these three operations, it is possible to reduce {\tt f} modulo +the differential ideal generated by {\tt g}. The general scheme is to +first reduce the order, then reduce the degree in the leader. First, +eliminate {\tt z.3} using the derivative of {\tt g}. + +\spadcommand{g1 := D g } +$$ +{2 \ {{z \sb {1}} \sp 3} \ {z \sb {2}} \ {z \sb {3}}} - +{w \sb {3}}+ +{3 \ {{z \sb {1}} \sp 2} \ {{z \sb {2}} \sp 3}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Find its leader. + +\spadcommand{lg1:= leader g1 } +$$ +z \sb {3} +$$ +\returnType{Type: OrderlyDifferentialVariable Symbol} + +Differentiate {\tt f} partially with respect to this leader. + +\spadcommand{pdf:=D(f, lg1) } +$$ +-{{w \sb {1}} \sp 2} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Compute the partial remainder of {\tt f} with respect to {\tt g}. + +\spadcommand{prf:=sg * f- pdf * g1 } +$$ +{2 \ {{z \sb {1}} \sp 3} \ {z \sb {2}} \ {w \sb {4}}} - +{{{w \sb {1}} \sp 2} \ {w \sb {3}}}+ +{3 \ {{w \sb {1}} \sp 2} \ {{z \sb {1}} \sp 2} \ {{z \sb {2}} \sp 3}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Note that high powers of {\tt lg} still appear in {\tt prf}. Compute +the leading coefficient of {\tt prf} as a polynomial in the leader of +{\tt g}. + +\spadcommand{lcf:=leadingCoefficient univariate(prf, lg) } +$$ +3 \ {{w \sb {1}} \sp 2} \ {{z \sb {1}} \sp 2} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +Finally, continue eliminating the high powers of {\tt lg} appearing in +{\tt prf} to obtain the (pseudo) remainder of {\tt f} modulo {\tt g} +and its derivatives. + +\spadcommand{ig * prf - lcf * g * lg } +$$ +{2 \ {{z \sb {1}} \sp 6} \ {z \sb {2}} \ {w \sb {4}}} - +{{{w \sb {1}} \sp 2} \ {{z \sb {1}} \sp 3} \ {w \sb {3}}}+ +{3 \ {{w \sb {1}} \sp 2} \ {{z \sb {1}} \sp 2} +\ {w \sb {2}} \ {z \sb {2}}} +$$ +\returnType{Type: OrderlyDifferentialPolynomial Fraction Integer} + +\section{PartialFraction} +\label{PartialFractionXmpPage} + +A {\it partial fraction} is a decomposition of a quotient into a sum +of quotients where the denominators of the summands are powers of +primes.\footnote{Most people first encounter partial fractions when +they are learning integral calculus. For a technical discussion of +partial fractions, see, for example, Lang's {\it Algebra.}} For +example, the rational number {\tt 1/6} is decomposed into {\tt 1/2-1/3}. +You can compute partial fractions of quotients of objects from +domains belonging to the category {\tt EuclideanDomain}. For example, +{\tt Integer}, {\tt Complex Integer}, and +{\tt UnivariatePolynomial(x, Fraction Integer)} +all belong to {\tt EuclideanDomain}. In the +examples following, we demonstrate how to decompose quotients of each +of these kinds of object into partial fractions. Issue the system +command {\tt )show PartialFraction} to display the full list of +operations defined by {\tt PartialFraction}. + +It is necessary that we know how to factor the denominator when we +want to compute a partial fraction. Although the interpreter can +often do this automatically, it may be necessary for you to include a +call to {\tt factor}. In these examples, it is not necessary to +factor the denominators explicitly. + +The main operation for computing partial fractions is called +\spadfunFrom{partialFraction}{PartialFraction} and we use this to +compute a decomposition of {\tt 1 / 10!}. The first argument to +\spadfunFrom{partialFraction}{PartialFraction} is the numerator of the +quotient and the second argument is the factored denominator. + +\spadcommand{partialFraction(1,factorial 10) } +$$ +{{159} \over {2 \sp 8}} -{{23} \over {3 \sp 4}} -{{12} \over {5 \sp 2}}+{1 +\over 7} +$$ +\returnType{Type: PartialFraction Integer} + +Since the denominators are powers of primes, it may be possible +to expand the numerators further with respect to those primes. Use the +operation \spadfunFrom{padicFraction}{PartialFraction} to do this. + +\spadcommand{f := padicFraction(\%) } +$$ +{1 \over 2}+{1 \over {2 \sp 4}}+{1 \over {2 \sp 5}}+{1 \over {2 \sp 6}}+{1 +\over {2 \sp 7}}+{1 \over {2 \sp 8}} -{2 \over {3 \sp 2}} -{1 \over {3 \sp +3}} -{2 \over {3 \sp 4}} -{2 \over 5} -{2 \over {5 \sp 2}}+{1 \over 7} +$$ +\returnType{Type: PartialFraction Integer} + +The operation \spadfunFrom{compactFraction}{PartialFraction} returns +an expanded fraction into the usual form. The compacted version is +used internally for computational efficiency. + +\spadcommand{compactFraction(f) } +$$ +{{159} \over {2 \sp 8}} -{{23} \over {3 \sp 4}} -{{12} \over {5 \sp 2}}+{1 +\over 7} +$$ +\returnType{Type: PartialFraction Integer} + +You can add, subtract, multiply and divide partial fractions. In +addition, you can extract the parts of the decomposition. +\spadfunFrom{numberOfFractionalTerms}{PartialFraction} computes the +number of terms in the fractional part. This does not include the +whole part of the fraction, which you get by calling +\spadfunFrom{wholePart}{PartialFraction}. In this example, the whole +part is just {\tt 0}. + +\spadcommand{numberOfFractionalTerms(f) } +$$ +12 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{nthFractionalTerm}{PartialFraction} returns +the individual terms in the decomposition. Notice that the object +returned is a partial fraction itself. +\spadfunFrom{firstNumer}{PartialFraction} and +\spadfunFrom{firstDenom}{PartialFraction} extract the numerator and +denominator of the first term of the fraction. + +\spadcommand{nthFractionalTerm(f,3) } +$$ +1 \over {2 \sp 5} +$$ +\returnType{Type: PartialFraction Integer} + +Given two gaussian integers (see +\ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage}), you can +decompose their quotient into a partial fraction. + +\spadcommand{partialFraction(1,- 13 + 14 * \%i) } +$$ +-{1 \over {1+{2 \ i}}}+{4 \over {3+{8 \ i}}} +$$ +\returnType{Type: PartialFraction Complex Integer} + +To convert back to a quotient, simply use a conversion. + +\spadcommand{\% :: Fraction Complex Integer } +$$ +-{i \over {{14}+{{13} \ i}}} +$$ +\returnType{Type: Fraction Complex Integer} + +To conclude this section, we compute the decomposition of +\begin{verbatim} + 1 + ------------------------------- + 2 3 4 + (x + 1)(x + 2) (x + 3) (x + 4) +\end{verbatim} + +The polynomials in this object have type +{\tt UnivariatePolynomial(x, Fraction Integer)}. + +We use the \spadfunFrom{primeFactor}{Factored} operation (see +\ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage}) +to create the denominator in factored form directly. + +\spadcommand{u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4]) } +$$ +{\left( x+1 \right)} +\ {{\left( x+2 \right)}\sp 2} +\ {{\left( x+3 \right)}\sp 3} +\ {{\left( x+4 \right)}\sp 4} +$$ +\returnType{Type: Factored UnivariatePolynomial(x,Fraction Integer)} + +These are the compact and expanded partial fractions for the quotient. + +\spadcommand{partialFraction(1,u) } +$$ +\begin{array}{@{}l} +{{1 \over {648}} \over {x+1}}+ +{{{{1 \over 4} \ x}+{7 \over {16}}} \over {{\left( x+2 \right)}\sp 2}}+ +{{-{{{17} \over 8} \ {x \sp 2}} -{{12} \ x} -{{139} \over 8}} +\over {{\left( x+3 \right)}\sp 3}}+ +\\ +\\ +\displaystyle +{{{{{607} \over {324}} \ {x \sp 3}}+ +{{{10115} \over {432}} \ {x \sp 2}}+ +{{{391} \over 4} \ x}+ +{{44179} \over {324}}} +\over {{\left( x+4 \right)}\sp 4}} +\end{array} +$$ +\returnType{Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)} + +\spadcommand{padicFraction \% } +$$ +\begin{array}{@{}l} +{{1 \over {648}} \over {x+1}}+ +{{1 \over 4} \over {x+2}} - +{{1 \over {16}} \over {{\left( x+2 \right)}\sp 2}} - +{{{17} \over 8} \over {x+3}}+ +{{3 \over 4} \over {{\left( x+3 \right)}\sp 2}} - +{{1 \over 2} \over {{\left( x+3 \right)}\sp 3}}+ +{{{607} \over {324}} \over {x+4}}+ +\\ +\\ +\displaystyle +{{{403} \over {432}} \over {{\left( x+4 \right)}\sp 2}}+ +{{{13} \over {36}} \over {{\left( x+4 \right)}\sp 3}}+ +{{1 \over {12}} \over {{\left( x+4 \right)}\sp 4}} +\end{array} +$$ +\returnType{Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)} + +All see \ref{FullPartialFractionExpansionXmpPage} on +page~\pageref{FullPartialFractionExpansionXmpPage} for examples of +factor-free conversion of quotients to full partial fractions. + +\section{Permanent} +\label{PermanentXmpPage} + +The package {\tt Permanent} provides the function +\spadfunFrom{permanent}{Permanent} for square matrices. The +\spadfunFrom{permanent}{Permanent} of a square matrix can be computed +in the same way as the determinant by expansion of minors except that +for the permanent the sign for each element is {\tt 1}, rather than +being {\tt 1} if the row plus column indices is positive and {\tt -1} +otherwise. This function is much more difficult to compute +efficiently than the \spadfunFrom{determinant}{Matrix}. An example of +the use of \spadfunFrom{permanent}{Permanent} is the calculation of +the $n$-th derangement number, defined to be the number of +different possibilities for {\tt n} couples to dance but never with +their own spouse. + +Consider an {\tt n} by {\tt n} matrix with entries {\tt 0} on the +diagonal and {\tt 1} elsewhere. Think of the rows as one-half of each +couple (for example, the males) and the columns the other half. The +permanent of such a matrix gives the desired derangement number. + +\begin{verbatim} +kn n == + r : MATRIX INT := new(n,n,1) + for i in 1..n repeat + r.i.i := 0 + r +\end{verbatim} +\returnType{Void} + +Here are some derangement numbers, which you see grow quite fast. + +\spadcommand{permanent(kn(5) :: SQMATRIX(5,INT)) } +\begin{verbatim} +Compiling function kn with type PositiveInteger -> Matrix Integer +\end{verbatim} +$$ +44 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{[permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13] } +\begin{verbatim} +Cannot compile conversion for types involving local variables. + In particular, could not compile the expression involving + :: SQMATRIX(n,INT) +AXIOM will attempt to step through and interpret the code. +\end{verbatim} +$$ +\begin{array}{@{}l} +\left[ +0, 1, 2, 9, {44}, {265}, {1854}, {14833}, {133496}, +\right. +\\ +\displaystyle +\left. +{1334961}, {14684570}, {176214841}, {2290792932} +\right] +\end{array} +$$ +\returnType{Type: List NonNegativeInteger} + +\section{Polynomial} +\label{PolynomialXmpPage} + +The domain constructor {\tt Polynomial} (abbreviation: {\tt POLY}) +provides polynomials with an arbitrary number of unspecified +variables. + +It is used to create the default polynomial domains in Axiom. +Here the coefficients are integers. + +\spadcommand{x + 1} +$$ +x+1 +$$ +\returnType{Type: Polynomial Integer} + +Here the coefficients have type {\tt Float}. + +\spadcommand{z - 2.3} +$$ +z -{2.3} +$$ +\returnType{Type: Polynomial Float} + +And here we have a polynomial in two variables with coefficients which +have type {\tt Fraction Integer}. + +\spadcommand{y**2 - z + 3/4} +$$ +-z+{y \sp 2}+{3 \over 4} +$$ +\returnType{Type: Polynomial Fraction Integer} + +The representation of objects of domains created by {\tt Polynomial} +is that of recursive univariate polynomials.\footnote{The term +{\tt univariate} means ``one variable.'' {\tt multivariate} means +``possibly more than one variable.''} + +This recursive structure is sometimes obvious from the display of +a polynomial. + +\spadcommand{y **2 + x*y + y } +$$ +{y \sp 2}+{{\left( x+1 +\right)} +\ y} +$$ +\returnType{Type: Polynomial Integer} + +In this example, you see that the polynomial is stored as a polynomial +in {\tt y} with coefficients that are polynomials in {\tt x} with +integer coefficients. In fact, you really don't need to worry about +the representation unless you are working on an advanced application +where it is critical. The polynomial types created from {\tt +DistributedMultivariatePolynomial} and +{\tt NewDistributedMultivariatePolynomial} (discussed in +\ref{DistributedMultivariatePolynomialXmpPage} on +page~\pageref{DistributedMultivariatePolynomialXmpPage}) are stored and +displayed in a non-recursive manner. + +You see a ``flat'' display of the above polynomial by converting to +one of those types. + +\spadcommand{\% :: DMP([y,x],INT) } +$$ +{y \sp 2}+{y \ x}+y +$$ +\returnType{Type: DistributedMultivariatePolynomial([y,x],Integer)} + +We will demonstrate many of the polynomial facilities by using two +polynomials with integer coefficients. + +By default, the interpreter expands polynomial expressions, even if they +are written in a factored format. + +\spadcommand{p := (y-1)**2 * x * z } +$$ +{\left( {x \ {y \sp 2}} -{2 \ x \ y}+x +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +See \ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage} +to see how to create objects in factored form directly. + +\spadcommand{q := (y-1) * x * (z+5) } +$$ +{{\left( {x \ y} -x +\right)} +\ z}+{5 \ x \ y} -{5 \ x} +$$ +\returnType{Type: Polynomial Integer} + +The fully factored form can be recovered by using +\spadfunFrom{factor}{Polynomial}. + +\spadcommand{factor(q) } +$$ +x \ {\left( y -1 +\right)} +\ {\left( z+5 +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +This is the same name used for the operation to factor integers. Such +reuse of names is called \index{overloading} and makes it much easier +to think of solving problems in general ways. Axiom facilities for +factoring polynomials created with {\tt Polynomial} are currently +restricted to the integer and rational number coefficient cases. +There are more complete facilities for factoring univariate +polynomials: see \ref{ugProblemFactorPage} on +page~\pageref{ugProblemFactorPage} in Section \ref{ugProblemFactorNumber} +on page~\pageref{ugProblemFactorNumber}. + +The standard arithmetic operations are available for polynomials. + +\spadcommand{p - q**2} +$$ +\begin{array}{@{}l} +{{\left( +-{{x \sp 2} \ {y \sp 2}}+ +{2 \ {x \sp 2} \ y} - +{x \sp 2} +\right)}\ {z \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( +{{\left( +-{{10} \ {x \sp 2}}+x +\right)}\ {y \sp 2}}+ +{{\left( +{{20} \ {x \sp 2}} - +{2 \ x} +\right)}\ y} +-{{10} \ {x \sp 2}}+ +x +\right)}\ z} - +\\ +\\ +\displaystyle +{{25} \ {x \sp 2} \ {y \sp 2}}+ +{{50} \ {x \sp 2} \ y} - +{{25} \ {x \sp 2}} +\end{array} +$$ +\returnType{Type: Polynomial Integer} + +The operation \spadfunFrom{gcd}{Polynomial} is used to compute the +greatest common divisor of two polynomials. + +\spadcommand{gcd(p,q) } +$$ +{x \ y} -x +$$ +\returnType{Type: Polynomial Integer} + +In the case of {\tt p} and {\tt q}, the gcd is obvious from their +definitions. We factor the gcd to show this relationship better. + +\spadcommand{factor \% } +$$ +x \ {\left( y -1 +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +The least common multiple is computed by using \spadfunFrom{lcm}{Polynomial}. + +\spadcommand{lcm(p,q) } +$$ +{{\left( {x \ {y \sp 2}} -{2 \ x \ y}+x +\right)} +\ {z \sp 2}}+{{\left( {5 \ x \ {y \sp 2}} -{{10} \ x \ y}+{5 \ x} +\right)} +\ z} +$$ +\returnType{Type: Polynomial Integer} + +Use \spadfunFrom{content}{Polynomial} to compute the greatest common +divisor of the coefficients of the polynomial. + +\spadcommand{content p } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Many of the operations on polynomials require you to specify a +variable. For example, \spadfunFrom{resultant}{Polynomial} requires +you to give the variable in which the polynomials should be expressed. + +This computes the resultant of the values of {\tt p} and {\tt q}, +considering them as polynomials in the variable {\tt z}. They do not +share a root when thought of as polynomials in {\tt z}. + +\spadcommand{resultant(p,q,z) } +$$ +{5 \ {x \sp 2} \ {y \sp 3}} -{{15} \ {x \sp 2} \ {y \sp 2}}+{{15} \ {x +\sp 2} \ y} -{5 \ {x \sp 2}} +$$ +\returnType{Type: Polynomial Integer} + +This value is {\tt 0} because as polynomials in {\tt x} the polynomials +have a common root. + +\spadcommand{resultant(p,q,x) } +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +The data type used for the variables created by {\tt Polynomial} is +{\tt Symbol}. As mentioned above, the representation used by {\tt +Polynomial} is recursive and so there is a main variable for +nonconstant polynomials. + +The operation \spadfunFrom{mainVariable}{Polynomial} returns this +variable. The return type is actually a union of {\tt Symbol} and +{\tt "failed"}. + +\spadcommand{mainVariable p } +$$ +z +$$ +\returnType{Type: Union(Symbol,...)} + +The latter branch of the union is be used if the polynomial has no +variables, that is, is a constant. + +\spadcommand{mainVariable(1 :: POLY INT)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +You can also use the predicate \spadfunFrom{ground?}{Polynomial} to test +whether a polynomial is in fact a member of its ground ring. + +\spadcommand{ground? p } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{ground?(1 :: POLY INT)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The complete list of variables actually used in a particular +polynomial is returned by \spadfunFrom{variables}{Polynomial}. For +constant polynomials, this list is empty. + +\spadcommand{variables p } +$$ +\left[ +z, y, x +\right] +$$ +\returnType{Type: List Symbol} + +The \spadfunFrom{degree}{Polynomial} operation returns the +degree of a polynomial in a specific variable. + +\spadcommand{degree(p,x) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{degree(p,y) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{degree(p,z) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +If you give a list of variables for the second argument, a list +of the degrees in those variables is returned. + +\spadcommand{degree(p,[x,y,z]) } +$$ +\left[ +1, 2, 1 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +The minimum degree of a variable in a polynomial is computed using +\spadfunFrom{minimumDegree}{Polynomial}. + +\spadcommand{minimumDegree(p,z) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The total degree of a polynomial is returned by +\spadfunFrom{totalDegree}{Polynomial}. + +\spadcommand{totalDegree p } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +It is often convenient to think of a polynomial as a leading monomial plus +the remaining terms. + +\spadcommand{leadingMonomial p } +$$ +x \ {y \sp 2} \ z +$$ +\returnType{Type: Polynomial Integer} + +The \spadfunFrom{reductum}{Polynomial} operation returns a polynomial +consisting of the sum of the monomials after the first. + +\spadcommand{reductum p } +$$ +{\left( -{2 \ x \ y}+x +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +These have the obvious relationship that the original polynomial +is equal to the leading monomial plus the reductum. + +\spadcommand{p - leadingMonomial p - reductum p } +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +The value returned by \spadfunFrom{leadingMonomial}{Polynomial} +includes the coefficient of that term. This is extracted by using +\spadfunFrom{leadingCoefficient}{Polynomial} on the original +polynomial. + +\spadcommand{leadingCoefficient p } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{eval}{Polynomial} is used to substitute a value +for a variable in a polynomial. + +\spadcommand{p } +$$ +{\left( {x \ {y \sp 2}} -{2 \ x \ y}+x +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +This value may be another variable, a constant or a polynomial. + +\spadcommand{eval(p,x,w) } +$$ +{\left( {w \ {y \sp 2}} -{2 \ w \ y}+w +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{eval(p,x,1) } +$$ +{\left( {y \sp 2} -{2 \ y}+1 +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +Actually, all the things being substituted are just polynomials, +some more trivial than others. + +\spadcommand{eval(p,x,y**2 - 1) } +$$ +{\left( {y \sp 4} -{2 \ {y \sp 3}}+{2 \ y} -1 +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +Derivatives are computed using the \spadfunFrom{D}{Polynomial} operation. + +\spadcommand{D(p,x) } +$$ +{\left( {y \sp 2} -{2 \ y}+1 +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +The first argument is the polynomial and the second is the variable. + +\spadcommand{D(p,y) } +$$ +{\left( {2 \ x \ y} -{2 \ x} +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +Even if the polynomial has only one variable, you must specify it. + +\spadcommand{D(p,z) } +$$ +{x \ {y \sp 2}} -{2 \ x \ y}+x +$$ +\returnType{Type: Polynomial Integer} + +Integration of polynomials is similar and the +\spadfunFrom{integrate}{Polynomial} operation is used. + +Integration requires that the coefficients support division. +Consequently, Axiom converts polynomials over the integers to +polynomials over the rational numbers before integrating them. + +\spadcommand{integrate(p,y) } +$$ +{\left( {{1 \over 3} \ x \ {y \sp 3}} -{x \ {y \sp 2}}+{x \ y} +\right)} +\ z +$$ +\returnType{Type: Polynomial Fraction Integer} + +It is not possible, in general, to divide two polynomials. In our +example using polynomials over the integers, the operation +\spadfunFrom{monicDivide}{Polynomial} divides a polynomial by a monic +polynomial (that is, a polynomial with leading coefficient equal to +1). The result is a record of the quotient and remainder of the +division. + +You must specify the variable in which to express the polynomial. + +\spadcommand{qr := monicDivide(p,x+1,x) } +$$ +\left[ +{quotient={{\left( {y \sp 2} -{2 \ y}+1 +\right)} +\ z}}, {remainder={{\left( -{y \sp 2}+{2 \ y} -1 +\right)} +\ z}} +\right] +$$ +\returnType{Type: Record(quotient: Polynomial Integer,remainder: Polynomial Integer)} + +The selectors of the components of the record are {\tt quotient} and +{\tt remainder}. Issue this to extract the remainder. + +\spadcommand{qr.remainder } +$$ +{\left( -{y \sp 2}+{2 \ y} -1 +\right)} +\ z +$$ +\returnType{Type: Polynomial Integer} + +Now that we can extract the components, we can demonstrate the +relationship among them and the arguments to our original expression +{\tt qr := monicDivide(p,x+1,x)}. + +\spadcommand{p - ((x+1) * qr.quotient + qr.remainder) } +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +If the \spadopFrom{/}{Fraction} operator is used with polynomials, a +fraction object is created. In this example, the result is an object +of type {\tt Fraction Polynomial Integer}. + +\spadcommand{p/q } +$$ +{{\left( y -1 +\right)} +\ z} \over {z+5} +$$ +\returnType{Type: Fraction Polynomial Integer} + +If you use rational numbers as polynomial coefficients, the +resulting object is of type {\tt Polynomial Fraction Integer}. + +\spadcommand{(2/3) * x**2 - y + 4/5 } +$$ +-y+{{2 \over 3} \ {x \sp 2}}+{4 \over 5} +$$ +\returnType{Type: Polynomial Fraction Integer} + +This can be converted to a fraction of polynomials and back again, if +required. + +\spadcommand{\% :: FRAC POLY INT } +$$ +{-{{15} \ y}+{{10} \ {x \sp 2}}+{12}} \over {15} +$$ +\returnType{Type: Fraction Polynomial Integer} + +\spadcommand{\% :: POLY FRAC INT } +$$ +-y+{{2 \over 3} \ {x \sp 2}}+{4 \over 5} +$$ +\returnType{Type: Polynomial Fraction Integer} + +To convert the coefficients to floating point, map the {\tt numeric} +operation on the coefficients of the polynomial. + +\spadcommand{map(numeric,\%) } +$$ +-{{1.0} \ y}+{{0.6666666666 6666666667} \ {x \sp 2}}+{0.8} +$$ +\returnType{Type: Polynomial Float} + +For more information on related topics, see +\ref{UnivariatePolynomialXmpPage} on +page~\pageref{UnivariatePolynomialXmpPage}, +\ref{MultivariatePolynomialXmpPage} on +page~\pageref{MultivariatePolynomialXmpPage}, and +\ref{DistributedMultivariatePolynomialXmpPage} on +page~\pageref{DistributedMultivariatePolynomialXmpPage}. You can also issue +the system command {\tt )show Polynomial} to display the full list +of operations defined by {\tt Polynomial}. + +\section{Quaternion} +\label{QuaternionXmpPage} + +The domain constructor {\tt Quaternion} implements quaternions over +commutative rings. For information on related topics, see +%\menuxmpref{CliffordAlgebra} +\ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} and +\ref{OctonionXmpPage} on page~\pageref{OctonionXmpPage}. +You can also issue the system command +{\tt )show Quaternion} to display the full list of operations +defined by {\tt Quaternion}. + +The basic operation for creating quaternions is +\spadfunFrom{quatern}{Quaternion}. +This is a quaternion over the rational numbers. + +\spadcommand{q := quatern(2/11,-8,3/4,1) } +$$ +{2 \over {11}} -{8 \ i}+{{3 \over 4} \ j}+k +$$ +\returnType{Type: Quaternion Fraction Integer} + +The four arguments are the real part, the {\tt i} imaginary part, the +{\tt j} imaginary part, and the {\tt k} imaginary part, respectively. + +\spadcommand{[real q, imagI q, imagJ q, imagK q] } +$$ +\left[ +{2 \over {11}}, -8, {3 \over 4}, 1 +\right] +$$ +\returnType{Type: List Fraction Integer} + +Because {\tt q} is over the rationals (and nonzero), you can invert it. + +\spadcommand{inv q } +$$ +{{352} \over {126993}}+{{{15488} \over {126993}} \ i} -{{{484} \over +{42331}} \ j} -{{{1936} \over {126993}} \ k} +$$ +\returnType{Type: Quaternion Fraction Integer} + +The usual arithmetic (ring) operations are available + +\spadcommand{q**6 } +$$ +-{{2029490709319345} \over {7256313856}} - +{{{48251690851} \over {1288408}} \ i}+ +{{{144755072553} \over {41229056}} \ j}+ +{{{48251690851} \over {10307264}} +\ k} +$$ +\returnType{Type: Quaternion Fraction Integer} + +\spadcommand{r := quatern(-2,3,23/9,-89); q + r } +$$ +-{{20} \over {11}} - +{5 \ i}+ +{{{119} \over {36}} \ j} - +{{88} \ k} +$$ +\returnType{Type: Quaternion Fraction Integer} + +In general, multiplication is not commutative. + +\spadcommand{q * r - r * q} +$$ +-{{{2495} \over {18}} \ i} -{{1418} \ j} -{{{817} \over {18}} \ k} +$$ +\returnType{Type: Quaternion Fraction Integer} + +There are no predefined constants for the imaginary {\tt i, j}, +and {\tt k} parts, but you can easily define them. + +\spadcommand{i:=quatern(0,1,0,0); j:=quatern(0,0,1,0); k:=quatern(0,0,0,1) } +$$ +k +$$ +\returnType{Type: Quaternion Integer} + +These satisfy the normal identities. + +\spadcommand{[i*i, j*j, k*k, i*j, j*k, k*i, q*i] } +$$ +\left[ +-1, -1, -1, k, i, j, {8+{{2 \over {11}} \ i}+j -{{3 \over +4} \ k}} +\right] +$$ +\returnType{Type: List Quaternion Fraction Integer} + +The norm is the quaternion times its conjugate. + +\spadcommand{norm q } +$$ +{126993} \over {1936} +$$ +\returnType{Type: Fraction Integer} + +\spadcommand{conjugate q } +$$ +{2 \over {11}}+{8 \ i} -{{3 \over 4} \ j} -k +$$ +\returnType{Type: Quaternion Fraction Integer} + +\spadcommand{q * \% } +$$ +{126993} \over {1936} +$$ +\returnType{Type: Quaternion Fraction Integer} + +\section{RadixExpansion} +\label{RadixExpansionXmpPage} + +It possible to expand numbers in general bases. + +Here we expand {\tt 111} in base {\tt 5}. +This means +$$10^2+10^1+10^0 = 4 \cdot 5^2+2 \cdot 5^1 + 5^0$$ + +\spadcommand{111::RadixExpansion(5)} +$$ +421 +$$ +\returnType{Type: RadixExpansion 5} + +You can expand fractions to form repeating expansions. + +\spadcommand{(5/24)::RadixExpansion(2)} +$$ +0.{001}{\overline {10}} +$$ +\returnType{Type: RadixExpansion 2} + +\spadcommand{(5/24)::RadixExpansion(3)} +$$ +0.0{\overline {12}} +$$ +\returnType{Type: RadixExpansion 3} + +\spadcommand{(5/24)::RadixExpansion(8)} +$$ +0.1{\overline {52}} +$$ +\returnType{Type: RadixExpansion 8} + +\spadcommand{(5/24)::RadixExpansion(10)} +$$ +0.{208}{\overline 3} +$$ +\returnType{Type: RadixExpansion 10} + +For bases from 11 to 36 the letters A through Z are used. + +\spadcommand{(5/24)::RadixExpansion(12)} +$$ +0.{26} +$$ +\returnType{Type: RadixExpansion 12} + +\spadcommand{(5/24)::RadixExpansion(16)} +$$ +0.3{\overline 5} +$$ +\returnType{Type: RadixExpansion 16} + +\spadcommand{(5/24)::RadixExpansion(36)} +$$ +0.{\rm 7I} +$$ +\returnType{Type: RadixExpansion 36} + +For bases greater than 36, the ragits are separated by blanks. + +\spadcommand{(5/24)::RadixExpansion(38)} +$$ +0 \ . \ 7 \ {34} \ {31} \ {\overline {{25} \ {12}}} +$$ +\returnType{Type: RadixExpansion 38} + +The {\tt RadixExpansion} type provides operations to obtain the +individual ragits. Here is a rational number in base {\tt 8}. + +\spadcommand{a := (76543/210)::RadixExpansion(8) } +$$ +{554}.3{\overline {7307}} +$$ +\returnType{Type: RadixExpansion 8} + +The operation \spadfunFrom{wholeRagits}{RadixExpansion} returns a list of the +ragits for the integral part of the number. + +\spadcommand{w := wholeRagits a } +$$ +\left[ +5, 5, 4 +\right] +$$ +\returnType{Type: List Integer} + +The operations \spadfunFrom{prefixRagits}{RadixExpansion} and +\spadfunFrom{cycleRagits}{RadixExpansion} return lists of the initial +and repeating ragits in the fractional part of the number. + +\spadcommand{f0 := prefixRagits a } +$$ +\left[ +3 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{f1 := cycleRagits a } +$$ +\left[ +7, 3, 0, 7 +\right] +$$ +\returnType{Type: List Integer} + +You can construct any radix expansion by giving the whole, prefix and +cycle parts. The declaration is necessary to let Axiom know the base +of the ragits. + +\spadcommand{u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1) } +$$ +{554}.3{\overline {7307}} +$$ +\returnType{Type: RadixExpansion 8} + +If there is no repeating part, then the list {\tt [0]} should be used. + +\spadcommand{v: RadixExpansion(12) := fractRadix([1,2,3,11], [0]) } +$$ +0.{\rm 123B}{\rm {\overline 0}} +$$ +\returnType{Type: RadixExpansion 12} + +If you are not interested in the repeating nature of the expansion, +an infinite stream of ragits can be obtained using +\spadfunFrom{fractRagits}{RadixExpansion}. + +\spadcommand{fractRagits(u) } +$$ +\left[ +3, 7, {\overline {3, 0, 7, 7}} +\right] +$$ +\returnType{Type: Stream Integer} + +Of course, it's possible to recover the fraction representation: + +\spadcommand{a :: Fraction(Integer) } +$$ +{76543} \over {210} +$$ +\returnType{Type: Fraction Integer} + +More examples of expansions are available in +\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, +\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, and +\ref{HexadecimalExpansionXmpPage} on +page~\pageref{HexadecimalExpansionXmpPage}. + +\section{RealClosure} +\label{RealClosureXmpPage} + +The Real Closure 1.0 package provided by Renaud Rioboo +(Renaud.Rioboo@lip6.fr) consists of different packages, categories and +domains : + +\begin{list}{} +\item The package {\tt RealPolynomialUtilitiesPackage} which needs a +{\tt Field} {\em F} and a {\tt UnivariatePolynomialCategory} domain +with coefficients in {\em F}. It computes some simple functions such +as Sturm and Sylvester sequences +(\spadfunFrom{sturmSequence}{RealPolynomialUtilitiesPackage}, +\spadfunFrom{sylvesterSequence}{RealPolynomialUtilitiesPackage}). + +\item The category {\tt RealRootCharacterizationCategory} provides abstract +functions to work with ``real roots'' of univariate polynomials. These +resemble variables with some functionality needed to compute important +operations. + +\item The category {\tt RealClosedField} provides common operations +available over real closed fiels. These include finding all the roots +of a univariate polynomial, taking square (and higher) roots, ... + +\item The domain {\tt RightOpenIntervalRootCharacterization} is the +main code that provides the functionality of {\tt +RealRootCharacterizationCategory} for the case of archimedean +fields. Abstract roots are encoded with a left closed right open +interval containing the root together with a defining polynomial for +the root. + +\item The {\tt RealClosure} domain is the end-user code. It provides +usual arithmetic with real algebraic numbers, along with the +functionality of a real closed field. It also provides functions to +approximate a real algebraic number by an element of the base +field. This approximation may either be absolute +(\spadfunFrom{approximate}{RealClosure}) or relative +(\spadfunFrom{relativeApprox}{RealClosure}). + +\end{list} + +\centerline{CAVEATS} + +Since real algebraic expressions are stored as depending on ``real +roots'' which are managed like variables, there is an ordering on +these. This ordering is dynamical in the sense that any new algebraic +takes precedence over older ones. In particular every creation +function raises a new ``real root''. This has the effect that when you +type something like {\tt sqrt(2) + sqrt(2)} you have two new variables +which happen to be equal. To avoid this name the expression such as in +{\tt s2 := sqrt(2) ; s2 + s2} + +Also note that computing times depend strongly on the ordering you +implicitly provide. Please provide algebraics in the order which seems +most natural to you. + +\centerline{LIMITATIONS} + +This packages uses algorithms which are published in [1] and [2] which +are based on field arithmetics, in particular for polynomial gcd +related algorithms. This can be quite slow for high degree polynomials +and subresultants methods usually work best. Beta versions of the +package try to use these techniques in a better way and work +significantly faster. These are mostly based on unpublished algorithms +and cannot be distributed. Please contact the author if you have a +particular problem to solve or want to use these versions. + +Be aware that approximations behave as post-processing and that all +computations are done exactly. They can thus be quite time consuming when +depending on several ``real roots''. + +\centerline{REFERENCES} + + +[1] R. Rioboo : Real Algebraic Closure of an ordered Field : Implementation + in Axiom. + In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215. + +[2] Z. Ligatsikas, R. Rioboo, M. F. Roy : Generic computation of the real + closure of an ordered field. + In Mathematics and Computers in Simulation Volume 42, Issue 4-6, + November 1996. + +\centerline{EXAMPLES} + +We shall work with the real closure of the ordered field of +rational numbers. + +\spadcommand{Ran := RECLOS(FRAC INT) } +$$ +\mbox{\rm RealClosure Fraction Integer} +$$ +\returnType{Type: Domain} + +Some simple signs for square roots, these correspond to an extension +of degree 16 of the rational numbers. Examples provided by J. Abbot. + +\spadcommand{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d) } +\begin{verbatim} +Function declaration fourSquares : (RealClosure Fraction Integer, + RealClosure Fraction Integer,RealClosure Fraction Integer, + RealClosure Fraction Integer) -> RealClosure Fraction Integer has + been added to workspace. +\end{verbatim} +\returnType{Void} + +These produce values very close to zero. + +\spadcommand{squareDiff1 := fourSquares(73,548,60,586) } +$$ +-{\sqrt {{586}}} -{\sqrt {{60}}}+{\sqrt {{548}}}+{\sqrt {{73}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff1)} +$$ +\begin{array}{@{}l} +{{\left( +{{\left( +{{54602} \ {\sqrt {{548}}}}+ +{{149602} \ {\sqrt {{73}}}} +\right)} +\ {\sqrt {{60}}}}+ +{{49502} \ {\sqrt {{73}}} \ {\sqrt {{548}}}}+ +{9900895} +\right)} +\ {\sqrt {{586}}}}+ +\\ +\\ +\displaystyle +{{\left( {{154702} \ {\sqrt {{73}}} \ {\sqrt {{548}}}}+ +{30941947} +\right)} +\ {\sqrt {{60}}}}+{{10238421} \ {\sqrt {{548}}}}+ +{{28051871} \ {\sqrt {{73}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff1)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{squareDiff2 := fourSquares(165,778,86,990) } +$$ +-{\sqrt {{990}}} -{\sqrt {{86}}}+{\sqrt {{778}}}+{\sqrt {{165}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff2)} +$$ +\begin{array}{@{}l} +\left( +\left( +{{556778} \ {\sqrt {{778}}}}+ +{{1209010} \ {\sqrt {{165}}}} +\right)\ {\sqrt {{86}}}+ +\right. +\\ +\\ +\displaystyle +\left. +{{401966} \ {\sqrt {{165}}} \ {\sqrt {{778}}}}+ +{144019431} +\right)\ {\sqrt {{990}}}+ +\\ +\\ +\displaystyle +{{\left( {{1363822} \ {\sqrt {{165}}} \ {\sqrt {{778}}}}+ +{488640503} +\right)} +\ {\sqrt {{86}}}}+ +\\ +\\ +\displaystyle +{{162460913} \ {\sqrt {{778}}}}+ +{{352774119} \ {\sqrt {{165}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff2)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{squareDiff3 := fourSquares(217,708,226,692) } +$$ +-{\sqrt {{692}}} -{\sqrt {{226}}}+{\sqrt {{708}}}+{\sqrt {{217}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff3)} +$$ +\begin{array}{@{}l} +\left( +\left( +-{{34102} \ {\sqrt {{708}}}} - +{{61598} \ {\sqrt {{217}}}} +\right)\ {\sqrt {{226}}} - +\right. +\\ +\\ +\displaystyle +\left. +{{34802} \ {\sqrt {{217}}} \ {\sqrt {{708}}}} - +{13641141} +\right)\ {\sqrt {{692}}}+ +\\ +\\ +\displaystyle +\left( - +{{60898} \ {\sqrt {{217}}} \ {\sqrt {{708}}}} - +{23869841} +\right)\ {\sqrt {{226}}} - +\\ +\\ +\displaystyle +{{13486123} \ {\sqrt {{708}}}} - +{{24359809} \ {\sqrt {{217}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff3)} +$$ +-1 +$$ +\returnType{Type: Integer} + +\spadcommand{squareDiff4 := fourSquares(155,836,162,820) } +$$ +-{\sqrt {{820}}} -{\sqrt {{162}}}+{\sqrt {{836}}}+{\sqrt {{155}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff4)} +$$ +\begin{array}{@{}l} +\left( +\left( +-{{37078} \ {\sqrt {{836}}}} - +{{86110} \ {\sqrt {{155}}}} +\right)\ {\sqrt {{162}}} - +\right. +\\ +\\ +\displaystyle +\left. +{{37906} \ {\sqrt {{155}}} \ {\sqrt {{836}}}} - +{13645107} +\right)\ {\sqrt {{820}}}+ +\\ +\\ +\displaystyle +\left( -{{85282} \ {\sqrt {{155}}} \ {\sqrt {{836}}}} - +{30699151} +\right)\ {\sqrt {{162}}} - +\\ +\\ +\displaystyle +{{13513901} \ {\sqrt {{836}}}} - +{{31384703} \ {\sqrt {{155}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff4)} +$$ +-1 +$$ +\returnType{Type: Integer} + +\spadcommand{squareDiff5 := fourSquares(591,772,552,818) } +$$ +-{\sqrt {{818}}} -{\sqrt {{552}}}+{\sqrt {{772}}}+{\sqrt {{591}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff5)} +$$ +\begin{array}{@{}l} +\left( +\left( +{{70922} \ {\sqrt {{772}}}}+ +{{81058} \ {\sqrt {{591}}}} +\right)\ {\sqrt {{552}}}+ +\right. +\\ +\\ +\displaystyle +\left. +{{68542} \ {\sqrt {{591}}} \ {\sqrt {{772}}}}+ +{46297673} +\right)\ {\sqrt {{818}}}+ +\\ +\\ +\displaystyle +\left( +{{83438} \ {\sqrt {{591}}} \ {\sqrt {{772}}}}+ +{56359389} +\right)\ {\sqrt {{552}}}+ +\\ +\\ +\displaystyle +{{47657051} \ {\sqrt {{772}}}}+ +{{54468081} \ {\sqrt {{591}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff5)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{squareDiff6 := fourSquares(434,1053,412,1088) } +$$ +-{\sqrt {{1088}}} - +{\sqrt {{412}}}+ +{\sqrt {{1053}}}+ +{\sqrt {{434}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff6)} +$$ +\begin{array}{@{}l} +\left( +\left( +{{115442} \ {\sqrt {{1053}}}}+ +{{179818} \ {\sqrt {{434}}}} +\right)\ {\sqrt {{412}}}+ +\right. +\\ +\\ +\displaystyle +\left. +{{112478} \ {\sqrt {{434}}} \ {\sqrt {{1053}}}}+ +{76037291} +\right)\ {\sqrt {{1088}}}+ +\\ +\\ +\displaystyle +\left( +{{182782} \ {\sqrt {{434}}} \ {\sqrt {{1053}}}}+ +{123564147} +\right)\ {\sqrt {{412}}}+ +\\ +\\ +\displaystyle +{{77290639} \ {\sqrt {{1053}}}}+ +{{120391609} \ {\sqrt {{434}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff6)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{squareDiff7 := fourSquares(514,1049,446,1152) } +$$ +-{\sqrt {{1152}}} -{\sqrt {{446}}}+{\sqrt {{1049}}}+{\sqrt {{514}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff7)} +$$ +\begin{array}{@{}l} +\left( +\left( +{{349522} \ {\sqrt {{1049}}}}+ +{{499322} \ {\sqrt {{514}}}} +\right)\ {\sqrt {{446}}}+ +\right. +\\ +\\ +\displaystyle +\left. +{{325582} \ {\sqrt {{514}}} \ {\sqrt {{1049}}}}+ +{239072537} +\right)\ {\sqrt {{1152}}}+ +\\ +\\ +\displaystyle +\left( +{{523262} \ {\sqrt {{514}}} \ {\sqrt {{1049}}}}+ +{384227549} +\right)\ {\sqrt {{446}}}+ +\\ +\\ +\displaystyle +{{250534873} \ {\sqrt {{1049}}}}+ +{{357910443} \ {\sqrt {{514}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff7)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{squareDiff8 := fourSquares(190,1751,208,1698) } +$$ +-{\sqrt {{1698}}} - +{\sqrt {{208}}}+ +{\sqrt {{1751}}}+ +{\sqrt {{190}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{recip(squareDiff8)} +$$ +\begin{array}{@{}l} +\left( +\left( +-{{214702} \ {\sqrt {{1751}}}} - +{{651782} \ {\sqrt {{190}}}} +\right)\ {\sqrt {{208}}} - +\right. +\\ +\\ +\displaystyle +\left. +{{224642} \ {\sqrt {{190}}} \ {\sqrt {{1751}}}} +-{129571901} +\right)\ {\sqrt {{1698}}}+ +\\ +\\ +\displaystyle +\left( +-{{641842} \ {\sqrt {{190}}} \ {\sqrt {{1751}}}} - +{370209881} +\right)\ {\sqrt {{208}}} - +\\ +\\ +\displaystyle +{{127595865} \ {\sqrt {{1751}}}} - +{{387349387} \ {\sqrt {{190}}}} +\end{array} +$$ +\returnType{Type: Union(RealClosure Fraction Integer,...)} + +\spadcommand{sign(squareDiff8)} +$$ +-1 +$$ +\returnType{Type: Integer} + +This should give three digits of precision + +\spadcommand{relativeApprox(squareDiff8,10**(-3))::Float } +$$ +-{0.2340527771\ 5937700123 E -10} +$$ +\returnType{Type: Float} + +The sum of these 4 roots is 0 + +\spadcommand{l := allRootsOf((x**2-2)**2-2)\$Ran } +$$ +\left[ +{ \%A{33}}, { \%A{34}}, { \%A{35}}, { \%A{36}} +\right] +$$ +\returnType{Type: List RealClosure Fraction Integer} + +Check that they are all roots of the same polynomial + +\spadcommand{removeDuplicates map(mainDefiningPolynomial,l) } +$$ +\left[ +{{? \sp 4} -{4 \ {? \sp 2}}+2} +\right] +$$ +\returnType{Type: +List Union(SparseUnivariatePolynomial RealClosure Fraction Integer,"failed")} + +We can see at a glance that they are separate roots + +\spadcommand{map(mainCharacterization,l) } +$$ +\left[ +{{[-2}, {-1[}}, {{[-1}, {0[}}, {{[0}, {1[}}, {{[1}, +{2[}} +\right] +$$ +\returnType{Type: +List Union( +RightOpenIntervalRootCharacterization( +RealClosure Fraction Integer, +SparseUnivariatePolynomial RealClosure Fraction Integer), +"failed")} + +Check the sum and product + +\spadcommand{[reduce(+,l),reduce(*,l)-2] } +$$ +\left[ +0, 0 +\right] +$$ +\returnType{Type: List RealClosure Fraction Integer} + +A more complicated test that involve an extension of degree 256. +This is a way of checking nested radical identities. + +\spadcommand{(s2, s5, s10) := (sqrt(2)\$Ran, sqrt(5)\$Ran, sqrt(10)\$Ran) } +$$ +\sqrt {{10}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10) } +$$ +{-{{\sqrt {{{\sqrt {{10}}} -3}}} +\ {\sqrt {{{\sqrt {5}} -2}}}}+ +{{\sqrt {{{\sqrt {{10}}}+3}}} +\ {\sqrt {{{\sqrt {5}}+2}}}}}= +{\sqrt {{{{10} \ {\sqrt {2}}}+{10}}}} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{eq1::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)} +$$ +{-{{\sqrt {{{\sqrt {5}} -2}}} +\ {\sqrt {{{\sqrt {2}} -1}}}}+ +{{\sqrt {{{\sqrt {5}}+2}}} +\ {\sqrt {{{\sqrt {2}}+1}}}}}= +{\sqrt {{{2 \ {\sqrt {{10}}}}+2}}} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{eq2::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +Some more examples from J. M. Arnaudies + +\spadcommand{s3 := sqrt(3)\$Ran } +$$ +\sqrt {3} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{s7:= sqrt(7)\$Ran } +$$ +\sqrt {7} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{e1 := sqrt(2*s7-3*s3,3) } +$$ +\root {3} \of {{{2 \ {\sqrt {7}}} -{3 \ {\sqrt {3}}}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{e2 := sqrt(2*s7+3*s3,3) } +$$ +\root {3} \of {{{2 \ {\sqrt {7}}}+{3 \ {\sqrt {3}}}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +This should be null + +\spadcommand{e2-e1-s3 } +$$ +0 +$$ +\returnType{Type: RealClosure Fraction Integer} + +A quartic polynomial + +\spadcommand{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3) } +$$ +{x \sp 4}+ +{{7 \over 3} \ {x \sp 2}}+ +{{30} \ x} - +{{100} \over 3} +$$ +\returnType{Type: UnivariatePolynomial(x,RealClosure Fraction Integer)} + +Add some cubic roots + +\spadcommand{r1 := sqrt(7633)\$Ran } +$$ +\sqrt {{7633}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{alpha := sqrt(5*r1-436,3)/3 } +$$ +{1 \over 3} \ {\root {3} \of {{{5 \ {\sqrt {{7633}}}} -{436}}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{beta := -sqrt(5*r1+436,3)/3 } +$$ +-{{1 \over 3} \ {\root {3} \of {{{5 \ {\sqrt {{7633}}}}+{436}}}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +this should be null + +\spadcommand{pol.(alpha+beta-1/3) } +$$ +0 +$$ +\returnType{Type: RealClosure Fraction Integer} + +A quintic polynomial + +\spadcommand{qol : UP(x,Ran) := x**5+10*x**3+20*x+22 } +$$ +{x \sp 5}+{{10} \ {x \sp 3}}+{{20} \ x}+{22} +$$ +\returnType{Type: UnivariatePolynomial(x,RealClosure Fraction Integer)} + +Add some cubic roots + +\spadcommand{r2 := sqrt(153)\$Ran } +$$ +\sqrt {{153}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{alpha2 := sqrt(r2-11,5) } +$$ +\root {5} \of {{{\sqrt {{153}}} -{11}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{beta2 := -sqrt(r2+11,5) } +$$ +-{\root {5} \of {{{\sqrt {{153}}}+{11}}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +this should be null + +\spadcommand{qol(alpha2+beta2) } +$$ +0 +$$ +\returnType{Type: RealClosure Fraction Integer} + +Finally, some examples from the book Computer Algebra by +Davenport, Siret and Tournier (page 77). +The last one is due to Ramanujan. + +\spadcommand{dst1:=sqrt(9+4*s2)=1+2*s2 } +$$ +{\sqrt {{{4 \ {\sqrt {2}}}+9}}}={{2 \ {\sqrt {2}}}+1} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{dst1::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{s6:Ran:=sqrt 6 } +$$ +\sqrt {6} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3 } +$$ +{{\sqrt {{-{2 \ {\sqrt {6}}}+5}}}+{\sqrt {{{2 \ {\sqrt {6}}}+5}}}}={2 \ +{\sqrt {3}}} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{dst2::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{s29:Ran:=sqrt 29 } +$$ +\sqrt {{29}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5 } +$$ +{\sqrt {{{2 \ {\sqrt {{-{{10} \ {\sqrt {{29}}}}+{55}}}}} - +{2 \ {\sqrt {{29}}}}+ +{16}}}}= +{-{\sqrt {{{2 \ {\sqrt {{29}}}}+{11}}}}+ +{\sqrt {{{2 \ {\sqrt {5}}}+ +{22}}}}+{\sqrt {5}}} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{dst4::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5 } +$$ +{\sqrt {{{{\left( {{34} \ {\sqrt {2}}}+{46} +\right)} +\ {\sqrt {5}}}+{{70} \ {\sqrt {2}}}+{112}}}}={{{\left( {\sqrt {2}}+3 +\right)} +\ {\sqrt {5}}}+{4 \ {\sqrt {2}}}+5} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{dst6::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{f3:Ran:=sqrt(3,5) } +$$ +\root {5} \of {3} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{f25:Ran:=sqrt(1/25,5) } +$$ +\root {5} \of {{1 \over {25}}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{f32:Ran:=sqrt(32/5,5) } +$$ +\root {5} \of {{{32} \over 5}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{f27:Ran:=sqrt(27/5,5) } +$$ +\root {5} \of {{{27} \over 5}} +$$ +\returnType{Type: RealClosure Fraction Integer} + +\spadcommand{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)} +$$ +{\root {3} \of {{-{\root {5} \of {{{27} \over 5}}}+{\root {5} \of {{{32} +\over 5}}}}}}={{\left( -{{\root {5} \of {3}} \sp 2}+{\root {5} \of {3}}+1 +\right)} +\ {\root {5} \of {{1 \over {25}}}}} +$$ +\returnType{Type: Equation RealClosure Fraction Integer} + +\spadcommand{dst5::Boolean } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\section{RegularTriangularSet} +\label{RegularTriangularSetXmpPage} + +The {\tt RegularTriangularSet} domain constructor implements regular +triangular sets. These particular triangular sets were introduced by +M. Kalkbrener (1991) in his PhD Thesis under the name regular chains. +Regular chains and their related concepts are presented in the paper +``On the Theories of Triangular sets'' By P. Aubry, D. Lazard and +M. Moreno Maza (to appear in the Journal of Symbolic Computation). +The {\tt RegularTriangularSet} constructor also provides a new method +(by the third author) for solving polynomial system by means of +regular chains. This method has two ways of solving. One has the +same specifications as Kalkbrener's algorithm (1991) and the other is +closer to Lazard's method (Discr. App. Math, 1991). Moreover, this +new method removes redundant component from the decompositions when +this is not {\em too expensive}. This is always the case with +square-free regular chains. So if you want to obtain decompositions +without redundant components just use the {\tt +SquareFreeRegularTriangularSet} domain constructor or the {\tt +LazardSetSolvingPackage} package constructor. See also the {\tt +LexTriangularPackage} and {\tt ZeroDimensionalSolvePackage} for the +case of algebraic systems with a finite number of (complex) solutions. + +One of the main features of regular triangular sets is that they +naturally define towers of simple extensions of a field. +This allows to perform with multivariate polynomials the +same kind of operations as one can do in an {\tt EuclideanDomain}. + +The {\tt RegularTriangularSet} constructor takes four arguments. The +first one, {\bf R}, is the coefficient ring of the polynomials; it +must belong to the category {\tt GcdDomain}. The second one, {\bf E}, +is the exponent monoid of the polynomials; it must belong to the +category {\tt OrderedAbelianMonoidSup}. the third one, {\bf V}, is +the ordered set of variables; it must belong to the category +{\tt OrderedSet}. The last one is the polynomial ring; it must belong to +the category {\tt RecursivePolynomialCategory(R,E,V)}. The +abbreviation for {\tt RegularTriangularSet} is {\tt REGSET}. See also +the constructor {\tt RegularChain} which only takes two arguments, the +coefficient ring and the ordered set of variables; in that case, +polynomials are necessarily built with the +{\tt NewSparseMultivariatePolynomial} domain constructor. + +We shall explain now how to use the constructor {\tt REGSET} and how +to read the decomposition of a polynomial system by means of regular +sets. + +Let us give some examples. We start with an easy one +(Donati-Traverso) in order to understand the two ways of solving +polynomial systems provided by the {\tt REGSET} constructor. + +Define the coefficient ring. + +\spadcommand{R := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the list of variables, + +\spadcommand{ls : List Symbol := [x,y,z,t] } +$$ +\left[ +x, y, z, t +\right] +$$ +\returnType{Type: List Symbol} + +and make it an ordered set; + +\spadcommand{V := OVAR(ls) } +$$ +\mbox{\rm OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +then define the exponent monoid. + +\spadcommand{E := IndexedExponents V } +$$ +\mbox{\rm IndexedExponents OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +Define the polynomial ring. + +\spadcommand{P := NSMP(R, V) } +$$ +\mbox{\rm NewSparseMultivariatePolynomial(Integer,OrderedVariableList +[x,y,z,t])} +$$ +\returnType{Type: Domain} + +Let the variables be polynomial. + +\spadcommand{x: P := 'x } +$$ +x +$$ +\returnType{Type: +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +\spadcommand{y: P := 'y } +$$ +y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +\spadcommand{z: P := 'z } +$$ +z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +\spadcommand{t: P := 't } +$$ +t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +Now call the {\tt RegularTriangularSet} domain constructor. + +\spadcommand{T := REGSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm RegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t]))} +\end{array} +$$ +\returnType{Type: Domain} + +Define a polynomial system. + +\spadcommand{p1 := x ** 31 - x ** 6 - x - y } +$$ +{x \sp {31}} -{x \sp 6} -x -y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p2 := x ** 8 - z } +$$ +{x \sp 8} -z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p3 := x ** 10 - t } +$$ +{x \sp {10}} -t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +\spadcommand{lp := [p1, p2, p3] } +$$ +\left[ +{{x \sp {31}} -{x \sp 6} -x -y}, {{x \sp 8} -z}, {{x \sp {10}} -t} +\right] +$$ +\returnType{Type: +List NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t])} + +First of all, let us solve this system in the sense of Kalkbrener. + +\spadcommand{zeroSetSplit(lp)\$T } +$$ +\left[ +{\left\{ +{{z \sp 5} - +{t \sp 4}}, +{{t \ z \ {y \sp 2}}+ +{2 \ {z \sp 3} \ y} - +{t \sp 8}+ +{2 \ {t \sp 5}}+ +{t \sp 3} - +{t \sp 2}}, +{{{\left( {t \sp 4} +-t +\right)}\ x} - +{t \ y} - +{z \sp 2}} +\right\}} +\right] +$$ +\returnType{Type: +List RegularTriangularSet( +Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t]))} + +And now in the sense of Lazard (or Wu and other authors). + +\spadcommand{lts := zeroSetSplit(lp,false)\$T } +$$ +\begin{array}{@{}l} +\left[ +{\left\{ +{{z \sp 5} - +{t \sp 4}}, +{{t \ z \ {y \sp 2}}+ +{2 \ {z \sp 3} \ y} - +{t \sp 8}+ +{2 \ {t \sp 5}}+ +{t \sp 3} - +{t \sp 2}}, +{{{\left( +{t \sp 4} -t +\right)}\ x} - +{t \ y} - +{z \sp 2}} +\right\}}, +\right. +\\ +\\ +\displaystyle +\left. +{\left\{ +{{t \sp 3} -1}, +{{z \sp 5} -t}, +{{t \ z \ {y \sp 2}}+ +{2 \ {z \sp 3} \ y}+1}, +{{z \ {x \sp 2}} -t} +\right\}}, +{\left\{ +t, z, y, x +\right\}} +\right] +\end{array} +$$ +\returnType{Type: +List RegularTriangularSet( +Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial( +Integer, +OrderedVariableList [x,y,z,t]))} + +We can see that the first decomposition is a subset of the second. +So how can both be correct ? + +Recall first that polynomials from a domain of the category +{\tt RecursivePolynomialCategory} are regarded as univariate polynomials in +their main variable. For instance the second polynomial in the first +set of each decomposition has main variable {\bf y} and its initial +(i.e. its leading coefficient w.r.t. its main variable) is {\bf t z}. + +Now let us explain how to read the second decomposition. Note that +the non-constant initials of the first set are $t^4-t$ and $t z$. +Then the solutions described by this first set are the common zeros of +its polynomials that do not cancel the polynomials $t^4-t$ and $ty z$. +Now the solutions of the input system {\bf lp} satisfying these +equations are described by the second and the third sets of the +decomposition. Thus, in some sense, they can be considered as +degenerated solutions. The solutions given by the first set are +called the generic points of the system; they give the general form of +the solutions. The first decomposition only provides these generic +points. This latter decomposition is useful when they are many +degenerated solutions (which is sometimes hard to compute) and when +one is only interested in general informations, like the dimension of +the input system. + +We can get the dimensions of each component +of a decomposition as follows. + +\spadcommand{[coHeight(ts) for ts in lts] } +$$ +\left[ +1, 0, 0 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +Thus the first set has dimension one. Indeed {\bf t} can take any +value, except {\bf 0} or any third root of {\bf 1}, whereas {\bf z} is +completely determined from {\bf t}, {\bf y} is given by {\bf z} and +{\bf t}, and finally {\bf x} is given by the other three variables. +In the second and the third sets of the second decomposition the four +variables are completely determined and thus these sets have dimension +zero. + +We give now the precise specifications of each decomposition. This +assume some mathematical knowledge. However, for the non-expert user, +the above explanations will be sufficient to understand the other +features of the {\tt RSEGSET} constructor. + +The input system {\bf lp} is decomposed in the sense of Kalkbrener as +finitely many regular sets {\bf T1,...,Ts} such that the radical ideal +generated by {\bf lp} is the intersection of the radicals of the +saturated ideals of {\bf T1,...,Ts}. In other words, the affine +variety associated with {\bf lp} is the union of the closures +(w.r.t. Zarisky topology) of the regular-zeros sets of +{\bf T1,...,Ts}. + +{\bf N. B.} The prime ideals associated with the radical of the +saturated ideal of a regular triangular set have all the same +dimension; moreover these prime ideals can be given by characteristic +sets with the same main variables. Thus a decomposition in the sense +of Kalkbrener is unmixed dimensional. Then it can be viewed as a {\em +lazy} decomposition into prime ideals (some of these prime ideals +being merged into unmixed dimensional ideals). + +Now we explain the other way of solving by means of regular triangular +sets. The input system {\bf lp} is decomposed in the sense of Lazard +as finitely many regular triangular sets {\bf T1,...,Ts} such that the +affine variety associated with {\bf lp} is the union of the +regular-zeros sets of {\bf T1,...,Ts}. Thus a decomposition in the +sense of Lazard is also a decomposition in the sense of Kalkbrener; +the converse is false as we have seen before. + +When the input system has a finite number of solutions, both ways of +solving provide similar decompositions as we shall see with this +second example (Caprasse). + +Define a polynomial system. + +\spadcommand{f1 := y**2*z+2*x*y*t-2*x-z } +$$ +{{\left( {2 \ t \ y} -2 +\right)} +\ x}+{z \ {y \sp 2}} -z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{f2 := -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2 } +$$ +-{z \ {x \sp 3}}+ +{{\left( +{4 \ t \ y}+4 +\right)}\ {x \sp 2}}+ +{{\left( +{4 \ z \ {y \sp 2}}+ +{4 \ z} +\right)}\ x}+ +{2 \ t \ {y \sp 3}} - +{{10} \ {y \sp 2}} - +{{10} \ t \ y}+2 +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{f3 := 2*y*z*t+x*t**2-x-2*z } +$$ +{{\left( +{t \sp 2} -1 +\right)}\ x}+ +{2 \ t \ z \ y} - +{2 \ z} +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{f4 := -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2} +$$ +{{\left( -{z \sp 3}+{{\left( {4 \ {t \sp 2}}+4 +\right)} +\ z} +\right)} +\ x}+{{\left( {4 \ t \ {z \sp 2}}+{2 \ {t \sp 3}} -{{10} \ t} +\right)} +\ y}+{4 \ {z \sp 2}} -{{10} \ {t \sp 2}}+2 +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{lf := [f1, f2, f3, f4]} +$$ +\begin{array}{@{}l} +\left[ +{{{\left( +{2 \ t \ y} -2 +\right)}\ x}+ +{z \ {y \sp 2}} -z}, +\right. +\\ +\\ +\displaystyle +{-{z \ {x \sp 3}}+ +{{\left( +{4 \ t \ y}+4 +\right)}\ {x \sp 2}}+ +{{\left( +{4 \ z \ {y \sp 2}}+ +{4 \ z} +\right)}\ x}+ +{2 \ t \ {y \sp 3}} - +{{10} \ {y \sp 2}} - +{{10} \ t \ y}+2}, +\\ +\\ +\displaystyle +{{{\left( +{t \sp 2} -1 +\right)}\ x}+ +{2 \ t \ z \ y} - +{2 \ z}}, +\\ +\\ +\displaystyle +\left. +{{{\left( +-{z \sp 3}+ +{{\left( +{4 \ {t \sp 2}}+4 +\right)}\ z} +\right)}\ x}+ +{{\left( +{4 \ t \ {z \sp 2}}+ +{2 \ {t \sp 3}} - +{{10} \ t} +\right)}\ y}+ +{4 \ {z \sp 2}} - +{{10} \ {t \sp 2}}+2} +\right] +\end{array} +$$ +\returnType{Type: List +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +First of all, let us solve this system in the sense of Kalkbrener. + +\spadcommand{zeroSetSplit(lf)\$T } +$$ +\begin{array}{@{}l} +\left[ +{\left\{ +{{t \sp 2} -1}, +{{z \sp 8} -{{16} \ {z \sp 6}}+{{256} \ {z \sp 2}} -{256}}, +{{t \ y} -1}, +{{{\left( {z \sp 3} -{8 \ z} \right)}\ x} -{8 \ {z \sp 2}}+{16}} +\right\}}, +\right. +\\ +\\ +\displaystyle +{\left\{ {{3 \ {t \sp 2}}+1}, +{{z \sp 2} -{7 \ {t \sp 2}} -1}, +{y+t}, +{x+z} +\right\}}, +\\ +\\ +\displaystyle +{\left\{ {{t \sp 8} -{{10} \ {t \sp 6}}+{{10} \ {t \sp 2}} -1}, +z, +{{{\left( {t \sp 3} -{5 \ t} \right)}\ y} -{5 \ {t \sp 2}}+1}, +x +\right\}}, +\\ +\\ +\displaystyle +\left. +{\left\{ {{t \sp 2}+3}, +{{z \sp 2} -4}, +{y+t}, +{x -z} +\right\}} +\right] +\end{array} +$$ +\returnType{Type: +List RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +And now in the sense of Lazard (or Wu and other authors). + +\spadcommand{lts2 := zeroSetSplit(lf,false)\$T } +$$ +\begin{array}{@{}l} +\left[ +{\left\{ +{{t \sp 8} -{{10} \ {t \sp 6}}+{{10} \ {t \sp 2}} -1}, +z, +{{{\left( {t \sp 3} -{5 \ t} \right)}\ y} -{5 \ {t \sp 2}}+1}, +x +\right\}}, +\right. +\\ +\\ +\displaystyle +{\left\{ +{{t \sp 2} -1}, +{{z \sp 8} -{{16} \ {z \sp 6}}+{{256} \ {z \sp 2}} -{256}}, +{{t \ y} -1}, +{{{\left( {z \sp 3} -{8 \ z} \right)}\ x} -{8 \ {z \sp 2}}+{16}} +\right\}}, +\\ +\\ +\displaystyle +{\left\{ +{{3 \ {t \sp 2}}+1}, +{{z \sp 2} -{7 \ {t \sp 2}} -1}, +{y+t}, +{x+z} +\right\}}, +\\ +\\ +\displaystyle +\left. +{\left\{ {{t \sp 2}+3}, +{{z \sp 2} -4}, +{y+t}, +{x -z} +\right\}} +\right] +\end{array} +$$ +\returnType{Type: +List RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +Up to the ordering of the components, both decompositions are identical. + +Let us check that each component has a finite number of solutions. + +\spadcommand{[coHeight(ts) for ts in lts2] } +$$ +\left[ +0, 0, 0, 0 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +Let us count the degrees of each component, + +\spadcommand{degrees := [degree(ts) for ts in lts2] } +$$ +\left[ +8, {16}, 4, 4 +\right] +$$ +\returnType{Type: List NonNegativeInteger} + +and compute their sum. + +\spadcommand{reduce(+,degrees) } +$$ +32 +$$ +\returnType{Type: PositiveInteger} + +We study now the options of the {\tt zeroSetSplit} operation. As we +have seen yet, there is an optional second argument which is a boolean +value. If this value is {\tt true} (this is the default) then the +decomposition is computed in the sense of Kalkbrener, otherwise it is +computed in the sense of Lazard. + +There is a second boolean optional argument that can be used (in that +case the first optional argument must be present). This second option +allows you to get some information during the computations. + +Therefore, we need to understand a little what is going on during the +computations. An important feature of the algorithm is that the +intermediate computations are managed in some sense like the processes +of a Unix system. Indeed, each intermediate computation may generate +other intermediate computations and the management of all these +computations is a crucial task for the efficiency. Thus any +intermediate computation may be suspended, killed or resumed, +depending on algebraic considerations that determine priorities for +these processes. The goal is of course to go as fast as possible +towards the final decomposition which means to avoid as much as +possible unnecessary computations. + +To follow the computations, one needs to set to {\tt true} the second +argument. Then a lot of numbers and letters are displayed. Between a +{\tt [} and a {\tt ]} one has the state of the processes at a given +time. Just after {\tt [} one can see the number of processes. Then +each process is represented by two numbers between {\tt <} and +{\tt >}. A process consists of a list of polynomial {\bf ps} and a +triangular set {\bf ts}; its goal is to compute the common zeros of +{\bf ps} that belong to the regular-zeros set of {\bf ts}. After the +processes, the number between pipes gives the total number of +polynomials in all the sets {\tt ps}. Finally, the number between +braces gives the number of components of a decomposition that are +already computed. This number may decrease. + +Let us take a third example (Czapor-Geddes-Wang) to see how this +information is displayed. + +Define a polynomial system. + +\spadcommand{u : R := 2 } +$$ +2 +$$ +\returnType{Type: Integer} + +\spadcommand{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2} +$$ +\begin{array}{@{}l} +{{\left( +{y \sp 2} -{2 \ t \ y}+{t \sp 2} +\right)}\ {x \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( +-{2 \ {y \sp 2}}+ +{{\left( {{\left( {2 \ t}+4 +\right)}\ z}+ +{2 \ t} +\right)}\ y}+ +{{\left( +-{2 \ {t \sp 2}}+2 +\right)}\ z} - +{4 \ {t \sp 2}} -2 +\right)}\ x}+ +\\ +\\ +\displaystyle +{y \sp 2}+ +{{\left( +-{2 \ t \ z} - +{4 \ t} +\right)}\ y}+ +{{\left( +{t \sp 2}+ +{10} +\right)}\ {z \sp 2}} - +{8 \ z}+ +{4 \ {t \sp 2}}+ +2 +\end{array} +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t} +$$ +{{\left( +-{3 \ z \ y}+ +{2 \ t \ z}+ +t +\right)}\ x}+ +{{\left( +z+4 +\right)}\ y}+ +{4 \ t \ {z \sp 2}} - +{7 \ t \ z} +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)} +$$ +{{\left( +-{2 \ z} -2 +\right)}\ x} - +{2 \ {z \sp 2}}+ +{8 \ z} - +2 +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)} +$$ +\begin{array}{@{}l} +{{\left( +{3 \ {y \sp 2}} - +{3 \ {t \sp 2}} - +4 +\right)}\ {x \sp 2}}+ +{{\left( +-{6 \ {y \sp 2}}+ +{{\left( {{12} \ t \ z}+ +{{12} \ t} +\right)}\ y}+ +{6 \ {t \sp 2} \ z} +\right)}\ x}+ +{3 \ {y \sp 2}}+ +\\ +\\ +\displaystyle +{{\left( +{{12} \ t \ z} - +{{12} \ t} +\right)}\ y}+ +{{\left( +{9 \ {t \sp 2}}+ +4 +\right)}\ {z \sp 2}}+ +{{\left( +-{{24} \ {t \sp 2}} - +4 +\right)}\ z}+ +{{12} \ {t \sp 2}}+ +4 +\end{array} +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{lq := [q1, q2, q3, q4]} +$$ +\begin{array}{@{}l} +\left[ +\left( +{y \sp 2} -{2 \ t \ y}+{t \sp 2} \right)\ {x \sp 2}+ +\right. +\\ +\\ +\displaystyle +\left( -{2 \ {y \sp 2}}+ +\left( +\left( {2 \ t}+4 +\right)\ z+{2 \ t} +\right)\ y+ +\left( -{2 \ {t \sp 2}}+2 +\right)\ z -{4 \ {t \sp 2}} -2 +\right)\ x+{y \sp 2}+ +\\ +\\ +\displaystyle +\left( -{2 \ t \ z} -{4 \ t} +\right)\ y+ +\left( {t \sp 2}+{10} +\right)\ {z \sp 2} - +{8 \ z}+ +{4 \ {t \sp 2}}+ +2, +\\ +\\ +\displaystyle +\left( -{3 \ z \ y}+{2 \ t \ z}+t +\right)\ x+ +\left( z+4 +\right)\ y+ +{4 \ t \ {z \sp 2}} -{7 \ t \ z}, +\\ +\\ +\displaystyle +\left( -{2 \ z} -2 +\right)\ x -{2 \ {z \sp 2}}+{8 \ z} -2, +\left( {3 \ {y \sp 2}} -{3 \ {t \sp 2}} -4 +\right)\ {x \sp 2}+ +\\ +\\ +\displaystyle +\left( -{6 \ {y \sp 2}}+ +\left( {{12} \ t \ z}+{{12} \ t} +\right)\ y+{6 \ {t \sp 2} \ z} +\right)\ x+{3 \ {y \sp 2}}+ +\\ +\\ +\displaystyle +\left. +\left( {{12} \ t \ z} -{{12} \ t} +\right)\ y+ +\left( {9 \ {t \sp 2}}+4 +\right)\ {z \sp 2}+ +\left( -{{24} \ {t \sp 2}} -4 +\right)\ z+{{12} \ {t \sp 2}}+4 +\right] +\end{array} +$$ +\returnType{Type: +List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +Let us try the information option. N.B. The timing should be between +1 and 10 minutes, depending on your machine. + +\spadcommand{zeroSetSplit(lq,true,true)\$T } +\begin{verbatim} +[1 <4,0> -> |4|; {0}]W[2 <5,0>,<3,1> -> |8|; {0}][2 <4,1>,<3,1> -> |7|; +{0}][1 <3,1> -> |3|; {0}]G[2 <4,1>,<4,1> -> |8|; {0}]W[3 <5,1>,<4,1>, +<3,2> -> |12|; {0}]GI[3 <4,2>,<4,1>,<3,2> -> |11|; {0}]GWw[3 <4,1>, +<3,2>,<5,2> -> |12|; {0}][3 <3,2>,<3,2>,<5,2> -> |11|; {0}]GIwWWWw +[4 <3,2>,<4,2>,<5,2>,<2,3> -> |14|; {0}][4 <2,2>,<4,2>,<5,2>,<2,3> -> +|13|; {0}]Gwww[5 <3,2>,<3,2>,<4,2>,<5,2>,<2,3> -> |17|; {0}]Gwwwwww +[8 <3,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}]Gwwwwww +[8 <4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |31|; {0}][8 +<3,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}][8 <2,3>, +\end{verbatim} +\begin{verbatim} +<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |29|; {0}][8 <1,3>,<4,2>, +<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |28|; {0}][7 <4,2>,<4,2>,<4,2>, +<4,2>,<4,2>,<5,2>,<2,3> -> |27|; {0}][6 <4,2>,<4,2>,<4,2>,<4,2>,<5,2>, +<2,3> -> |23|; {0}][5 <4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |19|; {0}] +GIGIWwww[6 <5,2>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |23|; {0}][6 <4,3>, +<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |22|; {0}]GIGI[6 <3,4>,<4,2>,<4,2>, +<5,2>,<3,3>,<2,3> -> |21|; {0}][6 <2,4>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> +-> |20|; {0}]GGG[5 <4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |18|; {0}]GIGIWwwwW +[6 <5,2>,<4,2>,<5,2>,<3,3>,<3,3>,<2,3> -> |22|; {0}][6 <4,3>,<4,2>, +<5,2>,<3,3>,<3,3>,<2,3> -> |21|; {0}]GIwwWwWWWWWWWwWWWWwwwww[8 <4,2>, +\end{verbatim} +\begin{verbatim} +<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |27|; {0}][8 <3,3>,<5,2>, +<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |26|; {0}][8 <2,3>,<5,2>,<3,3>, +<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |25|; {0}]Gwwwwwwwwwwwwwwwwwwww[9 +<5,2>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |29|; {0}] +GI[9 <4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |28|; +{0}][9 <3,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |27|; +{0}][9 <2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |26|; +{0}]GGwwwwwwwwwwwwWWwwwwwwww[11 <3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |33|; {0}][11 <2,3>,<3,3>,<3,3>,<3,3>, +<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |32|; {0}][11 <1,3>,<3,3>, +\end{verbatim} +\begin{verbatim} +<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |31|; {0}] +GGGwwwwwwwwwwwww[12 <2,3>,<2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>, +<3,3>,<3,3>,<3,4>,<3,4> -> |34|; {0}]GGwwwwwwwwwwwww[13 <3,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +|38|; {0}]Gwwwwwwwwwwwww[13 <2,3>,<3,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {0}]GGGwwwwwwwwwwwww[15 +<3,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>, +<3,3>,<3,4>,<3,4> -> |46|; {0}][14 <4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>, +<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |43|; {0}]GIGGGGIGGI +[14 <3,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>, +\end{verbatim} +\begin{verbatim} +<3,3>,<3,4>,<3,4> -> |42|; {0}]GGG[14 <2,4>,<3,3>,<3,3>,<3,3>,<3,3>, +<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |41|; {0}] +[14 <1,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>, +<3,3>,<3,4>,<3,4> -> |40|; {0}]GGG[13 <3,3>,<3,3>,<3,3>,<3,3>,<3,3>, +<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {0}] +Gwwwwwwwwwwwww[15 <3,3>,<3,3>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>, +<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |48|; {0}]Gwwwwwwwwwwwww +[15 <4,3>,<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>, +<3,3>,<3,3>,<3,4>,<3,4> -> |49|; {0}]GIGI[15 <3,4>,<4,3>,<3,3>,<4,3>, +<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +\end{verbatim} +\begin{verbatim} +|48|; {0}]G[14 <4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |45|; {0}][13 <3,3>,<4,3>,<4,3>, +<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |41|; +{0}]Gwwwwwwwwwwwww[13 <4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |42|; {0}]GIGGGGIGGI[13 <3,4>,<4,3>, +<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +|41|; {0}]GGGGGGGG[13 <2,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |40|; {0}][13 <1,4>,<4,3>,<4,3>,<3,3>, +<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {0}] +[13 <0,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>, +\end{verbatim} +\begin{verbatim} +<3,4>,<3,4> -> |38|; {0}][12 <4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |38|; {1}][11 <4,3>,<3,3>, +<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |34|; {1}] +[10 <3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +|30|; {1}][10 <2,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>, +<3,4> -> |29|; {1}]GGGwwwwwwwwwwwww[11 <3,3>,<3,3>,<4,3>,<3,3>, +<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |33|; {1}] +GGGwwwwwwwwwwwww[12 <4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<4,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |38|; {1}]Gwwwwwwwwwwwww +[12 <3,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>, +\end{verbatim} +\begin{verbatim} +<3,4>,<3,4> -> |39|; {1}]GGwwwwwwwwwwwww[13 <5,3>,<4,3>,<4,3>, +<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +|44|; {1}]GIGGGGIGGIW[13 <4,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>, +<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |43|; {1}]GGW[13 +<3,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>, +<3,4>,<3,4> -> |42|; {1}]GGG[12 <4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {1}]Gwwwwwwwwwwwww[12 +<4,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>, +<3,4> -> |40|; {1}]Gwwwwwwwwwwwww[13 <5,3>,<5,3>,<4,3>,<5,3>,<3,3>, +<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |46|; {1}]GIGIW +\end{verbatim} +\begin{verbatim} +[13 <4,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>, +<3,3>,<3,4>,<3,4> -> |45|; {1}][13 <3,4>,<5,3>,<4,3>,<5,3>,<3,3>, +<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |44|; {1}][13 +<2,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>, +<3,4>,<3,4> -> |43|; {1}]GG[12 <5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |41|; {1}]GIGGGGIGGIW[12 +<4,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>, +<3,4> -> |40|; {1}]GGGGGGW[12 <3,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>, +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {1}][12 <2,4>,<4,3>, +<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |38|; +\end{verbatim} +\begin{verbatim} +{1}][12 <1,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>, +<3,4>,<3,4> -> |37|; {1}]GGG[11 <4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |36|; {1}][10 <5,3>,<3,3>,<3,3>, +<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |32|; {1}][9 <3,3>, +<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |27|; {1}]W[9 +<2,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |26|; {1}] +[9 <1,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |25|; +{1}][8 <3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |24|; {1}] +W[8 <2,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |23|; {1}][8 +<1,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}][7 <4,3>, +\end{verbatim} +\begin{verbatim} +<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |21|; {1}]w[7 <3,4>,<2,3>, +<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |20|; {1}][7 <2,4>,<2,3>,<3,3>, +<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}][7 <1,4>,<2,3>,<3,3>,<3,3>, +<3,3>,<3,4>,<3,4> -> |18|; {1}][6 <2,3>,<3,3>,<3,3>,<3,3>,<3,4>, +<3,4> -> |17|; {1}]GGwwwwww[7 <3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>, +<3,4> -> |21|; {1}]GIW[7 <2,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> +-> |20|; {1}]GG[6 <3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}] +Gwwwwww[7 <4,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |23|; {1}] +GIW[7 <3,4>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}][6 +<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}]GIW[6 <3,4>,<3,3>, +\end{verbatim} +\begin{verbatim} +<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}]GGW[6 <2,4>,<3,3>,<3,3>,<3,3>, +<3,4>,<3,4> -> |17|; {1}][6 <1,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> +|16|; {1}]GGG[5 <3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |15|; {1}]GIW[5 +<2,4>,<3,3>,<3,3>,<3,4>,<3,4> -> |14|; {1}]GG[4 <3,3>,<3,3>,<3,4>, +<3,4> -> |12|; {1}][3 <3,3>,<3,4>,<3,4> -> |9|; {1}]W[3 <2,4>,<3,4>, +<3,4> -> |8|; {1}][3 <1,4>,<3,4>,<3,4> -> |7|; {1}]G[2 <3,4>,<3,4> +-> |6|; {1}]G[1 <3,4> -> |3|; {1}][1 <2,4> -> |2|; {1}][1 <1,4> -> +|1|; {1}] +\end{verbatim} +\begin{verbatim} + *** QCMPACK Statistics *** + Table size: 36 + Entries reused: 255 + + *** REGSETGCD: Gcd Statistics *** + Table size: 125 + Entries reused: 0 + + *** REGSETGCD: Inv Set Statistics *** + Table size: 30 + Entries reused: 0 +\end{verbatim} +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{{960725655771966} \ {t \sp {24}}}+ +{{386820897948702} \ {t \sp {23}}}+ +\right. +\right.\hbox{\hskip 4.0cm} +\\ +\displaystyle +{{8906817198608181} \ {t \sp {22}}}+ +{{2704966893949428} \ {t \sp {21}}}+ +\\ +\displaystyle +{{37304033340228264} \ {t \sp {20}}}+ +{{7924782817170207} \ {t \sp {19}}}+ +\\ +\displaystyle +{{93126799040354990} \ {t \sp {18}}}+ +{{13101273653130910} \ {t \sp {17}}}+ +\\ +\displaystyle +{{156146250424711858} \ {t \sp {16}}}+ +{{16626490957259119} \ {t \sp {15}}}+ +\\ +\displaystyle +{{190699288479805763} \ {t \sp {14}}}+ +{{24339173367625275} \ {t \sp {13}}}+ +\\ +\displaystyle +{{180532313014960135} \ {t \sp {12}}}+ +{{35288089030975378} \ {t \sp {11}}}+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{135054975747656285} \ {t \sp {10}}}+ +{{34733736952488540} \ {t \sp 9}}+\hbox{\hskip 3.7cm} +\\ +\displaystyle +{{75947600354493972} \ {t \sp 8}}+ +{{19772555692457088} \ {t \sp 7}}+ +\\ +\displaystyle +{{28871558573755428} \ {t \sp 6}}+ +{{5576152439081664} \ {t \sp 5}}+ +\\ +\displaystyle +{{6321711820352976} \ {t \sp 4}}+ +{{438314209312320} \ {t \sp 3}}+ +\\ +\displaystyle +{{581105748367008} \ {t \sp 2}} - +{{60254467992576} \ t}+ +\\ +\displaystyle +{1449115951104}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{26604210869491302385515265737052082361668474181372891857784} +\ {t \sp {23}}}+ +\right. +\\ +\displaystyle +{{443104378424686086067294899528296664238693556855017735265295} +\ {t \sp {22}}}+ +\\ +\displaystyle +{{279078393286701234679141342358988327155321305829547090310242} +\ {t \sp {21}}}+ +\\ +\displaystyle +{{3390276361413232465107617176615543054620626391823613392185226} +\ {t \sp {20}}}+ +\\ +\displaystyle +{{941478179503540575554198645220352803719793196473813837434129} +\ {t \sp {19}}}+ +\\ +\displaystyle +{{11547855194679475242211696749673949352585747674184320988144390} +\ {t \sp {18}}}+ +\\ +\displaystyle +{{1343609566765597789881701656699413216467215660333356417241432} +\ {t \sp {17}}}+ +\\ +\displaystyle +{{23233813868147873503933551617175640859899102987800663566699334} +\ {t \sp {16}}}+ +\\ +\displaystyle +{{869574020537672336950845440508790740850931336484983573386433} +\ {t \sp {15}}}+ +\\ +\displaystyle +{{31561554305876934875419461486969926554241750065103460820476969} +\ {t \sp {14}}}+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{1271400990287717487442065952547731879554823889855386072264931} +\ {t \sp {13}}}+ +\\ +\displaystyle +{{31945089913863736044802526964079540198337049550503295825160523} +\ {t \sp {12}}}+ +\\ +\displaystyle +{{3738735704288144509871371560232845884439102270778010470931960} +\ {t \sp {11}}}+ +\\ +\displaystyle +{{25293997512391412026144601435771131587561905532992045692885927} +\ {t \sp {10}}}+ +\\ +\displaystyle +{{5210239009846067123469262799870052773410471135950175008046524} +\ {t \sp 9}}+ +\\ +\displaystyle +{{15083887986930297166259870568608270427403187606238713491129188} +\ {t \sp 8}}+ +\\ +\displaystyle +{{3522087234692930126383686270775779553481769125670839075109000} +\ {t \sp 7}}+ +\\ +\displaystyle +{{6079945200395681013086533792568886491101244247440034969288588} +\ {t \sp 6}}+ +\\ +\displaystyle +{{1090634852433900888199913756247986023196987723469934933603680} +\ {t \sp 5}}+ +\\ +\displaystyle +{{1405819430871907102294432537538335402102838994019667487458352} +\ {t \sp 4}}+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{88071527950320450072536671265507748878347828884933605202432} +\ {t \sp 3}}+ +\\ +\displaystyle +{{135882489433640933229781177155977768016065765482378657129440} +\ {t \sp 2}} - +\\ +\displaystyle +{{13957283442882262230559894607400314082516690749975646520320} +\ t}+ +\\ +\displaystyle +\left. +{334637692973189299277258325709308472592117112855749713920} +\right)\ z+ +\\ +\displaystyle +{{8567175484043952879756725964506833932149637101090521164936} +\ {t \sp {23}}}+ +\\ +\displaystyle +{{149792392864201791845708374032728942498797519251667250945721} +\ {t \sp {22}}}+ +\\ +\displaystyle +{{77258371783645822157410861582159764138123003074190374021550} +\ {t \sp {21}}}+ +\\ +\displaystyle +{{1108862254126854214498918940708612211184560556764334742191654} +\ {t \sp {20}}}+ +\\ +\displaystyle +{{213250494460678865219774480106826053783815789621501732672327} +\ {t \sp {19}}}+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{3668929075160666195729177894178343514501987898410131431699882} +\ {t \sp {18}}}+ +\\ +\displaystyle +{{171388906471001872879490124368748236314765459039567820048872} +\ {t \sp {17}}}+ +\\ +\displaystyle +{{7192430746914602166660233477331022483144921771645523139658986} +\ {t \sp {16}}} - +\\ +\displaystyle +{{128798674689690072812879965633090291959663143108437362453385} +\ {t \sp {15}}}+ +\\ +\displaystyle +{{9553010858341425909306423132921134040856028790803526430270671} +\ {t \sp {14}}} - +\\ +\displaystyle +{{13296096245675492874538687646300437824658458709144441096603} +\ {t \sp {13}}}+ +\\ +\displaystyle +{{9475806805814145326383085518325333106881690568644274964864413} +\ {t \sp {12}}}+ +\\ +\displaystyle +{{803234687925133458861659855664084927606298794799856265539336} +\ {t \sp {11}}}+ +\\ +\displaystyle +{{7338202759292865165994622349207516400662174302614595173333825} +\ {t \sp {10}}}+ +\\ +\displaystyle +{{1308004628480367351164369613111971668880538855640917200187108} +\ {t \sp 9}}+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{4268059455741255498880229598973705747098216067697754352634748} +\ {t \sp 8}}+ +\\ +\displaystyle +{{892893526858514095791318775904093300103045601514470613580600} +\ {t \sp 7}}+ +\\ +\displaystyle +{{1679152575460683956631925852181341501981598137465328797013652} +\ {t \sp 6}}+ +\\ +\displaystyle +{{269757415767922980378967154143357835544113158280591408043936} +\ {t \sp 5}}+ +\\ +\displaystyle +{{380951527864657529033580829801282724081345372680202920198224} +\ {t \sp 4}}+ +\\ +\displaystyle +{{19785545294228495032998826937601341132725035339452913286656} +\ {t \sp 3}}+ +\\ +\displaystyle +{{36477412057384782942366635303396637763303928174935079178528} +\ {t \sp 2}} - +\\ +\displaystyle +{{3722212879279038648713080422224976273210890229485838670848} +\ t}+ +\\ +\displaystyle +{89079724853114348361230634484013862024728599906874105856}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{{\left( +{3 \ {z \sp 3}} - +{{11} \ {z \sp 2}}+ +{8 \ z}+4 +\right)}\ y}+ +{2 \ t \ {z \sp 3}}+ +{4 \ t \ {z \sp 2}} - +{5 \ t \ z} -t}, : +\\ +\\ +\displaystyle +\left. +\left. +\left( +z+1 +\right)\ x+ +{z \sp 2} - +{4 \ z}+1 +\right\} +\right] +\end{array} +$$ +\returnType{Type: +List RegularTriangularSet( +Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +Between a sequence of processes, thus between a {\tt ]} and a {\tt [} +you can see capital letters {\tt W, G, I} and lower case letters +{\tt i, w}. Each time a capital letter appears a non-trivial computation +has be performed and its result is put in a hash-table. Each time a +lower case letter appears a needed result has been found in an +hash-table. The use of these hash-tables generally speed up the +computations. However, on very large systems, it may happen that +these hash-tables become too big to be handle by your AXIOM +configuration. Then in these exceptional cases, you may prefer +getting a result (even if it takes a long time) than getting nothing. +Hence you need to know how to prevent the {\tt RSEGSET} constructor +from using these hash-tables. In that case you will be using the +{\tt zeroSetSplit} with five arguments. The first one is the input system +{\bf lp} as above. The second one is a boolean value {\tt hash?} +which is {\tt true} iff you want to use hash-tables. The third one is +boolean value {\tt clos?} which is {\tt true} iff you want to solve +your system in the sense of Kalkbrener, the other way remaining that +of Lazard. The fourth argument is boolean value {\tt info?} which is +{\tt true} iff you want to display information during the +computations. The last one is boolean value {\tt prep?} which is +{\tt true} iff you want to use some heuristics that are performed on the +input system before starting the real algorithm. The value of this +flag is {\tt true} when you are using {\tt zeroSetSplit} with less +than five arguments. Note that there is no available signature for +{\tt zeroSetSplit} with four arguments. + +We finish this section by some remarks about both ways of solving, in +the sense of Kalkbrener or in the sense of Lazard. For problems with +a finite number of solutions, there are theoretically equivalent and +the resulting decompositions are identical, up to the ordering of the +components. However, when solving in the sense of Lazard, the +algorithm behaves differently. In that case, it becomes more +incremental than in the sense of Kalkbrener. That means the +polynomials of the input system are considered one after another +whereas in the sense of Kalkbrener the input system is treated more +globally. + +This makes an important difference in positive dimension. Indeed when +solving in the sense of Kalkbrener, the {\em Primeidealkettensatz} of +Krull is used. That means any regular triangular containing more +polynomials than the input system can be deleted. This is not +possible when solving in the sense of Lazard. This explains why +Kalkbrener's decompositions usually contain less components than those +of Lazard. However, it may happen with some examples that the +incremental process (that cannot be used when solving in the sense of +Kalkbrener) provide a more efficient way of solving than the global +one even if the {\em Primeidealkettensatz} is used. Thus just try +both, with the various options, before concluding that you cannot +solve your favorite system with {\tt zeroSetSplit}. There exist more +options at the development level that are not currently available in +this public version. + +\section{RomanNumeral} +\label{RomanNumeralXmpPage} + +The Roman numeral package was added to Axiom in MCMLXXXVI for use in +denoting higher order derivatives. + +For example, let {\tt f} be a symbolic operator. + +\spadcommand{f := operator 'f } +$$ +f +$$ +\returnType{Type: BasicOperator} + +This is the seventh derivative of {\tt f} with respect to {\tt x}. + +\spadcommand{D(f x,x,7) } +$$ +{f \sb {{\ }} \sp {{\left( vii +\right)}}} +\left( +{x} +\right) +$$ +\returnType{Type: Expression Integer} + +You can have integers printed as Roman numerals by declaring variables to +be of type {\tt RomanNumeral} (abbreviation {\tt ROMAN}). + +\spadcommand{a := roman(1978 - 1965) } +$$ +XIII +$$ +\returnType{Type: RomanNumeral} + +This package now has a small but devoted group of followers that claim +this domain has shown its efficacy in many other contexts. They claim +that Roman numerals are every bit as useful as ordinary integers. + +In a sense, they are correct, because Roman numerals form a ring and you +can therefore construct polynomials with Roman numeral coefficients, +matrices over Roman numerals, etc.. + +\spadcommand{x : UTS(ROMAN,'x,0) := x } +$$ +x +$$ +\returnType{Type: UnivariateTaylorSeries(RomanNumeral,x,0)} + +Was Fibonacci Italian or ROMAN? + +\spadcommand{recip(1 - x - x**2) } +$$ +\begin{array}{@{}l} +I+x+ +{II \ {x \sp 2}}+ +{III \ {x \sp 3}}+ +{V \ {x \sp 4}}+ +{VIII \ {x \sp 5}}+ +{XIII \ {x \sp 6}}+ +{XXI \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{XXXIV \ {x \sp 8}}+ +{LV \ {x \sp 9}}+ +{LXXXIX \ {x \sp {10}}}+ +{O \left({{x \sp {11}}} \right)} +\end{array} +$$ +\returnType{Type: Union(UnivariateTaylorSeries(RomanNumeral,x,0),...)} + +You can also construct fractions with Roman numeral numerators and +denominators, as this matrix Hilberticus illustrates. + +\spadcommand{m : MATRIX FRAC ROMAN } +\returnType{Void} + +\spadcommand{m := matrix [ [1/(i + j) for i in 1..3] for j in 1..3] } +$$ +\left[ +\begin{array}{ccc} +{I \over II} & {I \over III} & {I \over IV} \\ +{I \over III} & {I \over IV} & {I \over V} \\ +{I \over IV} & {I \over V} & {I \over VI} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction RomanNumeral} + +Note that the inverse of the matrix has integral {\tt ROMAN} entries. + +\spadcommand{inverse m } +$$ +\left[ +\begin{array}{ccc} +LXXII & -CCXL & CLXXX \\ +-CCXL & CM & -DCCXX \\ +CLXXX & -DCCXX & DC +\end{array} +\right] +$$ +\returnType{Type: Union(Matrix Fraction RomanNumeral,...)} + +Unfortunately, the spoil-sports say that the fun stops when the +numbers get big---mostly because the Romans didn't establish +conventions about representing very large numbers. + +\spadcommand{y := factorial 10 } +$$ +3628800 +$$ +\returnType{Type: PositiveInteger} + +You work it out! + +\spadcommand{roman y } +$$ +\begin{array}{@{}l} +{\rm ((((I))))((((I))))((((I))))(((I)))(((I)))(((I)))(((I)))} +\\ +\displaystyle +{\rm (((I)))(((I))) ((I))((I)) MMMMMMMMDCCC} +\end{array} +$$ +\returnType{Type: RomanNumeral} + +Issue the system command {\tt )show RomanNumeral} to display the full +list of operations defined by {\tt RomanNumeral}. + +\section{Segment} +\label{SegmentXmpPage} + +The {\tt Segment} domain provides a generalized interval type. + +Segments are created using the {\tt ..} construct by indicating the +(included) end points. + +\spadcommand{s := 3..10 } +$$ +3..{10} +$$ +\returnType{Type: Segment PositiveInteger} + +The first end point is called the \spadfunFrom{lo}{Segment} and the +second is called \spadfunFrom{hi}{Segment}. + +\spadcommand{lo s } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +These names are used even though the end points might belong to an +unordered set. + +\spadcommand{hi s } +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +In addition to the end points, each segment has an integer ``increment.'' +An increment can be specified using the ``{\tt by}'' construct. + +\spadcommand{t := 10..3 by -2 } +$$ +{{10}..3} \mbox{\rm by\ } -2 +$$ +\returnType{Type: Segment PositiveInteger} + +This part can be obtained using the \spadfunFrom{incr}{Segment} function. + +\spadcommand{incr s } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Unless otherwise specified, the increment is {\tt 1}. + +\spadcommand{incr t } +$$ +-2 +$$ +\returnType{Type: Integer} + +A single value can be converted to a segment with equal end points. +This happens if segments and single values are mixed in a list. + +\spadcommand{l := [1..3, 5, 9, 15..11 by -1] } +$$ +\left[ +{1..3}, {5..5}, {9..9}, {{{15}..{11}} \mbox{\rm by } -1} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +If the underlying type is an ordered ring, it is possible to perform +additional operations. The \spadfunFrom{expand}{Segment} operation +creates a list of points in a segment. + +\spadcommand{expand s } +$$ +\left[ +3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: List Integer} + +If {\tt k > 0}, then {\tt expand(l..h by k)} creates the list +{\tt [l, l+k, ..., lN]} where {\tt lN <= h < lN+k}. +If {\tt k < 0}, then {\tt lN >= h > lN+k}. + +\spadcommand{expand t } +$$ +\left[ +{10}, 8, 6, 4 +\right] +$$ +\returnType{Type: List Integer} + +It is also possible to expand a list of segments. This is equivalent +to appending lists obtained by expanding each segment individually. + +\spadcommand{expand l } +$$ +\left[ +1, 2, 3, 5, 9, {15}, {14}, {13}, {12}, {11} +\right] +$$ +\returnType{Type: List Integer} + +For more information on related topics, see +\ref{SegmentBindingXmpPage} on page~\pageref{SegmentBindingXmpPage} and +\ref{UniversalSegmentXmpPage} on page~\pageref{UniversalSegmentXmpPage}. + +\section{SegmentBinding} +\label{SegmentBindingXmpPage} + +The {\tt SegmentBinding} type is used to indicate a range for a named +symbol. + +First give the symbol, then an {\tt =} and finally a segment of values. + +\spadcommand{x = a..b} +$$ +x={a..b} +$$ +\returnType{Type: SegmentBinding Symbol} + +This is used to provide a convenient syntax for arguments to certain +operations. + +\spadcommand{sum(i**2, i = 0..n)} +$$ +{{2 \ {n \sp 3}}+{3 \ {n \sp 2}}+n} \over 6 +$$ +\returnType{Type: Fraction Polynomial Integer} + +\spadcommand{draw(x**2, x = -2..2)} + +The left-hand side must be of type {\tt Symbol} but the +right-hand side can be a segment over any type. + +\spadcommand{sb := y = 1/2..3/2 } +$$ +y={{\left( 1 \over 2 +\right)}..{\left( +3 \over 2 +\right)}} +$$ +\returnType{Type: SegmentBinding Fraction Integer} + +The left- and right-hand sides can be obtained using the +\spadfunFrom{variable}{SegmentBinding} and +\spadfunFrom{segment}{SegmentBinding} operations. + +\spadcommand{variable(sb) } +$$ +y +$$ +\returnType{Type: Symbol} + +\spadcommand{segment(sb) } +$$ +{\left( 1 \over 2 +\right)}..{\left( +3 \over 2 +\right)} +$$ +\returnType{Type: Segment Fraction Integer} + +For more information on related topics, see +\ref{SegmentXmpPage} on page~\pageref{SegmentXmpPage} and +\ref{UniversalSegmentXmpPage} on page~\pageref{UniversalSegmentXmpPage}. + +\section{Set} +\label{SetXmpPage} + +The {\tt Set} domain allows one to represent explicit finite sets of values. +These are similar to lists, but duplicate elements are not allowed. + +Sets can be created by giving a fixed set of values \ldots + +\spadcommand{s := set [x**2-1, y**2-1, z**2-1] } +$$ +\left\{ +{{x \sp 2} -1}, {{y \sp 2} -1}, {{z \sp 2} -1} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +or by using a collect form, just as for lists. In either case, the +set is formed from a finite collection of values. + +\spadcommand{t := set [x**i - i+1 for i in 2..10 | prime? i] } +$$ +\left\{ +{{x \sp 2} -1}, {{x \sp 3} -2}, {{x \sp 5} -4}, {{x \sp 7} -6} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +The basic operations on sets are \spadfunFrom{intersect}{Set}, +\spadfunFrom{union}{Set}, \spadfunFrom{difference}{Set}, and +\spadfunFrom{symmetricDifference}{Set}. + +\spadcommand{i := intersect(s,t)} +$$ +\left\{ +{{x \sp 2} -1} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +\spadcommand{u := union(s,t)} +$$ +\left\{ +{{x \sp 2} -1}, {{x \sp 3} -2}, {{x \sp 5} -4}, {{x \sp 7} -6}, +{{y \sp 2} -1}, {{z \sp 2} -1} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +The set {\tt difference(s,t)} contains those members of {\tt s} which +are not in {\tt t}. + +\spadcommand{difference(s,t)} +$$ +\left\{ +{{y \sp 2} -1}, {{z \sp 2} -1} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +The set {\tt symmetricDifference(s,t)} contains those elements which are +in {\tt s} or {\tt t} but not in both. + +\spadcommand{symmetricDifference(s,t)} +$$ +\left\{ +{{x \sp 3} -2}, {{x \sp 5} -4}, {{x \sp 7} -6}, {{y \sp 2} -1}, +{{z \sp 2} -1} +\right\} +$$ +\returnType{Type: Set Polynomial Integer} + +Set membership is tested using the \spadfunFrom{member?}{Set} operation. + +\spadcommand{member?(y, s)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{member?((y+1)*(y-1), s)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The \spadfunFrom{subset?}{Set} function determines whether one set is +a subset of another. + +\spadcommand{subset?(i, s)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{subset?(u, s)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +When the base type is finite, the absolute complement of a set is +defined. This finds the set of all multiplicative generators of +{\tt PrimeField 11}---the integers mod {\tt 11.} + +\spadcommand{gs := set [g for i in 1..11 | primitive?(g := i::PF 11)] } +$$ +\left\{ +2, 6, 7, 8 +\right\} +$$ +\returnType{Type: Set PrimeField 11} + +The following values are not generators. + +\spadcommand{complement gs } +$$ +\left\{ +1, 3, 4, 5, 9, {10}, 0 +\right\} +$$ +\returnType{Type: Set PrimeField 11} + +Often the members of a set are computed individually; in addition, +values can be inserted or removed from a set over the course of a +computation. + +There are two ways to do this: + +\spadcommand{a := set [i**2 for i in 1..5] } +$$ +\left\{ +1, 4, 9, {16}, {25} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +One is to view a set as a data structure and to apply updating operations. + +\spadcommand{insert!(32, a) } +$$ +\left\{ +1, 4, 9, {16}, {25}, {32} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +\spadcommand{remove!(25, a) } +$$ +\left\{ +1, 4, 9, {16}, {32} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +\spadcommand{a } +$$ +\left\{ +1, 4, 9, {16}, {32} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +The other way is to view a set as a mathematical entity and to +create new sets from old. + +\spadcommand{b := b0 := set [i**2 for i in 1..5] } +$$ +\left\{ +1, 4, 9, {16}, {25} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +\spadcommand{b := union(b, {32})} +$$ +\left\{ +1, 4, 9, {16}, {25}, {32} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +\spadcommand{b := difference(b, {25})} +$$ +\left\{ +1, 4, 9, {16}, {32} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +\spadcommand{b0 } +$$ +\left\{ +1, 4, 9, {16}, {25} +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +For more information about lists, see \ref{ListXmpPage} on +page~\pageref{ListXmpPage}. + +\section{SingleInteger} +\label{SingleIntegerXmpPage} + +The {\tt SingleInteger} domain is intended to provide support in Axiom +for machine integer arithmetic. It is generally much faster than +(bignum) {\tt Integer} arithmetic but suffers from a limited range of +values. Since Axiom can be implemented on top of various dialects of +Lisp, the actual representation of small integers may not correspond +exactly to the host machines integer representation. + +In the CCL implementation of AXIOM (Release 2.1 onwards) the underlying +representation of {\tt SingleInteger} is the same as {\tt Integer}. +The underlying Lisp primitives treat machine-word sized computations +specially. + +You can discover the minimum and maximum values in your implementation +by using \spadfunFrom{min}{SingleInteger} and \spadfunFrom{max}{SingleInteger}. + +\spadcommand{min()\$SingleInteger} +$$ +-{134217728} +$$ +\returnType{Type: SingleInteger} + +\spadcommand{max()\$SingleInteger} +$$ +134217727 +$$ +\returnType{Type: SingleInteger} + +To avoid confusion with {\tt Integer}, which is the default type for +integers, you usually need to work with declared variables +(\ref{ugTypesDeclarePage} on page~\pageref{ugTypesDeclarePage} in Section +\ref{ugTypesDeclareNumber} on page~\pageref{ugTypesDeclareNumber}) +\ldots + +\spadcommand{a := 1234 :: SingleInteger } +$$ +1234 +$$ +\returnType{Type: SingleInteger} + +or use package calling +(\ref{ugTypesPkgCallPage} on page~\pageref{ugTypesPkgCallPage} in Section +\ref{ugTypesPkgCallNumber} on page~\pageref{ugTypesPkgCallNumber}). + +\spadcommand{b := 124\$SingleInteger } +$$ +124 +$$ +\returnType{Type: SingleInteger} + +You can add, multiply and subtract {\tt SingleInteger} objects, +and ask for the greatest common divisor ({\tt gcd}). + +\spadcommand{gcd(a,b) } +$$ +2 +$$ +\returnType{Type: SingleInteger} + +The least common multiple ({\tt lcm}) is also available. + +\spadcommand{lcm(a,b) } +$$ +76508 +$$ +\returnType{Type: SingleInteger} + +Operations \spadfunFrom{mulmod}{SingleInteger}, +\spadfunFrom{addmod}{SingleInteger}, +\spadfunFrom{submod}{SingleInteger}, and +\spadfunFrom{invmod}{SingleInteger} are similar---they provide +arithmetic modulo a given small integer. +Here is $5 * 6 {\tt mod} 13$. + +\spadcommand{mulmod(5,6,13)\$SingleInteger} +$$ +4 +$$ +\returnType{Type: SingleInteger} + +To reduce a small integer modulo a prime, use +\spadfunFrom{positiveRemainder}{SingleInteger}. + +\spadcommand{positiveRemainder(37,13)\$SingleInteger} +$$ +11 +$$ +\returnType{Type: SingleInteger} + +Operations +\spadfunFrom{And}{SingleInteger}, +\spadfunFrom{Or}{SingleInteger}, +\spadfunFrom{xor}{SingleInteger}, +and \spadfunFrom{Not}{SingleInteger} +provide bit level operations on small integers. + +\spadcommand{And(3,4)\$SingleInteger} +$$ +0 +$$ +\returnType{Type: SingleInteger} + +Use {\tt shift(int,numToShift)} to shift bits, where {\tt i} is +shifted left if {\tt numToShift} is positive, right if negative. + +\spadcommand{shift(1,4)\$SingleInteger} +$$ +16 +$$ +\returnType{Type: SingleInteger} + +\spadcommand{shift(31,-1)\$SingleInteger} +$$ +15 +$$ +\returnType{Type: SingleInteger} + +Many other operations are available for small integers, including many +of those provided for {\tt Integer}. To see the other operations, use +the Browse HyperDoc facility (\ref{ugBrowsePage} on +page~\pageref{ugBrowsePage} in Section +\ref{ugBrowseNumber} on page~\pageref{ugBrowseNumber}). + +\section{SparseTable} +\label{SparseTableXmpPage} + +The {\tt SparseTable} domain provides a general purpose table type +with default entries. + +Here we create a table to save strings under integer keys. The value +{\tt "Try again!"} is returned if no other value has been stored for a +key. + +\spadcommand{t: SparseTable(Integer, String, "Try again!") := table() } +$$ +table() +$$ +\returnType{Type: SparseTable(Integer,String,Try again!)} + +Entries can be stored in the table. + +\spadcommand{t.3 := "Number three" } +$$ +\mbox{\tt "Number three"} +$$ +\returnType{Type: String} + +\spadcommand{t.4 := "Number four" } +$$ +\mbox{\tt "Number four"} +$$ +\returnType{Type: String} + +These values can be retrieved as usual, but if a look up fails the +default entry will be returned. + +\spadcommand{t.3 } +$$ +\mbox{\tt "Number three"} +$$ +\returnType{Type: String} + +\spadcommand{t.2 } +$$ +\mbox{\tt "Try again!"} +$$ +\returnType{Type: String} + +To see which values are explicitly stored, the +\spadfunFrom{keys}{SparseTable} and \spadfunFrom{entries}{SparseTable} +functions can be used. + +\spadcommand{keys t } +$$ +\left[ +4, 3 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{entries t } +$$ +\left[ +\mbox{\tt "Number four"} , \mbox{\tt "Number three"} +\right] +$$ +\returnType{Type: List String} + +If a specific table representation is required, the +{\tt GeneralSparseTable} constructor should be used. The domain +{\tt SparseTable(K, E, dflt)} is equivalent to +{\tt GeneralSparseTable(K,E,Table(K,E), dflt)}. +For more information, see +\ref{TableXmpPage} on page~\pageref{TableXmpPage} and +\ref{GeneralSparseTableXmpPage} on page~\pageref{GeneralSparseTableXmpPage}. + +\section{SquareMatrix} +\label{SquareMatrixXmpPage} + +The top level matrix type in Axiom is {\tt Matrix} (see +\ref{MatrixXmpPage} on page~\pageref{MatrixXmpPage}), which provides +basic arithmetic and linear algebra functions. However, since the +matrices can be of any size it is not true that any pair can be added +or multiplied. Thus {\tt Matrix} has little algebraic structure. + +Sometimes you want to use matrices as coefficients for polynomials +or in other algebraic contexts. In this case, {\tt SquareMatrix} +should be used. The domain {\tt SquareMatrix(n,R)} gives the ring of +{\tt n} by {\tt n} square matrices over {\tt R}. + +Since {\tt SquareMatrix} is not normally exposed at the top level, +you must expose it before it can be used. + +\spadcommand{)set expose add constructor SquareMatrix } +\begin{verbatim} + SquareMatrix is now explicitly exposed in frame G82322 +\end{verbatim} + +Once {\tt SQMATRIX} has been exposed, values can be created using the +\spadfunFrom{squareMatrix}{SquareMatrix} function. + +\spadcommand{m := squareMatrix [ [1,-\%i],[\%i,4] ] } +$$ +\left[ +\begin{array}{cc} +1 & -i \\ +i & 4 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Complex Integer)} + +The usual arithmetic operations are available. + +\spadcommand{m*m - m } +$$ +\left[ +\begin{array}{cc} +1 & -{4 \ i} \\ +{4 \ i} & {13} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Complex Integer)} + +Square matrices can be used where ring elements are required. +For example, here is a matrix with matrix entries. + +\spadcommand{mm := squareMatrix [ [m, 1], [1-m, m**2] ] } +$$ +\left[ +\begin{array}{cc} +{\left[ +\begin{array}{cc} +1 & -i \\ +i & 4 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +1 & 0 \\ +0 & 1 +\end{array} +\right]} +\\ +{\left[ +\begin{array}{cc} +0 & i \\ +-i & -3 +\end{array} +\right]} +& {\left[ +\begin{array}{cc} +2 & -{5 \ i} \\ +{5 \ i} & {17} +\end{array} +\right]} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,SquareMatrix(2,Complex Integer))} + +Or you can construct a polynomial with square matrix coefficients. + +\spadcommand{p := (x + m)**2 } +$$ +{x \sp 2}+{{\left[ +\begin{array}{cc} +2 & -{2 \ i} \\ +{2 \ i} & 8 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +2 & -{5 \ i} \\ +{5 \ i} & {17} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Complex Integer)} + +This value can be converted to a square matrix with polynomial coefficients. + +\spadcommand{p::SquareMatrix(2, ?) } +$$ +\left[ +\begin{array}{cc} +{{x \sp 2}+{2 \ x}+2} & {-{2 \ i \ x} -{5 \ i}} \\ +{{2 \ i \ x}+{5 \ i}} & {{x \sp 2}+{8 \ x}+{17}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Complex Integer)} + +For more information on related topics, see +\ref{ugTypesWritingModesPage} on +page~\pageref{ugTypesWritingModesPage} in Section +\ref{ugTypesWritingModesNumber} on +page~\pageref{ugTypesWritingModesNumber}, \ref{ugTypesExposePage} on +page~\pageref{ugTypesExposePage} in Section \ref{ugTypesExposeNumber} +on page~\pageref{ugTypesExposeNumber}, and \ref{MatrixXmpPage} on +page~\pageref{MatrixXmpPage}. + +\section{SquareFreeRegularTriangularSet} +\label{SquareFreeRegularTriangularSetXmpPage} + +The {\tt SquareFreeRegularTriangularSet} domain constructor implements +square-free regular triangular sets. See the +{\tt RegularTriangularSet} domain constructor for general regular +triangular sets. Let {\em T} be a regular triangular set consisting +of polynomials {\em t1, ..., tm} ordered by increasing main variables. +The regular triangular set {\em T} is square-free if {\em T} is empty +or if {\em t1, ..., tm-1} is square-free and if the polynomial +{\em tm} is square-free as a univariate polynomial with coefficients in the +tower of simple extensions associated with {\em t1, ..., tm-1}. + +The main interest of square-free regular triangular sets is that their +associated towers of simple extensions are product of fields. +Consequently, the saturated ideal of a square-free regular triangular +set is radical. This property simplifies some of the operations +related to regular triangular sets. However, building square-free +regular triangular sets is generally more expensive than building +general regular triangular sets. + +As the {\tt RegularTriangularSet} domain constructor, the +{\tt SquareFreeRegularTriangularSet} domain constructor also implements a +method for solving polynomial systems by means of regular triangular +sets. This is in fact the same method with some adaptations to take +into account the fact that the computed regular chains are +square-free. Note that it is also possible to pass from a +decomposition into general regular triangular sets to a decomposition +into square-free regular triangular sets. This conversion is used +internally by the {\tt LazardSetSolvingPackage} package constructor. + +{\bf N.B.} When solving polynomial systems with the +{\tt SquareFreeRegularTriangularSet} domain constructor or the +{\tt LazardSetSolvingPackage} package constructor, decompositions have no +redundant components. See also {\tt LexTriangularPackage} and +{\tt ZeroDimensionalSolvePackage} for the case of algebraic systems with a +finite number of (complex) solutions. + +We shall explain now how to use the constructor +{\tt SquareFreeRegularTriangularSet}. + +This constructor takes four arguments. +The first one, {\bf R}, is the coefficient ring of the polynomials; +it must belong to the category {\tt GcdDomain}. +The second one, {\bf E}, is the exponent monoid of the polynomials; +it must belong to the category {\tt OrderedAbelianMonoidSup}. +the third one, {\bf V}, is the ordered set of variables; +it must belong to the category {\tt OrderedSet}. +The last one is the polynomial ring; +it must belong to the category {\tt RecursivePolynomialCategory(R,E,V)}. +The abbreviation for {\tt SquareFreeRegularTriangularSet} is +{\tt SREGSET}. + +Note that the way of understanding triangular decompositions +is detailed in the example of the {\tt RegularTriangularSet} +constructor. + +Let us illustrate the use of this constructor with one example +(Donati-Traverso). Define the coefficient ring. + +\spadcommand{R := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the list of variables, + +\spadcommand{ls : List Symbol := [x,y,z,t] } +$$ +\left[ +x, y, z, t +\right] +$$ +\returnType{Type: List Symbol} + +and make it an ordered set; + +\spadcommand{V := OVAR(ls) } +$$ +\mbox{\rm OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +then define the exponent monoid. + +\spadcommand{E := IndexedExponents V } +$$ +\mbox{\rm IndexedExponents OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +Define the polynomial ring. + +\spadcommand{P := NSMP(R, V)} +$$ +\mbox{\rm NewSparseMultivariatePolynomial(Integer,OrderedVariableList +[x,y,z,t])} +$$ +\returnType{Type: Domain} + +Let the variables be polynomial. + +\spadcommand{x: P := 'x } +$$ +x +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{y: P := 'y } +$$ +y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{z: P := 'z } +$$ +z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{t: P := 't } +$$ +t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +Now call the {\tt SquareFreeRegularTriangularSet} domain constructor. + +\spadcommand{ST := SREGSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm SquareFreeRegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t]))} +\end{array} +$$ +\returnType{Type: Domain} + +Define a polynomial system. + +\spadcommand{p1 := x ** 31 - x ** 6 - x - y} +$$ +{x \sp {31}} -{x \sp 6} -x -y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p2 := x ** 8 - z} +$$ +{x \sp 8} -z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p3 := x ** 10 - t} +$$ +{x \sp {10}} -t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{lp := [p1, p2, p3]} +$$ +\left[ +{{x \sp {31}} -{x \sp 6} -x -y}, {{x \sp 8} -z}, {{x \sp {10}} -t} +\right] +$$ +\returnType{Type: +List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +First of all, let us solve this system in the sense of Kalkbrener. + +\spadcommand{zeroSetSplit(lp)\$ST} +$$ +\left[ +{\left\{ {{z \sp 5} -{t \sp 4}}, {{t \ z \ {y \sp 2}}+{2 \ {z \sp 3} \ +y} -{t \sp 8}+{2 \ {t \sp 5}}+{t \sp 3} -{t \sp 2}}, {{{\left( {t \sp 4} +-t +\right)} +\ x} -{t \ y} -{z \sp 2}} +\right\}} +\right] +$$ +\returnType{Type: +List SquareFreeRegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +And now in the sense of Lazard (or Wu and other authors). + +\spadcommand{zeroSetSplit(lp,false)\$ST} +$$ +\begin{array}{@{}l} +\left[ +{\left\{ +{{z \sp 5} -{t \sp 4}}, +{{t \ z \ {y \sp 2}}+ +{2 \ {z \sp 3} \ y} - +{t \sp 8}+ +{2 \ {t \sp 5}}+ +{t \sp 3} - +{t \sp 2}}, +{{{\left( {t \sp 4} -t \right)}\ x} -{t \ y} -{z \sp 2}} \right\}}, +\right. +\\ +\\ +\displaystyle +\left. +{\left\{ {{t \sp 3} -1}, {{z \sp 5} -t}, {{t \ y}+{z \sp 2}}, +{{z \ {x \sp 2}} -t} \right\}}, +{\left\{ t, z, y, x \right\}} +\right] +\end{array} +$$ +\returnType{Type: +List SquareFreeRegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +Now to see the difference with the {\tt RegularTriangularSet} domain +constructor, we define: + +\spadcommand{T := REGSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm RegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm OrderedVariableList [x,y,z,t]))} +\end{array} +$$ +\returnType{Type: Domain} + +and compute: + +\spadcommand{lts := zeroSetSplit(lp,false)\$T} +$$ +\begin{array}{@{}l} +\left[ +{\left\{ {{z \sp 5} -{t \sp 4}}, +{{t \ z \ {y \sp 2}}+ +{2 \ {z \sp 3} \ y} - +{t \sp 8}+ +{2 \ {t \sp 5}}+ +{t \sp 3} - +{t \sp 2}}, +{{{\left( {t \sp 4} -t \right)}\ x} -{t \ y} -{z \sp 2}} \right\}}, +\right. +\\ +\\ +\displaystyle +\left. +{\left\{ {{t \sp 3} -1}, {{z \sp 5} -t}, +{{t \ z \ {y \sp 2}}+{2 \ {z \sp 3} \ y}+1}, +{{z \ {x \sp 2}} -t} \right\}}, +{\left\{ t, z, y, x \right\}} +\right] +\end{array} +$$ +\returnType{Type: +List RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +If you look at the second set in both decompositions in the sense of Lazard, +you will see that the polynomial with main variable {\bf y} is not the same. + +Let us understand what has happened. + +We define: + +\spadcommand{ts := lts.2} +$$ +\left\{ +{{t \sp 3} -1}, {{z \sp 5} -t}, {{t \ z \ {y \sp 2}}+{2 \ {z \sp 3} +\ y}+1}, {{z \ {x \sp 2}} -t} +\right\} +$$ +\returnType{Type: +RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +\spadcommand{pol := select(ts,'y)\$T} +$$ +{t \ z \ {y \sp 2}}+{2 \ {z \sp 3} \ y}+1 +$$ +\returnType{Type: +Union( +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]),...)} + +\spadcommand{tower := collectUnder(ts,'y)\$T} +$$ +\left\{ +{{t \sp 3} -1}, {{z \sp 5} -t} +\right\} +$$ +\returnType{Type: +RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +\spadcommand{pack := RegularTriangularSetGcdPackage(R,E,V,P,T)} +$$ +\begin{array}{@{}l} +{\rm RegularTriangularSetGcdPackage(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t]),} +\\ +\displaystyle +{\rm \ \ RegularTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t])))} +\end{array} +$$ +\returnType{Type: Domain} + +Then we compute: + +\spadcommand{toseSquareFreePart(pol,tower)\$pack} +$$ +\left[ +{\left[ +{val={{t \ y}+{z \sp 2}}}, +{tower={\left\{ {{t \sp 3} -1}, {{z \sp 5} -t} \right\}}} +\right]} +\right] +$$ +\returnType{Type: +List Record(val: +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]), +tower: RegularTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t])))} + +\section{Stream} +\label{StreamXmpPage} + +A {\tt Stream} object is represented as a list whose last element +contains the wherewithal to create the next element, should it ever be +required. + +Let {\tt ints} be the infinite stream of non-negative integers. + +\spadcommand{ints := [i for i in 0..] } +$$ +\left[ +0, 1, 2, 3, 4, 5, 6, 7, 8, 9, \ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +By default, ten stream elements are calculated. This number may be +changed to something else by the system command {\tt )set streams +calculate}. For the display purposes of this book, we have chosen a +smaller value. + +More generally, you can construct a stream by specifying its initial +value and a function which, when given an element, creates the next element. + +\spadcommand{f : List INT -> List INT } +\returnType{Void} + +\spadcommand{f x == [x.1 + x.2, x.1] } +\returnType{Void} + +\spadcommand{fibs := [i.2 for i in [generate(f,[1,1])]] } +\begin{verbatim} + Compiling function f with type List Integer -> List Integer +\end{verbatim} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +You can create the stream of odd non-negative integers by either filtering +them from the integers, or by evaluating an expression for each integer. + +\spadcommand{[i for i in ints | odd? i] } +$$ +\left[ +1, 3, 5, 7, 9, {11}, {13}, {15}, {17}, {19}, +\ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +\spadcommand{odds := [2*i+1 for i in ints]} +$$ +\left[ +1, 3, 5, 7, 9, {11}, {13}, {15}, {17}, {19}, +\ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +You can accumulate the initial segments of a stream using the +\spadfunFrom{scan}{StreamFunctions2} operation. + +\spadcommand{scan(0,+,odds) } +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, +{100}, \ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +The corresponding elements of two or more streams can be combined in +this way. + +\spadcommand{[i*j for i in ints for j in odds] } +$$ +\left[ +0, 3, {10}, {21}, {36}, {55}, {78}, {105}, {136}, +{171}, \ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +\spadcommand{map(*,ints,odds)} +$$ +\left[ +0, 3, {10}, {21}, {36}, {55}, {78}, {105}, {136}, +{171}, \ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +Many operations similar to those applicable to lists are available for +streams. + +\spadcommand{first ints } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\spadcommand{rest ints } +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream NonNegativeInteger} + +\spadcommand{fibs 20 } +$$ +6765 +$$ +\returnType{Type: PositiveInteger} + +The packages {\tt StreamFunctions1}, {\tt StreamFunctions2} and +{\tt StreamFunctions3} export some useful stream manipulation operations. +For more information, see \ref{ugLangItsPage} on +page~\pageref{ugLangItsPage} in Section \ref{ugLangItsNumber} on +page~\pageref{ugLangItsNumber}, \ref{ugProblemSeriesPage} on +page~\pageref{ugProblemSeriesPage} in Section +\ref{ugProblemSeriesNumber} on page~\pageref{ugProblemSeriesNumber}, +\ref{ContinuedFractionXmpPage} on +page~\pageref{ContinuedFractionXmpPage}, and \ref{ListXmpPage} on +page~\pageref{ListXmpPage}. + +\section{String} +\label{StringXmpPage} + +The type {\tt String} provides character strings. Character strings +provide all the operations for a one-dimensional array of characters, +plus additional operations for manipulating text. For more +information on related topics, see +\ref{CharacterXmpPage} on page~\pageref{CharacterXmpPage} and +\ref{CharacterClassXmpPage} on page~\pageref{CharacterClassXmpPage}. +You can also issue the system command +{\tt )show String} to display the full list of operations defined +by {\tt String}. + +String values can be created using double quotes. + +\spadcommand{hello := "Hello, I'm AXIOM!" } +$$ +\mbox{\tt "Hello, I'm AXIOM!"} +$$ +\returnType{Type: String} + +Note, however, that double quotes and underscores must be preceded by +an extra underscore. + +\spadcommand{said := "Jane said, \_"Look!\_"" } +$$ +\mbox{\tt "Jane said, "Look!""} +$$ +\returnType{Type: String} + +\spadcommand{saw := "She saw exactly one underscore: \_\_." } +$$ +\mbox{\tt "She saw exactly one underscore: \_."} +$$ +\returnType{Type: String} + +It is also possible to use \spadfunFrom{new}{String} to create a +string of any size filled with a given character. Since there are +many {\tt new} functions it is necessary to indicate the desired type. + +\spadcommand{gasp: String := new(32, char "x") } +$$ +\mbox{\tt "xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx"} +$$ +\returnType{Type: String} + +The length of a string is given by \spadopFrom{\#}{List}. + +\spadcommand{\#gasp } +$$ +32 +$$ +\returnType{Type: PositiveInteger} + +Indexing operations allow characters to be extracted or replaced in strings. +For any string {\tt s}, indices lie in the range {\tt 1..\#s}. + +\spadcommand{hello.2 } +$$ +e +$$ +\returnType{Type: Character} + +Indexing is really just the application of a string to a subscript, +so any application syntax works. + +\spadcommand{hello 2 } +$$ +e +$$ +\returnType{Type: Character} + +\spadcommand{hello(2) } +$$ +e +$$ +\returnType{Type: Character} + +If it is important not to modify a given string, it should be copied +before any updating operations are used. + +\spadcommand{hullo := copy hello } +$$ +\mbox{\tt "Hello, I'm AXIOM!"} +$$ +\returnType{Type: String} + +\spadcommand{hullo.2 := char "u"; [hello, hullo] } +$$ +\left[ +\mbox{\tt "Hello, I'm AXIOM!"} , \mbox{\tt "Hullo, I'm AXIOM!"} +\right] +$$ +\returnType{Type: List String} + +Operations are provided to split and join strings. The +\spadfunFrom{concat}{String} operation allows several strings to be +joined together. + +\spadcommand{saidsaw := concat ["alpha","---","omega"] } +$$ +\mbox{\tt "alpha---omega"} +$$ +\returnType{Type: String} + +There is a version of \spadfunFrom{concat}{String} that works with +two strings. + +\spadcommand{concat("hello ","goodbye")} +$$ +\mbox{\tt "hello goodbye"} +$$ +\returnType{Type: String} + +Juxtaposition can also be used to concatenate strings. + +\spadcommand{"This " "is " "several " "strings " "concatenated."} +$$ +\mbox{\tt "This is several strings concatenated."} +$$ +\returnType{Type: String} + +Substrings are obtained by giving an index range. + +\spadcommand{hello(1..5) } +$$ +\mbox{\tt "Hello"} +$$ +\returnType{Type: String} + +\spadcommand{hello(8..) } +$$ +\mbox{\tt "I'm AXIOM!"} +$$ +\returnType{Type: String} + +A string can be split into several substrings by giving a separation +character or character class. + +\spadcommand{split(hello, char " ")} +$$ +\left[ +\mbox{\tt "Hello,"} , \mbox{\tt "I'm"} , \mbox{\tt "AXIOM!"} +\right] +$$ +\returnType{Type: List String} + +\spadcommand{other := complement alphanumeric(); } +\returnType{Type: CharacterClass} + +\spadcommand{split(saidsaw, other)} +$$ +\left[ +\mbox{\tt "alpha"} , \mbox{\tt "omega"} +\right] +$$ +\returnType{Type: List String} + +Unwanted characters can be trimmed from the beginning or end of a string +using the operations \spadfunFrom{trim}{String}, \spadfunFrom{leftTrim}{String} +and \spadfunFrom{rightTrim}{String}. + +\spadcommand{trim("\#\# ++ relax ++ \#\#", char "\#")} +$$ +\mbox{\tt " ++ relax ++ "} +$$ +\returnType{Type: String} + +Each of these functions takes a string and a second argument to specify +the characters to be discarded. + +\spadcommand{trim("\#\# ++ relax ++ \#\#", other) } +$$ +\mbox{\tt "relax"} +$$ +\returnType{Type: String} + +The second argument can be given either as a single character or as a +character class. + +\spadcommand{leftTrim ("\#\# ++ relax ++ \#\#", other) } +$$ +\mbox{\tt "relax ++ \#\#"} +$$ +\returnType{Type: String} + +\spadcommand{rightTrim("\#\# ++ relax ++ \#\#", other) } +$$ +\mbox{\tt "\#\# ++ relax"} +$$ +\returnType{Type: String} + +Strings can be changed to upper case or lower case using the +operations \spadfunFrom{upperCase}{String}, and +\spadfunFrom{lowerCase}{String}. + +\spadcommand{upperCase hello } +$$ +\mbox{\tt "HELLO, I'M AXIOM!"} +$$ +\returnType{Type: String} + +The versions with the exclamation mark change the original string, +while the others produce a copy. + +\spadcommand{lowerCase hello } +$$ +\mbox{\tt "hello, i'm axiom!"} +$$ +\returnType{Type: String} + +Some basic string matching is provided. The function +\spadfunFrom{prefix?}{String} tests whether one string is an initial +prefix of another. + +\spadcommand{prefix?("He", "Hello")} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{prefix?("Her", "Hello")} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +A similar function, \spadfunFrom{suffix?}{String}, tests for suffixes. + +\spadcommand{suffix?("", "Hello")} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{suffix?("LO", "Hello")} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +The function \spadfunFrom{substring?}{String} tests for a substring +given a starting position. + +\spadcommand{substring?("ll", "Hello", 3)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{substring?("ll", "Hello", 4)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +A number of \spadfunFrom{position}{String} functions locate things in strings. +If the first argument to position is a string, then {\tt position(s,t,i)} +finds the location of {\tt s} as a substring of {\tt t} starting the +search at position {\tt i}. + +\spadcommand{n := position("nd", "underground", 1) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{n := position("nd", "underground", n+1) } +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +If {\tt s} is not found, then {\tt 0} is returned ({\tt minIndex(s)-1} +in {\tt IndexedString}). + +\spadcommand{n := position("nd", "underground", n+1) } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +To search for a specific character or a member of a character class, +a different first argument is used. + +\spadcommand{position(char "d", "underground", 1)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{position(hexDigit(), "underground", 1)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\section{StringTable} +\label{StringTableXmpPage} + +This domain provides a table type in which the keys are known to be +strings so special techniques can be used. Other than performance, +the type {\tt StringTable(S)} should behave exactly the same way as +{\tt Table(String,S)}. See \ref{TableXmpPage} on page~\pageref{TableXmpPage} +for general information about tables. + +This creates a new table whose keys are strings. + +\spadcommand{t: StringTable(Integer) := table() } +$$ +table() +$$ +\returnType{Type: StringTable Integer} + +The value associated with each string key is the number of characters +in the string. + +\begin{verbatim} +for s in split("My name is Ian Watt.",char " ") + repeat + t.s := #s +\end{verbatim} +\returnType{Void} + +\spadcommand{for key in keys t repeat output [key, t.key] } +\begin{verbatim} + ["Ian",3] + ["My",2] + ["Watt.",5] + ["name",4] + ["is",2] +\end{verbatim} +\returnType{Void} + +\section{Symbol} +\label{SymbolXmpPage} + +Symbols are one of the basic types manipulated by Axiom. The +{\tt Symbol} domain provides ways to create symbols of many varieties. + +The simplest way to create a symbol is to ``single quote'' an identifier. + +\spadcommand{X: Symbol := 'x } +$$ +x +$$ +\returnType{Type: Symbol} + +This gives the symbol even if {\tt x} has been assigned a value. If +{\tt x} has not been assigned a value, then it is possible to omit the +quote. + +\spadcommand{XX: Symbol := x} +$$ +x +$$ +\returnType{Type: Symbol} + +Declarations must be used when working with symbols, because otherwise +the interpreter tries to place values in a more specialized type +{\tt Variable}. + +\spadcommand{A := 'a} +$$ +a +$$ +\returnType{Type: Variable a} + +\spadcommand{B := b} +$$ +b +$$ +\returnType{Type: Variable b} + +The normal way of entering polynomials uses this fact. + +\spadcommand{x**2 + 1} +$$ +{x \sp 2}+1 +$$ +\returnType{Type: Polynomial Integer} + +Another convenient way to create symbols is to convert a string. +This is useful when the name is to be constructed by a program. + +\spadcommand{"Hello"::Symbol} +$$ +Hello +$$ +\returnType{Type: Symbol} + +Sometimes it is necessary to generate new unique symbols, for example, +to name constants of integration. The expression {\tt new()} +generates a symbol starting with {\tt \%}. + +\spadcommand{new()\$Symbol} +$$ + \%A +$$ +\returnType{Type: Symbol} + +Successive calls to \spadfunFrom{new}{Symbol} produce different symbols. + +\spadcommand{new()\$Symbol} +$$ + \%B +$$ +\returnType{Type: Symbol} + +The expression {\tt new("s")} produces a symbol starting with {\tt \%s}. + +\spadcommand{new("xyz")\$Symbol} +$$ + \%xyz0 +$$ +\returnType{Type: Symbol} + +A symbol can be adorned in various ways. The most basic thing is +applying a symbol to a list of subscripts. + +\spadcommand{X[i,j] } +$$ +x \sb {i, j} +$$ +\returnType{Type: Symbol} + +Somewhat less pretty is to attach subscripts, superscripts or arguments. + +\spadcommand{U := subscript(u, [1,2,1,2]) } +$$ +u \sb {1, 2, 1, 2} +$$ +\returnType{Type: Symbol} + +\spadcommand{V := superscript(v, [n]) } +$$ +v \sp {n} +$$ +\returnType{Type: Symbol} + +\spadcommand{P := argscript(p, [t]) } +$$ +{p \sb {}} +\left( +{t} +\right) +$$ +\returnType{Type: Symbol} + +It is possible to test whether a symbol has scripts using the +\spadfunFrom{scripted?}{Symbol} test. + +\spadcommand{scripted? U } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{scripted? X } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +If a symbol is not scripted, then it may be converted to a string. + +\spadcommand{string X } +$$ +\mbox{\tt "x"} +$$ +\returnType{Type: String} + +The basic parts can always be extracted using the +\spadfunFrom{name}{Symbol} and \spadfunFrom{scripts}{Symbol} +operations. + +\spadcommand{name U } +$$ +u +$$ +\returnType{Type: Symbol} + +\spadcommand{scripts U } +$$ +\left[ +{sub={\left[ 1, 2, 1, 2 +\right]}}, +{sup={\left[ +\right]}}, +{presup={\left[ +\right]}}, +{presub={\left[ +\right]}}, +{args={\left[ +\right]}} +\right] +$$ +\returnType{Type: +Record( +sub: List OutputForm, +sup: List OutputForm, +presup: List OutputForm, +presub: List OutputForm, +args: List OutputForm)} + +\spadcommand{name X } +$$ +x +$$ +\returnType{Type: Symbol} + +\spadcommand{scripts X } +$$ +\left[ +{sub={\left[ +\right]}}, +{sup={\left[ +\right]}}, +{presup={\left[ +\right]}}, +{presub={\left[ +\right]}}, +{args={\left[ +\right]}} +\right] +$$ +\returnType{Type: +Record( +sub: List OutputForm, +sup: List OutputForm, +presup: List OutputForm, +presub: List OutputForm, +args: List OutputForm)} + +The most general form is obtained using the +\spadfunFrom{script}{Symbol} operation. This operation takes an +argument which is a list containing, in this order, lists of +subscripts, superscripts, presuperscripts, presubscripts and arguments +to a symbol. + +\spadcommand{M := script(Mammoth, [ [i,j],[k,l],[0,1],[2],[u,v,w] ]) } +$$ +{{} \sb {2} \sp {{0, 1}}Mammoth \sb {{i, j}} \sp {{k, l}}} +\left( +{u, v, w} +\right) +$$ +\returnType{Type: Symbol} + +\spadcommand{scripts M } +$$ +\left[ +{sub={\left[ i, j +\right]}}, +{sup={\left[ k, l +\right]}}, +{presup={\left[ 0, 1 +\right]}}, +{presub={\left[ 2 +\right]}}, +{args={\left[ u, v, w +\right]}} +\right] +$$ +\returnType{Type: +Record( +sub: List OutputForm, +sup: List OutputForm, +presup: List OutputForm, +presub: List OutputForm, +args: List OutputForm)} + +If trailing lists of scripts are omitted, they are assumed to be empty. + +\spadcommand{N := script(Nut, [ [i,j],[k,l],[0,1] ]) } +$$ +{} \sp {{0, 1}}Nut \sb {{i, j}} \sp {{k, l}} +$$ +\returnType{Type: Symbol} + +\spadcommand{scripts N } +$$ +\left[ +{sub={\left[ i, j +\right]}}, +{sup={\left[ k, l +\right]}}, +{presup={\left[ 0, 1 +\right]}}, +{presub={\left[ +\right]}}, +{args={\left[ +\right]}} +\right] +$$ +\returnType{Type: +Record( +sub: List OutputForm, +sup: List OutputForm, +presup: List OutputForm, +presub: List OutputForm, +args: List OutputForm)} + +\section{Table} +\label{TableXmpPage} + +The {\tt Table} constructor provides a general structure for +associative storage. This type provides hash tables in which data +objects can be saved according to keys of any type. For a given +table, specific types must be chosen for the keys and entries. + +In this example the keys to the table are polynomials with integer +coefficients. The entries in the table are strings. + +\spadcommand{t: Table(Polynomial Integer, String) := table() } +$$ +table() +$$ +\returnType{Type: Table(Polynomial Integer,String)} + +To save an entry in the table, the \spadfunFrom{setelt}{Table} +operation is used. This can be called directly, giving the table a +key and an entry. + +\spadcommand{setelt(t, x**2 - 1, "Easy to factor") } +$$ +\mbox{\tt "Easy to factor"} +$$ +\returnType{Type: String} + +Alternatively, you can use assignment syntax. + +\spadcommand{t(x**3 + 1) := "Harder to factor" } +$$ +\mbox{\tt "Harder to factor"} +$$ +\returnType{Type: String} + +\spadcommand{t(x) := "The easiest to factor" } +$$ +\mbox{\tt "The easiest to factor"} +$$ +\returnType{Type: String} + +Entries are retrieved from the table by calling the +\spadfunFrom{elt}{Table} operation. + +\spadcommand{elt(t, x) } +$$ +\mbox{\tt "The easiest to factor"} +$$ +\returnType{Type: String} + +This operation is called when a table is ``applied'' to a key using +this or the following syntax. + +\spadcommand{t.x } +$$ +\mbox{\tt "The easiest to factor"} +$$ +\returnType{Type: String} + +\spadcommand{t x } +$$ +\mbox{\tt "The easiest to factor"} +$$ +\returnType{Type: String} + +Parentheses are used only for grouping. They are needed if the key is +an infixed expression. + +\spadcommand{t.(x**2 - 1) } +$$ +\mbox{\tt "Easy to factor"} +$$ +\returnType{Type: String} + +Note that the \spadfunFrom{elt}{Table} operation is used only when the +key is known to be in the table---otherwise an error is generated. + +\spadcommand{t (x**3 + 1) } +$$ +\mbox{\tt "Harder to factor"} +$$ +\returnType{Type: String} + +You can get a list of all the keys to a table using the +\spadfunFrom{keys}{Table} operation. + +\spadcommand{keys t } +$$ +\left[ +x, {{x \sp 3}+1}, {{x \sp 2} -1} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +If you wish to test whether a key is in a table, the +\spadfunFrom{search}{Table} operation is used. This operation returns +either an entry or {\tt "failed"}. + +\spadcommand{search(x, t) } +$$ +\mbox{\tt "The easiest to factor"} +$$ +\returnType{Type: Union(String,...)} + +\spadcommand{search(x**2, t) } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +The return type is a union so the success of the search can be tested +using {\tt case}. + +\spadcommand{search(x**2, t) case "failed" } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +The \spadfunFrom{remove}{Table} operation is used to delete values from a +table. + +\spadcommand{remove!(x**2-1, t) } +$$ +\mbox{\tt "Easy to factor"} +$$ +\returnType{Type: Union(String,...)} + +If an entry exists under the key, then it is returned. Otherwise +\spadfunFrom{remove}{Table} returns {\tt "failed"}. + +\spadcommand{remove!(x-1, t) } +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +The number of key-entry pairs can be found using the +\spadfunFrom{\#}{Table} operation. + +\spadcommand{\#t } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Just as \spadfunFrom{keys}{Table} returns a list of keys to the table, +a list of all the entries can be obtained using the +\spadfunFrom{members}{Table} operation. + +\spadcommand{members t } +$$ +\left[ +\mbox{\tt "The easiest to factor"} , \mbox{\tt "Harder to factor"} +\right] +$$ +\returnType{Type: List String} + +A number of useful operations take functions and map them on to the +table to compute the result. Here we count the entries which +have ``{\tt Hard}'' as a prefix. + +\spadcommand{count(s: String +-> prefix?("Hard", s), t) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Other table types are provided to support various needs. +\indent +\begin{list}{} +\item {\tt AssociationList} gives a list with a table view. +This allows new entries to be appended onto the front of the list +to cover up old entries. +This is useful when table entries need to be stacked or when +frequent list traversals are required. +See \ref{AssociationListXmpPage} on page~\pageref{AssociationListXmpPage} +for more information. +\item {\tt EqTable} gives tables in which keys are considered +equal only when they are in fact the same instance of a structure. +See \ref{EqTableXmpPage} on page~\pageref{EqTableXmpPage} for more information. +\item {\tt StringTable} should be used when the keys are known to +be strings. +See \ref{StringTableXmpPage} on page~\pageref{StringTableXmpPage} +for more information. +\item {\tt SparseTable} provides tables with default +entries, so +lookup never fails. The {\tt GeneralSparseTable} constructor +can be used to make any table type behave this way. +See \ref{SparseTableXmpPage} on page~\pageref{SparseTableXmpPage} +for more information. +\item {\tt KeyedAccessFile} allows values to be saved in a file, +accessed as a table. +See \ref{KeyedAccessFileXmpPage} on page~\pageref{KeyedAccessFileXmpPage} +for more information. +\end{list} +\noindent + +\section{TextFile} +\label{TextFileXmpPage} + +The domain {\tt TextFile} allows Axiom to read and write +character data and exchange text with other programs. +This type behaves in Axiom much like a {\tt File} of strings, +with additional operations to cause new lines. +We give an example of how to produce an upper case copy of a file. + +This is the file from which we read the text. + +\spadcommand{f1: TextFile := open("/etc/group", "input") } +$$ +\mbox{\tt "/etc/group"} +$$ +\returnType{Type: TextFile} + +This is the file to which we write the text. + +\spadcommand{f2: TextFile := open("/tmp/MOTD", "output") } +$$ +\mbox{\tt "/tmp/MOTD"} +$$ +\returnType{Type: TextFile} + +Entire lines are handled using the \spadfunFrom{readLine}{TextFile} and +\spadfunFrom{writeLine}{TextFile} operations. + +\spadcommand{l := readLine! f1 } +$$ +\mbox{\tt "root:x:0:root"} +$$ +\returnType{Type: String} + +\spadcommand{writeLine!(f2, upperCase l) } +$$ +\mbox{\tt "ROOT:X:0:ROOT"} +$$ +\returnType{Type: String} + +Use the \spadfunFrom{endOfFile?}{TextFile} operation to check if you +have reached the end of the file. + +\begin{verbatim} +while not endOfFile? f1 repeat + s := readLine! f1 + writeLine!(f2, upperCase s) +\end{verbatim} +\returnType{Void} + +The file {\tt f1} is exhausted and should be closed. + +\spadcommand{close! f1 } +$$ +\mbox{\tt "/etc/group"} +$$ +\returnType{Type: TextFile} + +It is sometimes useful to write lines a bit at a time. The +\spadfunFrom{write}{TextFile} operation allows this. + +\spadcommand{write!(f2, "-The-") } +$$ +\mbox{\tt "-The-"} +$$ +\returnType{Type: String} + +\spadcommand{write!(f2, "-End-") } +$$ +\mbox{\tt "-End-"} +$$ +\returnType{Type: String} + +This ends the line. This is done in a machine-dependent manner. + +\spadcommand{writeLine! f2} +$$ +\mbox{\tt ""} +$$ +\returnType{Type: String} + +\spadcommand{close! f2} +$$ +\mbox{\tt "/tmp/MOTD"} +$$ +\returnType{Type: TextFile} + +Finally, clean up. + +\spadcommand{)system rm /tmp/MOTD} + +For more information on related topics, see +\ref{FileXmpPage} on page~\pageref{FileXmpPage}, +\ref{KeyedAccessFileXmpPage} on page~\pageref{KeyedAccessFileXmpPage}, and +\ref{LibraryXmpPage} on page~\pageref{LibraryXmpPage}. + +\section{TwoDimensionalArray} +\label{TwoDimensionalArrayXmpPage} + +The {\tt TwoDimensionalArray} domain is used for storing data in a +two dimensional data structure indexed by row and by column. Such an array +is a homogeneous data structure in that all the entries of the array +must belong to the same Axiom domain (although see +\ref{ugTypesAnyNonePage} on page~\pageref{ugTypesAnyNonePage} in Section +\ref{ugTypesAnyNoneNumber} on page~\pageref{ugTypesAnyNoneNumber}). Each +array has a fixed number of rows and columns specified by the user and +arrays are not extensible. In Axiom, the indexing of two-dimensional +arrays is one-based. This means that both the ``first'' row of an +array and the ``first'' column of an array are given the index +{\tt 1}. Thus, the entry in the upper left corner of an array is in +position {\tt (1,1)}. + +The operation \spadfunFrom{new}{TwoDimensionalArray} creates an array +with a specified number of rows and columns and fills the components +of that array with a specified entry. The arguments of this operation +specify the number of rows, the number of columns, and the entry. + +This creates a five-by-four array of integers, all of whose entries are +zero. + +\spadcommand{arr : ARRAY2 INT := new(5,4,0) } +$$ +\left[ +\begin{array}{cccc} +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +The entries of this array can be set to other integers using the +operation \spadfunFrom{setelt}{TwoDimensionalArray}. + +Issue this to set the element in the upper left corner of this array to +{\tt 17}. + +\spadcommand{setelt(arr,1,1,17) } +$$ +17 +$$ +\returnType{Type: PositiveInteger} + +Now the first element of the array is {\tt 17.} + +\spadcommand{arr } +$$ +\left[ +\begin{array}{cccc} +{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +Likewise, elements of an array are extracted using the operation +\spadfunFrom{elt}{TwoDimensionalArray}. + +\spadcommand{elt(arr,1,1) } +$$ +17 +$$ +\returnType{Type: PositiveInteger} + +Another way to use these two operations is as follows. This sets the +element in position {\tt (3,2)} of the array to {\tt 15}. + +\spadcommand{arr(3,2) := 15 } +$$ +15 +$$ +\returnType{Type: PositiveInteger} + +This extracts the element in position {\tt (3,2)} of the array. + +\spadcommand{arr(3,2) } +$$ +15 +$$ +\returnType{Type: PositiveInteger} + +The operations \spadfunFrom{elt}{TwoDimensionalArray} and +\spadfunFrom{setelt}{TwoDimensionalArray} come equipped with an error +check which verifies that the indices are in the proper ranges. For +example, the above array has five rows and four columns, so if you ask +for the entry in position {\tt (6,2)} with {\tt arr(6,2)} Axiom +displays an error message. If there is no need for an error check, +you can call the operations \spadfunFrom{qelt}{TwoDimensionalArray} +and \spadfunFrom{qsetelt}{TwoDimensionalArray} which provide the same +functionality but without the error check. Typically, these +operations are called in well-tested programs. + +The operations \spadfunFrom{row}{TwoDimensionalArray} and +\spadfunFrom{column}{TwoDimensionalArray} extract rows and columns, +respectively, and return objects of {\tt OneDimensionalArray} with the +same underlying element type. + +\spadcommand{row(arr,1) } +$$ +\left[ +{17}, 0, 0, 0 +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +\spadcommand{column(arr,1) } +$$ +\left[ +{17}, 0, 0, 0, 0 +\right] +$$ +\returnType{Type: OneDimensionalArray Integer} + +You can determine the dimensions of an array by calling the operations +\spadfunFrom{nrows}{TwoDimensionalArray} and +\spadfunFrom{ncols}{TwoDimensionalArray}, which return the number of +rows and columns, respectively. + +\spadcommand{nrows(arr) } +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{ncols(arr) } +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +To apply an operation to every element of an array, use +\spadfunFrom{map}{TwoDimensionalArray}. This creates a new array. +This expression negates every element. + +\spadcommand{map(-,arr) } +$$ +\left[ +\begin{array}{cccc} +-{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & -{15} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +This creates an array where all the elements are doubled. + +\spadcommand{map((x +-> x + x),arr) } +$$ +\left[ +\begin{array}{cccc} +{34} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & {30} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +To change the array destructively, use +\spadfunFrom{map}{TwoDimensionalArray} instead of +\spadfunFrom{map}{TwoDimensionalArray}. If you need to make a copy of +any array, use \spadfunFrom{copy}{TwoDimensionalArray}. + +\spadcommand{arrc := copy(arr) } +$$ +\left[ +\begin{array}{cccc} +{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & {15} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +\spadcommand{map!(-,arrc) } +$$ +\left[ +\begin{array}{cccc} +-{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & -{15} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +\spadcommand{arrc } +$$ +\left[ +\begin{array}{cccc} +-{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & -{15} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +\spadcommand{arr } +$$ +\left[ +\begin{array}{cccc} +{17} & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & {15} & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 +\end{array} +\right] +$$ +\returnType{Type: TwoDimensionalArray Integer} + +Use \spadfunFrom{member?}{TwoDimensionalArray} to see if a given element +is in an array. + +\spadcommand{member?(17,arr) } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{member?(10317,arr) } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +To see how many times an element appears in an array, use +\spadfunFrom{count}{TwoDimensionalArray}. + +\spadcommand{count(17,arr) } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{count(0,arr) } +$$ +18 +$$ +\returnType{Type: PositiveInteger} + +For more information about the operations available for {\tt +TwoDimensionalArray}, issue {\tt )show TwoDimensionalArray}. For +information on related topics, see +\ref{MatrixXmpPage} on page~\pageref{MatrixXmpPage} and +\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage}. + +\section{UnivariatePolynomial} +\label{UnivariatePolynomialXmpPage} + +The domain constructor {\tt UnivariatePolynomial} (abbreviated {\tt +UP}) creates domains of univariate polynomials in a specified +variable. For example, the domain {\tt UP(a1,POLY FRAC INT)} provides +polynomials in the single variable {\tt a1} whose coefficients are +general polynomials with rational number coefficients. + +\boxed{4.6in}{ +\vskip 0.1cm +\noindent {\bf Restriction:} +\begin{quotation}\noindent +Axiom does not allow you to create types where +{\tt UnivariatePolynomial} is contained in the coefficient type of +{\tt Polynomial}. Therefore, +{\tt UP(x,POLY INT)} is legal but {\tt POLY UP(x,INT)} is not. +\end{quotation} +. +} + +{\tt UP(x,INT)} is the domain of polynomials in the single +variable {\tt x} with integer coefficients. + +\spadcommand{(p,q) : UP(x,INT) } +\returnType{Void} + +\spadcommand{p := (3*x-1)**2 * (2*x + 8) } +$$ +{{18} \ {x \sp 3}}+{{60} \ {x \sp 2}} -{{46} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +\spadcommand{q := (1 - 6*x + 9*x**2)**2 } +$$ +{{81} \ {x \sp 4}} -{{108} \ {x \sp 3}}+{{54} \ {x \sp 2}} -{{12} \ x}+1 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The usual arithmetic operations are available for univariate polynomials. + +\spadcommand{p**2 + p*q } +$$ +{{1458} \ {x \sp 7}}+{{3240} \ {x \sp 6}} -{{7074} \ {x \sp 5}}+{{10584} \ +{x \sp 4}} -{{9282} \ {x \sp 3}}+{{4120} \ {x \sp 2}} -{{878} \ x}+{72} +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The operation \spadfunFrom{leadingCoefficient}{UnivariatePolynomial} +extracts the coefficient of the term of highest degree. + +\spadcommand{leadingCoefficient p } +$$ +18 +$$ +\returnType{Type: PositiveInteger} + +The operation \spadfunFrom{degree}{UnivariatePolynomial} returns +the degree of the polynomial. +Since the polynomial has only one variable, the variable is not supplied +to operations like \spadfunFrom{degree}{UnivariatePolynomial}. + +\spadcommand{degree p } +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +The reductum of the polynomial, the polynomial obtained by subtracting +the term of highest order, is returned by +\spadfunFrom{reductum}{UnivariatePolynomial}. + +\spadcommand{reductum p } +$$ +{{60} \ {x \sp 2}} -{{46} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The operation \spadfunFrom{gcd}{UnivariatePolynomial} computes the +greatest common divisor of two polynomials. + +\spadcommand{gcd(p,q) } +$$ +{9 \ {x \sp 2}} -{6 \ x}+1 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The operation \spadfunFrom{lcm}{UnivariatePolynomial} computes the +least common multiple. + +\spadcommand{lcm(p,q) } +$$ +{{162} \ {x \sp 5}}+{{432} \ {x \sp 4}} -{{756} \ {x \sp 3}}+{{408} \ {x +\sp 2}} -{{94} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The operation \spadfunFrom{resultant}{UnivariatePolynomial} computes +the resultant of two univariate polynomials. In the case of {\tt p} +and {\tt q}, the resultant is {\tt 0} because they share a common +root. + +\spadcommand{resultant(p,q) } +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +To compute the derivative of a univariate polynomial with respect to its +variable, use \spadfunFrom{D}{UnivariatePolynomial}. + +\spadcommand{D p } +$$ +{{54} \ {x \sp 2}}+{{120} \ x} -{46} +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +Univariate polynomials can also be used as if they were functions. To +evaluate a univariate polynomial at some point, apply the polynomial +to the point. + +\spadcommand{p(2) } +$$ +300 +$$ +\returnType{Type: PositiveInteger} + +The same syntax is used for composing two univariate polynomials, i.e. +substituting one polynomial for the variable in another. This +substitutes {\tt q} for the variable in {\tt p}. + +\spadcommand{p(q) } +$$ +\begin{array}{@{}l} +{{9565938} \ {x \sp {12}}} - +{{38263752} \ {x \sp {11}}}+ +{{70150212} \ {x \sp {10}}} - +{{77944680} \ {x \sp 9}}+ +{{58852170} \ {x \sp 8}} - +\\ +\\ +\displaystyle +{{32227632} \ {x \sp 7}}+ +{{13349448} \ {x \sp 6}} - +{{4280688} \ {x \sp 5}}+ +{{1058184} \ {x \sp 4}} - +\\ +\\ +\displaystyle +{{192672} \ {x \sp 3}}+ +{{23328} \ {x \sp 2}} - +{{1536} \ x}+ +{40} +\end{array} +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +This substitutes {\tt p} for the variable in {\tt q}. + +\spadcommand{q(p) } +$$ +\begin{array}{@{}l} +{{8503056} \ {x \sp {12}}}+ +{{113374080} \ {x \sp {11}}}+ +{{479950272} \ {x \sp {10}}}+ +{{404997408} \ {x \sp 9}} - +\\ +\\ +\displaystyle +{{1369516896} \ {x \sp 8}} - +{{626146848} \ {x \sp 7}}+ +{{2939858712} \ {x \sp 6}} - +{{2780728704} \ {x \sp 5}}+ +\\ +\\ +\displaystyle +{{1364312160} \ {x \sp 4}} - +{{396838872} \ {x \sp 3}}+ +{{69205896} \ {x \sp 2}} - +{{6716184} \ x}+ +{279841} +\end{array} +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +To obtain a list of coefficients of the polynomial, use +\spadfunFrom{coefficients}{UnivariatePolynomial}. + +\spadcommand{l := coefficients p } +$$ +\left[ +{18}, {60}, -{46}, 8 +\right] +$$ +\returnType{Type: List Integer} + +From this you can use \spadfunFrom{gcd}{UnivariatePolynomial} and +\spadfunFrom{reduce}{List} to compute the content of the polynomial. + +\spadcommand{reduce(gcd,l) } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Alternatively (and more easily), you can just call +\spadfunFrom{content}{UnivariatePolynomial}. + +\spadcommand{content p } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Note that the operation +\spadfunFrom{coefficients}{UnivariatePolynomial} omits the zero +coefficients from the list. Sometimes it is useful to convert a +univariate polynomial to a vector whose $i$-th position contains the +degree {\tt i-1} coefficient of the polynomial. + +\spadcommand{ux := (x**4+2*x+3)::UP(x,INT) } +$$ +{x \sp 4}+{2 \ x}+3 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +To get a complete vector of coefficients, use the operation +\spadfunFrom{vectorise}{UnivariatePolynomial}, which takes a +univariate polynomial and an integer denoting the length of the +desired vector. + +\spadcommand{vectorise(ux,5) } +$$ +\left[ +3, 2, 0, 0, 1 +\right] +$$ +\returnType{Type: Vector Integer} + +It is common to want to do something to every term of a polynomial, +creating a new polynomial in the process. + +This is a function for iterating across the terms of a polynomial, +squaring each term. + +\spadcommand{squareTerms(p) == reduce(+,[t**2 for t in monomials p])} +\returnType{Void} + +Recall what {\tt p} looked like. + +\spadcommand{p } +$$ +{{18} \ {x \sp 3}}+{{60} \ {x \sp 2}} -{{46} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +We can demonstrate {\tt squareTerms} on {\tt p}. + +\spadcommand{squareTerms p } +\begin{verbatim} +Compiling function squareTerms with type + UnivariatePolynomial(x,Integer) -> + UnivariatePolynomial(x,Integer) +\end{verbatim} +$$ +{{324} \ {x \sp 6}}+{{3600} \ {x \sp 4}}+{{2116} \ {x \sp 2}}+{64} +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +When the coefficients of the univariate polynomial belong to a +field,\footnote{For example, when the coefficients are rational +numbers, as opposed to integers. The important property of a field is +that non-zero elements can be divided and produce another element. The +quotient of the integers 2 and 3 is not another integer.} it is +possible to compute quotients and remainders. + +\spadcommand{(r,s) : UP(a1,FRAC INT) } +\returnType{Void} + +\spadcommand{r := a1**2 - 2/3 } +$$ +{a1 \sp 2} -{2 \over 3} +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +\spadcommand{s := a1 + 4} +$$ +a1+4 +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +When the coefficients are rational numbers or rational expressions, the +operation \spadfunFrom{quo}{UnivariatePolynomial} computes the quotient +of two polynomials. + +\spadcommand{r quo s } +$$ +a1 -4 +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +The operation \spadfunFrom{rem}{UnivariatePolynomial} computes the +remainder. + +\spadcommand{r rem s } +$$ +{46} \over 3 +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +The operation \spadfunFrom{divide}{UnivariatePolynomial} can be used to +return a record of both components. + +\spadcommand{d := divide(r, s) } +$$ +\left[ +{quotient={a1 -4}}, {remainder={{46} \over 3}} +\right] +$$ +\returnType{Type: +Record( +quotient: UnivariatePolynomial(a1,Fraction Integer), +remainder: UnivariatePolynomial(a1,Fraction Integer))} + +Now we check the arithmetic! + +\spadcommand{r - (d.quotient * s + d.remainder) } +$$ +0 +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +It is also possible to integrate univariate polynomials when the +coefficients belong to a field. + +\spadcommand{integrate r } +$$ +{{1 \over 3} \ {a1 \sp 3}} -{{2 \over 3} \ a1} +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +\spadcommand{integrate s } +$$ +{{1 \over 2} \ {a1 \sp 2}}+{4 \ a1} +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Integer)} + +One application of univariate polynomials is to see expressions in terms +of a specific variable. + +We start with a polynomial in {\tt a1} whose coefficients +are quotients of polynomials in {\tt b1} and {\tt b2}. + +\spadcommand{t : UP(a1,FRAC POLY INT) } +\returnType{Void} + +Since in this case we are not talking about using multivariate +polynomials in only two variables, we use {\tt Polynomial}. +We also use {\tt Fraction} because we want fractions. + +\spadcommand{t := a1**2 - a1/b2 + (b1**2-b1)/(b2+3) } +$$ +{a1 \sp 2} -{{1 \over b2} \ a1}+{{{b1 \sp 2} -b1} \over {b2+3}} +$$ +\returnType{Type: UnivariatePolynomial(a1,Fraction Polynomial Integer)} + +We push all the variables into a single quotient of polynomials. + +\spadcommand{u : FRAC POLY INT := t } +$$ +{{{a1 \sp 2} \ {b2 \sp 2}}+{{\left( {b1 \sp 2} -b1+{3 \ {a1 \sp 2}} -a1 +\right)} +\ b2} -{3 \ a1}} \over {{b2 \sp 2}+{3 \ b2}} +$$ +\returnType{Type: Fraction Polynomial Integer} + +Alternatively, we can view this as a polynomial in the variable +This is a {\it mode-directed} conversion: you indicate +as much of the structure as you care about and let Axiom +decide on the full type and how to do the transformation. + +\spadcommand{u :: UP(b1,?) } +$$ +{{1 \over {b2+3}} \ {b1 \sp 2}} -{{1 \over {b2+3}} \ b1}+{{{{a1 \sp 2} \ +b2} -a1} \over b2} +$$ +\returnType{Type: UnivariatePolynomial(b1,Fraction Polynomial Integer)} + +See \ref{ugProblemFactorPage} on page~\pageref{ugProblemFactorPage} in Section +\ref{ugProblemFactorNumber} on page~\pageref{ugProblemFactorNumber} +for a discussion of the factorization facilities +in Axiom for univariate polynomials. +For more information on related topics, see +\ref{ugIntroVariablesPage} on page~\pageref{ugIntroVariablesPage} in Section +\ref{ugIntroVariablesNumber} on page~\pageref{ugIntroVariablesNumber}, +\ref{ugTypesConvertPage} on page~\pageref{ugTypesConvertPage} in Section +\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}, +\ref{PolynomialXmpPage} on page~\pageref{PolynomialXmpPage}, +\ref{MultivariatePolynomialXmpPage} on +page~\pageref{MultivariatePolynomialXmpPage}, and +\ref{DistributedMultivariatePolynomialXmpPage} on +page~\pageref{DistributedMultivariatePolynomialXmpPage}. + +\section{UniversalSegment} +\label{UniversalSegmentXmpPage} + +The {\tt UniversalSegment} domain generalizes {\tt Segment} +by allowing segments without a ``hi'' end point. + +\spadcommand{pints := 1..} +$$ +1.. +$$ +\returnType{Type: UniversalSegment PositiveInteger} + +\spadcommand{nevens := (0..) by -2 } +$$ +{0..} \mbox{\rm by\ } -2 +$$ +\returnType{Type: UniversalSegment NonNegativeInteger} + +Values of type {\tt Segment} are automatically converted to +type {\tt UniversalSegment} when appropriate. + +\spadcommand{useg: UniversalSegment(Integer) := 3..10 } +$$ +3..{10} +$$ +\returnType{Type: UniversalSegment Integer} + +The operation \spadfunFrom{hasHi}{UniversalSegment} is used to test +whether a segment has a {\tt hi} end point. + +\spadcommand{hasHi pints } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{hasHi nevens } +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{hasHi useg } +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +All operations available on type {\tt Segment} apply to {\tt +UniversalSegment}, with the proviso that expansions produce streams +rather than lists. This is to accommodate infinite expansions. + +\spadcommand{expand pints } +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\spadcommand{expand nevens } +$$ +\left[ +0, -2, -4, -6, -8, -{10}, -{12}, -{14}, -{16}, +-{18}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\spadcommand{expand [1, 3, 10..15, 100..]} +$$ +\left[ +1, 3, {10}, {11}, {12}, {13}, {14}, {15}, {100}, +{101}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +For more information on related topics, see +\ref{SegmentXmpPage} on page~\pageref{SegmentXmpPage}, +\ref{SegmentBindingXmpPage} on page~\pageref{SegmentBindingXmpPage}, +\ref{ListXmpPage} on page~\pageref{ListXmpPage}, and +\ref{StreamXmpPage} on page~\pageref{StreamXmpPage}. + +\section{Vector} +\label{VectorXmpPage} + +The {\tt Vector} domain is used for storing data in a one-dimensional +indexed data structure. A vector is a homogeneous data structure in +that all the components of the vector must belong to the same Axiom +domain. Each vector has a fixed length specified by the user; vectors +are not extensible. This domain is similar to the +{\tt OneDimensionalArray} domain, except that when the components of a +{\tt Vector} belong to a {\tt Ring}, arithmetic operations are provided. +For more examples of operations that are defined for both {\tt Vector} +and {\tt OneDimensionalArray}, see +\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage}. + +As with the {\tt OneDimensionalArray} domain, a {\tt Vector} can +be created by calling the operation \spadfunFrom{new}{Vector}, its components +can be accessed by calling the operations \spadfunFrom{elt}{Vector} and +\spadfunFrom{qelt}{Vector}, and its components can be reset by calling the +operations \spadfunFrom{setelt}{Vector} and +\spadfunFrom{qsetelt}{Vector}. + +This creates a vector of integers of length {\tt 5} all of whose +components are {\tt 12}. + +\spadcommand{u : VECTOR INT := new(5,12) } +$$ +\left[ +{12}, {12}, {12}, {12}, {12} +\right] +$$ +\returnType{Type: Vector Integer} + +This is how you create a vector from a list of its components. + +\spadcommand{v : VECTOR INT := vector([1,2,3,4,5]) } +$$ +\left[ +1, 2, 3, 4, 5 +\right] +$$ +\returnType{Type: Vector Integer} + +Indexing for vectors begins at {\tt 1}. The last element has index +equal to the length of the vector, which is computed by +\spadopFrom{\#}{Vector}. + +\spadcommand{\#(v) } +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +This is the standard way to use \spadfunFrom{elt}{Vector} to extract +an element. Functionally, it is the same as if you had typed {\tt +elt(v,2)}. + +\spadcommand{v.2 } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +This is the standard way to use \spadfunFrom{setelt}{Vector} to change +an element. It is the same as if you had typed {\tt setelt(v,3,99)}. + +\spadcommand{v.3 := 99 } +$$ +99 +$$ +\returnType{Type: PositiveInteger} + +Now look at {\tt v} to see the change. You can use +\spadfunFrom{qelt}{Vector} and \spadfunFrom{qsetelt}{Vector} (instead +of \spadfunFrom{elt}{Vector} and \spadfunFrom{setelt}{Vector}, +respectively) but {\it only} when you know that the index is within +the valid range. + +\spadcommand{v } +$$ +\left[ +1, 2, {99}, 4, 5 +\right] +$$ +\returnType{Type: Vector Integer} + +When the components belong to a {\tt Ring}, Axiom provides arithmetic +operations for {\tt Vector}. These include left and right scalar +multiplication. + +\spadcommand{5 * v } +$$ +\left[ +5, {10}, {495}, {20}, {25} +\right] +$$ +\returnType{Type: Vector Integer} + +\spadcommand{v * 7 } +$$ +\left[ +7, {14}, {693}, {28}, {35} +\right] +$$ +\returnType{Type: Vector Integer} + +\spadcommand{w : VECTOR INT := vector([2,3,4,5,6]) } +$$ +\left[ +2, 3, 4, 5, 6 +\right] +$$ +\returnType{Type: Vector Integer} + +Addition and subtraction are also available. + +\spadcommand{v + w } +$$ +\left[ +3, 5, {103}, 9, {11} +\right] +$$ +\returnType{Type: Vector Integer} + +Of course, when adding or subtracting, the two vectors must have the same +length or an error message is displayed. + +\spadcommand{v - w } +$$ +\left[ +-1, -1, {95}, -1, -1 +\right] +$$ +\returnType{Type: Vector Integer} + +For more information about other aggregate domains, see the following: +\ref{ListXmpPage} on page~\pageref{ListXmpPage}, +\ref{MatrixXmpPage} on page~\pageref{MatrixXmpPage}, +\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage}, +\ref{SetXmpPage} on page~\pageref{SetXmpPage}, +\ref{TableXmpPage} on page~\pageref{TableXmpPage}, and +\ref{TwoDimensionalArrayXmpPage} on page~\pageref{TwoDimensionalArrayXmpPage}. +Issue the system command {\tt )show Vector} to display the full list of +operations defined by {\tt Vector}. + +\section{Void} +\label{VoidXmpPage} + +When an expression is not in a value context, it is given type +{\tt Void}. For example, in the expression +\begin{verbatim} + r := (a; b; if c then d else e; f) +\end{verbatim} +values are used only from the +subexpressions {\tt c} and {\tt f}: all others are thrown away. The +subexpressions {\tt a}, {\tt b}, {\tt d} and {\tt e} are evaluated for +side-effects only and have type {\tt Void}. There is a unique value +of type {\tt Void}. + +You will most often see results of type {\tt Void} when you +declare a variable. + +\spadcommand{a : Integer} +\returnType{Void} + +Usually no output is displayed for {\tt Void} results. +You can force the display of a rather ugly object by issuing +{\tt )set message void on}. + +\spadcommand{)set message void on} + +\spadcommand{b : Fraction Integer} +$$ +\mbox{\tt "()"} +$$ +\returnType{Type: Void} + +\spadcommand{)set message void off} + +All values can be converted to type {\tt Void}. + +\spadcommand{3::Void } +\returnType{Void} + +Once a value has been converted to {\tt Void}, it cannot be recovered. + +\spadcommand{\% :: PositiveInteger } +\begin{verbatim} +Cannot convert from type Void to PositiveInteger for value "()" +\end{verbatim} + +\section{WuWenTsunTriangularSet} +\label{WuWenTsunTriangularSetXmpPage} + +The {\tt WuWenTsunTriangularSet} domain constructor implements the +characteristic set method of Wu Wen Tsun. This algorithm computes a +list of triangular sets from a list of polynomials such that the +algebraic variety defined by the given list of polynomials decomposes +into the union of the regular-zero sets of the computed triangular +sets. The constructor takes four arguments. The first one, {\bf R}, +is the coefficient ring of the polynomials; it must belong to the +category {\tt IntegralDomain}. The second one, {\bf E}, is the +exponent monoid of the polynomials; it must belong to the category +{\tt OrderedAbelianMonoidSup}. The third one, {\bf V}, is the ordered +set of variables; it must belong to the category {\tt OrderedSet}. +The last one is the polynomial ring; it must belong to the category +{\tt RecursivePolynomialCategory(R,E,V)}. The abbreviation for +{\tt WuWenTsunTriangularSet} is {\tt WUTSET}. + +Let us illustrate the facilities by an example. + +Define the coefficient ring. + +\spadcommand{R := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the list of variables, + +\spadcommand{ls : List Symbol := [x,y,z,t] } +$$ +\left[ +x, y, z, t +\right] +$$ +\returnType{Type: List Symbol} + +and make it an ordered set; + +\spadcommand{V := OVAR(ls)} +$$ +\mbox{\rm OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +then define the exponent monoid. + +\spadcommand{E := IndexedExponents V} +$$ +\mbox{\rm IndexedExponents OrderedVariableList [x,y,z,t]} +$$ +\returnType{Type: Domain} + +Define the polynomial ring. + +\spadcommand{P := NSMP(R, V)} +$$ +\mbox{\rm NewSparseMultivariatePolynomial(Integer,OrderedVariableList +[x,y,z,t])} +$$ +\returnType{Type: Domain} + +Let the variables be polynomial. + +\spadcommand{x: P := 'x} +$$ +x +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{y: P := 'y} +$$ +y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{z: P := 'z} +$$ +z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{t: P := 't} +$$ +t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +Now call the {\tt WuWenTsunTriangularSet} domain constructor. + +\spadcommand{T := WUTSET(R,E,V,P)} +$$ +\begin{array}{@{}l} +{\rm WuWenTsunTriangularSet(Integer,} +\\ +\displaystyle +{\rm \ \ IndexedExponents OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t],} +\\ +\displaystyle +{\rm \ \ NewSparseMultivariatePolynomial(Integer,} +\\ +\displaystyle +{\rm \ \ OrderedVariableList [x,y,z,t]))} +\end{array} +$$ +\returnType{Type: Domain} + +Define a polynomial system. + +\spadcommand{p1 := x ** 31 - x ** 6 - x - y} +$$ +{x \sp {31}} -{x \sp 6} -x -y +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p2 := x ** 8 - z} +$$ +{x \sp 8} -z +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{p3 := x ** 10 - t} +$$ +{x \sp {10}} -t +$$ +\returnType{Type: +NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +\spadcommand{lp := [p1, p2, p3]} +$$ +\left[ +{{x \sp {31}} -{x \sp 6} -x -y}, {{x \sp 8} -z}, {{x \sp {10}} -t} +\right] +$$ +\returnType{Type: +List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])} + +Compute a characteristic set of the system. + +\spadcommand{characteristicSet(lp)\$T} +$$ +\begin{array}{@{}l} +\left\{ +{{z \sp 5} -{t \sp 4}}, +\right. +\\ +\\ +\displaystyle +{{{t \sp 4} \ {z \sp 2} \ {y \sp 2}}+ +{2 \ {t \sp 3} \ {z \sp 4} \ y}+ +{{\left( -{t \sp 7}+{2 \ {t \sp 4}} -t \right)}\ {z \sp 6}}+ +{{t \sp 6} \ z}}, +\\ +\\ +\displaystyle +\left. +\left( +{t \sp 3} -1 +\right)\ {z \sp 3} \ x - +{{z \sp 3} \ y} - +{t \sp 3} +\right\} +\end{array} +$$ +\returnType{Type: +Union( +WuWenTsunTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t])),...)} + +Solve the system. + +\spadcommand{zeroSetSplit(lp)\$T} +$$ +\begin{array}{@{}l} +\left[ +{\left\{ t, z, y, x \right\}}, +{\left\{ +{{t \sp 3} -1}, +{{z \sp 5} -{t \sp 4}}, +{{{z \sp 3} \ y}+{t \sp 3}}, +{{z \ {x \sp 2}} -t} +\right\}}, +\right. +\\ +\\ +\displaystyle +\left\{ +{{z \sp 5} -{t \sp 4}}, +{{t \sp 4} \ {z \sp 2} \ {y \sp 2}}+ +{2 \ {t \sp 3} \ {z \sp 4} \ y}+ +\left( +-{t \sp 7}+{2 \ {t \sp 4}} -t +\right)\ {z \sp 6}+ +{{t \sp 6} \ z}, +\right. +\\ +\\ +\displaystyle +\left. +\left. +\left( +{t \sp 3} -1 \right)\ {z \sp 3} \ x - +{{z \sp 3} \ y} -{t \sp 3} +\right\} +\right] +\end{array} +$$ +\returnType{Type: +List WuWenTsunTriangularSet(Integer, +IndexedExponents OrderedVariableList [x,y,z,t], +OrderedVariableList [x,y,z,t], +NewSparseMultivariatePolynomial(Integer, +OrderedVariableList [x,y,z,t]))} + +The {\tt RegularTriangularSet} and +{\tt SquareFreeRegularTriangularSet} domain constructors, the +{\tt LazardSetSolvingPackage} package constructors as well as, +{\tt SquareFreeRegularTriangularSet} and +{\tt ZeroDimensionalSolvePackage} package constructors also provide +operations to compute triangular decompositions of algebraic +varieties. These five constructor use a special kind of +characteristic sets, called regular triangular sets. These special +characteristic sets have better properties than the general ones. +Regular triangular sets and their related concepts are presented in +the paper ``On the Theories of Triangular sets'' By P. Aubry, D. Lazard +and M. Moreno Maza (to appear in the Journal of Symbolic Computation). +The decomposition algorithm (due to the third author) available in the +four above constructors provide generally better timings than the +characteristic set method. In fact, the {\tt WUTSET} constructor +remains interesting for the purpose of manipulating characteristic +sets whereas the other constructors are more convenient for solving +polynomial systems. + +Note that the way of understanding triangular decompositions +is detailed in the example of the {\tt RegularTriangularSet} +constructor. + +\section{XPBWPolynomial} +\label{XPBWPolynomialXmpPage} + +Initialisations + +\spadcommand{a:Symbol := 'a } +$$ +a +$$ +\returnType{Type: Symbol} + +\spadcommand{b:Symbol := 'b } +$$ +b +$$ +\returnType{Type: Symbol} + +\spadcommand{RN := Fraction(Integer) } +$$ +\mbox{\rm Fraction Integer} +$$ +\returnType{Type: Domain} + +\spadcommand{word := OrderedFreeMonoid Symbol } +$$ +\mbox{\rm OrderedFreeMonoid Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{lword := LyndonWord(Symbol) } +$$ +\mbox{\rm LyndonWord Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{base := PoincareBirkhoffWittLyndonBasis Symbol } +$$ +\mbox{\rm PoincareBirkhoffWittLyndonBasis Symbol} +$$ +\returnType{Type: Domain} + +\spadcommand{dpoly := XDistributedPolynomial(Symbol, RN) } +$$ +\mbox{\rm XDistributedPolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{rpoly := XRecursivePolynomial(Symbol, RN) } +$$ +\mbox{\rm XRecursivePolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{lpoly := LiePolynomial(Symbol, RN) } +$$ +\mbox{\rm LiePolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{poly := XPBWPolynomial(Symbol, RN) } +$$ +\mbox{\rm XPBWPolynomial(Symbol,Fraction Integer)} +$$ +\returnType{Type: Domain} + +\spadcommand{liste : List lword := LyndonWordsList([a,b], 6) } +$$ +\begin{array}{@{}l} +\left[ +{\left[ a \right]}, +{\left[ b \right]}, +{\left[ a \ b \right]}, +{\left[ {a \sp 2} \ b \right]}, +{\left[ a \ {b \sp 2} \right]}, +{\left[ {a \sp 3} \ b \right]}, +{\left[ {a \sp 2} \ {b \sp 2} \right]}, +{\left[ a \ {b \sp 3} \right]}, +{\left[ {a \sp 4} \ b \right]}, +\right. +\\ +\\ +\displaystyle +{\left[ {a \sp 3} \ {b \sp 2} \right]}, +{\left[ {a \sp 2} \ b \ a \ b \right]}, +{\left[ {a \sp 2} \ {b \sp 3} \right]}, +{\left[ a \ b \ a \ {b \sp 2} \right]}, +{\left[ a \ {b \sp 4} \right]}, +{\left[ {a \sp 5} \ b \right]}, +{\left[ {a \sp 4} \ {b \sp 2} \right]}, +\\ +\\ +\displaystyle +\left. +{\left[ {a \sp 3} \ b \ a \ b \right]}, +{\left[ {a \sp 3} \ {b \sp 3} \right]}, +{\left[ {a \sp 2} \ b \ a \ {b \sp 2} \right]}, +{\left[ {a \sp 2} \ {b \sp 2} \ a \ b \right]}, +{\left[ {a \sp 2} \ {b \sp 4} \right]}, +{\left[ a \ b \ a \ {b \sp 3} \right]}, +{\left[ a \ {b \sp 5} \right]} +\right] +\end{array} +$$ +\returnType{Type: List LyndonWord Symbol} + +Let's make some polynomials + +\spadcommand{0\$poly } +$$ +0 +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{1\$poly } +$$ +1 +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{p : poly := a } +$$ +\left[ +a +\right] +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{q : poly := b } +$$ +\left[ +b +\right] +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{pq: poly := p*q } +$$ +{\left[ a \ b +\right]}+{{\left[ +b +\right]} +\ {\left[ a +\right]}} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +Coerce to distributed polynomial + +\spadcommand{pq :: dpoly } +$$ +a \ b +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Fraction Integer)} + +Check some polynomial operations + +\spadcommand{mirror pq } +$$ +{\left[ b +\right]} +\ {\left[ a +\right]} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{ListOfTerms pq } +$$ +\left[ +{\left[ {k={{\left[ b +\right]} +\ {\left[ a +\right]}}}, +{c=1} +\right]}, +{\left[ {k={\left[ a \ b +\right]}}, +{c=1} +\right]} +\right] +$$ +\returnType{Type: +List Record(k: PoincareBirkhoffWittLyndonBasis Symbol,c: Fraction Integer)} + +\spadcommand{reductum pq } +$$ +\left[ +a \ b +\right] +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{leadingMonomial pq } +$$ +{\left[ b +\right]} +\ {\left[ a +\right]} +$$ +\returnType{Type: PoincareBirkhoffWittLyndonBasis Symbol} + +\spadcommand{coefficients pq } +$$ +\left[ +1, 1 +\right] +$$ +\returnType{Type: List Fraction Integer} + +\spadcommand{leadingTerm pq } +$$ +\left[ +{k={{\left[ b +\right]} +\ {\left[ a +\right]}}}, +{c=1} +\right] +$$ +\returnType{Type: +Record(k: PoincareBirkhoffWittLyndonBasis Symbol,c: Fraction Integer)} + +\spadcommand{degree pq } +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{pq4:=exp(pq,4) } +$$ +\begin{array}{@{}l} +1+ +{\left[ a \ b \right]}+ +{{\left[b \right]}\ +{\left[ a \right]}}+ +{{1\over 2} \ {\left[ a \ b \right]}\ +{\left[ a \ b \right]}}+ +{{1\over 2} \ {\left[ a \ {b \sp 2} \right]}\ +{\left[ a \right]}}+ +{{1\over 2} \ {\left[ b \right]}\ +{\left[ {a \sp 2} \ b \right]}}+ +\\ +\\ +\displaystyle +{{3\over 2} \ {\left[ b \right]}\ +{\left[ a \ b \right]}\ +{\left[ a \right]}}+ +{{1\over 2} \ {\left[ b \right]}\ +{\left[ b \right]}\ +{\left[ a \right]}\ +{\left[ a \right]}} +\end{array} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{log(pq4,4) - pq } +$$ +0 +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +Calculations with verification in {\tt XDistributedPolynomial}. + +\spadcommand{lp1 :lpoly := LiePoly liste.10 } +$$ +\left[ +{a \sp 3} \ {b \sp 2} +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{lp2 :lpoly := LiePoly liste.11 } +$$ +\left[ +{a \sp 2} \ b \ a \ b +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{lp :lpoly := [lp1, lp2] } +$$ +\left[ +{a \sp 3} \ {b \sp 2} \ {a \sp 2} \ b \ a \ b +\right] +$$ +\returnType{Type: LiePolynomial(Symbol,Fraction Integer)} + +\spadcommand{lpd1: dpoly := lp1 } +$$ +{{a \sp 3} \ {b \sp 2}} -{2 \ {a \sp 2} \ b \ a \ b} -{{a \sp 2} \ {b +\sp 2} \ a}+{4 \ a \ b \ a \ b \ a} -{a \ {b \sp 2} \ {a \sp 2}} -{2 +\ b \ a \ b \ {a \sp 2}}+{{b \sp 2} \ {a \sp 3}} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Fraction Integer)} + +\spadcommand{lpd2: dpoly := lp2 } +$$ +\begin{array}{@{}l} +{{a \sp 2} \ b \ a \ b} - +{{a \sp 2} \ {b \sp 2} \ a} - +{3 \ a \ b \ {a \sp 2} \ b}+ +{4 \ a \ b \ a \ b \ a} - +\\ +\\ +\displaystyle +{a \ {b \sp 2} \ {a \sp 2}}+ +{2 \ b \ {a \sp 3} \ b} - +{3 \ b \ {a \sp 2} \ b \ a}+ +{b \ a \ b \ {a \sp 2}} +\end{array} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Fraction Integer)} + +\spadcommand{lpd : dpoly := lpd1 * lpd2 - lpd2 * lpd1 } +$$ +\begin{array}{@{}l} +{{a \sp 3} \ {b \sp 2} \ {a \sp 2} \ b \ a \ b} - +{{a \sp 3} \ {b \sp 2} \ {a \sp 2} \ {b \sp 2} \ a} - +{3 \ {a \sp 3} \ {b \sp 2} \ a \ b \ {a \sp 2} \ b}+ +{4 \ {a \sp 3} \ {b \sp 2} \ a \ b \ a \ b \ a} - +\\ +\displaystyle +{{a \sp 3} \ {b \sp 2} \ a \ {b \sp 2} \ {a \sp 2}}+ +{2 \ {a \sp 3} \ {b \sp 3} \ {a \sp 3} \ b} - +{3 \ {a \sp 3} \ {b \sp 3} \ {a \sp 2} \ b \ a}+ +{{a \sp 3} \ {b \sp 3} \ a \ b \ {a \sp 2}} - +\\ +\displaystyle +{{a \sp 2} \ b \ a \ b \ {a \sp 3} \ {b \sp 2}}+ +{3 \ {a \sp 2} \ b \ a \ b \ {a \sp 2} \ {b \sp 2} \ a}+ +{6 \ {a \sp 2} \ b \ a \ b \ a \ b \ {a \sp 2} \ b} - +{{12} \ {a \sp 2} \ b \ a \ b \ a \ b \ a \ b \ a}+ +\\ +\displaystyle +{3 \ {a \sp 2} \ b \ a \ b \ a \ {b \sp 2} \ {a \sp 2}} - +{4 \ {a \sp 2} \ b \ a \ {b \sp 2} \ {a \sp 3} \ b}+ +{6 \ {a \sp 2} \ b \ a \ {b \sp 2} \ {a \sp 2} \ b \ a} - +{{a \sp 2} \ b \ a \ {b \sp 3} \ {a \sp 3}}+ +\\ +\displaystyle +{{a \sp 2} \ {b \sp 2} \ {a \sp 4} \ {b \sp 2}} - +{3 \ {a \sp 2} \ {b \sp 2} \ {a \sp 3} \ b \ a \ b}+ +{3 \ {a \sp 2} \ {b \sp 2} \ {a \sp 2} \ b \ {a \sp 2} \ b} - +{2 \ {a \sp 2} \ {b \sp 2} \ a \ b \ {a \sp 3} \ b}+ +\\ +\displaystyle +{3 \ {a \sp 2} \ {b \sp 2} \ a \ b \ {a \sp 2} \ b \ a} - +{3 \ {a \sp 2} \ {b \sp 2} \ a \ b \ a \ b \ {a \sp 2}}+ +{{a \sp 2} \ {b \sp 2} \ a \ {b \sp 2} \ {a \sp 3}}+ +{3 \ a \ b \ {a \sp 2} \ b \ {a \sp 3} \ {b \sp 2}} - +\\ +\displaystyle +{6 \ a \ b \ {a \sp 2} \ b \ {a \sp 2} \ b \ a \ b} - +{3 \ a \ b \ {a \sp 2} \ b \ {a \sp 2} \ {b \sp 2} \ a}+ +{{12} \ a \ b \ {a \sp 2} \ b \ a \ b \ a \ b \ a} - +{3 \ a \ b \ {a \sp 2} \ b \ a \ {b \sp 2} \ {a \sp 2}} - +\\ +\displaystyle +{6 \ a \ b \ {a \sp 2} \ {b \sp 2} \ a \ b \ {a \sp 2}}+ +{3 \ a \ b \ {a \sp 2} \ {b \sp 3} \ {a \sp 3}} - +{4 \ a \ b \ a \ b \ {a \sp 4} \ {b \sp 2}}+ +{{12} \ a \ b \ a \ b \ {a \sp 3} \ b \ a \ b} - +\\ +\displaystyle +{{12} \ a \ b \ a \ b \ {a \sp 2} \ b \ {a \sp 2} \ b}+ +{8 \ a \ b \ a \ b \ a \ b \ {a \sp 3} \ b} - +{{12} \ a \ b \ a \ b \ a \ b \ {a \sp 2} \ b \ a}+ +\\ +\displaystyle +{{12} \ a \ b \ a \ b \ a \ b \ a \ b \ {a \sp 2}} - +{4 \ a \ b \ a \ b \ a \ {b \sp 2} \ {a \sp 3}}+ +{a \ {b \sp 2} \ {a \sp 5} \ {b \sp 2}} - +{3 \ a \ {b \sp 2} \ {a \sp 4} \ b \ a \ b}+ +\\ +\displaystyle +{3 \ a \ {b \sp 2} \ {a \sp 3} \ b \ {a \sp 2} \ b} - +{2 \ a \ {b \sp 2} \ {a \sp 2} \ b \ {a \sp 3} \ b}+ +{3 \ a \ {b \sp 2} \ {a \sp 2} \ b \ {a \sp 2} \ b \ a} - +{3 \ a \ {b \sp 2} \ {a \sp 2} \ b \ a \ b \ {a \sp 2}}+ +\\ +\displaystyle +{a \ {b \sp 2} \ {a \sp 2} \ {b \sp 2} \ {a \sp 3}} - +{2 \ b \ {a \sp 3} \ b \ {a \sp 3} \ {b \sp 2}}+ +{4 \ b \ {a \sp 3} \ b \ {a \sp 2} \ b \ a \ b}+ +{2 \ b \ {a \sp 3} \ b \ {a \sp 2} \ {b \sp 2} \ a} - +\\ +\displaystyle +{8 \ b \ {a \sp 3} \ b \ a \ b \ a \ b \ a}+ +{2 \ b \ {a \sp 3} \ b \ a \ {b \sp 2} \ {a \sp 2}}+ +{4 \ b \ {a \sp 3} \ {b \sp 2} \ a \ b \ {a \sp 2}} - +{2 \ b \ {a \sp 3} \ {b \sp 3} \ {a \sp 3}}+ +\\ +\displaystyle +{3 \ b \ {a \sp 2} \ b \ {a \sp 4} \ {b \sp 2}} - +{6 \ b \ {a \sp 2} \ b \ {a \sp 3} \ b \ a \ b} - +{3 \ b \ {a \sp 2} \ b \ {a \sp 3} \ {b \sp 2} \ a}+ +{{12} \ b \ {a \sp 2} \ b \ {a \sp 2} \ b \ a \ b \ a} - +\\ +\displaystyle +{3 \ b \ {a \sp 2} \ b \ {a \sp 2} \ {b \sp 2} \ {a \sp 2}} - +{6 \ b \ {a \sp 2} \ b \ a \ b \ a \ b \ {a \sp 2}}+ +{3 \ b \ {a \sp 2} \ b \ a \ {b \sp 2} \ {a \sp 3}} - +{b \ a \ b \ {a \sp 5} \ {b \sp 2}}+ +\\ +\displaystyle +{3 \ b \ a \ b \ {a \sp 4} \ {b \sp 2} \ a}+ +{6 \ b \ a \ b \ {a \sp 3} \ b \ {a \sp 2} \ b} - +{{12} \ b \ a \ b \ {a \sp 3} \ b \ a \ b \ a}+ +{3 \ b \ a \ b \ {a \sp 3} \ {b \sp 2} \ {a \sp 2}} - +\\ +\displaystyle +{4 \ b \ a \ b \ {a \sp 2} \ b \ {a \sp 3} \ b}+ +{6 \ b \ a \ b \ {a \sp 2} \ b \ {a \sp 2} \ b \ a} - +{b \ a \ b \ {a \sp 2} \ {b \sp 2} \ {a \sp 3}}+ +{{b \sp 2} \ {a \sp 5} \ b \ a \ b} - +\\ +\displaystyle +{{b \sp 2} \ {a \sp 5} \ {b \sp 2} \ a} - +{3 \ {b \sp 2} \ {a \sp 4} \ b \ {a \sp 2} \ b}+ +{4 \ {b \sp 2} \ {a \sp 4} \ b \ a \ b \ a} - +{{b \sp 2} \ {a \sp 4} \ {b \sp 2} \ {a \sp 2}}+ +\\ +\displaystyle +{2 \ {b \sp 2} \ {a \sp 3} \ b \ {a \sp 3} \ b} - +{3 \ {b \sp 2} \ {a \sp 3} \ b \ {a \sp 2} \ b \ a}+ +{{b \sp 2} \ {a \sp 3} \ b \ a \ b \ {a \sp 2}} +\end{array} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Fraction Integer)} + +\spadcommand{lp :: dpoly - lpd } +$$ +0 +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Fraction Integer)} + +Calculations with verification in {\tt XRecursivePolynomial}. + +\spadcommand{p := 3 * lp } +$$ +3 \ {\left[ {a \sp 3} \ {b \sp 2} \ {a \sp 2} \ b \ a \ b +\right]} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{q := lp1 } +$$ +\left[ +{a \sp 3} \ {b \sp 2} +\right] +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{pq:= p * q } +$$ +3 \ +{\left[ {a \sp 3} \ {b \sp 2} \ {a \sp 2} \ b \ a \ b +\right]}\ +{\left[ {a \sp 3} \ {b \sp 2} +\right]} +$$ +\returnType{Type: XPBWPolynomial(Symbol,Fraction Integer)} + +\spadcommand{pr:rpoly := p :: rpoly } +$$ +\begin{array}{@{}l} +a \ +\left( a \ +\left( a \ b \ b \ +\left( a \ +\left( a \ b \ +\left( a \ b \ 3+b \ a \ +\left( -3 +\right) +\right)+b\ +\right. +\right. +\right. +\right. +\\ +\displaystyle +\left. +\left( a \ +\left( a \ b \ +\left( -9 +\right)+b\ a \ {12} +\right)+b\ a \ a \ +\left( -3 +\right) +\right) +\right)+b\ a \ +\\ +\displaystyle +\left. +\left( a \ +\left( a \ b \ 6+b \ a \ +\left( -9 +\right) +\right)+b\ a \ a \ 3 +\right) +\right)+b\ +\left( a \ b \ +\left( a \ +\left( a \ +\right. +\right. +\right. +\\ +\displaystyle +\left( a \ b \ b \ +\left( -3 +\right)+b\ b \ a \ 9 +\right)+b\ +\left( a \ +\left( a \ b \ {18}+b \ a \ +\left( -{36} +\right) +\right)+b\ a \ a \ 9 +\right. +\\ +\displaystyle +\left. +\left. +\right) +\right)+b\ +\left( a \ a \ +\left( a \ b \ +\left( -{12} +\right)+b\ a \ {18} +\right)+b\ a \ a \ a \ +\left( -3 +\right) +\right) +\\ +\displaystyle +\left. +\right)+b\ a \ +\left( a \ +\left( a \ +\left( a \ b \ b \ 3+b \ a \ b \ +\left( -9 +\right) +\right)+b\ a \ a \ b \ 9 +\right)+b\ +\left( a \ +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left( a \ +\left( a \ b \ +\left( -6 +\right)+b\ a \ 9 +\right)+b\ a \ a \ +\left( -9 +\right) +\right)+b\ a \ a \ a \ 3 +\right) +\right) +\\ +\displaystyle +\left. +\left. +\right) +\right)+b\ +\left( a \ +\left( a \ b \ +\left( a \ +\left( a \ +\left( a \ b \ b \ 9+b \ +\left( a \ b \ +\left( -{18} +\right)+b\ a \ +\right. +\right. +\right. +\right. +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left( -9 +\right) +\right) +\right)+b\ +\left( a \ b \ a \ {36}+b \ a \ a \ +\left( -9 +\right) +\right) +\right)+b\ +\left( a \ b \ a \ a \ +\right. +\\ +\displaystyle +\left. +\left. +\left( -{18} +\right)+b\ a \ a \ a \ 9 +\right) +\right)+b\ a \ +\left( a \ +\left( a \ +\left( a \ b \ b \ +\left( -{12} +\right)+b\ a \ b \ {36} +\right)+b\ a \ a \ b \ +\right. +\right. +\\ +\displaystyle +\left. +\left( -{36} +\right) +\right)+b\ +\left( a \ +\left( a \ +\left( a \ b \ {24}+b \ a \ +\left( -{36} +\right) +\right)+b\ a \ a \ {36} +\right)+b\ a \ a \ a \ +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left( -{12} +\right) +\right) +\right) +\right)+b\ a \ a \ +\left( a \ +\left( a \ +\left( a \ b \ b \ 3+b \ a \ b \ +\left( -9 +\right) +\right. +\right. +\right. +\\ +\displaystyle +\left. +\left. +\right)+b\ a \ a \ b \ 9 +\right)+b\ +\left( a \ +\left( a \ +\left( a \ b \ +\left( -6 +\right)+b\ a \ 9 +\right)+b\ a \ a \ +\left( -9 +\right) +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left. +\left. +\right)+b\ a \ a \ a \ 3 +\right) +\right) +\right) +\right)+b\ +\left( a \ +\left( a \ +\left( a \ b \ +\left( a \ +\left( a \ +\right. +\right. +\right. +\right. +\right. +\\ +\displaystyle +\left( a \ b \ b \ +\left( -6 +\right)+b\ +\left( a \ b \ {12}+b \ a \ 6 +\right) +\right)+b\ +\left( a \ b \ a \ +\left( -{24} +\right)+b\ a \ a \ 6 +\right) +\\ +\displaystyle +\left. +\left. +\right)+b\ +\left( a \ b \ a \ a \ {12}+b \ a \ a \ a \ +\left( -6 +\right) +\right) +\right)+b\ a \ +\left( a \ +\left( a \ +\left( a \ b \ b \ 9+b \ +\left( a \ b \ +\left( -{18} +\right. +\right. +\right. +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left. +\right)+b\ a \ +\left( -9 +\right) +\right) +\right)+b\ +\left( a \ b \ a \ {36}+b \ a \ a \ +\left( -9 +\right) +\right) +\right)+b\ +\left( a \ b \ a \ a \ +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left( -{18} +\right)+b\ a \ a \ a \ 9 +\right) +\right) +\right)+b\ a \ a \ +\left( a \ +\left( a \ +\left( a \ b \ b \ +\left( -3 +\right)+b\ b \ a \ 9 +\right. +\right. +\right. +\\ +\displaystyle +\left. +\left. +\right)+b\ +\left( a \ +\left( a \ b \ {18}+b \ a \ +\left( -{36} +\right) +\right)+b\ a \ a \ 9 +\right) +\right)+b\ +\left( a \ a \ +\left( a \ b \ +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left. +\left( -{12} +\right)+b\ a \ {18} +\right)+b\ a \ a \ a \ +\left( -3 +\right) +\right) +\right) +\right)+b\ a \ a \ a \ +\left( a \ +\right. +\\ +\displaystyle +\left( a \ b \ +\left( a \ b \ 3+b \ a \ +\left( -3 +\right) +\right)+b\ +\left( a \ +\left( a \ b \ +\left( -9 +\right)+b\ a \ {12} +\right)+b\ a \ a \ +\right. +\right. +\\ +\displaystyle +\left. +\left. +\left. +\left. +\left( -3 +\right) +\right) +\right)+b\ a \ +\left( a \ +\left( a \ b \ 6+b \ a \ +\left( -9 +\right) +\right)+b\ a \ a \ 3 +\right) +\right) +\right) +\end{array} +$$ +\returnType{Type: XRecursivePolynomial(Symbol,Fraction Integer)} + +\spadcommand{qr:rpoly := q :: rpoly } +$$ +\begin{array}{@{}l} +a \ +\left( a \ +\left( a \ b \ b \ 1+b \ +\left( a \ b \ +\left( -2 +\right)+b\ a \ +\left( -1 +\right) +\right) +\right)+ +\right. +\\ +\\ +\displaystyle +\left. +b\ +\left( a \ b \ a \ 4+b \ a \ a \ +\left( -1 +\right) +\right) +\right)+ +\\ +\\ +\displaystyle +b\ +\left( a \ b \ a \ a \ +\left( -2 +\right)+b\ a \ a \ a \ 1 +\right) +\end{array} +$$ +\returnType{Type: XRecursivePolynomial(Symbol,Fraction Integer)} + +\spadcommand{pq :: rpoly - pr*qr } +$$ +0 +$$ +\returnType{Type: XRecursivePolynomial(Symbol,Fraction Integer)} + +\section{XPolynomial} +\label{XPolynomialXmpPage} + +The {\tt XPolynomial} domain constructor implements multivariate +polynomials whose set of variables is {\tt Symbol}. These variables +do not commute. The only parameter of this construtor is the +coefficient ring which may be non-commutative. However, coefficients +and variables commute. The representation of the polynomials is +recursive. The abbreviation for {\tt XPolynomial} is {\tt XPOLY}. + +Other constructors like {\tt XPolynomialRing}, +{\tt XRecursivePolynomial} as well as {\tt XDistributedPolynomial}, +{\tt LiePolynomial} and {\tt XPBWPolynomial} implement multivariate +polynomials in non-commutative variables. + +We illustrate now some of the facilities of the +{\tt XPOLY} domain constructor. + +Define a polynomial ring over the integers. + +\spadcommand{poly := XPolynomial(Integer) } +$$ +\mbox{\rm XPolynomial Integer} +$$ +\returnType{Type: Domain} + +Define a first polynomial, + +\spadcommand{pr: poly := 2*x + 3*y-5 } +$$ +-5+{x \ 2}+{y \ 3} +$$ +\returnType{Type: XPolynomial Integer} + +and a second one. + +\spadcommand{pr2: poly := pr*pr } +$$ +{25}+{x \ {\left( -{20}+{x \ 4}+{y \ 6} +\right)}}+{y +\ {\left( -{30}+{x \ 6}+{y \ 9} +\right)}} +$$ +\returnType{Type: XPolynomial Integer} + +Rewrite {\bf pr} in a distributive way, + +\spadcommand{pd := expand pr} +$$ +-5+{2 \ x}+{3 \ y} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Integer)} + +compute its square, + +\spadcommand{pd2 := pd*pd } +$$ +{25} -{{20} \ x} -{{30} \ y}+{4 \ {x \sp 2}}+{6 \ x \ y}+{6 \ y \ +x}+{9 \ {y \sp 2}} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Integer)} + +and checks that: + +\spadcommand{expand(pr2) - pd2 } +$$ +0 +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Integer)} + +We define: + +\spadcommand{qr := pr**3 } +$$ +\begin{array}{@{}l} +-{125}+{x \ +{\left( {150}+{x \ +{\left( -{60}+{x \ 8}+{y \ {12}} +\right)}}+{y\ +{\left( -{90}+{x \ {12}}+{y \ {18}} +\right)}} +\right)}}+ +\\ +\\ +\displaystyle +{y\ +{\left( {225}+{x \ +{\left( -{90}+{x \ {12}}+{y \ {18}} +\right)}}+{y\ +{\left( -{135}+{x \ {18}}+{y \ {27}} +\right)}} +\right)}} +\end{array} +$$ +\returnType{Type: XPolynomial Integer} + +and: + +\spadcommand{qd := pd**3 } +$$ +\begin{array}{@{}l} +-{125}+ +{{150} \ x}+ +{{225} \ y} - +{{60} \ {x \sp 2}} - +{{90} \ x \ y} - +{{90} \ y \ x} - +{{135} \ {y \sp 2}}+ +{8 \ {x \sp 3}}+ +{{12} \ {x \sp 2} \ y}+ +\\ +\\ +\displaystyle +{{12} \ x \ y \ x}+ +{{18} \ x \ {y \sp 2}}+ +{{12} \ y \ {x \sp 2}}+ +{{18} \ y \ x \ y}+ +{{18} \ {y \sp 2} \ x}+ +{{27} \ {y \sp 3}} +\end{array} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Integer)} + +We truncate {\bf qd} at degree {\bf 3}: + +\spadcommand{ trunc(qd,2) } +$$ +-{125}+{{150} \ x}+{{225} \ y} -{{60} \ {x \sp 2}} -{{90} \ x \ y} +-{{90} \ y \ x} -{{135} \ {y \sp 2}} +$$ +\returnType{Type: XDistributedPolynomial(Symbol,Integer)} + +The same for {\bf qr}: + +\spadcommand{trunc(qr,2) } +$$ +-{125}+{x \ {\left( {150}+{x \ {\left( -{60} +\right)}}+{y +\ {\left( -{90} +\right)}} +\right)}}+{y +\ {\left( {225}+{x \ {\left( -{90} +\right)}}+{y +\ {\left( -{135} +\right)}} +\right)}} +$$ +\returnType{Type: XPolynomial Integer} + +We define: + +\spadcommand{Word := OrderedFreeMonoid Symbol } +$$ +\mbox{\rm OrderedFreeMonoid Symbol} +$$ +\returnType{Type: Domain} + +and: + +\spadcommand{w: Word := x*y**2 } +$$ +x \ {y \sp 2} +$$ +\returnType{Type: OrderedFreeMonoid Symbol} + +We can compute the right-quotient of {\bf qr} by {\bf r}: + +\spadcommand{rquo(qr,w) } +$$ +18 +$$ +\returnType{Type: XPolynomial Integer} + +and the shuffle-product of {\bf pr} by {\bf r}: + +\spadcommand{sh(pr,w::poly) } +$$ +{x \ {\left( {x \ y \ y \ 4}+{y \ {\left( {x \ y \ 2}+{y \ {\left( +-5+{x \ 2}+{y \ 9} +\right)}} +\right)}} +\right)}}+{y +\ x \ y \ y \ 3} +$$ +\returnType{Type: XPolynomial Integer} + +\section{XPolynomialRing} +\label{XPolynomialRingXmpPage} + +The {\tt XPolynomialRing} domain constructor implements generalized +polynomials with coefficients from an arbitrary {\tt Ring} (not +necessarily commutative) and whose exponents are words from an +arbitrary {\tt OrderedMonoid} (not necessarily commutative too). Thus +these polynomials are (finite) linear combinations of words. + +This constructor takes two arguments. The first one is a {\tt Ring} +and the second is an {\tt OrderedMonoid}. The abbreviation for +{\tt XPolynomialRing} is {\tt XPR}. + +Other constructors like {\tt XPolynomial}, {\tt XRecursivePolynomial} +{\tt XDistributedPolynomial}, {\tt LiePolynomial} and +{\tt XPBWPolynomial} implement multivariate polynomials in non-commutative +variables. + +We illustrate now some of the facilities of the {\tt XPR} domain constructor. + +Define the free ordered monoid generated by the symbols. + +\spadcommand{Word := OrderedFreeMonoid(Symbol) } +$$ +\mbox{\rm OrderedFreeMonoid Symbol} +$$ +\returnType{Type: Domain} + +Define the linear combinations of these words with integer coefficients. + +\spadcommand{poly:= XPR(Integer,Word) } +$$ +\mbox{\rm XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} +$$ +\returnType{Type: Domain} + +Then we define a first element from {\bf poly}. + +\spadcommand{p:poly := 2 * x - 3 * y + 1 } +$$ +1+{2 \ x} -{3 \ y} +$$ +\returnType{Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} + +And a second one. + +\spadcommand{q:poly := 2 * x + 1 } +$$ +1+{2 \ x} +$$ +\returnType{Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} + +We compute their sum, + +\spadcommand{p + q} +$$ +2+{4 \ x} -{3 \ y} +$$ +\returnType{Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} + +their product, + +\spadcommand{p * q} +$$ +1+{4 \ x} -{3 \ y}+{4 \ {x \sp 2}} -{6 \ y \ x} +$$ +\returnType{Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} + +and see that variables do not commute. + +\spadcommand{(p+q)**2-p**2-q**2-2*p*q} +$$ +-{6 \ x \ y}+{6 \ y \ x} +$$ +\returnType{Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)} + +Now we define a ring of square matrices, + +\spadcommand{M := SquareMatrix(2,Fraction Integer) } +$$ +\mbox{\rm SquareMatrix(2,Fraction Integer)} +$$ +\returnType{Type: Domain} + +and the linear combinations of words with these matrices as coefficients. + +\spadcommand{poly1:= XPR(M,Word) } +$$ +\begin{array}{@{}l} +{\rm XPolynomialRing(SquareMatrix(2,Fraction Integer),} +\\ +\displaystyle +{\rm \ \ OrderedFreeMonoid Symbol)} +\end{array} +$$ +\returnType{Type: Domain} + +Define a first matrix, + +\spadcommand{m1:M := matrix [ [i*j**2 for i in 1..2] for j in 1..2] } +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +4 & 8 +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Integer)} + +a second one, + +\spadcommand{m2:M := m1 - 5/4 } +$$ +\left[ +\begin{array}{cc} +-{1 \over 4} & 2 \\ +4 & {{27} \over 4} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Integer)} + +and a third one. + +\spadcommand{m3: M := m2**2 } +$$ +\left[ +\begin{array}{cc} +{{129} \over {16}} & {13} \\ +{26} & {{857} \over {16}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Integer)} + +Define a polynomial, + +\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3 } +$$ +{\left[ +\begin{array}{cc} +-{2 \over 3} & 0 \\ +0 & -{2 \over 3} +\end{array} +\right]}+{{\left[ +\begin{array}{cc} +1 & 2 \\ +4 & 8 +\end{array} +\right]} +\ x}+{{\left[ +\begin{array}{cc} +-{1 \over 4} & 2 \\ +4 & {{27} \over 4} +\end{array} +\right]} +\ y}+{{\left[ +\begin{array}{cc} +{{129} \over {16}} & {13} \\ +{26} & {{857} \over {16}} +\end{array} +\right]} +\ z} +$$ +\returnType{Type: +XPolynomialRing( +SquareMatrix(2,Fraction Integer), +OrderedFreeMonoid Symbol)} + +a second one, + +\spadcommand{qm:poly1 := pm - m1*x } +$$ +{\left[ +\begin{array}{cc} +-{2 \over 3} & 0 \\ +0 & -{2 \over 3} +\end{array} +\right]}+{{\left[ +\begin{array}{cc} +-{1 \over 4} & 2 \\ +4 & {{27} \over 4} +\end{array} +\right]} +\ y}+{{\left[ +\begin{array}{cc} +{{129} \over {16}} & {13} \\ +{26} & {{857} \over {16}} +\end{array} +\right]} +\ z} +$$ +\returnType{Type: +XPolynomialRing( +SquareMatrix(2,Fraction Integer), +OrderedFreeMonoid Symbol)} + +and the following power. + +\spadcommand{qm**3 } +$$ +\begin{array}{@{}l} +{\left[ +\begin{array}{cc} +-{8 \over {27}} & 0 \\ +0 & -{8 \over {27}} +\end{array} +\right]}+ +{{\left[ +\begin{array}{cc} +-{1 \over 3} & {8 \over 3} \\ +{{16} \over 3} & 9 +\end{array} +\right]}\ y}+ +{{\left[ +\begin{array}{cc} +{{43} \over 4} & {{52} \over 3} \\ +{{104} \over 3} & {{857} \over {12}} +\end{array} +\right]}\ z}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +-{{129} \over 8} & -{26} \\ +-{52} & -{{857} \over 8} +\end{array} +\right]}\ {y \sp 2}}+ +{{\left[ +\begin{array}{cc} +-{{3199} \over {32}} & -{{831} \over 4} \\ +-{{831} \over 2} & -{{26467} \over {32}} +\end{array} +\right]}\ y \ z}+ +{{\left[ +\begin{array}{cc} +-{{3199} \over {32}} & -{{831} \over 4} \\ +-{{831} \over 2} & -{{26467} \over {32}} +\end{array} +\right]}\ z \ y}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +-{{103169} \over {128}} & -{{6409} \over 4} \\ +-{{6409} \over 2} & -{{820977} \over {128}} +\end{array} +\right]}\ {z \sp 2}}+ +{{\left[ +\begin{array}{cc} +{{3199} \over {64}} & {{831} \over 8} \\ +{{831} \over 4} & {{26467} \over {64}} +\end{array} +\right]}\ {y \sp 3}}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +{{103169} \over {256}} & {{6409} \over 8} \\ +{{6409} \over 4} & {{820977} \over {256}} +\end{array} +\right]}\ {y \sp 2} \ z}+ +{{\left[ +\begin{array}{cc} +{{103169} \over {256}} & {{6409} \over 8} \\ +{{6409} \over 4} & {{820977} \over {256}} +\end{array} +\right]}\ y \ z \ y}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +{{3178239} \over {1024}} & {{795341} \over {128}} \\ +{{795341} \over {64}} & {{25447787} \over {1024}} +\end{array} +\right]}\ y \ {z \sp 2}}+ +{{\left[ +\begin{array}{cc} +{{103169} \over {256}} & {{6409} \over 8} \\ +{{6409} \over 4} & {{820977} \over {256}} +\end{array} +\right]}\ z \ {y \sp 2}}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +{{3178239} \over {1024}} & {{795341} \over {128}} \\ +{{795341} \over {64}} & {{25447787} \over {1024}} +\end{array} +\right]}\ z \ y \ z}+ +{{\left[ +\begin{array}{cc} +{{3178239} \over {1024}} & {{795341} \over {128}} \\ +{{795341} \over {64}} & {{25447787} \over {1024}} +\end{array} +\right]}\ {z \sp 2} \ y}+ +\\ +\\ +\displaystyle +{{\left[ +\begin{array}{cc} +{{98625409} \over {4096}} & {{12326223} \over {256}} \\ +{{12326223} \over {128}} & {{788893897} \over {4096}} +\end{array} +\right]}\ {z \sp 3}} +\end{array} +$$ +\returnType{Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)} + +\section{ZeroDimensionalSolvePackage} +\label{ZeroDimensionalSolvePackageXmpPage} + +The {\tt ZeroDimensionalSolvePackage} package constructor provides +operations for computing symbolically the complex or real roots of +zero-dimensional algebraic systems. + +The package provides {\bf no} multiplicity information (i.e. some +returned roots may be double or higher) but only distinct roots are +returned. + +Complex roots are given by means of univariate representations of +irreducible regular chains. These representations are computed by the +\spadfunFrom{univariateSolve}{ZeroDimensionalSolvePackage} operation +(by calling the {\tt InternalRationalUnivariateRepresentationPackage} +package constructor which does the job). + +Real roots are given by means of tuples of coordinates lying in the +{\tt RealClosure} of the coefficient ring. They are computed by the +\spadfunFrom{realSolve}{ZeroDimensionalSolvePackage} and +\spadfunFrom{positiveSolve}{ZeroDimensionalSolvePackage} operations. +The former computes all the solutions of the input system with real +coordinates whereas the later concentrate on the solutions with +(strictly) positive coordinates. In both cases, the computations are +performed by the {\tt RealClosure} constructor. + +Both computations of complex roots and real roots rely on triangular +decompositions. These decompositions can be computed in two different +ways. First, by a applying the +\spadfunFrom{zeroSetSplit}{RegularTriangularSet} operation from the +{\tt REGSET} domain constructor. In that case, no Groebner bases are +computed. This strategy is used by default. Secondly, by applying +the \spadfunFrom{zeroSetSplit}{LexTriangularPackage} from +{\tt LEXTRIPK}. To use this later strategy with the operations +\spadfunFrom{univariateSolve}{ZeroDimensionalSolvePackage}, +\spadfunFrom{realSolve}{ZeroDimensionalSolvePackage} and +\spadfunFrom{positiveSolve}{ZeroDimensionalSolvePackage} one just +needs to use an extra boolean argument. + +Note that the way of understanding triangular decompositions +is detailed in the example of the {\tt RegularTriangularSet} +constructor. + +The {\tt ZeroDimensionalSolvePackage} constructor takes three +arguments. The first one {\bf R} is the coefficient ring; it must +belong to the categories {\tt OrderedRing}, {\tt EuclideanDomain}, +{\tt CharacteristicZero} and {\tt RealConstant}. This means +essentially that {\bf R} is {\tt Integer} or {\tt Fraction(Integer)}. +The second argument {\bf ls} is the list of variables involved in the +systems to solve. The third one MUST BE {\bf concat(ls,s)} where +{\bf s} is an additional symbol used for the univariate representations. +The abbreviation for {\tt ZeroDimensionalSolvePackage} is {\tt ZDSOLVE}. + +We illustrate now how to use the constructor {\tt ZDSOLVE} by two +examples: the {\em Arnborg and Lazard} system and the {\em L-3} system +(Aubry and Moreno Maza). Note that the use of this package is also +demonstrated in the example of the {\tt LexTriangularPackage} +constructor. + +Define the coefficient ring. + +\spadcommand{R := Integer } +$$ +Integer +$$ +\returnType{Type: Domain} + +Define the lists of variables: + +\spadcommand{ls : List Symbol := [x,y,z,t] } +$$ +\left[ +x, y, z, t +\right] +$$ +\returnType{Type: List Symbol} + +and: + +\spadcommand{ls2 : List Symbol := [x,y,z,t,new()\$Symbol] } +$$ +\left[ +x, y, z, t, \%A +\right] +$$ +\returnType{Type: List Symbol} + +Call the package: + +\spadcommand{pack := ZDSOLVE(R,ls,ls2)} +$$ +ZeroDimensionalSolvePackage(Integer,[x,y,z,t],[x,y,z,t,%A]) +$$ +\returnType{Type: Domain} + +Define a polynomial system (Arnborg-Lazard) + +\spadcommand{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z } +$$ +{x \ y \ {z \sp 2}}+{{\left( {x \ {y \sp 2}}+{{\left( {x \sp 2}+x+1 +\right)} +\ y}+x +\right)} +\ z}+{x \ y} +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z } +$$ +{x \ {y \sp 2} \ {z \sp 2}}+{{\left( {{x \sp 2} \ {y \sp 2}}+{{\left( {x +\sp 2}+x+1 +\right)} +\ y}+1 +\right)} +\ z}+x +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1 } +$$ +{{x \sp 2} \ {y \sp 2} \ {z \sp 2}}+{{\left( {{\left( {x \sp 2}+x +\right)} +\ {y \sp 2}}+{x \ y}+x+1 +\right)} +\ z}+1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{lp := [p1, p2, p3]} +$$ +\begin{array}{@{}l} +\left[ +{{x \ y \ {z \sp 2}}+{{\left( {x \ {y \sp 2}}+{{\left( {x \sp 2}+x+1 +\right)} +\ y}+x +\right)} +\ z}+{x \ y}}, +\right. +\\ +\\ +\displaystyle +{{x \ {y \sp 2} \ {z \sp 2}}+{{\left( {{x \sp 2} \ {y +\sp 2}}+{{\left( {x \sp 2}+x+1 +\right)} +\ y}+1 +\right)} +\ z}+x}, +\\ +\\ +\displaystyle +\left. +{{{x \sp 2} \ {y \sp 2} \ {z \sp 2}}+{{\left( {{\left( {x \sp +2}+x +\right)} +\ {y \sp 2}}+{x \ y}+x+1 +\right)} +\ z}+1} +\right] +\end{array} +$$ +\returnType{Type: List Polynomial Integer} + +Note that these polynomials do not involve the variable {\bf t}; +we will use it in the second example. + +First compute a decomposition into regular chains (i.e. regular +triangular sets). + +\spadcommand{triangSolve(lp)\$pack } +$$ +\begin{array}{@{}l} +\left[ +\left\{ +{z \sp {20}} - +{6 \ {z \sp {19}}} - +{{41} \ {z \sp {18}}}+ +{{71} \ {z \sp {17}}}+ +{{106} \ {z \sp {16}}}+ +{{92} \ {z \sp {15}}}+ +{{197} \ {z \sp {14}}}+ +\right. +\right. +\\ +\\ +\displaystyle +{{145} \ {z \sp {13}}}+ +{{257} \ {z \sp {12}}}+ +{{278} \ {z \sp {11}}}+ +{{201} \ {z \sp {10}}}+ +{{278} \ {z \sp 9}}+ +{{257} \ {z \sp 8}}+ +{{145} \ {z \sp 7}}+ +\\ +\\ +\displaystyle +{{197} \ {z \sp 6}}+ +{{92} \ {z \sp 5}}+ +{{106} \ {z \sp 4}}+ +{{71} \ {z \sp 3}} - +{{41} \ {z \sp 2}} - +{6 \ z}+1, +\\ +\\ +\displaystyle +\left( {{14745844} \ {z \sp {19}}}+ +{{50357474} \ {z \sp {18}}} - +{{130948857} \ {z \sp {17}}} - +{{185261586} \ {z \sp {16}}} - +\right. +\\ +\\ +\displaystyle +{{180077775} \ {z \sp {15}}} - +{{338007307} \ {z \sp {14}}} - +{{275379623} \ {z \sp {13}}} - +{{453190404} \ {z \sp {12}}} - +\\ +\\ +\displaystyle +{{474597456} \ {z \sp {11}}} - +{{366147695} \ {z \sp {10}}} - +{{481433567} \ {z \sp 9}} - +{{430613166} \ {z \sp 8}} - +\\ +\\ +\displaystyle +{{261878358} \ {z \sp 7}} - +{{326073537} \ {z \sp 6}} - +{{163008796} \ {z \sp 5}} - +{{177213227} \ {z \sp 4}} - +\\ +\\ +\displaystyle +\left. +{{104356755} \ {z \sp 3}}+ +{{65241699} \ {z \sp 2}}+ +{{9237732} \ z} - +{1567348} +\right)\ y+ +\\ +\\ +\displaystyle +{{1917314} \ {z \sp {19}}}+ +{{6508991} \ {z \sp {18}}} - +{{16973165} \ {z \sp {17}}} - +{{24000259} \ {z \sp {16}}} - +\\ +\\ +\displaystyle +{{23349192} \ {z \sp {15}}} - +{{43786426} \ {z \sp {14}}} - +{{35696474} \ {z \sp {13}}} - +{{58724172} \ {z \sp {12}}} - +\\ +\\ +\displaystyle +{{61480792} \ {z \sp {11}}} - +{{47452440} \ {z \sp {10}}} - +{{62378085} \ {z \sp 9}} - +{{55776527} \ {z \sp 8}} - +\\ +\\ +\displaystyle +{{33940618} \ {z \sp 7}} - +{{42233406} \ {z \sp 6}} - +{{21122875} \ {z \sp 5}} - +{{22958177} \ {z \sp 4}} - +\\ +\\ +\displaystyle +{{13504569} \ {z \sp 3}}+ +{{8448317} \ {z \sp 2}}+ +{{1195888} \ z} - +{202934}, +\\ +\\ +\displaystyle +\left. +\left. +\left( +\left( {z \sp 3} -{2 \ z} +\right)\ {y \sp 2}+ +\left( -{z \sp 3} - +{z \sp 2} - +{2 \ z} - +1 +\right)\ y - +{z \sp 2} -z+1 +\right)\ x+ +{z \sp 2} -1 +\right\} +\right] +\end{array} +$$ +\returnType{Type: List RegularChain(Integer,[x,y,z,t])} + +We can see easily from this decomposition (consisting of a single +regular chain) that the input system has 20 complex roots. + +Then we compute a univariate representation of this regular chain. + +\spadcommand{univariateSolve(lp)\$pack} +$$ +\begin{array}{@{}l} +\left[ +\left[ +complexRoots= +{? \sp {12}} - +{{12} \ {? \sp {11}}}+ +{{24} \ {? \sp {10}}}+ +{4 \ {? \sp 9}} - +{9 \ {? \sp 8}}+ +{{27} \ {? \sp 7}} - +\right. +\right. +\\ +\\ +\displaystyle +{{21} \ {? \sp 6}}+ +{{27} \ {? \sp 5}} - +{9 \ {? \sp 4}}+ +{4 \ {? \sp 3}}+ +{{24} \ {? \sp 2}} - +{{12} \ ?}+1, +\\ +\\ +\displaystyle +coordinates= +\\ +\displaystyle +\left[ +{{63} \ x}+ +{{62} \ { \%A \sp {11}}} - +{{721} \ { \%A \sp {10}}}+ +{{1220} \ { \%A \sp 9}}+ +{{705} \ { \%A \sp 8}} - +{{285} \ { \%A \sp 7}}+ +\right. +\\ +\\ +\displaystyle +{{1512} \ { \%A \sp 6}} - +{{735} \ { \%A \sp 5}}+ +{{1401} \ { \%A \sp 4}} - +{{21} \ { \%A \sp 3}}+ +{{215} \ { \%A \sp 2}}+ +{{1577} \ \%A} -{142}, +\\ +\\ +\displaystyle +{63} \ y - +{{75} \ { \%A \sp {11}}}+ +{{890} \ { \%A \sp {10}}} - +{{1682} \ { \%A \sp 9}} - +{{516} \ { \%A \sp 8}}+ +{{588} \ { \%A \sp 7}} - +{{1953} \ { \%A \sp 6}}+ +\\ +\\ +\displaystyle +{{1323} \ { \%A \sp 5}} - +{{1815} \ { \%A \sp 4}}+ +{{426} \ { \%A \sp 3}} - +{{243} \ { \%A \sp 2}} - +{{1801} \ \%A}+{679}, +\\ +\\ +\displaystyle +\left. +\left. +{z - \%A} +\right] +\right], +\\ +\\ +\displaystyle +\left[ +complexRoots={{? \sp 6}+{? \sp 5}+{? \sp 4}+{? \sp 3}+{? \sp 2}+?+1}, +\right. +\\ +\displaystyle +\left. +\left. +{coordinates= +{\left[ {x -{ \%A \sp 5}}, {y -{ \%A \sp 3}}, {z - \%A} +\right]}} +\right], +\right. +\\ +\\ +\displaystyle +\left. +\left. +{\left[ +{complexRoots={{? \sp 2}+{5 \ ?}+1}}, +{coordinates={\left[ {x -1}, {y -1}, {z - \%A} +\right]}} +\right]} +\right] +\right. +\end{array} +$$ +\returnType{Type: +List Record( +complexRoots: SparseUnivariatePolynomial Integer, +coordinates: List Polynomial Integer)} + +We see that the zeros of our regular chain are split into three components. +This is due to the use of univariate polynomial factorization. + +Each of these components consist of two parts. The first one is an +irreducible univariate polynomial {\bf p(?)} which defines a simple +algebraic extension of the field of fractions of {\bf R}. The second +one consists of multivariate polynomials {\bf pol1(x,\%A)}, +{\bf pol2(y,\%A)} and {\bf pol3(z,\%A)}. Each of these polynomials involve +two variables: one is an indeterminate {\bf x}, {\bf y} or {\bf z} of +the input system {\bf lp} and the other is {\bf \%A} which represents +any root of {\bf p(?)}. Recall that this {\bf \%A} is the last +element of the third parameter of {\tt ZDSOLVE}. Thus any complex +root {\bf ?} of {\bf p(?)} leads to a solution of the input system +{\bf lp} by replacing {\bf \%A} by this {\bf ?} in {\bf pol1(x,\%A)}, +{\bf pol2(y,\%A)} and {\bf pol3(z,\%A)}. Note that the polynomials +{\bf pol1(x,\%A)}, {\bf pol2(y,\%A)} and {\bf pol3(z,\%A)} have degree +one w.r.t. {\bf x}, {\bf y} or {\bf z} respectively. This is always +the case for all univariate representations. Hence the operation +{\bf univariateSolve} replaces a system of multivariate polynomials by a +list of univariate polynomials, what justifies its name. Another +example of univariate representations illustrates the +{\tt LexTriangularPackage} package constructor. + +We now compute the solutions with real coordinates: + +\spadcommand{lr := realSolve(lp)\$pack } +$$ +\begin{array}{@{}l} +\left[ +\left[ +{ \%B1}, +\right. +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B1} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B1} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B1} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B1} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B1} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B1} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B1} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B1} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B1} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B1} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B1} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B1} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B1} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B1} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B1} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B1} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B1} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B1} \sp 2}} - +{{{8270} \over {343}} \ { \%B1}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B1} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B1} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B1} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B1} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B1} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B1} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B1} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B1} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B1} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B1} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B1} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B1} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B1} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B1} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B1} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B1} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B1} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B1} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B1}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B2}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B2} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B2} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B2} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B2} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B2} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B2} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B2} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B2} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B2} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B2} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B2} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B2} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B2} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B2} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B2} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B2} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B2} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B2} \sp 2}} - +{{{8270} \over {343}} \ { \%B2}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B2} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B2} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B2} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B2} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B2} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B2} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B2} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B2} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B2} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B2} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B2} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B2} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B2} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B2} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B2} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B2} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B2} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B2} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B2}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B3}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B3} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B3} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B3} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B3} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B3} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B3} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B3} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B3} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B3} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B3} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B3} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B3} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B3} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B3} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B3} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B3} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B3} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B3} \sp 2}} - +{{{8270} \over {343}} \ { \%B3}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B3} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B3} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B3} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B3} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B3} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B3} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B3} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B3} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B3} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B3} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B3} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B3} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B3} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B3} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B3} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B3} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B3} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B3} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B3}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B4}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B4} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B4} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B4} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B4} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B4} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B4} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B4} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B4} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B4} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B4} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B4} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B4} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B4} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B4} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B4} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B4} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B4} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B4} \sp 2}} - +{{{8270} \over {343}} \ { \%B4}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B4} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B4} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B4} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B4} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B4} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B4} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B4} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B4} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B4} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B4} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B4} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B4} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B4} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B4} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B4} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B4} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B4} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B4} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B4}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B5}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B5} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B5} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B5} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B5} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B5} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B5} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B5} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B5} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B5} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B5} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B5} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B5} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B5} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B5} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B5} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B5} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B5} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B5} \sp 2}} - +{{{8270} \over {343}} \ { \%B5}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B5} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B5} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B5} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B5} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B5} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B5} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B5} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B5} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B5} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B5} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B5} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B5} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B5} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B5} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B5} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B5} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B5} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B5} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B5}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B6}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B6} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B6} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B6} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B6} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B6} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B6} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B6} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B6} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B6} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B6} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B6} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B6} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B6} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B6} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B6} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B6} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B6} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B6} \sp 2}} - +{{{8270} \over {343}} \ { \%B6}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B6} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B6} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B6} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B6} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B6} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B6} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B6} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B6} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B6} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B6} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B6} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B6} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B6} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B6} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B6} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B6} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B6} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B6} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B6}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B7}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B7} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B7} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B7} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B7} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B7} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B7} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B7} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B7} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B7} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B7} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B7} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B7} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B7} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B7} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B7} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B7} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B7} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B7} \sp 2}} - +{{{8270} \over {343}} \ { \%B7}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B7} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B7} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B7} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B7} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B7} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B7} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B7} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B7} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B7} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B7} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B7} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B7} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B7} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B7} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B7} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B7} \sp 4}} - +\\ +\displaystyle +\left. +{{{801511} \over {26117}} \ {{ \%B7} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B7} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B7}}+ +{{377534} \over {705159}} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B8}, +\right. +\\ +\\ +\displaystyle +{{{1184459} \over {1645371}} \ {{ \%B8} \sp {19}}} - +{{{2335702} \over {548457}} \ {{ \%B8} \sp {18}}} - +{{{5460230} \over {182819}} \ {{ \%B8} \sp {17}}}+ +{{{79900378} \over {1645371}} \ {{ \%B8} \sp {16}}}+ +\\ +\displaystyle +{{{43953929} \over {548457}} \ {{ \%B8} \sp {15}}}+ +{{{13420192} \over {182819}} \ {{ \%B8} \sp {14}}}+ +{{{553986} \over {3731}} \ {{ \%B8} \sp {13}}}+ +{{{193381378} \over {1645371}} \ {{ \%B8} \sp {12}}}+ +\\ +\displaystyle +{{{35978916} \over {182819}} \ {{ \%B8} \sp {11}}}+ +{{{358660781} \over {1645371}} \ {{ \%B8} \sp {10}}}+ +{{{271667666} \over {1645371}} \ {{ \%B8} \sp 9}}+ +{{{118784873} \over {548457}} \ {{ \%B8} \sp 8}}+ +\\ +\displaystyle +{{{337505020} \over {1645371}} \ {{ \%B8} \sp 7}}+ +{{{1389370} \over {11193}} \ {{ \%B8} \sp 6}}+ +{{{688291} \over {4459}} \ {{ \%B8} \sp 5}}+ +{{{3378002} \over {42189}} \ {{ \%B8} \sp 4}}+ +\\ +\displaystyle +{{{140671876} \over {1645371}} \ {{ \%B8} \sp 3}}+ +{{{32325724} \over {548457}} \ {{ \%B8} \sp 2}} - +{{{8270} \over {343}} \ { \%B8}} - +{{9741532} \over {1645371}}, +\\ +\\ +\displaystyle +-{{{91729} \over {705159}} \ {{ \%B8} \sp {19}}}+ +{{{487915} \over {705159}} \ {{ \%B8} \sp {18}}}+ +{{{4114333} \over {705159}} \ {{ \%B8} \sp {17}}} - +{{{1276987} \over {235053}} \ {{ \%B8} \sp {16}}} - +\\ +\displaystyle +{{{13243117} \over {705159}} \ {{ \%B8} \sp {15}}} - +{{{16292173} \over {705159}} \ {{ \%B8} \sp {14}}} - +{{{26536060} \over {705159}} \ {{ \%B8} \sp {13}}} - +{{{722714} \over {18081}} \ {{ \%B8} \sp {12}}} - +\\ +\displaystyle +{{{5382578} \over {100737}} \ {{ \%B8} \sp {11}}} - +{{{15449995} \over {235053}} \ {{ \%B8} \sp {10}}} - +{{{14279770} \over {235053}} \ {{ \%B8} \sp 9}} - +{{{6603890} \over {100737}} \ {{ \%B8} \sp 8}} - +\\ +\displaystyle +{{{409930} \over {6027}} \ {{ \%B8} \sp 7}} - +{{{37340389} \over {705159}} \ {{ \%B8} \sp 6}} - +{{{34893715} \over {705159}} \ {{ \%B8} \sp 5}} - +{{{26686318} \over {705159}} \ {{ \%B8} \sp 4}} - +\\ +\displaystyle +\left. +\left. +{{{801511} \over {26117}} \ {{ \%B8} \sp 3}} - +{{{17206178} \over {705159}} \ {{ \%B8} \sp 2}} - +{{{4406102} \over {705159}} \ { \%B8}}+ +{{377534} \over {705159}} +\right] +\right] +\end{array} +$$ +\returnType{Type: List List RealClosure Fraction Integer} +The number of real solutions for the input system is: + +\spadcommand{\# lr } +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +Each of these real solutions is given by a list of elements in +{\tt RealClosure(R)}. In these 8 lists, the first element is a value of +{\bf z}, the second of {\bf y} and the last of {\bf x}. This is +logical since by setting the list of variables of the package to +{\bf [x,y,z,t]} we mean that the elimination ordering on the variables is +{\bf t < z < y < x }. Note that each system treated by the +{\tt ZDSOLVE} package constructor needs only to be zero-dimensional +w.r.t. the variables involved in the system it-self and not +necessarily w.r.t. all the variables used to define the package. + +We can approximate these real numbers as follows. +This computation takes between 30 sec. and 5 min, depending on your machine. + +\spadcommand{[ [approximate(r,1/1000000) for r in point] for point in lr] } +$$ +\begin{array}{@{}l} +\left[ +\left[ +-{{10048059} \over {2097152}}, +\right. +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +450305731698538794352439791383896641459673197621176821933588120838 +\\ +\displaystyle +551631405892456717609142362969577740309983336076104889822891657813 +\\ +\displaystyle +709430983859733113720258484693913237615701950676035760116591745498 +\\ +\displaystyle +681538209878909485152342039281129312614132985654697714546466149548 +\\ +\displaystyle +782591994118844704172244049192156726354215802806143775884436463441 +\\ +\displaystyle +0045253024786561923163288214175 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +450305728302524548851651180698582663508310069375732046528055470686 +\\ +\displaystyle +564494957750991686720188943809040835481793171859386279762455151898 +\\ +\displaystyle +357079304877442429148870882984032418920030143612331486020082144373 +\\ +\displaystyle +379075531124363291986489542170422894957129001611949880795702366386 +\\ +\displaystyle +544306939202714897968826671232335604349152343406892427528041733857 +\\ +\displaystyle +4817381189277066143312396681216, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +210626076882347507389479868048601659624960714869068553876368371502 +\\ +\displaystyle +063968085864965079005588950564689330944709709993780218732909532589 +\\ +\displaystyle +878524724902071750498366048207515661873872451468533306001120296463 +\\ +\displaystyle +516638135154325598220025030528398108683711061484230702609121129792 +\\ +\displaystyle +987689628568183047905476005638076266490561846205530604781619178201 +\\ +\displaystyle +15887037891389881895 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +210626060949846419247211380481647417534196295329643410241390314236 +\\ +\displaystyle +875796768527388858559097596521177886218987288195394364024629735706 +\\ +\displaystyle +195981232610365979902512686325867656720234210687703171018424748418 +\\ +\displaystyle +142328892183768123706270847029570621848592886740077193782849920092 +\\ +\displaystyle +376059331416890100066637389634759811822855673103707202647449677622 +\\ +\displaystyle +83837629939232800768 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +-{{2563013} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +-261134617679192778969861769323775771923825996306354178192275233 +\\ +\displaystyle +044018989966807292833849076862359320744212592598673381593224350480 +\\ +\displaystyle +9294837523030237337236806668167446173001727271353311571242897 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +11652254005052225305839819160045891437572266102768589900087901348 +\\ +\displaystyle +199149409224137539839713940195234333204081399281531888294957554551 +\\ +\displaystyle +63963417619308395977544797140231469234269034921938055593984, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +3572594550275917221096588729615788272998517054675603239578198141 +\\ +\displaystyle +006034091735282826590621902304466963941971038923304526273329316373 +\\ +\displaystyle +7574500619789892286110976997087250466235373 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +10395482693455989368770712448340260558008145511201705922005223665 +\\ +\displaystyle +917594096594864423391410294529502651799899601048118758225302053465 +\\ +\displaystyle +051315812439017247289173865014702966308864 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +-{{1715967} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +-421309353378430352108483951797708239037726150396958622482899843 +\\ +\displaystyle +660603065607635937456481377349837660312126782256580143620693951995 +\\ +\displaystyle +146518222580524697287410022543952491 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +94418141441853744586496920343492240524365974709662536639306419607 +\\ +\displaystyle +958058825854931998401916999176594432648246411351873835838881478673 +\\ +\displaystyle +4019307857605820364195856822304768, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +7635833347112644222515625424410831225347475669008589338834162172 +\\ +\displaystyle +501904994376346730876809042845208919919925302105720971453918982731 +\\ +\displaystyle +3890725914035 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +26241887640860971997842976104780666339342304678958516022785809785 +\\ +\displaystyle +037845492057884990196406022669660268915801035435676250390186298871 +\\ +\displaystyle +4128491675648 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +-{{437701} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +1683106908638349588322172332654225913562986313181951031452750161 +\\ +\displaystyle +441497473455328150721364868355579646781603507777199075077835213366 +\\ +\displaystyle +48453365491383623741304759 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +16831068680952133890017099827059136389630776687312261111677851880 +\\ +\displaystyle +049074252262986803258878109626141402985973669842648879989083770687 +\\ +\displaystyle +9999845423381649008099328, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +4961550109835010186422681013422108735958714801003760639707968096 +\\ +\displaystyle +64691282670847283444311723917219104249213450966312411133 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +49615498727577383155091920782102090298528971186110971262363840408 +\\ +\displaystyle +2937659261914313170254867464792718363492160482442215424 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{{222801} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +-899488488040242826510759512197069142713604569254197827557300186 +\\ +\displaystyle +521375992158813771669612634910165522019514299493229913718324170586 +\\ +\displaystyle +7672383477 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +11678899986650263721777651006918885827089699602299347696908357524 +\\ +\displaystyle +570777794164352094737678665077694058889427645877185424342556259924 +\\ +\displaystyle +56372224, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +-238970488813315687832080154437380839561277150920849101984745299 +\\ +\displaystyle +188550954651952546783901661359399969388664003628357055232115503787 +\\ +\displaystyle +1291458703265 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +53554872736450963260904032866899319059882254446854114332215938336 +\\ +\displaystyle +811929575628336714686542903407469936562859255991176021204461834431 +\\ +\displaystyle +45479421952 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{{765693} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +8558969219816716267873244761178198088724698958616670140213765754 +\\ +\displaystyle +322002303251685786118678330840203328837654339523418704917749518340 +\\ +\displaystyle +772512899000391009630373148561 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +29414424455330107909764284113763934998155802159458569179064525354 +\\ +\displaystyle +957230138568189417023302287798901412962367211381542319972389173221 +\\ +\displaystyle +567119652444639331719460159488, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +-205761823058257210124765032486024256111130258154358880884392366 +\\ +\displaystyle +276754938224165936271229077761280019292142057440894808519374368858 +\\ +\displaystyle +27622246433251878894899015 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +26715982033257355380979523535014502205763137598908350970917225206 +\\ +\displaystyle +427101987719026671839489062898637147596783602924839492046164715377 +\\ +\displaystyle +77775324180661095366656 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{{5743879} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +1076288816968906847955546394773570208171456724942618614023663123 +\\ +\displaystyle +574768960850434263971398072546592772662158833449797698617455397887 +\\ +\displaystyle +562900072984768000608343553189801693408727205047612559889232757563 +\\ +\displaystyle +830528688953535421809482771058917542602890060941949620874083007858 +\\ +\displaystyle +36666945350176624841488732463225 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +31317689570803179466484619400235520441903766134585849862285496319 +\\ +\displaystyle +161966016162197817656155325322947465296482764305838108940793745664 +\\ +\displaystyle +607578231468885811955560292085152188388832003186584074693994260632 +\\ +\displaystyle +605898286123092315966691297079864813198515719429272303406229340239 +\\ +\displaystyle +234867030420681530440845099008, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left( +\begin{array}{@{}l} +-211328669918575091836412047556545843787017248986548599438982813 +\\ +\displaystyle +533526444466528455752649273493169173140787270143293550347334817207 +\\ +\displaystyle +609872054584900878007756416053431789468836611952973998050294416266 +\\ +\displaystyle +855009812796195049621022194287808935967492585059442776850225178975 +\\ +\displaystyle +8706752831632503615 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +16276155849379875802429066243471045808891444661684597180431538394 +\\ +\displaystyle +083725255333098080703636995855022160112110871032636095510260277694 +\\ +\displaystyle +140873911481262211681397816825874380753225914661319399754572005223 +\\ +\displaystyle +498385689642856344480185620382723787873544601061061415180109356172 +\\ +\displaystyle +051706396253618176 +\end{array} +\right) +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{{19739877} \over {2097152}}, +\right. +\\ +\\ +\displaystyle +\left( +\begin{array}{@{}l} +-299724993683270330379901580486152094921504038750070717770128576 +\\ +\displaystyle +672019253057942247895356602435986014310154780163808277161116037221 +\\ +\displaystyle +287484777803580987284314922548423836585801362934170532170258233335 +\\ +\displaystyle +091800960178993702398593530490046049338987383703085341034708990888 +\\ +\displaystyle +081485398113201846458245880061539477074169948729587596021075021589 +\\ +\displaystyle +194881447685487103153093129546733219013370267109820090228230051075 +\\ +\displaystyle +18607185928457030277807397796525813862762239286996106809728023675 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +23084332748522785907289100811918110239065041413214326461239367948 +\\ +\displaystyle +739333192706089607021381934176478983606202295191766329376317868514 +\\ +\displaystyle +550147660272062590222525055517418236888968838066366025744317604722 +\\ +\displaystyle +402920931967294751602472688341211418933188487286618444349272872851 +\\ +\displaystyle +128970807675528648950565858640331785659103870650061128015164035227 +\\ +\displaystyle +410373609905560544769495270592270708095930494912575195547088792595 +\\ +\displaystyle +9552929920110858560812556635485429471554031675979542656381353984, +\end{array} +\right) +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left. +\left. +\left( +\begin{array}{@{}l} +-512818926354822848909627639786894008060093841066308045940796633 +\\ +\displaystyle +584500926410949052045982531625008472301004703502449743652303892581 +\\ +\displaystyle +895928931293158470135392762143543439867426304729390912285013385199 +\\ +\displaystyle +069649023156609437199433379507078262401172758774998929661127731837 +\\ +\displaystyle +229462420711653791043655457414608288470130554391262041935488541073 +\\ +\displaystyle +594015777589660282236457586461183151294397397471516692046506185060 +\\ +\displaystyle +376287516256195847052412587282839139194642913955 +\end{array} +\right) +\over +\left( +\begin{array}{@{}l} +22882819397784393305312087931812904711836310924553689903863908242 +\\ +\displaystyle +435094636442362497730806474389877391449216077946826538517411890917 +\\ +\displaystyle +117418681451149783372841918224976758683587294866447308566225526872 +\\ +\displaystyle +092037244118004814057028371983106422912756761957746144438159967135 +\\ +\displaystyle +026293917497835900414708601277523729964886277426724876224800632688 +\\ +\displaystyle +088893248918508424949343473376030759399802682084829048596781777514 +\\ +\displaystyle +4465749979827872616963053217673201717237252096 +\end{array} +\right) +\right] +\right] +\end{array} +$$ +\returnType{Type: List List Fraction Integer} + +We can also concentrate on the solutions with real (strictly) positive +coordinates: + +\spadcommand{lpr := positiveSolve(lp)\$pack } +$$ +\left[ +\right] +$$ +\returnType{Type: List List RealClosure Fraction Integer} + +Thus we have checked that the input system has no solution with +strictly positive coordinates. + +Let us define another polynomial system ({\em L-3}). + +\spadcommand{f0 := x**3 + y + z + t- 1 } +$$ +z+y+{x \sp 3}+t -1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{f1 := x + y**3 + z + t -1 } +$$ +z+{y \sp 3}+x+t -1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{f2 := x + y + z**3 + t-1 } +$$ +{z \sp 3}+y+x+t -1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{f3 := x + y + z + t**3 -1 } +$$ +z+y+x+{t \sp 3} -1 +$$ +\returnType{Type: Polynomial Integer} + +\spadcommand{lf := [f0, f1, f2, f3]} +$$ +\begin{array}{@{}l} +\left[ +{z+y+{x \sp 3}+t -1}, +{z+{y \sp 3}+x+t -1}, +\right. +\\ +\\ +\displaystyle +\left. +{{z \sp 3}+y+x+t -1}, +{z+y+x+{t \sp 3} -1} +\right] +\end{array} +$$ +\returnType{Type: List Polynomial Integer} + +First compute a decomposition into regular chains (i.e. regular +triangular sets). + +\spadcommand{lts := triangSolve(lf)\$pack } +$$ +\begin{array}{@{}l} +\left[ +\left\{ {{t \sp 2}+t+1}, {{z \sp 3} -z -{t \sp 3}+t}, +\left( {3 \ z}+{3 \ {t \sp 3}} -3 +\right)\ {y \sp 2}+ +\left( {3 \ {z \sp 2}}+ +\left( {6 \ {t \sp 3}} -6 +\right)\ z+ +{3 \ {t \sp 6}} - +\right. +\right. +\right. +\\ +\displaystyle +\left. +{6 \ {t \sp 3}}+3 +\right)\ y+ +{{\left( {3 \ {t \sp 3}} -3 +\right)}\ {z \sp 2}}+ +{{\left( {3 \ {t \sp 6}} -{6 \ {t \sp 3}}+3 +\right)}\ z}+ +{t \sp 9} - +{3 \ {t \sp 6}}+ +{5 \ {t \sp 3}} - +{3 \ t}, +\\ +\displaystyle +\left. +{x+y+z} +\right\}, +\left\{ +{t \sp {16}} +-{6 \ {t \sp {13}}}+ +{9 \ {t \sp {10}}}+ +{4 \ {t \sp 7}}+ +{{15} \ {t \sp 4}} - +{{54} \ {t \sp 2}}+ +{27}, +\right. +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{4907232} \ {t \sp {15}}}+ +{{40893984} \ {t \sp {14}}} - +{{115013088} \ {t \sp {13}}}+ +{{22805712} \ {t \sp {12}}}+ +{{36330336} \ {t \sp {11}}}+ +\right. +\\ +\displaystyle +{{162959040} \ {t \sp {10}}} - +{{159859440} \ {t \sp 9}} - +{{156802608} \ {t \sp 8}}+ +{{117168768} \ {t \sp 7}}+ +\\ +\displaystyle +{{126282384} \ {t \sp 6}} - +{{129351600} \ {t \sp 5}}+ +{{306646992} \ {t \sp 4}}+ +{{475302816} \ {t \sp 3}} - +\\ +\displaystyle +\left. +{{1006837776} \ {t \sp 2}} - +{{237269088} \ t}+ +{480716208} +\right)\ z+ +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{48} \ {t \sp {54}}} - +{{912} \ {t \sp {51}}}+ +{{8232} \ {t \sp {48}}} - +{{72} \ {t \sp {46}}} - +{{46848} \ {t \sp {45}}}+ +{{1152} \ {t \sp {43}}}+ +{{186324} \ {t \sp {42}}} -\hbox{\hskip 1.0cm} +\\ +\displaystyle +{{3780} \ {t \sp {40}}} - +{{543144} \ {t \sp {39}}} - +{{3168} \ {t \sp {38}}} - +{{21384} \ {t \sp {37}}}+ +{{1175251} \ {t \sp {36}}}+ +{{41184} \ {t \sp {35}}}+ +\\ +\displaystyle +{{278003} \ {t \sp {34}}}- +{{1843242} \ {t \sp {33}}} - +{{301815} \ {t \sp {32}}} - +{{1440726} \ {t \sp {31}}}+ +{{1912012} \ {t \sp {30}}}+ +\\ +\displaystyle +{{1442826} \ {t \sp {29}}}+ +{{4696262} \ {t \sp {28}}} - +{{922481} \ {t \sp {27}}} - +{{4816188} \ {t \sp {26}}} - +{{10583524} \ {t \sp {25}}} - +\\ +\displaystyle +{{208751} \ {t \sp {24}}}+ +{{11472138} \ {t \sp {23}}}+ +{{16762859} \ {t \sp {22}}} - +{{857663} \ {t \sp {21}}} - +{{19328175} \ {t \sp {20}}} - +\\ +\displaystyle +{{18270421} \ {t \sp {19}}}+ +{{4914903} \ {t \sp {18}}}+ +{{22483044} \ {t \sp {17}}}+ +{{12926517} \ {t \sp {16}}} - +{{8605511} \ {t \sp {15}}} - +\\ +\displaystyle +{{17455518} \ {t \sp {14}}} - +{{5014597} \ {t \sp {13}}}+ +{{8108814} \ {t \sp {12}}}+ +{{8465535} \ {t \sp {11}}}+ +{{190542} \ {t \sp {10}}} - +\\ +\displaystyle +{{4305624} \ {t \sp 9}} - +{{2226123} \ {t \sp 8}}+ +{{661905} \ {t \sp 7}}+ +{{1169775} \ {t \sp 6}}+ +{{226260} \ {t \sp 5}} - +\\ +\displaystyle +{{209952} \ {t \sp 4}} - +{{141183} \ {t \sp 3}}+ +{{27216} \ t}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( {3 \ z}+{3 \ {t \sp 3}} -3 \right)\ {y \sp 2}+ +\left( {3 \ {z \sp 2}}+ +\left( {6 \ {t \sp 3}} -6 \right)\ z+ +{3 \ {t \sp 6}} -{6 \ {t \sp 3}}+3 \right)\ y+ +\left( {3 \ {t \sp 3}} -3 \right)\ {z \sp 2}+ +\\ +\displaystyle +\left. +\left( {3 \ {t \sp 6}} -{6 \ {t \sp 3}}+3 \right)\ z+ +{t \sp 9} -{3 \ {t \sp 6}}+{5 \ {t \sp 3}} -{3 \ t}, +{x+y+z+{t \sp 3} -1} +\right\}, +\\ +\displaystyle +{\left\{ t, {z -1}, {{y \sp 2} -1}, {x+y} \right\}}, +{\left\{ {t -1}, z, {{y \sp 2} -1}, {x+y} \right\}}, +{\left\{ {t -1}, {{z \sp 2} -1}, {{z \ y}+1}, x \right\}}, +\\ +\displaystyle +\left\{ +{t \sp {16}} -{6 \ {t \sp {13}}}+ +{9 \ {t \sp {10}}}+ +{4 \ {t \sp 7}}+ +{{15} \ {t \sp 4}} - +{{54} \ {t \sp 2}}+ +{27}, +\right. +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{4907232} \ {t \sp {29}}}+ +{{40893984} \ {t \sp {28}}} - +{{115013088} \ {t \sp {27}}} - +{{1730448} \ {t \sp {26}}} - +{{168139584} \ {t \sp {25}}}+ +\right. +\\ +\displaystyle +{{738024480} \ {t \sp {24}}} - +{{195372288} \ {t \sp {23}}}+ +{{315849456} \ {t \sp {22}}} - +{{2567279232} \ {t \sp {21}}}+ +\\ +\displaystyle +{{937147968} \ {t \sp {20}}}+ +{{1026357696} \ {t \sp {19}}}+ +{{4780488240} \ {t \sp {18}}} - +{{2893767696} \ {t \sp {17}}} - +\\ +\displaystyle +{{5617160352} \ {t \sp {16}}} - +{{3427651728} \ {t \sp {15}}}+ +{{5001100848} \ {t \sp {14}}}+ +{{8720098416} \ {t \sp {13}}}+ +\\ +\displaystyle +{{2331732960} \ {t \sp {12}}} - +{{499046544} \ {t \sp {11}}} - +{{16243306272} \ {t \sp {10}}} - +{{9748123200} \ {t \sp 9}}+ +\\ +\displaystyle +{{3927244320} \ {t \sp 8}}+ +{{25257280896} \ {t \sp 7}}+ +{{10348032096} \ {t \sp 6}} - +{{17128672128} \ {t \sp 5}} - +\\ +\displaystyle +{{14755488768} \ {t \sp 4}}+ +{{544086720} \ {t \sp 3}}+ +{{10848188736} \ {t \sp 2}}+ +{{1423614528} \ t} - +\\ +\displaystyle +\left. +{2884297248} +\right) z - +\end{array} +$$ +$$ +\begin{array}{@{}l} +{{48} \ {t \sp {68}}}+ +{{1152} \ {t \sp {65}}} - +{{13560} \ {t \sp {62}}}+ +{{360} \ {t \sp {60}}}+ +{{103656} \ {t \sp {59}}} - +{{7560} \ {t \sp {57}}} - +{{572820} \ {t \sp {56}}}+ +\\ +\displaystyle +{{71316} \ {t \sp {54}}}+ +{{2414556} \ {t \sp {53}}}+ +{{2736} \ {t \sp {52}}} - +{{402876} \ {t \sp {51}}} - +{{7985131} \ {t \sp {50}}} - +{{49248} \ {t \sp {49}}}+ +\\ +\displaystyle +{{1431133} \ {t \sp {48}}}+ +{{20977409} \ {t \sp {47}}}+ +{{521487} \ {t \sp {46}}} - +{{2697635} \ {t \sp {45}}} - +{{43763654} \ {t \sp {44}}} - +\\ +\displaystyle +{{3756573} \ {t \sp {43}}} - +{{2093410} \ {t \sp {42}}}+ +{{71546495} \ {t \sp {41}}}+ +{{19699032} \ {t \sp {40}}}+ +{{35025028} \ {t \sp {39}}} - +\\ +\displaystyle +{{89623786} \ {t \sp {38}}} - +{{77798760} \ {t \sp {37}}} - +{{138654191} \ {t \sp {36}}}+ +{{87596128} \ {t \sp {35}}}+ +{{235642497} \ {t \sp {34}}}+ +\\ +\displaystyle +{{349607642} \ {t \sp {33}}} +-{{93299834} \ {t \sp {32}}} - +{{551563167} \ {t \sp {31}}} - +{{630995176} \ {t \sp {30}}}+ +\\ +\displaystyle +{{186818962} \ {t \sp {29}}}+ +{{995427468} \ {t \sp {28}}}+ +{{828416204} \ {t \sp {27}}} - +{{393919231} \ {t \sp {26}}} - +\\ +\displaystyle +{{1076617485} \ {t \sp {25}}} - +{{1609479791} \ {t \sp {24}}}+ +{{595738126} \ {t \sp {23}}}+ +{{1198787136} \ {t \sp {22}}}+ +\\ +\displaystyle +{{4342832069} \ {t \sp {21}}} - +{{2075938757} \ {t \sp {20}}} - +{{4390835799} \ {t \sp {19}}} +-{{4822843033} \ {t \sp {18}}}+ +\\ +\displaystyle +{{6932747678} \ {t \sp {17}}}+ +{{6172196808} \ {t \sp {16}}}+ +{{1141517740} \ {t \sp {15}}} - +{{4981677585} \ {t \sp {14}}} - +\\ +\displaystyle +{{9819815280} \ {t \sp {13}}} - +{{7404299976} \ {t \sp {12}}} - +{{157295760} \ {t \sp {11}}}+ +{{29124027630} \ {t \sp {10}}}+ +\\ +\displaystyle +{{14856038208} \ {t \sp 9}} - +{{16184101410} \ {t \sp 8}} - +{{26935440354} \ {t \sp 7}} - +{{3574164258} \ {t \sp 6}}+ +\\ +\displaystyle +{{10271338974} \ {t \sp 5}}+ +{{11191425264} \ {t \sp 4}}+ +{{6869861262} \ {t \sp 3}} - +{{9780477840} \ {t \sp 2}} - +\\ +\displaystyle +{{3586674168} \ t}+ +{2884297248}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( {3 \ {z \sp 3}}+ +{{\left( {6 \ {t \sp 3}} -6 +\right)}\ {z \sp 2}}+ +\left( {6 \ {t \sp 6}} -{{12} \ {t \sp 3}}+3 +\right)\ z+{2 \ {t \sp 9}} -{6 \ {t \sp 6}}+{t \sp 3}+{3 \ t} +\right)\ y+ +\hbox{\hskip 1.0cm} +\\ +\\ +\displaystyle +{{\left( {3 \ {t \sp 3}} -3 \right)}\ {z \sp 3}}+ +{{\left( {6 \ {t \sp 6}} -{{12} \ {t \sp 3}}+6 \right)}\ {z \sp 2}}+ +{{\left( {4 \ {t \sp 9}} -{{12} \ {t \sp 6}}+{{11} \ {t \sp 3}} -3 +\right)}\ z}+ +\\ +\\ +\displaystyle +\left. +{t \sp {12}} - +{4 \ {t \sp 9}}+ +{5 \ {t \sp 6}} - +{2 \ {t \sp 3}}, +{x+y+z+{t \sp 3} -1} +\right\}, +\\ +\\ +\displaystyle +\left\{ {t -1}, {{z \sp 2} -1}, y, {x+z} \right\}, +\left\{ +{t \sp 8}+ +{t \sp 7}+ +{t \sp 6} - +{2 \ {t \sp 5}} - +{2 \ {t \sp 4}} - +{2 \ {t \sp 3}}+ +{{19} \ {t \sp 2}}+ +{{19} \ t} -8, +\right. +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( +{{2395770} \ {t \sp 7}}+ +{{3934440} \ {t \sp 6}} - +{{3902067} \ {t \sp 5}} - +{{10084164} \ {t \sp 4}} - +{{1010448} \ {t \sp 3}}+ +{{32386932} \ {t \sp 2}}+ +\right. +\\ +\\ +\displaystyle +\left. +{{22413225} \ t} - +{10432368} +\right)\ z - +{{463519} \ {t \sp 7}}+ +{{3586833} \ {t \sp 6}}+ +{{9494955} \ {t \sp 5}} - +{{8539305} \ {t \sp 4}} - +\\ +\\ +\displaystyle +{{33283098} \ {t \sp 3}}+ +{{35479377} \ {t \sp 2}}+ +{{46263256} \ t} - +{17419896}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left( {3 \ {z \sp 4}}+ +\left( {9 \ {t \sp 3}} -9 \right)\ {z \sp 3}+ +\left( {{12} \ {t \sp 6}} -{{24} \ {t \sp 3}}+9 \right)\ {z \sp 2}+ +\left( -{{152} \ {t \sp 3}}+{{219} \ t} -{67} \right)\ z - +\right. +\\ +\\ +\displaystyle +\left. +{{41} \ {t \sp 6}}+{{57} \ {t \sp 4}}+{{25} \ {t \sp 3}} -{{57} \ t}+{16} +\right)\ y+ +{{\left( {3 \ {t \sp 3}} -3 \right)}\ {z \sp 4}}+ +{{\left( {9 \ {t \sp 6}} -{{18} \ {t \sp 3}}+9 \right)}\ {z \sp 3}}+ +\\ +\\ +\displaystyle +{{\left( -{{181} \ {t \sp 3}}+{{270} \ t} -{89} \right)}\ {z \sp 2}}+ +{{\left( -{{92} \ {t \sp 6}}+{{135} \ {t \sp 4}}+ +{{49} \ {t \sp 3}} -{{135} \ t}+{43} \right)}\ z}+ +\\ +\\ +\displaystyle +\left. +{{27} \ {t \sp 7}} - +{{27} \ {t \sp 6}} - +{{54} \ {t \sp 4}}+ +{{396} \ {t \sp 3}} - +{{486} \ t}+{144}, +{x+y+z+{t \sp 3} -1} +\right\}, +\\ +\\ +\displaystyle +{\left\{ t, {z -{t \sp 3}+1}, {y -1}, {x -1} \right\}}, +{\left\{ {t -1}, z, y, x \right\}}, +{\left\{ t, {z -1}, y, x \right\}}, +{\left\{ t, z, {y -1}, x \right\}}, +\\ +\\ +\displaystyle +\left. +{\left\{ t, z, y, {x -1} \right\}} +\right] +\end{array} +$$ +\returnType{Type: List RegularChain(Integer,[x,y,z,t])} + +Then we compute a univariate representation. + +\spadcommand{univariateSolve(lf)\$pack } +$$ +\begin{array}{@{}l} +\left[ +{\left[ {complexRoots=?}, {coordinates= +\left[ {x -1}, {y -1}, +{z+1}, {t - \%A} +\right]} +\right]}, +\right. +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates={\left[ x, {y -1}, z, +{t - \%A} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={? -1}}, {coordinates={\left[ x, y, z, +{t - \%A} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates={\left[ {x -1}, y, z, +{t - \%A} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates={\left[ x, y, {z -1}, +{t - \%A} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={? -2}}, {coordinates={\left[ {x -1}, {y+1}, +z, {t -1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates={\left[ {x+1}, {y -1}, z, +{t -1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={? -1}}, {coordinates={\left[ {x -1}, {y+1}, +{z -1}, t +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={?+1}}, {coordinates={\left[ {x+1}, {y -1}, +{z -1}, t +\right]}} +\right]}, +\\ +\displaystyle +\left[ {complexroots={{? \sp 6} -{2 \ {? \sp 3}}+{3 \ {? \sp 2}} -3}}, +coordinates= +\left[ {{2 \ x}+{ \%A \sp 3}+ \%A -1}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{{2 \ y}+{ \%A \sp 3}+ \%A -1}, {z - \%A}, {t - \%A} +\right] +\right], +\\ +\displaystyle +\left[ +{complexRoots={{? \sp 5}+{3 \ {? \sp 3}} -{2 \ {? \sp 2}}+{3 \ ?} -3}}, +coordinates= +\left[ {x - \%A}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{y - \%A}, {z+{ \%A \sp 3}+{2 \ \%A} -1}, {t - \%A} +\right] +\right], +\\ +\displaystyle +\left[ +{complexRoots={{? \sp 4} -{? \sp 3} -{2 \ {? \sp 2}}+3}}, +coordinates= +\left[ {x+{ \%A \sp 3} - \%A -1}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{y+{ \%A \sp 3} - \%A -1}, +{z -{ \%A \sp 3}+{2 \ \%A}+1}, +{t - \%A} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots={?+1}}, coordinates= +\left[ {x -1}, {y -1}, z, {t - \%A} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots={{? \sp 6}+{2 \ {? \sp 3}}+{3 \ {? \sp 2}} -3}}, +coordinates= +\left[ +{{2 \ x} -{ \%A \sp 3} - \%A -1}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{y+ \%A}, +{{2 \ z} -{ \%A \sp 3} - \%A -1}, {t+ \%A} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots={{? \sp 6}+{{12} \ {? \sp 4}}+{{20} \ {? \sp 3}} +-{{45} \ {? \sp 2}} -{{42} \ ?} -{953}}}, +coordinates= +\right. +\\ +\displaystyle +\left. +\left[ +{{{12609} \ x}+ +{{23} \ { \%A \sp 5}}+ +{{49} \ { \%A \sp 4}} - +{{46} \ { \%A \sp 3}}+ +{{362} \ { \%A \sp 2}} - +{{5015} \ \%A} -{8239}}, +\right. +\right. +\\ +\displaystyle +{{{25218} \ y}+ +{{23} \ { \%A \sp 5}}+ +{{49} \ { \%A \sp 4}} - +{{46} \ { \%A \sp 3}}+ +{{362} \ { \%A \sp 2}}+ +{{7594} \ \%A} - +{8239}}, +\\ +\displaystyle +{{{25218} \ z}+ +{{23} \ { \%A \sp 5}}+ +{{49} \ { \%A \sp 4}} - +{{46} \ { \%A \sp 3}}+ +{{362} \ { \%A \sp 2}}+ +{{7594} \ \%A} -{8239}}, +\\ +\displaystyle +\left. +\left. +{{{12609} \ t}+ +{{23} \ { \%A \sp 5}}+ +{{49} \ { \%A \sp 4}} - +{{46} \ { \%A \sp 3}}+ +{{362} \ { \%A \sp 2}} - +{{5015} \ \%A} - +{8239}} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 5}+{{12} \ {? \sp 3}} -{{16} \ {? \sp 2}}+{{48} \ ?} -{96}}}, +coordinates= +\left[ {8 \ x}+{ \%A \sp 3}+ +\right. +\right. +\\ +\displaystyle +\left. +\left. +{8 \ \%A} -8, {{2 \ y} - \%A}, {{2 \ z} - \%A}, {{2 \ t} - \%A} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 5}+{? \sp 4} -{5 \ {? \sp 3}} -{3 \ {? \sp 2}}+{9 \ ?}+3}}, +coordinates= +\left[ {2 \ x} -{ \%A \sp 3}+ +\right. +\right. +\\ +\displaystyle +\left. +\left. +{2 \ \%A} -1, +{{2 \ y}+{ \%A \sp 3} -{4 \ \%A}+1}, {{2 \ z} -{ \%A \sp 3}+ +{2 \ \%A} -1}, {{2 \ t} -{ \%A \sp 3}+{2 \ \%A} -1} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 4} -{3 \ {? \sp 3}}+{4 \ {? \sp 2}} -{6 \ ?}+{13}}}, +coordinates= +\left[ {9 \ x} -{2 \ { \%A \sp 3}}+ +\right. +\right. +\\ +\displaystyle +{4 \ { \%A \sp 2}} - \%A+2, +{9 \ y}+{ \%A \sp 3} - +{2 \ { \%A \sp 2}}+ +{5 \ \%A} -1, +{9 \ z}+ +{ \%A \sp 3} - +{2 \ { \%A \sp 2}}+ +\\ +\displaystyle +\left. +\left. +{5 \ \%A} -1, {{9 \ t}+{ \%A \sp 3} -{2 \ { \%A \sp 2}} -{4 \ \%A} -1} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots={{? \sp 4} -{{11} \ {? \sp 2}}+{37}}}, +coordinates= +\left[ {{3 \ x} -{ \%A \sp 2}+7}, {6 \ y}+{ \%A \sp 2}+ +\right. +\right. +\\ +\displaystyle +\left. +\left. +{3 \ \%A} -7, {{3 \ z} -{ \%A \sp 2}+7}, +{{6 \ t}+{ \%A \sp 2} -{3 \ \%A} -7} +\right] +\right], +\\ +\displaystyle +{\left[ {complexRoots={?+1}}, {coordinates= +{\left[ {x -1}, y, {z +-1}, {t+1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={?+2}}, {coordinates= +{\left[ x, {y -1}, {z -1}, {t+1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={? -2}}, {coordinates= +{\left[ x, {y -1}, {z+1}, {t -1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates= +{\left[ x, {y+1}, {z -1}, {t -1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots={? -2}}, {coordinates= +{\left[ {x -1}, y, {z+1}, {t -1} +\right]}} +\right]}, +\\ +\displaystyle +{\left[ {complexRoots=?}, {coordinates= +{\left[ {x+1}, y, {z -1}, {t -1} +\right]}} +\right]}, +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 4}+{5 \ {? \sp 3}}+{{16} \ {? \sp 2}}+{{30} \ ?}+{57}}}, +coordinates= +\left[ {{{151} \ x}+{15} \ { \%A \sp 3}}+ +\right. +\right. +\\ +\displaystyle +{{54} \ { \%A \sp 2}}+ +{{104} \ \%A}+{93}, +{{{151} \ y} -{{10} \ { \%A \sp 3}} - +{{36} \ { \%A \sp 2}} -{{19} \ \%A} - +{62}}, +\\ +\displaystyle +\left. +\left. +{{{151} \ z} -{5 \ { \%A \sp 3}} - +{{18} \ { \%A \sp 2}} -{{85} \ \%A} - +{31}}, +{{{151} \ t} -{5 \ { \%A \sp 3}} - +{{18} \ { \%A \sp 2}} -{{85} \ \%A} +-{31}} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots={{? \sp 4} -{? \sp 3} -{2 \ {? \sp 2}}+3}}, +coordinates= +\left[ {x -{ \%A \sp 3}+{2 \ \%A}+1}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{y+{ \%A \sp 3} - \%A -1}, {z - \%A}, {t+{ \%A \sp 3} - \%A -1} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 4}+{2 \ {? \sp 3}} -{8 \ {? \sp 2}}+{48}}}, +coordinates= +\left[ {{8 \ x} -{ \%A \sp 3}+{4 \ \%A} -8}, +\right. +\right. +\\ +\displaystyle +\left. +\left. +{{2 \ y}+ \%A}, {{8 \ z}+{ \%A \sp 3} -{8 \ \%A}+8}, +{{8 \ t} -{ \%A \sp 3}+{4 \ \%A} -8} +\right] +\right], +\\ +\displaystyle +\left[ {complexRoots= +{{? \sp 5}+{? \sp 4} -{2 \ {? \sp 3}} -{4 \ {? \sp 2}}+{5 \ ?}+8}}, +\right. +\\ +\displaystyle +\left. +coordinates= +\left[ +{{3 \ x}+{ \%A \sp 3} -1}, +{{3 \ y}+{ \%A \sp 3} -1}, +{{3 \ z}+{ \%A \sp 3} -1}, +{t - \%A} +\right] +\right], +\\ +\displaystyle +\left. +\left[ +{complexRoots={{? \sp 3}+{3 \ ?} -1}}, +coordinates= +\left[ {x - \%A}, {y - \%A}, {z - \%A}, {t - \%A} +\right] +\right] +\right] +\end{array} +$$ +\returnType{Type: +List Record( +complexRoots: SparseUnivariatePolynomial Integer, +coordinates: List Polynomial Integer)} + +Note that this computation is made from the input system {\bf lf}. + +However it is possible to reuse a pre-computed regular chain as follows: + +\spadcommand{ts := lts.1 } +$$ +\begin{array}{@{}l} +\left\{ +{{t \sp 2}+t+1}, +{{z \sp 3} -z -{t \sp 3}+t}, +\right. +\\ +\\ +\displaystyle +\left( {3 \ z}+{3 \ {t \sp 3}} -3 +\right)\ {y \sp 2}+ +\left( {3 \ {z \sp 2}}+ +\left( {6 \ {t \sp 3}} -6 +\right)\ z+{3 \ {t \sp 6}} -{6 \ {t \sp 3}}+3 +\right)\ y+ +\\ +\\ +\displaystyle +\left. +\left( {3 \ {t \sp 3}} -3 +\right)\ {z \sp 2}+ +\left( {3 \ {t \sp 6}} -{6 \ {t \sp 3}}+3 +\right)\ z+ +{t \sp 9} -{3 \ {t \sp 6}}+{5 \ {t \sp 3}} -{3 \ t}, +{x+y+z} +\right\} +\end{array} +$$ +\returnType{Type: RegularChain(Integer,[x,y,z,t])} + +\spadcommand{univariateSolve(ts)\$pack } +$$ +\begin{array}{@{}l} +\left[ +\left[ +{complexRoots= +{{? \sp 4}+{5 \ {? \sp 3}}+{{16} \ {? \sp 2}}+{{30} \ ?}+{57}}}, +p\right. +\right. +\\ +\displaystyle +coordinates= +\left[ +{{{151} \ x}+ +{{15} \ { \%A \sp 3}}+ +{{54} \ { \%A \sp 2}}+ +{{104} \ \%A}+{93}}, +\right. +\\ +\displaystyle +{{151} \ y} - +{{10} \ { \%A \sp 3}} - +{{36} \ { \%A \sp 2}} - +{{19} \ \%A} -{62}, +\\ +\displaystyle +{{{151} \ z} - +{5 \ { \%A \sp 3}} - +{{18} \ { \%A \sp 2}} - +{{85} \ \%A} -{31}}, +\\ +\displaystyle +\left. +\left. +{{{151} \ t} - +{5 \ { \%A \sp 3}} - +{{18} \ { \%A \sp 2}} - +{{85} \ \%A} -{31}} +\right] +\right], +\\ +\\ +\displaystyle +\left[ {complexRoots={{? \sp 4} -{? \sp 3} -{2 \ {? \sp 2}}+3}}, +\right. +\\ +\displaystyle +coordinates= +\left[ +{x -{ \%A \sp 3}+{2 \ \%A}+1}, +{y+{ \%A \sp 3} - \%A -1}, +\right. +\\ +\displaystyle +\left. +\left. +{z - \%A}, +{t+{ \%A \sp 3} - \%A -1} +\right] +\right], +\\ +\\ +\displaystyle +\left[ +{complexRoots={{? \sp 4}+{2 \ {? \sp 3}} -{8 \ {? \sp 2}}+{48}}}, +\right. +\\ +\displaystyle +coordinates= +\left[ +{{8 \ x} -{ \%A \sp 3}+{4 \ \%A} -8}, +{{2 \ y}+ \%A}, +\right. +\\ +\displaystyle +\left. +\left. +\left. +{{8 \ z}+{ \%A \sp 3} -{8 \ \%A}+8}, +{{8 \ t} -{ \%A \sp 3}+{4 \ \%A} -8} +\right] +\right] +\right] +\end{array} +$$ +\returnType{Type: List Record( +complexRoots: SparseUnivariatePolynomial Integer, +coordinates: List Polynomial Integer)} + +\spadcommand{realSolve(ts)\$pack } +$$ +\left[ +\right] +$$ +\returnType{Type: List List RealClosure Fraction Integer} + +We compute now the full set of points with real coordinates: + +\spadcommand{lr2 := realSolve(lf)\$pack } +$$ +\begin{array}{@{}l} +\left[ +{\left[ 0, -1, 1, 1 \right]}, +{\left[ 0, 0, 1, 0 \right]}, +{\left[ 1, 0, 0, 0 \right]}, +{\left[ 0, 0, 0, 1 \right]}, +{\left[ 0, 1, 0, 0 \right]},\hbox{\hskip 4.5cm} +\right. +\\ +\\ +\displaystyle +{\left[ 1, 0, { \%B{37}}, -{ \%B{37}} \right]}, +{\left[ 1, 0, { \%B{38}}, -{ \%B{38}} \right]}, +\\ +\\ +\displaystyle +{\left[ 0, 1, { \%B{35}}, -{ \%B{35}} \right]}, +{\left[ 0, 1, { \%B{36}}, -{ \%B{36}} \right]}, +{\left[ -1, 0, 1, 1 \right]}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{32}}, +{{1 \over {27}} \ {{ \%B{32}} \sp {15}}}+ +{{2 \over {27}} \ {{ \%B{32}} \sp {14}}}+ +{{1 \over {27}} \ {{ \%B{32}} \sp {13}}} - +{{4 \over {27}} \ {{ \%B{32}} \sp {12}}} - +{{{11} \over {27}} \ {{ \%B{32}} \sp {11}}} - +\right. +\\ +\\ +\displaystyle +{{4 \over {27}} \ {{ \%B{32}} \sp {10}}}+ +{{1 \over {27}} \ {{ \%B{32}} \sp 9}}+ +{{{14} \over {27}} \ {{ \%B{32}} \sp 8}}+ +{{1 \over {27}} \ {{ \%B{32}} \sp 7}}+ +{{2 \over 9} \ {{ \%B{32}} \sp 6}}+ +\\ +\\ +\displaystyle +{{1 \over 3} \ {{ \%B{32}} \sp 5}}+ +{{2 \over 9} \ {{ \%B{32}} \sp 4}}+ +{{ \%B{32}} \sp 3}+ +{{4 \over 3} \ {{ \%B{32}} \sp 2}} - +{ \%B{32}} +-2, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{32}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{32}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{32}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{32}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{32}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{32}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{32}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{32}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{32}} \sp 7}} - +{{1 \over 9} \ {{ \%B{32}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{32}} \sp 5}} - +{{1 \over 9} \ {{ \%B{32}} \sp 4}} - +{{ \%B{32}} \sp 3} -{{2 \over 3} \ +{{ \%B{32}} \sp 2}}+ +{{1 \over 2} \ { \%B{32}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{32}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{32}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{32}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{32}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{32}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{32}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{32}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{32}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{32}} \sp 7}} - +{{1 \over 9} \ {{ \%B{32}} \sp 6}} - +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{32}} \sp 5}} - +{{1 \over 9} \ {{ \%B{32}} \sp 4}} - +{{ \%B{32}} \sp 3} - +{{2 \over 3} \ {{ \%B{32}} \sp 2}}+ +{{1 \over 2} \ { \%B{32}}}+ +{3 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{33}}, +{{1 \over {27}} \ {{ \%B{33}} \sp {15}}}+ +{{2 \over {27}} \ {{ \%B{33}} \sp {14}}}+ +{{1 \over {27}} \ {{ \%B{33}} \sp {13}}} - +{{4 \over {27}} \ {{ \%B{33}} \sp {12}}} - +{{{11} \over {27}} \ {{ \%B{33}} \sp {11}}} - +\right. +\\ +\\ +\displaystyle +{{4 \over {27}} \ {{ \%B{33}} \sp {10}}}+ +{{1 \over {27}} \ {{ \%B{33}} \sp 9}}+ +{{{14} \over {27}} \ {{ \%B{33}} \sp 8}}+ +{{1 \over {27}} \ {{ \%B{33}} \sp 7}}+ +{{2 \over 9} \ {{ \%B{33}} \sp 6}}+ +\\ +\\ +\displaystyle +{{1\over 3} \ {{ \%B{33}} \sp 5}}+ +{{2 \over 9} \ {{ \%B{33}} \sp 4}}+ +{{ \%B{33}} \sp 3}+ +{{4 \over 3} \ {{ \%B{33}} \sp 2}} - +{ \%B{33}} -2, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{33}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{33}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{33}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{33}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{33}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{33}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{33}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{33}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{33}} \sp 7}} - +{{1 \over 9} \ {{ \%B{33}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{33}} \sp 5}} - +{{1 \over 9} \ {{ \%B{33}} \sp 4}} - +{{ \%B{33}} \sp 3} - +{{2 \over 3} \ {{ \%B{33}} \sp 2}}+ +{{1 \over 2} \ { \%B{33}}}+{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{33}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{33}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{33}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{33}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{33}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{33}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{33}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{33}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{33}} \sp 7}} - +{{1 \over 9} \ {{ \%B{33}} \sp 6}} - +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{33}} \sp 5}} - +{{1 \over 9} \ {{ \%B{33}} \sp 4}} - +{{ \%B{33}} \sp 3} - +{{2 \over 3} \ {{ \%B{33}} \sp 2}}+ +{{1 \over 2} \ { \%B{33}}}+ +{3 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{34}}, +{{1 \over {27}} \ {{ \%B{34}} \sp {15}}}+ +{{2 \over {27}} \ {{ \%B{34}} \sp {14}}}+ +{{1 \over {27}} \ {{ \%B{34}} \sp {13}}} - +{{4 \over {27}} \ {{ \%B{34}} \sp {12}}} - +{{{11} \over {27}} \ {{ \%B{34}} \sp {11}}} - +\right. +\\ +\\ +\displaystyle +{{4 \over {27}} \ {{ \%B{34}} \sp {10}}}+ +{{1 \over {27}} \ {{ \%B{34}} \sp 9}}+ +{{{14} \over {27}} \ {{ \%B{34}} \sp 8}}+ +{{1 \over {27}} \ {{ \%B{34}} \sp 7}}+ +{{2 \over 9} \ {{ \%B{34}} \sp 6}}+ +\\ +\\ +\displaystyle +{{1 \over 3} \ {{ \%B{34}} \sp 5}}+ +{{2 \over 9} \ {{ \%B{34}} \sp 4}}+ +{{ \%B{34}} \sp 3}+ +{{4 \over 3} \ {{ \%B{34}} \sp 2}} - +{ \%B{34}} -2, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{34}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{34}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{34}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{34}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{34}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{34}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{34}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{34}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{34}} \sp 7}} - +{{1 \over 9} \ {{ \%B{34}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{34}} \sp 5}} - +{{1 \over 9} \ {{ \%B{34}} \sp 4}} - +{{ \%B{34}} \sp 3} - +{{2 \over 3} \ {{ \%B{34}} \sp 2}}+ +{{1 \over 2} \ { \%B{34}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +-{{1 \over {54}} \ {{ \%B{34}} \sp {15}}} -\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{34}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{34}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{34}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{34}} \sp {11}}}+ +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{34}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{34}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{34}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{34}} \sp 7}} - +{{1 \over 9} \ {{ \%B{34}} \sp 6}} - +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{34}} \sp 5}} - +{{1 \over 9} \ {{ \%B{34}} \sp 4}} - +{{ \%B{34}} \sp 3} - +{{2 \over 3} \ {{ \%B{34}} \sp 2}}+ +{{1 \over 2} \ { \%B{34}}}+ +{3 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +{\left[ -1, 1, 0, 1 \right]}, +{\left[ -1, 1, 1, 0 \right]}, +\\ +\\ +\displaystyle +\left[ +{ \%B{23}}, +-{{1 \over {54}} \ {{ \%B{23}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{23}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{23}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{23}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{23}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{23}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{23}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{23}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{23}} \sp 7}} - +{{1 \over 9} \ {{ \%B{23}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{23}} \sp 5}} - +{{1 \over 9} \ {{ \%B{23}} \sp 4}} - +{{ \%B{23}} \sp 3} - +{{2 \over 3} \ {{ \%B{23}} \sp 2}}+ +{{1 \over 2} \ { \%B{23}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{30}}, +-{ \%B{30}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp {15}}}+\hbox{\hskip 1.0cm} +{{1 \over {27}} \ {{ \%B{23}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{23}} \sp {12}}} - +{{{11} \over {54}} \ {{ \%B{23}} \sp {11}}} - +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{23}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{23}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp 7}}+ +{{1 \over 9} \ {{ \%B{23}} \sp 6}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{23}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{23}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{23}} \sp 2}} - +{{1 \over 2} \ { \%B{23}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{23}}, +-{{1 \over {54}} \ {{ \%B{23}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{23}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{23}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{23}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{23}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{23}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{23}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{23}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{23}} \sp 7}} - +{{1 \over 9} \ {{ \%B{23}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{23}} \sp 5}} - +{{1 \over 9} \ {{ \%B{23}} \sp 4}} - +{{ \%B{23}} \sp 3} - +{{2 \over 3} \ {{ \%B{23}} \sp 2}}+ +{{1 \over 2} \ { \%B{23}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{31}}, +-{ \%B{31}}+{{1 \over {54}} \ {{ \%B{23}} \sp {15}}}+ +{{1 \over {27}} \ {{ \%B{23}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{23}} \sp {12}}} - +\\ +\\ +\displaystyle +{{{11} \over {54}} \ {{ \%B{23}} \sp {11}}} - +{{2 \over {27}} \ {{ \%B{23}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{23}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{23}} \sp 7}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 9} \ {{ \%B{23}} \sp 6}}+ +{{1 \over 6} \ {{ \%B{23}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{23}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{23}} \sp 2}} - +{{1 \over 2} \ { \%B{23}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{24}}, +-{{1 \over {54}} \ {{ \%B{24}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{24}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{24}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{24}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{24}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{24}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{24}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{24}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{24}} \sp 7}} - +{{1 \over 9} \ {{ \%B{24}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{24}} \sp 5}} - +{{1 \over 9} \ {{ \%B{24}} \sp 4}} - +{{ \%B{24}} \sp 3} - +{{2 \over 3} \ {{ \%B{24}} \sp 2}}+ +{{1 \over 2} \ { \%B{24}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{28}}, +-{ \%B{28}}+{{1 \over {54}} \ {{ \%B{24}} \sp {15}}}+ +{{1 \over {27}} \ {{ \%B{24}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{24}} \sp {12}}} - +{{{11} \over {54}} \ {{ \%B{24}} \sp {11}}} - +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{24}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{24}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp 7}}+ +{{1 \over 9} \ {{ \%B{24}} \sp 6}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{24}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{24}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{24}} \sp 2}} - +{{1 \over 2} \ { \%B{24}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{24}}, +-{{1 \over {54}} \ {{ \%B{24}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{24}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{24}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{24}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{24}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{24}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{24}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{24}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{24}} \sp 7}} - +{{1 \over 9} \ {{ \%B{24}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{24}} \sp 5}} - +{{1 \over 9} \ {{ \%B{24}} \sp 4}} - +{{ \%B{24}} \sp 3} - +{{2 \over 3} \ {{ \%B{24}} \sp 2}}+ +{{1 \over 2} \ { \%B{24}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{29}}, +-{ \%B{29}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp {15}}}+ +{{1 \over {27}} \ {{ \%B{24}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{24}} \sp {12}}} - +{{{11} \over {54}} \ {{ \%B{24}} \sp {11}}} - +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{24}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{24}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{24}} \sp 7}}+ +{{1 \over 9} \ {{ \%B{24}} \sp 6}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{24}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{24}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{24}} \sp 2}} - +{{1 \over 2} \ { \%B{24}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{25}}, +-{{1 \over {54}} \ {{ \%B{25}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{25}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{25}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{25}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{25}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{25}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{25}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{25}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{25}} \sp 7}} - +{{1 \over 9} \ {{ \%B{25}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{25}} \sp 5}} - +{{1 \over 9} \ {{ \%B{25}} \sp 4}} - +{{ \%B{25}} \sp 3} - +{{2 \over 3} \ {{ \%B{25}} \sp 2}}+ +{{1 \over 2} \ { \%B{25}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{26}}, +-{ \%B{26}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp {15}}}+ +{{1 \over {27}} \ {{ \%B{25}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{25}} \sp {12}}} - +{{{11} \over {54}} \ {{ \%B{25}} \sp {11}}} - +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{25}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{25}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp 7}}+ +{{1 \over 9} \ {{ \%B{25}} \sp 6}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{25}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{25}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{25}} \sp 2}} - +{{1 \over 2} \ { \%B{25}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +\left[ +{ \%B{25}}, +-{{1 \over {54}} \ {{ \%B{25}} \sp {15}}} - +{{1 \over {27}} \ {{ \%B{25}} \sp {14}}} - +{{1 \over {54}} \ {{ \%B{25}} \sp {13}}}+ +{{2 \over {27}} \ {{ \%B{25}} \sp {12}}}+ +{{{11} \over {54}} \ {{ \%B{25}} \sp {11}}}+ +\right. +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{25}} \sp {10}}} - +{{1 \over {54}} \ {{ \%B{25}} \sp 9}} - +{{7 \over {27}} \ {{ \%B{25}} \sp 8}} - +{{1 \over {54}} \ {{ \%B{25}} \sp 7}} - +{{1 \over 9} \ {{ \%B{25}} \sp 6}} - +\\ +\\ +\displaystyle +{{1 \over 6} \ {{ \%B{25}} \sp 5}} - +{{1 \over 9} \ {{ \%B{25}} \sp 4}} - +{{ \%B{25}} \sp 3} - +{{2 \over 3} \ {{ \%B{25}} \sp 2}}+ +{{1 \over 2} \ { \%B{25}}}+ +{3 \over 2}, +\end{array} +$$ +$$ +\begin{array}{@{}l} +{ \%B{27}}, +-{ \%B{27}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp {15}}}+ +{{1 \over {27}} \ {{ \%B{25}} \sp {14}}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp {13}}} - +{{2 \over {27}} \ {{ \%B{25}} \sp {12}}} - +{{{11} \over {54}} \ {{ \%B{25}} \sp {11}}} - +\\ +\\ +\displaystyle +{{2 \over {27}} \ {{ \%B{25}} \sp {10}}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp 9}}+ +{{7 \over {27}} \ {{ \%B{25}} \sp 8}}+ +{{1 \over {54}} \ {{ \%B{25}} \sp 7}}+ +{{1 \over 9} \ {{ \%B{25}} \sp 6}}+ +\\ +\\ +\displaystyle +\left. +{{1 \over 6} \ {{ \%B{25}} \sp 5}}+ +{{1 \over 9} \ {{ \%B{25}} \sp 4}}+ +{{2 \over 3} \ {{ \%B{25}} \sp 2}} - +{{1 \over 2} \ { \%B{25}}} - +{1 \over 2} +\right], +\end{array} +$$ +$$ +\begin{array}{@{}l} +{\left[ 1, { \%B{21}}, -{ \%B{21}}, 0 \right]}, +{\left[ 1, { \%B{22}}, -{ \%B{22}}, 0 \right]}, +{\left[ 1, { \%B{19}}, 0, -{ \%B{19}} \right]}, +{\left[ 1, { \%B{20}}, 0, -{ \%B{20}} \right]}, +\\ +\\ +\displaystyle +\left[ +{ \%B{17}}, +-{{1 \over 3} \ {{ \%B{17}} \sp 3}}+ +{1 \over 3}, +-{{1 \over 3} \ {{ \%B{17}} \sp 3}}+ +{1 \over 3}, +-{{1 \over 3} \ {{ \%B{17}} \sp 3}}+ +{1 \over 3} +\right], +\\ +\\ +\displaystyle +\left. +\left[ +{ \%B{18}}, +{-{{1 \over 3} \ {{ \%B{18}} \sp 3}}+{1 \over 3}}, +{-{{1 \over 3} \ {{ \%B{18}} \sp 3}}+{1 \over 3}}, +{-{{1 \over 3} \ {{ \%B{18}} \sp 3}}+{1 \over 3}} +\right] +\right] +\end{array} +$$ +\returnType{Type: List List RealClosure Fraction Integer} + +The number of real solutions for the input system is: + +\spadcommand{\#lr2 } +$$ +27 +$$ +\returnType{Type: PositiveInteger} + +Another example of computation of real solutions illustrates the +{\tt LexTriangularPackage} package constructor. + +We concentrate now on the solutions with real (strictly) positive +coordinates: + +\spadcommand{lpr2 := positiveSolve(lf)\$pack } +$$ +\left[ +{\left[ { \%B{40}}, {-{{1 \over 3} \ {{ \%B{40}} \sp 3}}+{1 \over 3}}, +{-{{1 \over 3} \ {{ \%B{40}} \sp 3}}+{1 \over 3}}, {-{{1 \over 3} \ {{ + \%B{40}} \sp 3}}+{1 \over 3}} +\right]} +\right] +$$ +\returnType{Type: List List RealClosure Fraction Integer} + +Finally, we approximate the coordinates of this point with 20 exact digits: + +\spadcommand{[approximate(r,1/10**21)::Float for r in lpr2.1] } +$$ +\begin{array}{@{}l} +\left[ +{0.3221853546\ 2608559291}, +{0.3221853546\ 2608559291}, +\right. +\\ +\displaystyle +\left. +{0.3221853546\ 2608559291}, +{0.3221853546 2608559291} +\right] +\end{array} +$$ +\returnType{Type: List Float} + + +\chapter{Interactive Programming} +\label{ugIntProg} + +Programming in the interpreter is easy. +So is the use of Axiom's graphics facility. +Both are rather flexible and allow you to use them for many +interesting applications. +However, both require learning some basic ideas and skills. + +All graphics examples in the gallery section are either +produced directly by interactive commands or by interpreter +programs. +Four of these programs are introduced here. +By the end of this chapter you will know enough about graphics and +programming in the interpreter to not only understand all these +examples, but to tackle interesting and difficult problems on your +own. +The appendix on graphics lists all the remaining commands and +programs used to create these images. + +\section{Drawing Ribbons Interactively} +\label{ugIntProgDrawing} + +We begin our discussion of interactive graphics with the creation +of a useful facility: plotting ribbons of two-graphs in +three-space. +Suppose you want to draw the two-di\-men\-sion\-al graphs of $n$ +functions $f_i(x), 1 \leq i \leq n,$ all over some fixed range of $x$. +One approach is to create a two-di\-men\-sion\-al graph for each one, then +superpose one on top of the other. +What you will more than likely get is a jumbled mess. +Even if you make each function a different color, the result is +likely to be confusing. + +A better approach is to display each of the $f_i(x)$ in three +\index{ribbon} +dimensions as a ``ribbon'' of some appropriate width along the +$y$-direction, laying down each ribbon next to the +previous one. +A ribbon is simply a function of $x$ and $y$ depending +only on $x$. + +We illustrate this for $f_i(x)$ defined as simple powers of +$x$ for $x$ ranging between $-1$ and $1$. + + +Draw the ribbon for $z = x^2$. + +\spadgraph{draw(x**2,x=-1..1,y=0..1)} + +%\epsffile[0 0 295 295]{ps/ribbon1.ps} + +Now that was easy! +What you get is a ``wire-mesh'' rendition of the ribbon. +That's fine for now. +Notice that the mesh-size is small in both the $x$ and the +$y$ directions. +Axiom normally computes points in both these directions. +This is unnecessary. +One step is all we need in the $y$-direction. +To have Axiom economize on $y$-points, we re-draw the +ribbon with option $var2Steps == 1$. + +Re-draw the ribbon, but with option $var2Steps == 1$ +so that only $1$ step is computed in the +$y$ direction. + +\spadgraph{vp := draw(x**2,x=-1..1,y=0..1,var2Steps==1) } + +%\epsffile[0 0 295 295]{ps/ribbon2.ps} + +The operation has created a viewport, that is, a graphics window +on your screen. +We assigned the viewport to $vp$ and now we manipulate +its contents. + + +Graphs are objects, like numbers and algebraic expressions. +You may want to do some experimenting with graphs. +For example, say +\begin{verbatim} +showRegion(vp, "on") +\end{verbatim} +to put a bounding box around the ribbon. +Try it! +Issue $rotate(vp, -45, 90)$ to rotate the +figure $-45$ longitudinal degrees and $90$ latitudinal +degrees. + +Here is a different rotation. +This turns the graph so you can view it along the $y$-axis. + +\spadcommand{rotate(vp, 0, -90)} + +%\epsffile[0 0 295 295]{ps/ribbon2r.ps} + + +There are many other things you can do. +In fact, most everything you can do interactively using the +three-di\-men\-sion\-al control panel (such as translating, zooming, resizing, +coloring, perspective and lighting selections) can also be done +directly by operations (see Chapter +\ref{ugGraph} on page~\pageref{ugGraph} for more details). + +When you are done experimenting, say $reset(vp)$ to restore the +picture to its original position and settings. + + +Let's add another ribbon to our picture---one +for $x^3$. +Since $y$ ranges from $0$ to $1$ for the +first ribbon, now let $y$ range from $1$ to +$2$. +This puts the second ribbon next to the first one. + +How do you add a second ribbon to the viewport? +One method is +to extract the ``space'' component from the +viewport using the operation +\spadfunFrom{subspace}{ThreeDimensionalViewport}. +You can think of the space component as the object inside the +window (here, the ribbon). +Let's call it $sp$. +To add the second ribbon, you draw the second ribbon using the +option $space == sp$. + +Extract the space component of $vp$. + +\spadcommand{sp := subspace(vp)} + +Add the ribbon for +$x^3$ alongside that for +$x^2$. + +\spadgraph{vp := draw(x**3,x=-1..1,y=1..2,var2Steps==1, space==sp)} + +%\epsffile[0 0 295 295]{ps/ribbons.ps} + +Unless you moved the original viewport, the new viewport covers +the old one. +You might want to check that the old object is still there by +moving the top window. + +Let's show quadrilateral polygon outlines on the ribbons and then +enclose the ribbons in a box. + +Show quadrilateral polygon outlines. + +\spadcommand{drawStyle(vp,"shade");outlineRender(vp,"on")} + +%\epsffile[0 0 295 295]{ps/ribbons2.ps} + +Enclose the ribbons in a box. + +\spadcommand{rotate(vp,20,-60); showRegion(vp,"on")} + +%\epsffile[0 0 295 295]{ps/ribbons2b.ps} + +This process has become tedious! +If we had to add two or three more ribbons, we would have to +repeat the above steps several more times. +It is time to write an interpreter program to help us take care of +the details. + +\section{A Ribbon Program} +\label{ugIntProgRibbon} + +The above approach creates a new viewport for each additional ribbon. +A better approach is to build one object composed of all ribbons +before creating a viewport. To do this, use {\bf makeObject} rather +than {\bf draw}. The operations have similar formats, but {\bf draw} +returns a viewport and {\bf makeObject} returns a space object. + +We now create a function {\bf drawRibbons} of two arguments: +$flist$, a list of formulas for the ribbons you want to draw, +and $xrange$, the range over which you want them drawn. +Using this function, you can just say +\begin{verbatim} +drawRibbons([x**2, x**3], x=-1..1) +\end{verbatim} +to do all of the work required in the last section. +Here is the {\bf drawRibbons} program. +Invoke your favorite editor and create a file called {\bf ribbon.input} +containing the following program. + +\begin{figure} +\begin{verbatim} +drawRibbons(flist, xrange) == + sp := createThreeSpace() Create empty space $sp$. + y0 := 0 The initial ribbon position. + for f in flist repeat For each function $f$, + makeObject(f, xrange, y=y0..y0+1, create and add a ribbon + space==sp, var2Steps == 1) for $f$ to the space $sp$. + y0 := y0 + 1 The next ribbon position. + vp := makeViewport3D(sp, "Ribbons") Create viewport. + drawStyle(vp, "shade") Select shading style. + outlineRender(vp, "on") Show polygon outlines. + showRegion(vp,"on") Enclose in a box. + n := # flist The number of ribbons + zoom(vp,n,1,n) Zoom in x- and z-directions. + rotate(vp,0,75) Change the angle of view. + vp Return the viewport. +\end{verbatim} +\caption{The first {\bf drawRibbons} function.} +\label{fig-ribdraw1} +\end{figure} + +Here are some remarks on the syntax used in the {\bf drawRibbons} function +(consult Chapter \ref{ugUser} on page~\pageref{ugUser} for more details). +Unlike most other programming languages which use semicolons, +parentheses, or {\it begin}--{\it end} brackets to delineate the +structure of programs, the structure of an Axiom program is +determined by indentation. +The first line of the function definition always begins in column 1. +All other lines of the function are indented with respect to the first +line and form a {\it pile} (see \ref{ugLangBlocks} on +page~\pageref{ugLangBlocks}). + +The definition of {\bf drawRibbons} +consists of a pile of expressions to be executed one after +another. +Each expression of the pile is indented at the same level. +Lines 4-7 designate one single expression: +since lines 5-7 are indented with respect to the others, these +lines are treated as a continuation of line 4. +Also since lines 5 and 7 have the same indentation level, these +lines designate a pile within the outer pile. + +The last line of a pile usually gives the value returned by the +pile. +Here it is also the value returned by the function. +Axiom knows this is the last line of the function because it +is the last line of the file. +In other cases, a new expression beginning in column one signals +the end of a function. + +The line {\bf drawStyle}{\tt (vp,"shade")} is given after the viewport +has been created to select the draw style. +We have also used the \spadfunFrom{zoom}{ThreeDimensionalViewport} +option. +Without the zoom, the viewport region would be scaled equally in +all three coordinate directions. + +Let's try the function {\bf drawRibbons}. +First you must read the file to give Axiom the function definition. + +Read the input file. + +\spadcommand{)read ribbon } + +Draw ribbons for $x, x^2,\dots, x^5$ +for $-1 \leq x \leq 1$ + +\spadgraph{drawRibbons([x**i for i in 1..5],x=-1..1) } + +%\epsffile[0 0 295 295]{ps/ribbons5.ps} + +\section{Coloring and Positioning Ribbons} +\label{ugIntProgColor} +% + +Before leaving the ribbon example, we make two improvements. +Normally, the color given to each point in the space is a +function of its height within a bounding box. +The points at the bottom of the +box are red, those at the top are purple. + +To change the normal coloring, you can give +an option $colorFunction == {\it function}$. +When Axiom goes about displaying the data, it +determines the range of colors used for all points within the box. +Axiom then distributes these numbers uniformly over the number of hues. +Here we use the simple color function +$(x,y) \mapsto i$ for the +$i$-th ribbon. + +Also, we add an argument $yrange$ so you can give the range of +$y$ occupied by the ribbons. +For example, if the $yrange$ is given as +$y=0..1$ and there are $5$ ribbons to be displayed, each +ribbon would have width $0.2$ and would appear in the +range $0 \leq y \leq 1$. + +Refer to lines 4-9. +Line 4 assigns to $yVar$ the variable part of the +$yrange$ (after all, it need not be $y$). +Suppose that $yrange$ is given as $t = a..b$ where $a$ and +$b$ have numerical values. +Then line 5 assigns the value of $a$ to the variable $y0$. +Line 6 computes the width of the ribbon by dividing the difference of +$a$ and $b$ by the number, $num$, of ribbons. +The result is assigned to the variable $width$. +Note that in the for-loop in line 7, we are iterating in parallel; it is +not a nested loop. + +\begin{figure} +\hrule +\begin{verbatim} +drawRibbons(flist, xrange, yrange) == + sp := createThreeSpace() Create empty space $sp$. + num := # flist The number of ribbons. + yVar := variable yrange The ribbon variable. + y0:Float := lo segment yrange The first ribbon coordinate. + width:Float := (hi segment yrange - y0)/num The width of a ribbon. + for f in flist for color in 1..num repeat For each function $f$, + makeObject(f, xrange, yVar = y0..y0+width, create and add ribbon to + var2Steps == 1, colorFunction == (x,y) +-> color, _ + space == sp) $sp$ of a different color. + y0 := y0 + width The next ribbon coordinate. + vp := makeViewport3D(sp, "Ribbons") Create viewport. + drawStyle(vp, "shade") Select shading style. + outlineRender(vp, "on") Show polygon outlines. + showRegion(vp, "on") Enclose in a box. + vp Return the viewport. +\end{verbatim} +\hrule +\caption{The final {\bf drawRibbons} function.} +\label{fig-ribdraw2} +\end{figure} + +\section{Points, Lines, and Curves} +\label{ugIntProgPLC} +% +What you have seen so far is a high-level program using the +graphics facility. +We now turn to the more basic notions of points, lines, and curves +in three-di\-men\-sion\-al graphs. +These facilities use small floats (objects +of type {\tt DoubleFloat}) for data. +Let us first give names to the small float values $0$ and +$1$. + +The small float 0. + +\spadcommand{zero := 0.0@DFLOAT } + +The small float 1. + +\spadcommand{one := 1.0@DFLOAT } + +The {\tt @} sign means ``of the type.'' Thus $zero$ is +$0.0$ of the type {\tt DoubleFloat}. +You can also say $0.0::DFLOAT$. + +Points can have four small float components: $x, y, z$ coordinates and an +optional color. +A ``curve'' is simply a list of points connected by straight line +segments. + +Create the point $origin$ with color zero, that is, the lowest color +on the color map. + +\spadcommand{origin := point [zero,zero,zero,zero] } + +Create the point $unit$ with color zero. + +\spadcommand{unit := point [one,one,one,zero] } + +Create the curve (well, here, a line) from +$origin$ to $unit$. + +\spadcommand{line := [origin, unit] } + + +We make this line segment into an arrow by adding an arrowhead. +The arrowhead extends to, +say, $p3$ on the left, and to, say, $p4$ on the right. +To describe an arrow, you tell Axiom to draw the two curves +$[p1, p2, p3]$ and $[p2, p4].$ +We also decide through experimentation on +values for $arrowScale$, the ratio of the size of +the arrowhead to the stem of the arrow, and $arrowAngle$, +the angle between the arrowhead and the arrow. + +Invoke your favorite editor and create +an input file called {\bf arrows.input}. +This input file first defines the values of +%$origin$,$unit$, +$arrowAngle$ and $arrowScale$, then +defines the function {\bf makeArrow}$(p_1, p_2)$ to +draw an arrow from point $p_1$ to $p_2$. + +\begin{verbatim} +arrowAngle := \%pi-\%pi/10.0@DFLOAT The angle of the arrowhead. +arrowScale := 0.2@DFLOAT The size of the arrowhead + relative to the stem. +makeArrow(p1, p2) == + delta := p2 - p1 The arrow. + len := arrowScale * length delta The length of the arrowhead. + theta := atan(delta.1, delta.2) The angle from the x-axis + c1 := len*cos(theta + arrowAngle) The x-coord of left endpoint + s1 := len*sin(theta + arrowAngle) The y-coord of left endpoint + c2 := len*cos(theta - arrowAngle) The x-coord of right endpoint + s2 := len*sin(theta - arrowAngle) The y-coord of right endpoint + z := p2.3*(1 - arrowScale) The z-coord of both endpoints + p3 := point [p2.1 + c1, p2.2 + s1, z, p2.4] The left endpoint of head + p4 := point [p2.1 + c2, p2.2 + s2, z, p2.4] The right endpoint of head + [ [p1, p2, p3], [p2, p4] ] The arrow as a list of curves +\end{verbatim} + +Read the file and then create +an arrow from the point $origin$ to the point $unit$. + +Read the input file defining {\bf makeArrow}. + +\spadcommand{)read arrows} + +Construct the arrow (a list of two curves). + +\spadcommand{arrow := makeArrow(origin,unit)} + +Create an empty object $sp$ of type $ThreeSpace$. + +\spadcommand{sp := createThreeSpace()} + +Add each curve of the arrow to the space $sp$. + +\spadcommand{for a in arrow repeat sp := curve(sp,a)} + +Create a three-di\-men\-sion\-al viewport containing that space. + +\spadgraph{vp := makeViewport3D(sp,"Arrow")} + +%\epsffile[0 0 295 295]{ps/arrow.ps} + +Here is a better viewing angle. + +\spadcommand{rotate(vp,200,-60)} + +%\epsffile[0 0 295 295]{ps/arrowr.ps} + +\section{A Bouquet of Arrows} +\label{ugIntProgColorArr} + +%Axiom gathers up all the points of a graph and looks at the range +%of color values given as integers. +%If theses color values range from a minimum value of $a$ to a maximum +%value of $b$, then the $a$ values are colored red (the +%lowest color in our spectrum), and $b$ values are colored +%purple (the highest color), and those in the middle are colored +%green. +%When all the points are the same color as above, Axiom +%chooses green. + +Let's draw a ``bouquet'' of arrows. +Each arrow is identical. The arrowheads are +uniformly placed on a circle parallel to the $xy$-plane. +Thus the position of each arrow differs only +by the angle $\theta$, +$0 \leq \theta < 2\pi$, +between the arrow and +the $x$-axis on the $xy$-plane. + +Our bouquet is rather special: each arrow has a different +color (which won't be evident here, unfortunately). +This is arranged by letting the color of each successive arrow be +denoted by $\theta$. +In this way, the color of arrows ranges from red to green to violet. +Here is a program to draw a bouquet of $n$ arrows. + +\begin{verbatim} +drawBouquet(n,title) == + angle := 0.0@DFLOAT The initial angle + sp := createThreeSpace() Create empty space $sp$ + for i in 0..n-1 repeat For each index i, create: + start := point [0.0@DFLOAT,0.0@DFLOAT,0.0@DFLOAT,angle] + the point at base of arrow; + end := point [cos angle, sin angle, 1.0@DFLOAT, angle] + the point at tip of arrow; + arrow := makeArrow(start,end) the $i$th arrow + for a in makeArrow(start,end) repeat For each arrow component, + curve(sp,a) add the component to $sp$ + angle := angle + 2*\%pi/n The next angle + makeViewport3D(sp,title) Create the viewport from $sp$ +\end{verbatim} + +Read the input file. + +\spadcommand{)read bouquet} + +A bouquet of a dozen arrows. + +\spadgraph{drawBouquet(12,"A Dozen Arrows")} + +%\epsffile[0 0 295 295]{ps/bouquet.ps} + +\section{Diversion: When Things Go Wrong} +\label{ugIntProgDivTwo} +% +%Up to now, if you have typed in all the programs exactly as they are in +%the book, you have encountered no errors. +%In practice, however, it is easy to make mistakes. +%Computers are unforgiving: your program must be letter-for-letter correct +%or you will encounter some error. +% +%One thing that can go wrong is that you can create a syntactically +%incorrect program. +%As pointed out in Diversion 1, the meaning of Axiom programs is +%affected by indentation. +% +%The Axiom parser will ensure that all parentheses, brackets, and +%braces balance, and that commas and operators appear in the correct +%context. +%For example, change line ?? +%to ?? +%and run. +% +%A common mistake is to misspell an identifier or operation name. +%These are generally easy to spot since the interpreter will tell you the +%name of the operation together with the type and number of arguments which +%it is trying to find. +% +%Another mistake is to either to omit an argument or to give too many. +%Again Axiom will notify you of the offending operation. +% +%Indentation makes your programs more readable. +%However there are several ways to create a syntactically valid program. +%A most common problem occurs when a line is either indented improperly. +%% either or what? +%If this is a first line of a pile, then all the other lines will act as an +%inner pile to the first line. +%If it is a line of the pile other than the first line, Axiom then +%thinks that this line is a continuation of the previous line. +%More frequently than not, a syntactically correct expression is created. +%Almost never however will this be a semantically correct. +%Only when the program is run will an error be discovered. +%For example, change line ?? +%to ?? +%and run. + +\section{Drawing Complex Vector Fields} +\label{ugIntProgVecFields} + +We now put our arrows to good use drawing complex vector fields. +These vector fields give a representation of complex-valued +functions of complex variables. +Consider a Cartesian coordinate grid of points $(x, y)$ in +the plane, and some complex-valued function $f$ defined on +this grid. +At every point on this grid, compute the value of $f(x + +iy)$ and call it $z$. +Since $z$ has both a real and imaginary value for a given +$(x,y)$ grid point, there are four dimensions to plot. +What do we do? +We represent the values of $z$ by arrows planted at each +grid point. +Each arrow represents the value of $z$ in polar coordinates +$(r,\theta)$. +The length of the arrow is proportional to $r$. +Its direction is given by $\theta$. + +The code for drawing vector fields is in the file {\bf vectors.input}. +We discuss its contents from top to bottom. + +Before showing you the code, we have two small +matters to take care of. +First, what if the function has large spikes, say, ones that go off +to infinity? +We define a variable $clipValue$ for this purpose. When +$r$ exceeds the value of $clipValue$, then the value of +$clipValue$ is used instead of that for $r$. +For convenience, we define a function $clipFun(x)$ which uses +$clipValue$ to ``clip'' the value of $x$. + +% +\begin{verbatim} +clipValue : DFLOAT := 6 Maximum value allowed +clipFun(x) == min(max(x,-clipValue),clipValue) +\end{verbatim} + +Notice that we identify $clipValue$ as a small float but do +not declare the type of the function {\bf clipFun}. +As it turns out, {\bf clipFun} is called with a +small float value. +This declaration ensures that {\bf clipFun} never does a +conversion when it is called. + +The second matter concerns the possible ``poles'' of a +function, the actual points where the spikes have infinite +values. +Axiom uses normal {\tt DoubleFloat} arithmetic which +does not directly handle infinite values. +If your function has poles, you must adjust your step size to +avoid landing directly on them (Axiom calls {\bf error} +when asked to divide a value by $0$, for example). + +We set the variables $realSteps$ and $imagSteps$ to +hold the number of steps taken in the real and imaginary +directions, respectively. +Most examples will have ranges centered around the origin. +To avoid a pole at the origin, the number of points is taken +to be odd. + +\begin{verbatim} +realSteps: INT := 25 Number of real steps +imagSteps: INT := 25 Number of imaginary steps +)read arrows +\end{verbatim} + +Now define the function {\bf drawComplexVectorField} to draw the arrows. +It is good practice to declare the type of the main function in +the file. +This one declaration is usually sufficient to ensure that other +lower-level functions are compiled with the correct types. + +\begin{verbatim} +C := Complex DoubleFloat +S := Segment DoubleFloat +drawComplexVectorField: (C -> C, S, S) -> VIEW3D +\end{verbatim} + +The first argument is a function mapping complex small floats into +complex small floats. +The second and third arguments give the range of real and +imaginary values as segments like $a..b$. +The result is a three-di\-men\-sion\-al viewport. +Here is the full function definition: + +\begin{verbatim} +drawComplexVectorField(f, realRange,imagRange) == + delReal := (hi(realRange)-lo(realRange))/realSteps The real step size + delImag := (hi(imagRange)-lo(imagRange))/imagSteps The imaginary step size + sp := createThreeSpace() Create empty space $sp$ + real := lo(realRange) The initial real value + for i in 1..realSteps+1 repeat Begin real iteration + imag := lo(imagRange) initial imaginary value + for j in 1..imagSteps+1 repeat Begin imaginary iteration + z := f complex(real,imag) value of $f$ at the point + arg := argument z direction of the arrow + len := clipFun sqrt norm z length of the arrow + p1 := point [real, imag, 0.0@DFLOAT, arg] base point of the arrow + scaleLen := delReal * len scaled length of the arrow + p2 := point [p1.1 + scaleLen*cos(arg), tip point of the arrow + p1.2 + scaleLen*sin(arg),0.0@DFLOAT, arg] + arrow := makeArrow(p1, p2) Create the arrow + for a in arrow repeat curve(sp, a) Add arrow to space $sp$ + imag := imag + delImag The next imaginary value + real := real + delReal The next real value + makeViewport3D(sp, "Complex Vector Field") Draw it +\end{verbatim} + +As a first example, let us draw $f(z) == sin(z)$. +There is no need to create a user function: just pass the +\spadfunFrom{sin}{Complex DoubleFloat} from {\tt Complex DoubleFloat}. + +Read the file. + +\spadcommand{)read vectors } + +Draw the complex vector field of $sin(x)$. + +\spadgraph{drawComplexVectorField(sin,-2..2,-2..2) } + +%\epsffile[0 0 295 295]{ps/vectorsin.ps} + +\section{Drawing Complex Functions} +\label{ugIntProgCompFuns} + +Here is another way to graph a complex function of complex +arguments. +For each complex value $z$, compute $f(z)$, again +expressing the value in polar coordinates $(r,\theta{})$. +We draw the complex valued function, again considering the +$(x,y)$-plane as the complex plane, using $r$ as the +height (or $z$-coordinate) and $\theta$ as the color. +This is a standard plot---we learned how to do this in +Chapter \ref{ugGraph} on page~\pageref{ugGraph}--- +but here we write a new program to illustrate +the creation of polygon meshes, or grids. + +Call this function {\bf drawComplex}. +It displays the points using the ``mesh'' of points. +The function definition is in three parts. + +\begin{verbatim} +drawComplex: (C -> C, S, S) -> VIEW3D +drawComplex(f, realRange, imagRange) == The first part + delReal := (hi(realRange)-lo(realRange))/realSteps The real step size + delImag := (hi(imagRange)-lo(imagRange))/imagSteps The imaginary step size + Initial list of list of points $llp$ + llp:List List Point DFLOAT := [] +\end{verbatim} + +Variables $delReal$ and $delImag$ give the step +sizes along the real and imaginary directions as computed by the values +of the global variables $realSteps$ and $imagSteps$. +The mesh is represented by a list of lists of points $llp$, +initially empty. +Now $[ ]$ alone is ambiguous, so +to set this initial value +you have to tell Axiom what type of empty list it is. +Next comes the loop which builds $llp$. + +\begin{verbatim} + real := lo(realRange) The initial real value + for i in 1..realSteps+1 repeat Begin real iteration + imag := lo(imagRange) initial imaginary value + lp := []$(List Point DFLOAT) initial list of points $lp$ + for j in 1..imagSteps+1 repeat Begin imaginary iteration + z := f complex(real,imag) value of $f$ at the point + pt := point [real,imag, clipFun sqrt norm z, Create a point + argument z] + lp := cons(pt,lp) Add the point to $lp$ + imag := imag + delImag The next imaginary value + real := real + delReal The next real value + llp := cons(lp, llp) Add $lp$ to $llp$ +\end{verbatim} + +The code consists of both an inner and outer loop. +Each pass through the inner loop adds one list $lp$ of points +to the list of lists of points $llp$. +The elements of $lp$ are collected in reverse order. + +\begin{verbatim} + makeViewport3D(mesh(llp), "Complex Function") Create a mesh and display +\end{verbatim} + +The operation {\bf mesh} then creates an object of type +{\tt ThreeSpace(DoubleFloat)} from the list of lists of points. +This is then passed to {\bf makeViewport3D} to display the +image. + +Now add this function directly to your {\bf vectors.input} +file and re-read the file using read vectors. +We try {\bf drawComplex} using +a user-defined function $f$. + +Read the file. + +\spadcommand{)read vectors } + +This one has a pole at $z=0$. + +\spadcommand{f(z) == exp(1/z)} + +Draw it with an odd number of steps to avoid the pole. + +\spadgraph{drawComplex(f,-2..2,-2..2)} + +%\epsffile[0 0 295 295]{ps/complexexp.ps} + +\section{Functions Producing Functions} +\label{ugIntProgFunctions} + +In \ref{ugUserMake} on page~\pageref{ugUserMake}, +you learned how to use the operation +{\bf function} to create a function from symbolic formulas. +Here we introduce a similar operation which not only +creates functions, but functions from functions. + +The facility we need is provided by the package +{\tt MakeUnaryCompiledFunction(E,S,T)}. +\index{MakeUnaryCompiledFunction} +This package produces a unary (one-argument) compiled +function from some symbolic data +generated by a previous computation.\footnote{% +{\tt MakeBinaryCompiledFunction} is available for binary +functions.} +\index{MakeBinaryCompiledFunction} +The $E$ tells where the symbolic data comes from; +the $S$ and $T$ give Axiom the +source and target type of the function, respectively. +The compiled function produced has type +\spadsig{$S$}{$T$}. +To produce a compiled function with definition $p(x) == expr$, call +$compiledFunction(expr, x)$ from this package. +The function you get has no name. +You must to assign the function to the variable $p$ to give it that name. +% + +Do some computation. + +\spadcommand{(x+1/3)**5} + +Convert this to an anonymous function of $x$. +Assign it to the variable $p$ to give the function a name. + +\spadcommand{p := compiledFunction(\%,x)\$MakeUnaryCompiledFunction(POLY FRAC INT,DFLOAT,DFLOAT)} + +Apply the function. + +\spadcommand{p(sin(1.3))} + +For a more sophisticated application, read on. + +\section{Automatic Newton Iteration Formulas} +\label{ugIntProgNewton} + +This setting is needed to get Newton's iterations to converge. + +\spadcommand{)set streams calculate 10} + +We resume +our continuing saga of arrows and complex functions. +Suppose we want to investigate the behavior of Newton's iteration function +\index{Newton iteration} +in the complex plane. +Given a function $f$, we want to find the complex values +$z$ such that $f(z) = 0$. + +The first step is to produce a Newton iteration formula for +a given $f$: +$x_{n+1} = x_n - {{f(x_n)}\over{f'(x_n)}}.$ +We represent this formula by a function $g$ +that performs the computation on the right-hand side, that is, +$x_{n+1} = {g}(x_n)$. + +The type {\tt Expression Integer} (abbreviated {\tt EXPR +INT}) is used to represent general symbolic expressions in +Axiom. +\index{Expression} +To make our facility as general as possible, we assume +$f$ has this type. +Given $f$, we want +to produce a Newton iteration function $g$ which, +given a complex point $x_n$, delivers the next +Newton iteration point $x_{n+1}$. + +This time we write an input file called {\bf newton.input}. +We need to import {\tt MakeUnaryCompiledFunction} (discussed +in the last section), call it with appropriate types, and then define +the function $newtonStep$ which references it. +Here is the function $newtonStep$: + +\begin{verbatim} +C := Complex DoubleFloat The complex numbers +complexFunPack:=MakeUnaryCompiledFunction(EXPR INT,C,C) + Package for making functions + +newtonStep(f) == Newton's iteration function + fun := complexNumericFunction f Function for $f$ + deriv := complexDerivativeFunction(f,1) Function for $f'$ + (x:C):C +-> Return the iterator function + x - fun(x)/deriv(x) + +complexNumericFunction f == Turn an expression $f$ into a + v := theVariableIn f function + compiledFunction(f, v)$complexFunPack + +complexDerivativeFunction(f,n) == Create an nth derivative + v := theVariableIn f function + df := D(f,v,n) + compiledFunction(df, v)\$complexFunPack + +theVariableIn f == Returns the variable in $f$ + vl := variables f The list of variables + nv := # vl The number of variables + nv > 1 => error "Expression is not univariate." + nv = 0 => 'x Return a dummy variable + first vl +\end{verbatim} + +Do you see what is going on here? +A formula $f$ is passed into the function {\bf newtonStep}. +First, the function turns $f$ into a compiled program mapping +complex numbers into complex numbers. Next, it does the same thing +for the derivative of $f$. Finally, it returns a function which +computes a single step of Newton's iteration. + +The function {\bf complexNumericFunction} extracts the variable +from the expression $f$ and then turns $f$ into a function +which maps complex numbers into complex numbers. The function +{\bf complexDerivativeFunction} does the same thing for the +derivative of $f$. The function {\bf theVariableIn} +extracts the variable from the expression $f$, calling the function +{\bf error} if $f$ has more than one variable. +It returns the dummy variable $x$ if $f$ has no variables. + +Let's now apply {\bf newtonStep} to the formula for computing +cube roots of two. + +Read the input file with the definitions. + +\spadcommand{)read newton} + +\spadcommand{)read vectors } + +The cube root of two. + +\spadcommand{f := x**3 - 2} + +Get Newton's iteration formula. + +\spadcommand{g := newtonStep f} + +Let $a$ denote the result of +applying Newton's iteration once to the complex number $1 + \%i$. + +\spadcommand{a := g(1.0 + \%i)} + +Now apply it repeatedly. How fast does it converge? + +\spadcommand{[(a := g(a)) for i in 1..]} + +Check the accuracy of the last iterate. + +\spadcommand{a**3} + +In MappingPackage1, we show how functions can be +manipulated as objects in Axiom. +A useful operation to consider here is $*$, which means +composition. +For example $g*g$ causes the Newton iteration formula +to be applied twice. +Correspondingly, $g**n$ means to apply the iteration formula +$n$ times. + +Apply $g$ twice to the point $1 + \%i$. +\spadcommand{(g*g) (1.0 + \%i)} + +Apply $g$ 11 times. + +\spadcommand{(g**11) (1.0 + \%i)} + +Look now at the vector field and surface generated +after two steps of Newton's formula for the cube root of two. +The poles in these pictures represent bad starting values, and the +flat areas are the regions of convergence to the three roots. +% + +The vector field. + +\spadgraph{drawComplexVectorField(g**3,-3..3,-3..3)} + +%\epsffile[0 0 295 295]{ps/vectorroot.ps} + +The surface. + +\spadgraph{drawComplex(g**3,-3..3,-3..3)} + +%\epsffile[0 0 295 295]{ps/complexroot.ps} + +\setcounter{chapter}{10} % Chapter 11 + +\hyphenation{ +In-dexed-Aggre-gate +Lin-ear-Aggre-gate +shallowly-Mutable +draw-Vector-Field +set-Real-Steps +set-Imag-Steps +set-Clip-Value +} + +% Here and throughout the book we should use the terminology +% "type of a function", rather than talking about source and target. +% This is how the brave new world of SMWATT regards them. A function +% is just an object that has a mapping type. +% +\chapter{Packages} +\label{ugPackages} + +Packages provide the bulk of +\index{package} +Axiom's algorithmic library, from numeric packages for computing +special functions to symbolic facilities for +\index{constructor!package} +differential equations, symbolic integration, and limits. +\index{package!constructor} + +In Chapter \ref{ugIntProg} on page~\pageref{ugIntProg}, +we developed several useful functions for drawing +vector fields and complex functions. +We now show you how you can add these functions to the +Axiom library to make them available for general use. + +The way we created the functions in Chapter \ref{ugIntProg} +on page~\pageref{ugIntProg} is typical of how +you, as an advanced Axiom user, may interact with Axiom. +You have an application. +You go to your editor and create an input file defining some +functions for the application. +Then you run the file and try the functions. +Once you get them all to work, you will often want to extend them, +add new features, perhaps write additional functions. + +Eventually, when you have a useful set of functions for your application, +you may want to add them to your local Axiom library. +To do this, you embed these function definitions in a package and add +that package to the library. + +To introduce new packages, categories, and domains into the system, +you need to use the Axiom compiler to convert the constructors +into executable machine code. +An existing compiler in Axiom is available on an ``as-is'' +basis. +A new, faster compiler will be available in version 2.0 +of Axiom. + +\begin{figure} +\label{pak-cdraw} +\begin{verbatim} +C ==> Complex DoubleFloat All constructors used in a file +S ==> Segment DoubleFloat must be spelled out in full +INT ==> Integer unless abbreviated by macros +DFLOAT ==> DoubleFloat like these at the top of +VIEW3D ==> ThreeDimensionalViewport a file +CURVE ==> List List Point DFLOAT + +)abbrev package DRAWCX DrawComplex Identify kinds and abbreviations +DrawComplex(): Exports == Implementation where Type definition begins here + + Exports == with Export part begins + drawComplex: (C -> C,S,S,Boolean) -> VIEW3D Exported Operations + drawComplexVectorField: (C -> C,S,S) -> VIEW3D + setRealSteps: INT -> INT + setImagSteps: INT -> INT + setClipValue: DFLOAT-> DFLOAT + + Implementation == add Implementation part begins + arrowScale : DFLOAT := (0.2)::DFLOAT --relative size Local variable 1 + arrowAngle : DFLOAT := pi()-pi()/(20::DFLOAT) Local variable 2 + realSteps : INT := 11 --# real steps Local variable 3 + imagSteps : INT := 11 --# imaginary steps Local variable 4 + clipValue : DFLOAT := 10::DFLOAT --maximum vector length + Local variable 5 + + setRealSteps(n) == realSteps := n Exported function definition 1 + setImagSteps(n) == imagSteps := n Exported function definition 2 + setClipValue(c) == clipValue := c Exported function definition 3 + + clipFun: DFLOAT -> DFLOAT --Clip large magnitudes. + clipFun(x) == min(max(x, -clipValue), clipValue) + Local function definition 1 + + makeArrow: (Point DFLOAT,Point DFLOAT,DFLOAT,DFLOAT) -> CURVE + makeArrow(p1, p2, len, arg) == ... Local function definition 2 + + drawComplex(f, realRange, imagRange, arrows?) == ... + Exported function definition 4 +\end{verbatim} +\caption{The DrawComplex package.} +\label{fig-pak-cdraw} +\end{figure} + +\section{Names, Abbreviations, and File Structure} +\label{ugPackagesNames} +% +Each package has a name and an abbreviation. +For a package of the complex draw functions from Chapter +\ref{ugIntProg} on page~\pageref{ugIntProg}, +we choose the name {\tt DrawComplex} +and +\index{abbreviation!constructor} +abbreviation {\tt DRAWCX}.\footnote{An abbreviation can be any string +of +\index{constructor!abbreviation} +between two and seven capital letters and digits, beginning with a letter. +See \ref{ugTypesWritingAbbr} on page~\pageref{ugTypesWritingAbbr} +for more information.} +To be sure that you have not chosen a name or abbreviation already used by +the system, issue the system command {\tt )show} for both the name and +the abbreviation. +\index{show} + +Once you have named the package and its abbreviation, you can choose any new +filename you like with extension ``{\bf .spad}'' to hold the +definition of your package. +We choose the name {\bf drawpak.spad}. +If your application involves more than one package, you +can put them all in the same file. +Axiom assumes no relationship between the name of a library file, and +the name or abbreviation of a package. + +Near the top of the ``{\bf .spad}'' file, list all the +abbreviations for the packages +using {\tt )abbrev}, each command beginning in column one. +Macros giving names to Axiom expressions can also be placed near the +top of the file. +The macros are only usable from their point of definition until the +end of the file. + +Consider the definition of +{\tt DrawComplex} in +Figure \ref{fig-pak-cdraw} on page~\pageref{fig-pak-cdraw}. +After the macro +\index{macro} +definition +\begin{verbatim} +S ==> Segment DoubleFloat +\end{verbatim} +the name +{\tt S} can be used in the file as a +shorthand for {\tt Segment DoubleFloat}.\footnote{The interpreter also allows +{\tt macro} for macro definitions.} +The abbreviation command for the package +\begin{verbatim} +)abbrev package DRAWCX DrawComplex +\end{verbatim} +is given after the macros (although it could precede them). + +\section{Syntax} +\label{ugPackagesSyntax} +% +The definition of a package has the syntax: +\begin{center} +\frenchspacing{\it PackageForm {\tt :} Exports\quad{\tt ==}\quad Implementation} +\end{center} +The syntax for defining a package constructor is the same as that +\index{syntax} +for defining any function in Axiom. +In practice, the definition extends over many lines so that this syntax is +not practical. +Also, the type of a package is expressed by the operator $with$ +\index{with} +followed by an explicit list of operations. +A preferable way to write the definition of a package is with a $where$ +\index{where} +expression: + +The definition of a package usually has the form: \newline +{\tt% +{\it PackageForm} : Exports == Implementation where \newline +\hspace*{.75pc} {\it optional type declarations}\newline +\hspace*{.75pc} Exports == with \newline +\hspace*{2.0pc} {\it list of exported operations}\newline +\hspace*{.75pc} Implementation == add \newline +\hspace*{2.0pc} {\it list of function definitions for exported operations} +} + +The {\tt DrawComplex} package takes no parameters and exports five +operations, each a separate item of a {\it pile}. +Each operation is described as a {\it declaration}: a name, followed +by a colon ({\tt :}), followed by the type of the operation. +All operations have types expressed as mappings with +the syntax +\begin{center} +{\it +source\quad{\tt ->}\quad target +} +\end{center} + +\section{Abstract Datatypes} +\label{ugPackagesAbstract} + +A constructor as defined in Axiom is called an {\it abstract +datatype} in the computer science literature. +Abstract datatypes separate ``specification'' (what operations are +provided) from ``implementation'' (how the operations are implemented). +The {\tt Exports} (specification) part of a constructor is said to be ``public'' (it +provides the user interface to the package) whereas the {\tt Implementation} +part is ``private'' (information here is effectively hidden---programs +cannot take advantage of it). + +The {\tt Exports} part specifies what operations the package provides to users. +As an author of a package, you must ensure that +the {\tt Implementation} part provides a function for each +operation in the {\tt Exports} part.\footnote{The {\tt DrawComplex} +package enhances the facility +described in Chapter +\ref{ugIntProgCompFuns} on page~\pageref{ugIntProgCompFuns} by allowing a +complex function to have +arrows emanating from the surface to indicate the direction of the +complex argument.} + +An important difference between interactive programming and the +use of packages is in the handling of global variables such as +$realSteps$ and $imagSteps$. +In interactive programming, you simply change the values of +variables by {\it assignment}. +With packages, such variables are local to the package---their +values can only be set using functions exported by the package. +In our example package, we provide two functions +{\bf setRealSteps} and {\bf setImagSteps} for +this purpose. + +Another local variable is $clipValue$ which can be changed using +the exported operation {\bf setClipValue}. +This value is referenced by the internal function {\bf clipFun} that +decides whether to use the computed value of the function at a point or, +if the magnitude of that value is too large, the +value assigned to $clipValue$ (with the +appropriate sign). + +\section{Capsules} +\label{ugPackagesCapsules} +% +The part to the right of {\tt add} in the {\tt Implementation} +\index{add} +part of the definition is called a {\it capsule}. +The purpose of a capsule is: +\begin{itemize} +\item to define a function for each exported operation, and +\item to define a {\it local environment} for these functions to run. +\end{itemize} + +What is a local environment? +First, what is an environment? +\index{environment} +Think of the capsule as an input file that Axiom reads from top to +bottom. +Think of the input file as having a {\bf )clear all} at the top +so that initially no variables or functions are defined. +When this file is read, variables such as $realSteps$ and +$arrowSize$ in {\tt DrawComplex} are set to initial values. +Also, all the functions defined in the capsule are compiled. +These include those that are exported (like $drawComplex$), and +those that are not (like $makeArrow$). +At the end, you get a set of name-value pairs: +variable names (like $realSteps$ and $arrowSize$) +are paired with assigned values, while +operation names (like $drawComplex$ and $makeArrow$) +are paired with function values. + +This set of name-value pairs is called an {\it environment}. +Actually, we call this environment the ``initial environment'' of a package: +it is the environment that exists immediately after the package is +first built. +Afterwards, functions of this capsule can +access or reset a variable in the environment. +The environment is called {\it local} since any changes to the value of a +variable in this environment can be seen {\it only} by these functions. + +Only the functions from the package can change the variables in the local +environment. +When two functions are called successively from a package, +any changes caused by the first function called +are seen by the second. + +Since the environment is local to the package, its names +don't get mixed +up with others in the system or your workspace. +If you happen to have a variable called $realSteps$ in your +workspace, it does not affect what the +{\tt DrawComplex} functions do in any way. + +The functions in a package are compiled into machine code. +Unlike function definitions in input files that may be compiled repeatedly +as you use them with varying argument types, +functions in packages have a unique type (generally parameterized by +the argument parameters of a package) and a unique compilation residing on disk. + +The capsule itself is turned into a compiled function. +This so-called {\it capsule function} is what builds the initial environment +spoken of above. +If the package has arguments (see below), then each call to the package +constructor with a distinct pair of arguments +builds a distinct package, each with its own local environment. + +\section{Input Files vs. Packages} +\label{ugPackagesInputFiles} +% +A good question at this point would be ``Is writing a package more difficult than +writing an input file?'' + +The programs in input files are designed for flexibility and ease-of-use. +Axiom can usually work out all of your types as it reads your program +and does the computations you request. +Let's say that you define a one-argument function without giving its type. +When you first apply the function to a value, this +value is understood by Axiom as identifying the type for the +argument parameter. +Most of the time Axiom goes through the body of your function and +figures out the target type that you have in mind. +Axiom sometimes fails to get it right. +Then---and only then---do you need a declaration to tell Axiom what +type you want. + +Input files are usually written to be read by Axiom---and by you. +\index{file!input!vs. package} +Without suitable documentation and declarations, your input files +\index{package!vs. input file} +are likely incomprehensible to a colleague---and to you some +months later! + +Packages are designed for legibility, as well as +run-time efficiency. +There are few new concepts you need to learn to write +packages. Rather, you just have to be explicit about types +and type conversions. +The types of all functions are pre-declared so that Axiom---and the reader--- +knows precisely what types of arguments can be passed to and from +the functions (certainly you don't want a colleague to guess or to +have to work this out from context!). +The types of local variables are also declared. +Type conversions are explicit, never automatic.\footnote{There +is one exception to this rule: conversions from a subdomain to a +domain are automatic. +After all, the objects both have the domain as a common type.} + +In summary, packages are more tedious to write than input files. +When writing input files, you can casually go ahead, giving some +facts now, leaving others for later. +Writing packages requires forethought, care and discipline. + +\section{Compiling Packages} +\label{ugPackagesPackages} +% + +Once you have defined the package {\tt DrawComplex}, +you need to compile and test it. +To compile the package, issue the system command {\tt )compile drawpak}. +Axiom reads the file {\bf drawpak.spad} +and compiles its contents into machine binary. +If all goes well, the file {\tt DRAWCX.nrlib} is created in your +local directory for the package. +To test the package, you must load the package before trying an +operation. + +Compile the package. + +)compile drawpak + +Expose the package. + +\spadcommand{)expose DRAWCX } + +Use an odd step size to avoid +a pole at the origin. + +\spadcommand{setRealSteps 51 } + +\spadcommand{setImagSteps 51 } + +Define {\bf f} to be the Gamma function. + +\spadcommand{f(z) == Gamma(z) } + +Clip values of function with magnitude larger than 7. + +\spadcommand{setClipValue 7} + +Draw the {\bf Gamma} function. + +\spadgraph{drawComplex(f,-\%pi..\%pi,-\%pi..\%pi, false) } + +%\epsffile[0 0 300 300]{ps/3dgamma11.ps} + +\section{Parameters} +\label{ugPackagesParameters} +% +The power of packages becomes evident when packages have parameters. +Usually these parameters are domains and the exported operations have types +involving these parameters. + +In Chapter \ref{ugTypes} on page~\pageref{ugTypes}, +you learned that categories denote classes of domains. +Although we cover this notion in detail in the next +chapter, we now give you a sneak preview of its usefulness. + +In \ref{ugUserBlocks} on page~\pageref{ugUserBlocks}, +we defined functions $bubbleSort(m)$ and +$insertionSort(m)$ to sort a list of integers. +If you look at the code for these functions, you see that they may be +used to sort {\it any} structure $m$ with the right properties. +Also, the functions can be used to sort lists of {\it any} elements---not +just integers. +Let us now recall the code for $bubbleSort$. + +\begin{verbatim} +bubbleSort(m) == + n := #m + for i in 1..(n-1) repeat + for j in n..(i+1) by -1 repeat + if m.j < m.(j-1) then swap!(m,j,j-1) + m +\end{verbatim} + +What properties of ``lists of integers'' are assumed by the sorting +algorithm? +In the first line, the operation {\bf \#} computes the maximum index of +the list. +The first obvious property is that $m$ must have a finite number of +elements. +In Axiom, this is done +by your telling Axiom that $m$ has +the ``attribute'' {\bf finiteAggregate}. +An {\it attribute} is a property +that a domain either has or does not have. +As we show later in +\ref{ugCategoriesAttributes} on page~\pageref{ugCategoriesAttributes}, +programs can query domains as to the presence or absence of an attribute. + +The operation {\bf swap} swaps elements of $m$. +Using Browse, you find that {\bf swap} requires its +elements to come from a domain of category +{\tt IndexedAggregate} with attribute +{\tt shallowlyMutable}. +This attribute means that you can change the internal components +of $m$ without changing its external structure. +Shallowly-mutable data structures include lists, streams, one- and +two-dimensional arrays, vectors, and matrices. + +The category {\tt IndexedAggregate} designates the class of +aggregates whose elements can be accessed by the notation +$m.s$ for suitable selectors $s$. +The category {\tt IndexedAggregate} takes two arguments: +$Index$, a domain of selectors for the aggregate, and +$Entry$, a domain of entries for the aggregate. +Since the sort functions access elements by integers, we must +choose $Index = ${\tt Integer}. +The most general class of domains for which $bubbleSort$ and +$insertionSort$ are defined are those of +category {\tt IndexedAggregate(Integer,Entry)} with the two +attributes {\bf shallowlyMutable} and +{\bf finiteAggregate}. + +Using Browse, you can also discover that Axiom has many kinds of domains +with attribute {\bf shallowlyMutable}. +Those of class {\tt IndexedAggregate(Integer,Entry)} include +{\tt Bits}, {\tt FlexibleArray}, {\tt OneDimensionalArray}, +{\tt List}, {\tt String}, and {\tt Vector}, and also +{\tt HashTable} and {\tt EqTable} with integer keys. +Although you may never want to sort all such structures, we +nonetheless demonstrate Axiom's +ability to do so. + +Another requirement is that {\tt Entry} has an +operation {\tt <}. +One way to get this operation is to assume that +{\tt Entry} has category {\tt OrderedSet}. +By definition, will then export a {\tt <} operation. +A more general approach is to allow any comparison function +$f$ to be used for sorting. +This function will be passed as an argument to the sorting +functions. + +Our sorting package then takes two arguments: a domain $S$ +of objects of {\it any} type, and a domain $A$, an aggregate +of type {\tt IndexedAggregate(Integer, S)} with the above +two attributes. +Here is its definition using what are close to the original +definitions of $bubbleSort$ and $insertionSort$ for +sorting lists of integers. +The symbol {\tt !} is added to the ends of the operation +names. +This uniform naming convention is used for Axiom operation +names that destructively change one or more of their arguments. + +\begin{verbatim} +SortPackage(S,A) : Exports == Implementation where + S: Object + A: IndexedAggregate(Integer,S) + with (finiteAggregate; shallowlyMutable) + + Exports == with + bubbleSort!: (A,(S,S) -> Boolean) -> A + insertionSort!: (A, (S,S) -> Boolean) -> A + + Implementation == add + bubbleSort!(m,f) == + n := #m + for i in 1..(n-1) repeat + for j in n..(i+1) by -1 repeat + if f(m.j,m.(j-1)) then swap!(m,j,j-1) + m + insertionSort!(m,f) == + for i in 2..#m repeat + j := i + while j > 1 and f(m.j,m.(j-1)) repeat + swap!(m,j,j-1) + j := (j - 1) pretend PositiveInteger + m +\end{verbatim} + +\section{Conditionals} +\label{ugPackagesConds} +% +When packages have parameters, you can say that an operation is or is not +\index{conditional} +exported depending on the values of those parameters. +When the domain of objects $S$ has an {\tt <} +operation, we can supply one-argument versions of +$bubbleSort$ and $insertionSort$ which use this operation +for sorting. +The presence of the +operation {\tt <} is guaranteed when $S$ is an ordered set. + +\begin{verbatim} +Exports == with + bubbleSort!: (A,(S,S) -> Boolean) -> A + insertionSort!: (A, (S,S) -> Boolean) -> A + + if S has OrderedSet then + bubbleSort!: A -> A + insertionSort!: A -> A +\end{verbatim} + +In addition to exporting the one-argument sort operations +\index{sort!bubble} +conditionally, we must provide conditional definitions for the +\index{sort!insertion} +operations in the {\tt Implementation} part. +This is easy: just have the one-argument functions call the +corresponding two-argument functions with the operation +{\tt <} from $S$. + +\begin{verbatim} + Implementation == add + ... + if S has OrderedSet then + bubbleSort!(m) == bubbleSort!(m,<$S) + insertionSort!(m) == insertionSort!(m,<$S) +\end{verbatim} + +In \ref{ugUserBlocks} on page~\pageref{ugUserBlocks}, +we give an alternative definition of +{\bf bubbleSort} using \spadfunFrom{first}{List} and +\spadfunFrom{rest}{List} that is more efficient for a list (for +which access to any element requires traversing the list from its +first node). +To implement a more efficient algorithm for lists, we need the +operation {\bf setelt} which allows us to destructively change +the {\bf first} and {\bf rest} of a list. +Using Browse, you find that these operations come from category +{\tt UnaryRecursiveAggregate}. +Several aggregate types are unary recursive aggregates including +those of {\tt List} and {\tt AssociationList}. +We provide two different implementations for +{\bf bubbleSort!} and {\bf insertionSort!}: one +for list-like structures, another for array-like structures. + +\begin{verbatim} +Implementation == add + ... + if A has UnaryRecursiveAggregate(S) then + bubbleSort!(m,fn) == + empty? m => m + l := m + while not empty? (r := l.rest) repeat + r := bubbleSort! r + x := l.first + if fn(r.first,x) then + l.first := r.first + r.first := x + l.rest := r + l := l.rest + m + insertionSort!(m,fn) == + ... +\end{verbatim} + +The ordering of definitions is important. +The standard definitions come first and +then the predicate +\begin{verbatim} +A has UnaryRecursiveAggregate(S) +\end{verbatim} +is evaluated. +If {\tt true}, the special definitions cover up the standard ones. + +Another equivalent way to write the capsule is to use an +$if-then-else$ expression: +\index{if} + +\begin{verbatim} + if A has UnaryRecursiveAggregate(S) then + ... + else + ... +\end{verbatim} + +\section{Testing} +\label{ugPackagesCompiling} +% +Once you have written the package, embed it in a file, for example, {\bf +sortpak.spad}. +\index{testing} +Be sure to include an {\bf )abbrev} command at the top of the file: +\begin{verbatim} +)abbrev package SORTPAK SortPackage +\end{verbatim} +Now compile the file (using {\tt )compile sortpak.spad}). + +Expose the constructor. +You are then ready to begin testing. + +\spadcommand{)expose SORTPAK} + +Define a list. + +\spadcommand{l := [1,7,4,2,11,-7,3,2]} + +Since the integers are an ordered set, +a one-argument operation will do. + +\spadcommand{bubbleSort!(l)} + +Re-sort it using ``greater than.'' + +\spadcommand{bubbleSort!(l,(x,y) +-> x > y)} + +Now sort it again using {\tt <} on integers. + +\spadcommand{bubbleSort!(l, <\$Integer)} + +A string is an aggregate of characters so we can sort them as well. + +\spadcommand{bubbleSort! "Mathematical Sciences"} + +Is {\tt <} defined on booleans? + +\spadcommand{false < true} + +Good! Create a bit string representing ten consecutive +boolean values {\tt true}. + +\spadcommand{u : Bits := new(10,true)} + +Set bits 3 through 5 to {\tt false}, then display the result. + +\spadcommand{u(3..5) := false; u} + +Now sort these booleans. + +\spadcommand{bubbleSort! u} + +Create an ``eq-table'', a +table having integers as keys +and strings as values. + +\spadcommand{t : EqTable(Integer,String) := table()} + +Give the table a first entry. + +\spadcommand{t.1 := "robert"} + +And a second. + +\spadcommand{t.2 := "richard"} + +What does the table look like? + +\spadcommand{t} + +Now sort it. + +\spadcommand{bubbleSort! t} + +\section{How Packages Work} +\label{ugPackagesHow} +% +Recall that packages as abstract datatypes are compiled independently +and put into the library. +The curious reader may ask: ``How is the interpreter able to find an +operation such as {\bf bubbleSort!}? +Also, how is a single compiled function such as {\bf bubbleSort!} able +to sort data of different types?'' + +After the interpreter loads the package {\tt SortPackage}, the four +operations from the package become known to the interpreter. +Each of these operations is expressed as a {\it modemap} in which the type +\index{modemap} +of the operation is written in terms of symbolic domains. + +See the modemaps for {\bf bubbleSort!}. + +)display op bubbleSort! + +\begin{verbatim} +There are 2 exposed functions called bubbleSort! : + + [1] D1 -> D1 from SortPackage(D2,D1) + if D2 has ORDSET and D2 has OBJECT and D1 has + IndexedAggregate(Integer, D2) with + finiteAggregate + shallowlyMutable + + [2] (D1,((D3,D3) -> Boolean)) -> D1 from SortPackage(D3,D1) + if D3 has OBJECT and D1 has + IndexedAggregate(Integer,D3) with + finiteAggregate + shallowlyMutable +\end{verbatim} + +What happens if you ask for $bubbleSort!([1,-5,3])$? +There is a unique modemap for an operation named +{\bf bubbleSort!} with one argument. +Since $[1,-5,3]$ is a list of integers, the symbolic domain +$D1$ is defined as {\tt List(Integer)}. +For some operation to apply, it must satisfy the predicate for +some $D2$. +What $D2$? +The third expression of the {\tt and} requires {\tt D1 has +IndexedAggregate(Integer, D2) with} two attributes. +So the interpreter searches for an {\tt IndexedAggregate} +among the ancestors of {\tt List (Integer)} (see +\ref{ugCategoriesHier} on page~\pageref{ugCategoriesHier}). +It finds one: {\tt IndexedAggregate(Integer, Integer)}. +The interpreter tries defining $D2$ as {\tt Integer}. +After substituting for $D1$ and $D2$, the predicate +evaluates to {\tt true}. +An applicable operation has been found! + +Now Axiom builds the package +{\tt SortPackage(List(Integer), Integer)}. +According to its definition, this package exports the required +operation: {\bf bubbleSort!}: \spadsig{List Integer}{List +Integer}. +The interpreter then asks the package for a function implementing +this operation. +The package gets all the functions it needs (for example, +{\bf rest} and {\bf swap}) from the appropriate +domains and then it +returns a {\bf bubbleSort!} to the interpreter together with +the local environment for {\bf bubbleSort!}. +The interpreter applies the function to the argument $[1,-5,3]$. +The {\bf bubbleSort!} function is executed in its local +environment and produces the result. + + +\setcounter{chapter}{11} % Chapter 12 + +\chapter{Categories} +\label{ugCategories} + +This chapter unravels the mysteries of categories---what +\index{category} +they are, how they are related to domains and packages, +\index{category!constructor} +how they are defined in Axiom, and how you can extend the +\index{constructor!category} +system to include new categories of your own. + +We assume that you have read the introductory material on domains +and categories in \ref{ugTypesBasicDomainCons} on +page~\pageref{ugTypesBasicDomainCons}. +There you learned that the notion of packages covered in the +previous chapter are special cases of domains. +While this is in fact the case, it is useful here to regard domains +as distinct from packages. + +Think of a domain as a datatype, a collection of objects (the +objects of the domain). +From your ``sneak preview'' in the previous chapter, you might +conclude that categories are simply named clusters of operations +exported by domains. +As it turns out, categories have a much deeper meaning. +Categories are fundamental to the design of Axiom. +They control the interactions between domains and algorithmic +packages, and, in fact, between all the components of Axiom. + +Categories form hierarchies as shown on the inside cover pages of +this book. +The inside front-cover pages illustrate the basic +algebraic hierarchy of the Axiom programming language. +The inside back-cover pages show the hierarchy for data +structures. + +Think of the category structures of Axiom as a foundation +for a city on which superstructures (domains) are built. +The algebraic hierarchy, for example, serves as a foundation for +constructive mathematical algorithms embedded in the domains of +Axiom. +Once in place, domains can be constructed, either independently or +from one another. + +Superstructures are built for quality---domains are compiled into +machine code for run-time efficiency. +You can extend the foundation in directions beyond the space +directly beneath the superstructures, then extend selected +superstructures to cover the space. +Because of the compilation strategy, changing components of the +foundation generally means that the existing superstructures +(domains) built on the changed parts of the foundation +(categories) have to be rebuilt---that is, recompiled. + +Before delving into some of the interesting facts about categories, let's see +how you define them in Axiom. + +\section{Definitions} +\label{ugCategoriesDefs} + +A category is defined by a function with exactly the same format as +\index{category!definition} +any other function in Axiom. + +The definition of a category has the syntax: +\begin{center} +{\it CategoryForm} : {\tt Category\quad{}==\quad{}} {\it Extensions} {\tt [ with} {\it Exports} {\tt ]} +\end{center} + +The brackets {\tt [ ]} here indicate optionality. + + +The first example of a category definition is +{\tt SetCategory}, +the most basic of the algebraic categories in Axiom. +\index{SetCategory} + +\begin{verbatim} +SetCategory(): Category == + Join(Type,CoercibleTo OutputForm) with + "=" : ($, $) -> Boolean +\end{verbatim} + +The definition starts off with the name of the +category ({\tt SetCategory}); this is +always in column one in the source file. +%% maybe talk about naming conventions for source files? .spad or .ax? +All parts of a category definition are then indented with respect to this +\index{indentation} +first line. + +In Chapter \ref{ugTypes} on page~\pageref{ugTypes}, +we talked about {\tt Ring} as denoting the +class of all domains that are rings, in short, the class of all +rings. +While this is the usual naming convention in Axiom, it is also +common to use the word ``Category'' at the end of a category name for clarity. +The interpretation of the name {\tt SetCategory} is, then, ``the +category of all domains that are (mathematical) sets.'' + +The name {\tt SetCategory} is followed in the definition by its +formal parameters enclosed in parentheses {\tt ()}. +Here there are no parameters. +As required, the type of the result of this category function is the +distinguished name {\sf Category}. + +Then comes the {\tt ==}. +As usual, what appears to the right of the {\tt ==} is a +definition, here, a category definition. +A category definition always has two parts separated by the reserved word +\index{with} +$with$. +%\footnote{Debugging hint: it is very easy to forget +%the $with$!} + +The first part tells what categories the category extends. +Here, the category extends two categories: {\tt Type}, the +category of all domains, and +{\tt CoercibleTo(OutputForm)}. +%\footnote{{\tt CoercibleTo(OutputForm)} +%can also be written (and is written in the definition above) without +%parentheses.} +The operation $Join$ is a system-defined operation that +\index{Join} +forms a single category from two or more other categories. + +Every category other than {\tt Type} is an extension of some other +category. +If, for example, {\tt SetCategory} extended only the category +{\tt Type}, the definition here would read ``{\tt Type with +...}''. +In fact, the {\tt Type} is optional in this line; ``{\tt with +...}'' suffices. + +\section{Exports} +\label{ugCategoriesExports} + + +To the right of the $with$ is a list of +\index{with} +all the exports of the category. +Each exported operation has a name and a type expressed by a +{\it declaration} of the form +``{\frenchspacing\tt {\it name}: {\it type}}''. + +Categories can export symbols, as well as +{\tt 0} and {\tt 1} which denote +domain constants.\footnote{The +numbers {\tt 0} and {\tt 1} are operation names in Axiom.} +In the current implementation, all other exports are operations with +types expressed as mappings with the syntax +\begin{center} +{\it +source\quad{\tt ->}\quad target +} +\end{center} + +The category {\tt SetCategory} has a single export: the operation +$=$ whose type is given by the mapping {\tt (\$, \$) -> Boolean}. +The {\tt \$} in a mapping type always means ``the domain.'' Thus +the operation $=$ takes two arguments from the domain and +returns a value of type {\tt Boolean}. + +The source part of the mapping here is given by a {\it tuple} +\index{tuple} +consisting of two or more types separated by commas and enclosed in +parentheses. +If an operation takes only one argument, you can drop the parentheses +around the source type. +If the mapping has no arguments, the source part of the mapping is either +left blank or written as {\tt ()}. +Here are examples of formats of various operations with some +contrived names. + +\begin{verbatim} +someIntegerConstant : $ +aZeroArgumentOperation: () -> Integer +aOneArgumentOperation: Integer -> $ +aTwoArgumentOperation: (Integer,$) -> Void +aThreeArgumentOperation: ($,Integer,$) -> Fraction($) +\end{verbatim} + +\section{Documentation} +\label{ugCategoriesDoc} + +The definition of {\tt SetCategory} above is missing +an important component: its library documentation. +\index{documentation} +Here is its definition, complete with documentation. + +\begin{verbatim} +++ Description: +++ \bs{}axiomType\{SetCategory\} is the basic category +++ for describing a collection of elements with +++ \bs{}axiomOp\{=\} (equality) and a \bs{}axiomFun\{coerce\} +++ to \bs{}axiomType\{OutputForm\}. + +SetCategory(): Category == + Join(Type, CoercibleTo OutputForm) with + "=": ($, $) -> Boolean + ++ \bs{}axiom\{x = y\} tests if \bs{}axiom\{x\} and + ++ \bs{}axiom\{y\} are equal. +\end{verbatim} + +Documentary comments are an important part of constructor definitions. +Documentation is given both for the category itself and for +each export. +A description for the category precedes the code. +Each line of the description begins in column one with {\tt ++}. +The description starts with the word {\tt Description:}.\footnote{Other +information such as the author's name, date of creation, and so on, +can go in this +area as well but are currently ignored by Axiom.} +All lines of the description following the initial line are +indented by the same amount. + +{\sloppy +Surround the name of any constructor (with or without parameters) with an +\verb+{\bf }+. +Similarly, surround an +operator name with \verb+{\tt }+, +an Axiom operation with \verb+{\bf }+, and a +variable or Axiom expression with +\verb+$$+. +Library documentation is given in a \TeX{}-like language so that +it can be used both for hard-copy and for Browse. +These different wrappings cause operations and types to have +mouse-active buttons in Browse. +For hard-copy output, wrapped expressions appear in a different font. +The above documentation appears in hard-copy as: + +} +% +\begin{quotation} +% +{\tt SetCategory} is the basic category +for describing a collection of elements with {\tt =} +(equality) and a {\bf coerce} to {\tt OutputForm}. +% +\end{quotation} +% +and +% +\begin{quotation} +% +$x = y$ tests if $x$ and $y$ are equal. +% +\end{quotation} +% + +For our purposes in this chapter, we omit the documentation from further +category descriptions. + +\section{Hierarchies} +\label{ugCategoriesHier} + +A second example of a category is +{\tt SemiGroup}, defined by: +\index{SemiGroup} + +\begin{verbatim} +SemiGroup(): Category == SetCategory with + "*": ($,$) -> $ + "**": ($, PositiveInteger) -> $ +\end{verbatim} + +This definition is as simple as that for {\tt SetCategory}, +except that there are two exported operations. +Multiple exported operations are written as a {\it pile}, +that is, they all begin in the same column. +Here you see that the category mentions another type, +{\tt PositiveInteger}, in a signature. +Any domain can be used in a signature. + +Since categories extend one another, they form hierarchies. +Each category other than {\tt Type} has one or more parents given +by the one or more categories mentioned before the $with$ part of +the definition. +{\tt SemiGroup} extends {\tt SetCategory} and +{\tt SetCategory} extends both {\tt Type} and +{\tt CoercibleTo (OutputForm)}. +Since {\tt CoercibleTo (OutputForm)} also extends {\tt Type}, +the mention of {\tt Type} in the definition is unnecessary but +included for emphasis. + +\section{Membership} +\label{ugCategoriesMembership} + +We say a category designates a class of domains. +What class of domains? +\index{category!membership} +That is, how does Axiom know what domains belong to what categories? +The simple answer to this basic question is key to the design of +Axiom: + +\begin{center} +{\bf Domains belong to categories by assertion.} +\end{center} + +When a domain is defined, it is asserted to belong to one or more +categories. +Suppose, for example, that an author of domain {\tt String} wishes to +use the binary operator $*$ to denote concatenation. +Thus $"hello " * "there"$ would produce the string +$"hello there"$\footnote{Actually, concatenation of strings in +Axiom is done by juxtaposition or by using the operation +\spadfunFrom{concat}{String}. +The expression $"hello " "there"$ produces the string +$"hello there"$.}. +The author of {\tt String} could then assert that {\tt String} +is a member of {\tt SemiGroup}. +According to our definition of {\tt SemiGroup}, strings +would then also have the operation $**$ defined automatically. +Then $"--" ** 4$ would produce a string of eight dashes +$"--------"$. +Since {\tt String} is a member of {\tt SemiGroup}, it also is +a member of {\tt SetCategory} and thus has an operation +$=$ for testing that two strings are equal. + +Now turn to the algebraic category hierarchy inside the +front cover of this book. +Any domain that is a member of a +category extending {\tt SemiGroup} is a member of +{\tt SemiGroup} (that is, it {\it is} a semigroup). +In particular, any domain asserted to be a {\tt Ring} is a +semigroup since {\tt Ring} extends {\tt Monoid}, that, +in turn, extends {\tt SemiGroup}. +The definition of {\tt Integer} in Axiom asserts that +{\tt Integer} is a member of category +{\tt IntegerNumberSystem}, that, in turn, asserts that it is +a member of {\tt EuclideanDomain}. +Now {\tt EuclideanDomain} extends +{\tt PrincipalIdealDomain} and so on. +If you trace up the hierarchy, you see that +{\tt EuclideanDomain} extends {\tt Ring}, and, +therefore, {\tt SemiGroup}. +Thus {\tt Integer} is a semigroup and also exports the +operations $*$ and $**$. + +\section{Defaults} +\label{ugCategoriesDefaults} + +We actually omitted the last \index{category!defaults} part of the +definition of \index{default definitions} {\tt SemiGroup} in +\ref{ugCategoriesHier} on page~\pageref{ugCategoriesHier}. Here now +is its complete Axiom definition. + +\begin{verbatim} +SemiGroup(): Category == SetCategory with + "*": ($, $) -> $ + "**": ($, PositiveInteger) -> $ + add + import RepeatedSquaring($) + x: $ ** n: PositiveInteger == expt(x,n) +\end{verbatim} + +The $add$ part at the end is used to give ``default definitions'' for +\index{add} +exported operations. +Once you have a multiplication operation $*$, you can +define exponentiation +for positive integer exponents +using repeated multiplication: + +$x^n = {\underbrace{x \, x \, x \, \cdots \, +x}_{\displaystyle n \hbox{\ times}}}$ + +This definition for $**$ is called a {\it default} definition. +In general, a category can give default definitions for any +operation it exports. +Since {\tt SemiGroup} and all its category descendants in the hierarchy +export $**$, any descendant category may redefine $**$ as well. + +A domain of category {\tt SemiGroup} +(such as {\tt Integer}) may or may not choose to +define its own $**$ operation. +If it does not, a default definition that is closest (in a ``tree-distance'' +sense of the hierarchy) to the domain is chosen. + +The part of the category definition following an $add$ operation +is a {\it capsule}, as discussed in +the previous chapter. +The line +\begin{verbatim} +import RepeatedSquaring($) +\end{verbatim} +references the package +{\tt RepeatedSquaring(\$)}, that is, the package +{\tt RepeatedSquaring} that takes ``this domain'' as its +parameter. +For example, if the semigroup {\tt Polynomial (Integer)} +does not define its own exponentiation operation, the +definition used may come from the package +{\tt RepeatedSquaring (Polynomial (Integer))}. +The next line gives the definition in terms of {\bf expt} from that +package. + +The default definitions are collected to form a ``default +package'' for the category. +The name of the package is the same as the category but with an +ampersand ({\tt \&}) added at the end. +A default package always takes an additional argument relative to the +category. +Here is the definition of the default package {\tt SemiGroup\&} as +automatically generated by Axiom from the above definition of +{\tt SemiGroup}. + +\begin{verbatim} +SemiGroup_&($): Exports == Implementation where + $: SemiGroup + Exports == with + "**": ($, PositiveInteger) -> $ + Implementation == add + import RepeatedSquaring($) + x:$ ** n:PositiveInteger == expt(x,n) +\end{verbatim} + +\section{Axioms} +\label{ugCategoriesAxioms} + +In the previous section you saw the +complete Axiom program defining \index{axiom} +{\tt SemiGroup}. +According to this definition, semigroups (that is, are sets with +the operations \spadopFrom{*}{SemiGroup} and +\spadopFrom{**}{SemiGroup}. +\index{SemiGroup} + +You might ask: ``Aside from the notion of default packages, isn't +a category just a {\it macro}, that is, a shorthand +equivalent to the two operations $*$ and $**$ with +their types?'' If a category were a macro, every time you saw the +word {\tt SemiGroup}, you would rewrite it by its list of +exported operations. +Furthermore, every time you saw the exported operations of +{\tt SemiGroup} among the exports of a constructor, you could +conclude that the constructor exported {\tt SemiGroup}. + +A category is {\it not} a macro and here is why. +The definition for {\tt SemiGroup} has documentation that states: + +\begin{quotation} + Category {\tt SemiGroup} denotes the class of all multiplicative + semigroups, that is, a set with an associative operation $*$. + + \vskip .5\baselineskip + {Axioms:} + + {\small\tt associative("*" : (\$,\$)->\$) -- (x*y)*z = x*(y*z)} +\end{quotation} + +According to the author's remarks, the mere +exporting of an operation named $*$ and $**$ is not +enough to qualify the domain as a {\tt SemiGroup}. +In fact, a domain can be a semigroup only if it explicitly +exports a $**$ and +a $*$ satisfying the associativity axiom. + +In general, a category name implies a set of axioms, even mathematical +theorems. +There are numerous axioms from {\tt Ring}, for example, +that are well-understood from the literature. +No attempt is made to list them all. +Nonetheless, all such mathematical facts are implicit by the use of the +name {\tt Ring}. + +\section{Correctness} +\label{ugCategoriesCorrectness} + +While such statements are only comments, +\index{correctness} +Axiom can enforce their intention simply by shifting the burden of +responsibility onto the author of a domain. +A domain belongs to category $Ring$ only if the +author asserts that the domain belongs to {\tt Ring} or +to a category that extends {\tt Ring}. + +This principle of assertion is important for large user-extendable +systems. +Axiom has a large library of operations offering facilities in +many areas. +Names such as {\bf norm} and {\bf product}, for example, have +diverse meanings in diverse contexts. +An inescapable hindrance to users would be to force those who wish to +extend Axiom to always invent new names for operations. +%>> I don't think disambiguate is really a word, though I like it +Axiom allows you to reuse names, and then use context to disambiguate one +from another. + +Here is another example of why this is important. +Some languages, such as {\bf APL}, +\index{APL} +denote the {\tt Boolean} constants {\tt true} and +{\tt false} by the integers $1$ and $0$. +You may want to let infix operators $+$ and $*$ serve as the logical +operators {\bf or} and {\bf and}, respectively. +But note this: {\tt Boolean} is not a ring. +The {\it inverse axiom} for {\tt Ring} states: +% +\begin{center} +Every element $x$ has an additive inverse $y$ such that +$x + y = 0$. +\end{center} +% +{\tt Boolean} is not a ring since {\tt true} has +no inverse---there is no inverse element $a$ such that +$1 + a = 0$ (in terms of booleans, {\tt (true or a) = false}). +Nonetheless, Axiom {\it could} easily and correctly implement +{\tt Boolean} this way. +{\tt Boolean} simply would not assert that it is of category +{\tt Ring}. +Thus the ``{\tt +}'' for {\tt Boolean} values +is not confused with the one for {\tt Ring}. +Since the {\tt Polynomial} constructor requires its argument +to be a ring, Axiom would then refuse to build the +domain {\tt Polynomial(Boolean)}. Also, Axiom would refuse to +wrongfully apply algorithms to {\tt Boolean} elements that presume that the +ring axioms for ``{\tt +}'' hold. + +\section{Attributes} +\label{ugCategoriesAttributes} + +Most axioms are not computationally useful. +Those that are can be explicitly expressed by what Axiom calls an +{\it attribute}. +The attribute {\bf commutative("*")}, for example, is used to assert +that a domain has commutative multiplication. +Its definition is given by its documentation: + +\begingroup \parindent=1pc \narrower\noindent% + A domain $R$ has {\bf commutative("*")} + if it has an operation "*": \spadsig{(R,R)}{R} such that $x * y = y * x$. +\par\endgroup + +Just as you can test whether a domain has the category {\tt Ring}, you +can test that a domain has a given attribute. + + +Do polynomials over the integers +have commutative multiplication? + +\spadcommand{Polynomial Integer has commutative("*")} + +Do matrices over the integers +have commutative multiplication? + +\spadcommand{Matrix Integer has commutative("*")} + +Attributes are used to conditionally export and define operations for +a domain (see \ref{ugDomainsAssertions} on +page~\pageref{ugDomainsAssertions}). Attributes can also be asserted +in a category definition. + +After mentioning category {\tt Ring} many times in this book, +it is high time that we show you its definition: +\index{Ring} + +\begin{verbatim} +Ring(): Category == + Join(Rng,Monoid,LeftModule($: Rng)) with + characteristic: -> NonNegativeInteger + coerce: Integer -> $ + unitsKnown + add + n:Integer + coerce(n) == n * 1$$ +\end{verbatim} + +There are only two new things here. +First, look at the {\tt \$\$} on the last line. +This is not a typographic error! +The first {\tt \$} says that the $1$ is to come from some +domain. +The second {\tt \$} says that the domain is ``this domain.'' +If {\tt \$} is {\tt Fraction(Integer)}, this line reads {\tt +coerce(n) == n * 1\$Fraction(Integer)}. + +The second new thing is the presence of attribute ``$unitsKnown$''. +Axiom can always distinguish an attribute from an operation. +An operation has a name and a type. An attribute has no type. +The attribute {\bf unitsKnown} asserts a rather subtle mathematical +fact that is normally taken for granted when working with +rings.\footnote{With this axiom, the units of a domain are the set of +elements $x$ that each have a multiplicative +inverse $y$ in the domain. +Thus $1$ and $-1$ are units in domain {\tt Integer}. +Also, for {\tt Fraction Integer}, the domain of rational numbers, +all non-zero elements are units.} +Because programs can test for this attribute, Axiom can +correctly handle rather more complicated mathematical structures (ones +that are similar to rings but do not have this attribute). + +\section{Parameters} +\label{ugCategoriesParameters} + +Like domain constructors, category constructors can also have +parameters. +For example, category {\tt MatrixCategory} is a parameterized +category for defining matrices over a ring $R$ so that the +matrix domains can have +different representations and indexing schemes. +Its definition has the form: + +\begin{verbatim} +MatrixCategory(R,Row,Col): Category == + TwoDimensionalArrayCategory(R,Row,Col) with ... +\end{verbatim} + +The category extends {\tt TwoDimensionalArrayCategory} with +the same arguments. +You cannot find {\tt TwoDimensionalArrayCategory} in the +algebraic hierarchy listing. +Rather, it is a member of the data structure hierarchy, +given inside the back cover of this book. +In particular, {\tt TwoDimensionalArrayCategory} is an extension of +{\tt HomogeneousAggregate} since its elements are all one type. + +The domain {\tt Matrix(R)}, the class of matrices with coefficients +from domain $R$, asserts that it is a member of category +{\tt MatrixCategory(R, Vector(R), Vector(R))}. +The parameters of a category must also have types. +The first parameter to {\tt MatrixCategory} +$R$ is required to be a ring. +The second and third are required to be domains of category +{\tt FiniteLinearAggregate(R)}.\footnote{% +This is another extension of +{\tt HomogeneousAggregate} that you can see in +the data structure hierarchy.} +In practice, examples of categories having parameters other than +domains are rare. + +Adding the declarations for parameters to the definition for +{\tt MatrixCategory}, we have: + +\begin{verbatim} +R: Ring +(Row, Col): FiniteLinearAggregate(R) + +MatrixCategory(R, Row, Col): Category == + TwoDimensionalArrayCategory(R, Row, Col) with ... +\end{verbatim} + +\section{Conditionals} +\label{ugCategoriesConditionals} + +As categories have parameters, the actual operations exported by a +\index{conditional} +category can depend on these parameters. +As an example, the operation \spadfunFrom{determinant}{MatrixCategory} +from category {\tt MatrixCategory} is only exported when the +underlying domain $R$ has commutative multiplication: + +\begin{verbatim} +if R has commutative("*") then + determinant: $ -> R +\end{verbatim} + +Conditionals can also define conditional extensions of a category. +Here is a portion of the definition of {\tt QuotientFieldCategory}: +\index{QuotientFieldCategory} + +\begin{verbatim} +QuotientFieldCategory(R) : Category == ... with ... + if R has OrderedSet then OrderedSet + if R has IntegerNumberSystem then + ceiling: $ -> R + ... +\end{verbatim} + +Think of category {\tt QuotientFieldCategory(R)} as +denoting the domain {\tt Fraction(R)}, the +class of all fractions of the form $a/b$ for elements of $R$. +The first conditional means in English: +``If the elements of $R$ are totally ordered ($R$ +is an {\tt OrderedSet}), then so are the fractions $a/b$''. +\index{Fraction} + +The second conditional is used to conditionally export an +operation {\bf ceiling} which returns the smallest integer +greater than or equal to its argument. +Clearly, ``ceiling'' makes sense for integers but not for +polynomials and other algebraic structures. +Because of this conditional, +the domain {\tt Fraction(Integer)} exports +an operation +{\bf ceiling}: \spadsig{Fraction Integer}{Integer}, but +{\tt Fraction Polynomial Integer} does not. + +Conditionals can also appear in the default definitions for the +operations of a category. +For example, a default definition for \spadfunFrom{ceiling}{Field} +within the part following the $add$ reads: + +\begin{verbatim} +if R has IntegerNumberSystem then + ceiling x == ... +\end{verbatim} + +Here the predicate used is identical to the predicate in the {\tt +Exports} part. This need not be the case. See \ref{ugPackagesConds} +on page~\pageref{ugPackagesConds} for a more complicated example. + +\section{Anonymous Categories} +\label{ugCategoriesAndPackages} + +The part of a category to the right of a {\tt with} is also regarded +as a category---an ``anonymous category.'' Thus you have already seen +a category definition \index{category!anonymous} in Chapter +\ref{ugPackages} on page~\pageref{ugPackages}. The {\tt Exports} part +of the package {\tt DrawComplex} (\ref{ugPackagesAbstract} on +page~\pageref{ugPackagesAbstract}) is an anonymous category. This is +not necessary. We could, instead, give this category a name: + +% +\begin{verbatim} +DrawComplexCategory(): Category == with + drawComplex: (C -> C,S,S,Boolean) -> VIEW3D + drawComplexVectorField: (C -> C,S,S) -> VIEW3D + setRealSteps: INT -> INT + setImagSteps: INT -> INT + setClipValue: DFLOAT-> DFLOAT +\end{verbatim} +% +and then define {\tt DrawComplex} by: +% +\begin{verbatim} +DrawComplex(): DrawComplexCategory == Implementation + where + ... +\end{verbatim} +% + +There is no reason, however, to give this list of exports a name +since no other domain or package exports it. +In fact, it is rare for a package to export a named category. +As you will see in the next chapter, however, it is very common +for the definition of domains to mention one or more category +before the {\tt with}. +\index{with} + +\setcounter{chapter}{12} % Chapter 13 + +\hyphenation{ +Quad-rat-ic-Form +} +\spadcommand{)read alql.boot} +\spadcommand{)load DLIST ICARD DBASE QEQUAT MTHING OPQUERY )update} + +\chapter{Domains} +\label{ugDomains} + +We finally come to the {\it domain constructor}. +A few subtle differences between packages and +domains turn up some interesting issues. +We first discuss these differences then +describe the resulting issues by illustrating a program +for the {\tt QuadraticForm} constructor. +After a short example of an algebraic constructor, +{\tt CliffordAlgebra}, we show how you use domain constructors to build +a database query facility. + +\section{Domains vs. Packages} +\label{ugPackagesDoms} +% +Packages are special cases of domains. +What is the difference between a package and a domain that is not a +package? +By definition, there is only one difference: a domain that is not a package +has the symbol {\tt \$} appearing +somewhere among the types of its exported operations. +The {\tt \$} denotes ``this domain.'' If the {\tt \$} +appears before the {\tt ->} in the type of a signature, it means +the operation takes an element from the domain as an argument. +If it appears after the {\tt ->}, then the operation returns an +element of the domain. + +If no exported operations mention {\tt \$}, then evidently there is +nothing of interest to do with the objects of the domain. You might +then say that a package is a ``boring'' domain! But, as you saw in +Chapter \ref{ugPackages} on page~\pageref{ugPackages}, packages are a +very useful notion indeed. The exported operations of a package +depend solely on the parameters to the package constructor and other +explicit domains. + +To summarize, domain constructors are versatile structures that serve two +distinct practical purposes: +Those like {\tt Polynomial} and {\tt List} +describe classes of computational objects; +others, like {\tt SortPackage}, describe packages of useful +operations. +As in the last chapter, we focus here on the first kind. + +\section{Definitions} +\label{ugDomainsDefs} +% + +The syntax for defining a domain constructor is the same as for any +function in Axiom: +\begin{center} +\frenchspacing{\tt {\it DomainForm} : {\it Exports} == {\it Implementation}} +\end{center} +As this definition usually extends over many lines, a +$where$ expression is generally used instead. +\index{where} + +A recommended format for the definition of a domain is:\newline +{\tt% +{\it DomainForm} : Exports == Implementation where \newline +\hspace*{.75pc} {\it optional type declarations} \newline +\hspace*{.75pc} Exports == [{\it Category Assertions}] with \newline +\hspace*{2.0pc} {\it list of exported operations} \newline +\hspace*{.75pc} Implementation == [{\it Add Domain}] add \newline +\hspace*{2.0pc} [Rep := {\it Representation}] \newline +\hspace*{2.0pc} {\it list of function definitions for exported operations} +} + +\vskip 4pt +Note: The brackets {\tt [ ]} here denote optionality. + +A complete domain constructor definition for {\tt QuadraticForm} is +shown in Figure \ref{fig-quadform} on page~\pageref{fig-quadform}. +Interestingly, this little domain illustrates all the new concepts you +need to learn. + +\begin{figure} +\begin{verbatim} +)abbrev domain QFORM QuadraticForm + +++ Description: +++ This domain provides modest support for +++ quadratic forms. +QuadraticForm(n, K): Exports == Implementation where + n: PositiveInteger + K: Field + + Exports == AbelianGroup with --The exports + quadraticForm: SquareMatrix(n,K) -> $ --export this + ++ \bs{}axiom\{quadraticForm(m)\} creates a quadratic + ++ quadratic form from a symmetric, + ++ square matrix \bs{}axiom\{m\}. + matrix: $ -> SquareMatrix(n,K) -- export matrix + ++ \bs{}axiom\{matrix(qf)\} creates a square matrix + ++ from the quadratic form \bs{}axiom\{qf\}. + elt: ($, DirectProduct(n,K)) -> K -- export elt + ++ \bs{}axiom\{qf(v)\} evaluates the quadratic form + ++ \bs{}axiom\{qf\} on the vector \bs{}axiom\{v\}, + ++ producing a scalar. + + Implementation == SquareMatrix(n,K) add --The exports + Rep := SquareMatrix(n,K) --representation + quadraticForm m == --definition + not symmetric? m => error + "quadraticForm requires a symmetric matrix" + m :: $ + matrix q == q :: Rep --definition + elt(q,v) == dot(v, (matrix q * v)) --definition + +\end{verbatim} +\caption{The {\tt QuadraticForm} domain.}\label{fig-quadform} +\end{figure} + +A domain constructor can take any number and type of parameters. +{\tt QuadraticForm} takes a positive integer $n$ and a field +$K$ as arguments. +Like a package, a domain has a set of explicit exports and an +implementation described by a capsule. +Domain constructors are documented in the same way as package constructors. + +Domain {\tt QuadraticForm(n, K)}, for a given positive integer +$n$ and domain $K$, explicitly exports three operations: +% +\begin{itemize} +\item$quadraticForm(A)$ creates a quadratic form from a matrix +$A$. +\item$matrix(q)$ returns the matrix $A$ used to create +the quadratic form $q$. +\item$q.v$ computes the scalar $v^TAv$ +for a given vector $v$. +\end{itemize} + +Compared with the corresponding syntax given for the definition of a +package, you see that a domain constructor has three optional parts to +its definition: {\it Category Assertions}, {\it Add Domain}, and +{\it Representation}. + +\section{Category Assertions} +\label{ugDomainsAssertions} +% + +The {\it Category Assertions} part of your domain constructor +definition lists those categories of which all domains created by the +constructor are unconditionally members. The word ``unconditionally'' +means that membership in a category does not depend on the values of +the parameters to the domain constructor. This part thus defines the +link between the domains and the category hierarchies given on the +inside covers of this book. As described in +\ref{ugCategoriesCorrectness} on +page~\pageref{ugCategoriesCorrectness}, it is this link that makes it +possible for you to pass objects of the domains as arguments to other +operations in Axiom. + +Every {\tt QuadraticForm} domain is declared +to be unconditionally a member of category {\tt AbelianGroup}. +An abelian group is a collection of elements closed under +addition. +Every object {\it x} of an abelian group has an additive inverse +{\it y} such that $x + y = 0$. +The exports of an abelian group include $0$, +{\tt +}, {\tt -}, and scalar multiplication by an integer. +After asserting that {\tt QuadraticForm} domains are abelian +groups, it is possible to pass quadratic forms to algorithms that +only assume arguments to have these abelian group +properties. + +In \ref{ugCategoriesConditionals} on +page~\pageref{ugCategoriesConditionals}, you saw that {\tt +Fraction(R)}, a member of {\tt QuotientFieldCategory(R)}, is a member +of {\tt OrderedSet} if $R$ is a member of {\tt OrderedSet}. Likewise, +from the {\tt Exports} part of the definition of {\tt ModMonic(R, S)}, + +\begin{verbatim} +UnivariatePolynomialCategory(R) with + if R has Finite then Finite + ... +\end{verbatim} +you see that {\tt ModMonic(R, S)} is a member of +{\tt Finite} is $R$ is. + +The {\tt Exports} part of a domain definition is +the same kind of +expression that can appear to the right of an +{\tt ==} in a category definition. +If a domain constructor is unconditionally a member of two or more +categories, a $Join$ form is used. +\index{Join} +The {\tt Exports} part of the definition of +{\tt FlexibleArray(S)} reads, for example: +\begin{verbatim} +Join(ExtensibleLinearAggregate(S), + OneDimensionalArrayAggregate(S)) with... +\end{verbatim} + +\section{A Demo} +\label{ugDomainsDemo} +% +Before looking at the {\it Implementation} part of {\tt QuadraticForm}, +let's try some examples. + +\vskip 2pc + +Build a domain $QF$. + +\spadcommand{QF := QuadraticForm(2,Fraction Integer)} + +Define a matrix to be used to construct +a quadratic form. + +\spadcommand{A := matrix [ [-1,1/2],[1/2,1] ]} + +Construct the quadratic form. +A package call {\tt \$QF} is necessary since there +are other {\tt QuadraticForm} domains. + +\spadcommand{q : QF := quadraticForm(A)} + +Looks like a matrix. Try computing +the number of rows. +Axiom won't let you. + +\spadcommand{nrows q} + +Create a direct product element $v$. +A package call is again necessary, but Axiom +understands your list as denoting a vector. + +\spadcommand{v := directProduct([2,-1])\$DirectProduct(2,Fraction Integer)} + +Compute the product $v^TAv$. + +\spadcommand{q.v} + +What is 3 times $q$ minus $q$ plus $q$? + +\spadcommand{3*q-q+q} + +\section{Browse} +\label{ugDomainsBrowse} + +The Browse facility of HyperDoc is useful for +investigating +the properties of domains, packages, and categories. +From the main HyperDoc menu, move your mouse to {\bf Browse} and +click on the left mouse button. +This brings up the Browse first page. +Now, with your mouse pointer somewhere in this window, enter the +string ``quadraticform'' into the input area (all lower case +letters will do). +Move your mouse to {\bf Constructors} and click. +Up comes a page describing {\tt QuadraticForm}. + +From here, click on {\bf Description}. +This gives you a page that includes a part labeled by ``{\it +Description:}''. +You also see the types for arguments $n$ and $K$ +displayed as well as the fact that {\tt QuadraticForm} +returns an {\tt AbelianGroup}. +You can go and experiment a bit by selecting {\tt Field} with +your mouse. +Eventually, use the ``UP'' button +several times to return to the first page on +{\tt QuadraticForm}. + +Select {\bf Operations} to get a list of operations for +{\tt QuadraticForm}. +You can select an operation by clicking on it +to get an individual page with information about that operation. +Or you can select the buttons along the bottom to see alternative +views or get additional information on the operations. +Then return to the page on {\tt QuadraticForm}. + +Select {\bf Cross Reference} to get another menu. +This menu has buttons for {\bf Parents}, {\bf Ancestors}, and +others. +Clicking on {\bf Parents}, you see that {\tt QuadraticForm} +has one parent {\tt AbelianMonoid}. + +\section{Representation} +\label{ugDomainsRep} +% +The {\tt Implementation} part of an Axiom capsule for a +domain constructor uses the special variable $Rep$ to +\index{Rep @ {\tt Rep}} +identify the lower level data type used to represent the objects +\index{representation!of a domain} +of the domain. +\index{domain!representation} +The $Rep$ for quadratic forms is {\tt SquareMatrix(n, K)}. +This means that all objects of the domain are required to be +$n$ by $n$ matrices with elements from {\bf K}. + +The code for {\tt quadraticForm} in Figure \ref{fig-quadform} +on page \pageref{fig-quadform} +checks that the matrix is symmetric and then converts it to +{\tt \$}, which means, as usual, ``this domain.'' Such explicit +conversions \index{conversion} are generally required by the +compiler. +Aside from checking that the matrix is symmetric, the code for +this function essentially does nothing. +The {\frenchspacing\tt m :: \$} on line 28 coerces $m$ to a +quadratic form. +In fact, the quadratic form you created in step (3) of +\ref{ugDomainsDemo} on page~\pageref{ugDomainsDemo} +is just the matrix you passed it in +disguise! +Without seeing this definition, you would not know that. +Nor can you take advantage of this fact now that you do know! +When we try in the next step of \ref{ugDomainsDemo} on +page~\pageref{ugDomainsDemo} to regard +$q$ as a matrix by asking for {\bf nrows}, the number of +its rows, Axiom gives you an error message saying, in +effect, ``Good try, but this won't work!'' + +The definition for the \spadfunFrom{matrix}{QuadraticForm} +function could hardly be simpler: +it just returns its argument after explicitly +coercing its argument to a matrix. +Since the argument is already a matrix, this coercion does no computation. + +Within the context of a capsule, an object of {\tt \$} is +regarded both as a quadratic form {\it and} as a +matrix.\footnote{In case each of {\tt \$} and $Rep$ +have the same named operation available, +the one from $\$$ takes precedence. +Thus, if you want the one from {\tt Rep}, you must +package call it using a {\tt \$Rep} suffix.} +This makes the definition of $q.v$ easy---it +just calls the \spadfunFrom{dot}{DirectProduct} product from +{\tt DirectProduct} to perform the indicated operation. + +\section{Multiple Representations} +\label{ugDomainsMultipleReps} +% + +To write functions that implement the operations of a domain, you +want to choose the most computationally efficient +data structure to represent the elements of your domain. + +A classic problem in computer algebra is the optimal choice for an +internal representation of polynomials. +If you create a polynomial, say $3x^2+ 5$, how +does Axiom hold this value internally? +There are many ways. +Axiom has nearly a dozen different representations of +polynomials, one to suit almost any purpose. +Algorithms for solving polynomial equations work most +efficiently with polynomials represented one way, whereas those for +factoring polynomials are most efficient using another. +One often-used representation is a list of terms, each term +consisting of exponent-coefficient records written in the order +of decreasing exponents. +For example, the polynomial $3x^2+5$ is +%>> I changed the k's in next line to e's as I thought that was +%>> clearer. +represented by the list $[ [e:2, c:3], [e:0, c:5] ]$. + +What is the optimal data structure for a matrix? +It depends on the application. +For large sparse matrices, a linked-list structure of records +holding only the non-zero elements may be optimal. +If the elements can be defined by a simple formula +$f(i,j)$, then a compiled function for +$f$ may be optimal. +Some programmers prefer to represent ordinary matrices as vectors +of vectors. +Others prefer to represent matrices by one big linear array where +elements are accessed with linearly computable indexes. + +While all these simultaneous structures tend to be confusing, +Axiom provides a helpful organizational tool for such a purpose: +categories. +{\tt PolynomialCategory}, for example, provides a uniform user +interface across all polynomial types. +Each kind of polynomial implements functions for +all these operations, each in its own way. +If you use only the top-level operations in +{\tt PolynomialCategory} you usually do not care what kind +of polynomial implementation is used. + +%>> I've often thought, though, that it would be nice to be +%>> be able to use conditionals for representations. +Within a given domain, however, you define (at most) one +representation.\footnote{You can make that representation a +{\tt Union} type, however. +See \ref{ugTypesUnions} on page~\pageref{ugTypesUnions} +for examples of unions.} +If you want to have multiple representations (that is, several +domains, each with its own representation), use a category to +describe the {\tt Exports}, then define separate domains for each +representation. + +\section{Add Domain} +\label{ugDomainsAddDomain} +% + +The capsule part of {\tt Implementation} defines functions that +implement the operations exported by the domain---usually only +some of the operations. +In our demo in \ref{ugDomainsDemo} on page~\pageref{ugDomainsDemo}, +we asked for the value of +$3*q-q+q$. +Where do the operations {\tt *}, {\tt +}, and +{\tt -} come from? +There is no definition for them in the capsule! + +The {\tt Implementation} part of a definition can +\index{domain!add} +optionally specify an ``add-domain'' to the left of an {\tt add} +\index{add} +(for {\tt QuadraticForm}, defines +{\tt SquareMatrix(n,K)} is the add-domain). +The meaning of an add-domain is simply this: if the capsule part +of the {\tt Implementation} does not supply a function for an +operation, Axiom goes to the add-domain to find the +function. +So do $*$, $+$ and $-$ (from QuadraticForm) come from +{\tt SquareMatrix(n,K)}? +%Read on! + +\section{Defaults} +\label{ugDomainsDefaults} +% +In Chapter \ref{ugPackages} on page~\pageref{ugPackages}, +we saw that categories can provide +default implementations for their operations. +How and when are they used? +When Axiom finds that {\tt QuadraticForm(2, Fraction +Integer)} does not implement the operations {\tt *}, +{\tt +}, and {\tt -}, it goes to +{\tt SquareMatrix(2,Fraction Integer)} to find it. +As it turns out, {\tt SquareMatrix(2, Fraction Integer)} does +not implement {\it any} of these operations! + +What does Axiom do then? +Here is its overall strategy. +First, Axiom looks for a function in the capsule for the domain. +If it is not there, Axiom looks in the add-domain for the +operation. +If that fails, Axiom searches the add-domain of the add-domain, +and so on. +If all those fail, it then searches the default packages for the +categories of which the domain is a member. +In the case of {\tt QuadraticForm}, it searches +{\tt AbelianGroup}, then its parents, grandparents, and +so on. +If this fails, it then searches the default packages of the +add-domain. +Whenever a function is found, the search stops immediately and the +function is returned. +When all fails, the system calls {\bf error} to report this +unfortunate news to you. +To find out the actual order of constructors searched for +{\tt QuadraticForm}, consult Browse: from the +{\tt QuadraticForm}, click on {\tt Cross Reference}, then on +{\tt Lineage}. + +Let's apply this search strategy for our example $3*q-q+q$. +The scalar multiplication comes first. +Axiom finds a default implementation in +{\tt AbelianGroup\&}. +Remember from \ref{ugCategoriesDefaults} on +page~\pageref{ugCategoriesDefaults} that +{\tt SemiGroup} provides a default definition for +$x^n$ by repeated squaring? +{\tt AbelianGroup} similarly provides a definition for +$n x$ by repeated doubling. + +But the search of the defaults for {\tt QuadraticForm} fails +to find any {\tt +} or {\tt *} in the default packages for +the ancestors of {\tt QuadraticForm}. +So it now searches among those for {\tt SquareMatrix}. +Category {\tt MatrixCategory}, which provides a uniform interface +for all matrix domains, +is a grandparent of {\tt SquareMatrix} and +has a capsule defining many functions for matrices, including +matrix addition, subtraction, and scalar multiplication. +The default package {\tt MatrixCategory\&} is where the +functions for $+$ and $-$ (from QuadraticForm) come from. + +You can use Browse to discover where the operations for +{\tt QuadraticForm} are implemented. +First, get the page describing {\tt QuadraticForm}. +With your mouse somewhere in this window, type a ``2'', press the +\fbox{\bf Tab} key, and then enter ``Fraction +Integer'' to indicate that you want the domain +{\tt QuadraticForm(2, Fraction Integer)}. +Now click on {\bf Operations} to get a table of operations and on +{\tt *} to get a page describing the {\tt *} operation. +Finally, click on {\bf implementation} at the bottom. + +\section{Origins} +\label{ugDomainsOrigins} +% + +Aside from the notion of where an operation is implemented, +\index{operation!origin} +a useful notion is the {\it origin} or ``home'' of an operation. +When an operation (such as +\spadfunFrom{quadraticForm}{QuadraticForm}) is explicitly exported by +a domain (such as {\tt QuadraticForm}), you can say that the +origin of that operation is that domain. +If an operation is not explicitly exported from a domain, it is inherited +from, and has as origin, the (closest) category that explicitly exports it. +The operations $+$ and $-$ (from AbelianMonoid) of {\tt QuadraticForm}, +for example, are inherited from {\tt AbelianMonoid}. +As it turns out, {\tt AbelianMonoid} is the origin of virtually every +{\tt +} operation in Axiom! + +Again, you can use Browse to discover the origins of +operations. +From the Browse page on {\tt QuadraticForm}, click on {\bf +Operations}, then on {\bf origins} at the bottom of the page. + +The origin of the operation is the {\it only} place where on-line +documentation is given. +However, you can re-export an operation to give it special +documentation. +Suppose you have just invented the world's fastest algorithm for +inverting matrices using a particular internal representation for +matrices. +If your matrix domain just declares that it exports +{\tt MatrixCategory}, it exports the {\bf inverse} +operation, but the documentation the user gets from Browse is +the standard one from {\tt MatrixCategory}. +To give your version of {\bf inverse} the attention it +deserves, simply export the operation explicitly with new +documentation. +This redundancy gives {\bf inverse} a new origin and tells +Browse to present your new documentation. + +\section{Short Forms} +\label{ugDomainsShortForms} +% +In Axiom, a domain could be defined using only an add-domain +and no capsule. +Although we talk about rational numbers as quotients of integers, +there is no type {\tt RationalNumber} in Axiom. +To create such a type, you could compile the following +``short-form'' definition: + +\begin{verbatim} +RationalNumber() == Fraction(Integer) +\end{verbatim} + +The {\tt Exports} part of this definition is missing and is taken +to be equivalent to that of {\tt Fraction(Integer)}. +Because of the add-domain philosophy, you get precisely +what you want. +The effect is to create a little stub of a domain. +When a user asks to add two rational numbers, Axiom would +ask {\tt RationalNumber} for a function implementing this +{\tt +}. +Since the domain has no capsule, the domain then immediately +sends its request to {\tt Fraction (Integer)}. + +The short form definition for domains is used to +define such domains as {\tt MultivariatePolynomial}: +\index{MultivariatePolynomial} + +\begin{verbatim} +MultivariatePolynomial(vl: List Symbol, R: Ring) == + SparseMultivariatePolynomial(R, + OrderedVariableList vl) +\end{verbatim} + +\section{Example 1: Clifford Algebra} +\label{ugDomainsClifford} +% + +Now that we have {\tt QuadraticForm} available, +let's put it to use. +Given some quadratic form $Q$ described by an +$n$ by $n$ matrix over a field $K$, the domain +{\tt CliffordAlgebra(n, K, Q)} defines a vector space of +dimension $2^n$ over $K$. +This is an interesting domain since complex numbers, quaternions, +exterior algebras and spin algebras are all examples of Clifford +algebras. + +The basic idea is this: +the quadratic form $Q$ defines a basis +$e_1,e_2\ldots,e_n$ for the +vector space $K^n$---the direct product of $K$ +with itself $n$ times. +From this, the Clifford algebra generates a basis of +$2^n$ elements given by all the possible products +of the $e_i$ in order without duplicates, that is, + +1, +$e_1$, +$e_2$, +$e_1e_2$, +$e_3$, +$e_1e_3$, +$e_2e_3$, +$e_1e_2,e_3$, +and so on. + +The algebra is defined by the relations +$$ +\begin{array}{lclc} +e_i \ e_i & = & Q(e_i) \\ +e_i \ e_j & = & -e_j \ e_i & \hbox{for } i \neq j +\end{array} +$$ + +Now look at the snapshot of its definition given in Figure +\ref{fig-clifalg} on page~\pageref{fig-clifalg}. +Lines 9-10 show part of the definitions of the +{\tt Exports}. A Clifford algebra over a field $K$ is asserted to be +a ring, an algebra over $K$, and a vector space over $K$. Its +explicit exports include $e(n),$ which returns the $n$-th unit +element. + +\begin{figure} +\begin{verbatim} +NNI ==> NonNegativeInteger +PI ==> PositiveInteger + +CliffordAlgebra(n,K,q): Exports == Implementation where + n: PI + K: Field + q: QuadraticForm(n, K) + + Exports == Join(Ring,Algebra(K),VectorSpace(K)) with + e: PI -> $ + ... + + Implementation == add + Qeelist := + [q.unitVector(i::PI) for i in 1..n] + dim := 2**n + Rep := PrimitiveArray K + New ==> new(dim, 0$K)$Rep + x + y == + z := New + for i in 0..dim-1 repeat z.i := x.i + y.i + z + addMonomProd: (K, NNI, K, NNI, $) -> $ + addMonomProd(c1, b1, c2, b2, z) == ... + x * y == + z := New + for ix in 0..dim-1 repeat + if x.ix \notequal{} 0 then for iy in 0..dim-1 repeat + if y.iy \notequal{} 0 + then addMonomProd(x.ix,ix,y.iy,iy,z) + z + ... +\end{verbatim} +\caption{Part of the {\tt CliffordAlgebra} domain.}\label{fig-clifalg} +\end{figure} + +The {\tt Implementation} part begins by defining a local variable +$Qeelist$ to hold the list of all $q.v$ where $v$ +runs over the unit vectors from 1 to the dimension $n$. +Another local variable $dim$ is set to $2^n$, +computed once and for all. +The representation for the domain is +{\tt PrimitiveArray(K)}, +which is a basic array of elements from domain $K$. +Line 18 defines $New$ as shorthand for the more lengthy +expression $new(dim, 0\$K)\$Rep$, which computes a primitive +array of length $2^n$ filled with $0$'s from +domain $K$. + +Lines 19-22 define the sum of two elements $x$ and $y$ +straightforwardly. +First, a new array of all $0$'s is created, then filled with +the sum of the corresponding elements. +Indexing for primitive arrays starts at 0. +The definition of the product of $x$ and $y$ first requires +the definition of a local function {\bf addMonomProd}. +Axiom knows it is local since it is not an exported function. +The types of all local functions must be declared. + +\section{Example 2: Building A Query Facility} +\label{ugDomsinsDatabase} +% +We now turn to an entirely different kind of application, +building a query language for a database. + +Here is the practical problem to solve. +The Browse facility of Axiom has a +database for all operations and constructors which is +stored on disk and accessed by HyperDoc. +For our purposes here, we regard each line of this file as having +eight fields: +{\tt class, name, type, nargs, exposed, kind, origin,} and {\tt condition.} +Here is an example entry: + +\begin{verbatim} +o`determinant`$->R`1`x`d`Matrix(R)`has(R,commutative("*")) +\end{verbatim} + +In English, the entry means: +\begin{quotation} +\raggedright +The operation {\bf determinant}: \spadsig{\$}{R} with {\it 1} argument, is +{\it exposed} and is exported by {\it domain} {\tt Matrix(R)} +if {\tt R has commutative("*")}. +\end{quotation} + +Our task is to create a little query language that allows us +to get useful information from this database. + +\subsection{A Little Query Language} +\label{ugDomainsQueryLanguage} + +First we design a simple language for accessing information from +the database. +We have the following simple model in mind for its design. +Think of the database as a box of index cards. +There is only one search operation---it +takes the name of a field and a predicate +\index{predicate} +(a boolean-valued function) defined on the fields of the +index cards. +When applied, the search operation goes through the entire box +selecting only those index cards for which the predicate is {\tt true}. +The result of a search is a new box of index cards. +This process can be repeated again and again. + +The predicates all have a particularly simple form: {\it symbol} +{\tt =} {\it pattern}, where {\it symbol} designates one of the +fields, and {\it pattern} is a ``search string''---a string +that may contain a ``{\tt *}'' as a +wildcard. +Wildcards match any substring, including the empty string. +Thus the pattern ``{\tt "*ma*t}'' matches +``{\tt "mat"},''{\tt doormat}'' and ``{\tt smart}''. + +To illustrate how queries are given, we give you a sneak preview +of the facility we are about to create. + +Extract the database of all Axiom operations. + +\spadcommand{ops := getDatabase("o")} + +How many exposed three-argument {\bf map} operations involving streams? + +\spadcommand{ops.(name="map").(nargs="3").(type="*Stream*")} + +As usual, the arguments of {\bf elt} ({\tt .}) +associate to the left. +The first {\bf elt} produces the set of all operations with +name {\tt map}. +The second {\bf elt} produces the set of all map operations +with three arguments. +The third {\bf elt} produces the set of all three-argument map +operations having a type mentioning {\tt Stream}. + +Another thing we'd like to do is to extract one field from each of +the index cards in the box and look at the result. +Here is an example of that kind of request. + +What constructors explicitly export a {\bf determinant} operation? + +\spadcommand{elt(elt(elt(elt(ops,name="determinant"),origin),sort),unique)} + +The first {\bf elt} produces the set of all index cards with +name {\tt determinant}. +The second {\bf elt} extracts the {\tt origin} component from +each index card. Each origin component +is the name of a constructor which directly +exports the operation represented by the index card. +Extracting a component from each index card produces what we call +a {\it datalist}. +The third {\bf elt}, {\tt sort}, causes the datalist of +origins to be sorted in alphabetic +order. +The fourth, {\tt unique}, causes duplicates to be removed. + +Before giving you a more extensive demo of this facility, +we now build the necessary domains and packages to implement it. +%We will introduce a few of our minor conveniences. + +\subsection{The Database Constructor} +\label{ugDomainsDatabaseConstructor} + +We work from the top down. First, we define a database, +our box of index cards, as an abstract datatype. +For sake of illustration and generality, +we assume that an index card is some type $S$, and +that a database is a box of objects of type $S$. +Here is the Axiom program defining the {\tt Database} +domain. + +\begin{verbatim} +PI ==> PositiveInteger +Database(S): Exports == Implementation where + S: Object with + elt: ($, Symbol) -> String + display: $ -> Void + fullDisplay: $ -> Void + + Exports == with + elt: ($,QueryEquation) -> $ Select by an equation + elt: ($, Symbol) -> DataList String Select by a field name + "+": ($,$) -> $ Combine two databases + "-": ($,$) -> $ Subtract one from another + display: $ -> Void A brief database display + fullDisplay: $ -> Void A full database display + fullDisplay: ($,PI,PI) -> Void A selective display + coerce: $ -> OutputForm Display a database + Implementation == add + ... +\end{verbatim} + +The domain constructor takes a parameter $S$, which +stands for the class of index cards. +We describe an index card later. +Here think of an index card as a string which has +the eight fields mentioned above. + +First, we tell Axiom what operations we are going to require +from index cards. +We need an {\bf elt} to extract the contents of a field +(such as {\tt name} and {\tt type}) as a string. +For example, +$c.name$ returns a string that is the content of the +$name$ field on the index card $c$. +We need to display an index card in two ways: +{\bf display} shows only the name and type of an +operation; +{\bf fullDisplay} displays all fields. +The display operations return no useful information and thus have +return type {\tt Void}. + +Next, we tell Axiom what operations the user can apply +to the database. +This part defines our little query language. +The most important operation is +{\frenchspacing\tt db . field = pattern} which +returns a new database, consisting of all index +cards of {\tt db} such that the $field$ part of the index +card is matched by the string pattern called $pattern$. +The expression {\tt field = pattern} is an object of type +{\tt QueryEquation} (defined in the next section). + +Another {\bf elt} is needed to produce a {\tt DataList} +object. +Operation {\tt +} is to merge two databases together; +{\tt -} is used to subtract away common entries in a second +database from an initial database. +There are three display functions. +The {\bf fullDisplay} function has two versions: one +that prints all the records, the other that prints only a fixed +number of records. +A {\bf coerce} to {\tt OutputForm} creates a display +object. + +The {\tt Implementation} part of {\tt Database} is straightforward. +\begin{verbatim} + Implementation == add + s: Symbol + Rep := List S + elt(db,equation) == ... + elt(db,key) == [x.key for x in db]::DataList(String) + display(db) == for x in db repeat display x + fullDisplay(db) == for x in db repeat fullDisplay x + fullDisplay(db, n, m) == for x in db for i in 1..m + repeat + if i >= n then fullDisplay x + x+y == removeDuplicates! merge(x,y) + x-y == mergeDifference(copy(x::Rep), + y::Rep)$MergeThing(S) + coerce(db): OutputForm == (#db):: OutputForm +\end{verbatim} + +The database is represented by a list of elements of $S$ (index cards). +We leave the definition of the first {\bf elt} operation +(on line 4) until the next section. +The second {\bf elt} collects all the strings with field name +{\it key} into a list. +The {\bf display} function and first {\bf fullDisplay} function +simply call the corresponding functions from $S$. +The second {\bf fullDisplay} function provides an efficient way of +printing out a portion of a large list. +The {\tt +} is defined by using the existing +\spadfunFrom{merge}{List} operation defined on lists, then +removing duplicates from the result. +The {\tt -} operation requires writing a corresponding +subtraction operation. +A package {\tt MergeThing} (not shown) provides this. + +The {\bf coerce} function converts the database to an +{\tt OutputForm} by computing the number of index cards. +This is a good example of the independence of +the representation of an Axiom object from how it presents +itself to the user. We usually do not want to look at a database---but +do care how many ``hits'' we get for a given query. +So we define the output representation of a database to be simply +the number of index cards our query finds. + +\subsection{Query Equations} +\label{ugDomainsQueryEquations} + +The predicate for our search is given by an object of type +{\tt QueryEquation}. +Axiom does not have such an object yet so we +have to invent it. + +\begin{verbatim} +QueryEquation(): Exports == Implementation where + Exports == with + equation: (Symbol, String) -> $ + variable: $ -> Symbol + value: $ -> String + + Implementation == add + Rep := Record(var:Symbol, val:String) + equation(x, s) == [x, s] + variable q == q.var + value q == q.val +\end{verbatim} + +Axiom converts an input expression of the form +${\it a} = {\it b}$ to $equation({\it a, b})$. +Our equations always have a symbol on the left and a string +on the right. +The {\tt Exports} part thus specifies an operation +{\bf equation} to create a query equation, and +{\bf variable} and {\bf value} to select the left- and +right-hand sides. +The {\tt Implementation} part uses {\tt Record} for a +space-efficient representation of an equation. + +Here is the missing definition for the {\bf elt} function of +{\tt Database} in the last section: + +\begin{verbatim} + elt(db,eq) == + field\ := variable eq + value := value eq + [x for x in db | matches?(value,x.field)] +\end{verbatim} + +Recall that a database is represented by a list. +Line 4 simply runs over that list collecting all elements +such that the pattern (that is, $value$) +matches the selected field of the element. + +\subsection{DataLists} +\label{ugDomainsDataLists} + +Type {\tt DataList} is a new type invented to hold the result +of selecting one field from each of the index cards in the box. +It is useful to make datalists extensions of lists---lists that +have special {\bf elt} operations defined on them for +sorting and removing duplicates. + +\begin{verbatim} +DataList(S:OrderedSet) : Exports == Implementation where + Exports == ListAggregate(S) with + elt: ($,"unique") -> $ + elt: ($,"sort") -> $ + elt: ($,"count") -> NonNegativeInteger + coerce: List S -> $ + + Implementation == List(S) add + Rep := List S + elt(x,"unique") == removeDuplicates(x) + elt(x,"sort") == sort(x) + elt(x,"count") == #x + coerce(x:List S) == x :: $ +\end{verbatim} + +The {\tt Exports} part asserts that datalists belong to the +category {\tt ListAggregate}. +Therefore, you can use all the usual list operations on datalists, +such as \spadfunFrom{first}{List}, \spadfunFrom{rest}{List}, and +\spadfunFrom{concat}{List}. +In addition, datalists have four explicit operations. +Besides the three {\bf elt} operations, there is a +{\bf coerce} operation that creates datalists from lists. + +The {\tt Implementation} part needs only to define four functions. +All the rest are obtained from {\tt List(S)}. + +\subsection{Index Cards} +\label{ugDomainsDatabase} + +An index card comes from a file as one long string. +We define functions that extract substrings from the long +string. +Each field has a name that +is passed as a second argument to {\bf elt}. + +\begin{verbatim} +IndexCard() == Implementation where + Exports == with + elt: ($, Symbol) -> String + display: $ -> Void + fullDisplay: $ -> Void + coerce: String -> $ + Implementation == String add ... +\end{verbatim} + +We leave the {\tt Implementation} part to the reader. +All operations involve straightforward string manipulations. + +\subsection{Creating a Database} +\label{ugDomainsCreating} + +We must not forget one important operation: one that builds the database in the +first place! +We'll name it {\bf getDatabase} and put it in a package. +This function is implemented by calling the Common Lisp function +$getBrowseDatabase(s)$ to get appropriate information from +Browse. +This operation takes a string indicating which lines you +want from the database: ``{\tt o}'' gives you all operation +lines, and ``{\tt k}'', all constructor lines. +Similarly, ``{\tt c}'', ``{\tt d}'', and ``{\tt p}'' give +you all category, domain and package lines respectively. +% +\begin{verbatim} +OperationsQuery(): Exports == Implementation where + Exports == with + getDatabase: String -> Database(IndexCard) + + Implementation == add + getDatabase(s) == getBrowseDatabase(s)$Lisp +\end{verbatim} + +We do not bother creating a special name for databases of index +cards. +{\tt Database (IndexCard)} will do. +Notice that we used the package {\tt OperationsQuery} to +create, in effect, +a new kind of domain: {\tt Database(IndexCard)}. + +\subsection{Putting It All Together} +\label{ugDomainsPutting} + +To create the database facility, you put all these constructors +into one file.\footnote{You could use separate files, but we +are putting them all together because, organizationally, that is +the logical thing to do.} +At the top of the file put {\tt )abbrev} commands, giving the +constructor abbreviations you created. + +\begin{verbatim} +)abbrev domain ICARD IndexCard +)abbrev domain QEQUAT QueryEquation +)abbrev domain MTHING MergeThing +)abbrev domain DLIST DataList +)abbrev domain DBASE Database +)abbrev package OPQUERY OperationsQuery +\end{verbatim} + +With all this in {\bf alql.spad}, for example, compile it using +\index{compile} +\begin{verbatim} +)compile alql +\end{verbatim} +and then load each of the constructors: +\begin{verbatim} +)load ICARD QEQUAT MTHING DLIST DBASE OPQUERY +\end{verbatim} +\index{load} +You are ready to try some sample queries. + +\subsection{Example Queries} +\label{ugDomainsExamples} + +Our first set of queries give some statistics on constructors in +the current Axiom system. + +How many constructors does Axiom have? + +\spadcommand{ks := getDatabase "k"} + +Break this down into the number of categories, domains, and packages. + +\spadcommand{[ks.(kind=k) for k in ["c","d","p"] ]} + +What are all the domain constructors that take no parameters? + +\spadcommand{elt(ks.(kind="d").(nargs="0"),name)} + +How many constructors have ``Matrix'' in their name? + +\spadcommand{mk := ks.(name="*Matrix*")} + +What are the names of those that are domains? + +\spadcommand{elt(mk.(kind="d"),name)} + +How many operations are there in the library? + +\spadcommand{o := getDatabase "o"} + +Break this down into categories, domains, and packages. + +\spadcommand{[o.(kind=k) for k in ["c","d","p"] ]} + + +The query language is helpful in getting information about a +particular operation you might like to apply. +While this information can be obtained with +Browse, the use of the query database gives you data that you +can manipulate in the workspace. + +How many operations have ``eigen'' in the name? + +\spadcommand{eigens := o.(name="*eigen*")} + +What are their names? + +\spadcommand{elt(eigens,name)} + +Where do they come from? + +\spadcommand{elt(elt(elt(eigens,origin),sort),unique) } + +The operations {\tt +} and {\tt -} are useful for +constructing small databases and combining them. +However, remember that the only matching you can do is string +matching. +Thus a pattern such as {\tt "*Matrix*"} on the type field +matches +any type containing {\tt Matrix}, {\tt MatrixCategory}, +{\tt SquareMatrix}, and so on. + +How many operations mention ``Matrix'' in their type? + +\spadcommand{tm := o.(type="*Matrix*")} + +How many operations come from constructors with ``Matrix'' in +their name? + +\spadcommand{fm := o.(origin="*Matrix*")} + +How many operations are in $fm$ but not in $tm$? + +\spadcommand{fm-tm } + +Display the operations that both mention ``Matrix'' in their type +and come from a constructor having ``Matrix'' in their name. + +\spadcommand{fullDisplay(fm-\%) } + +How many operations involve matrices? + +\spadcommand{m := tm+fm } + +Display 4 of them. + +\spadcommand{fullDisplay(m, 202, 205) } + +How many distinct names of operations involving matrices are there? + +\spadcommand{elt(elt(elt(m,name),unique),count) } + +%following definition should go into ug.sty +\gdef\aliascon#1#2{{\bf #1}} +\setcounter{chapter}{13} % Chapter 14 +% + +\chapter{Browse} +\label{ugBrowse} + +This chapter discusses the Browse +\index{Browse@Browse} +component of HyperDoc. +\index{HyperDoc@{HyperDoc}} +We suggest you invoke Axiom and work through this +chapter, section by section, following our examples to gain some +familiarity with Browse. + +\section{The Front Page: Searching the Library} +\label{ugBrowseStart} +To enter Browse, click on {\bf Browse} on the top level page +of HyperDoc to get the {\it front page} of Browse. +% +%324pt is 4.5",180pt is 2.5",432pt is 6"=textwidth,54=(432-324)/2 +%ps files are 4.5"x2.5" except source 4.5"x2.5" +% +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-brfront.ps} +\end{picture} +\caption{The Browse front page.} +\end{figure} + +To use this page, you first enter a {\it search string} into +the input area at the top, then click on one of the buttons below. +We show the use of each of the buttons by example. + +\subsubsection{Constructors} + +First enter the search string {\tt Matrix} into the input area and +click on {\bf Constructors}. +What you get is the {\it constructor page} for {\tt Matrix}. +We show and describe this page in detail in +\ref{ugBrowseDomain} on page~\pageref{ugBrowseDomain}. +By convention, Axiom does a case-insensitive search for a +match. +Thus {\tt matrix} is just as good as {\tt Matrix}, has the same +effect as {\tt MaTrix}, and so on. +We recommend that you generally use small letters for names +however. +A search string with only capital letters has a special meaning +(see \ref{ugBrowseCapitalizationConvention} on +page~\pageref{ugBrowseCapitalizationConvention}). + + +Click on \UpBitmap{} to return to the Browse front page. + +Use the symbol ``{\tt *}'' in search strings as a {\it wild +card}. +A wild card matches any substring, including the empty string. +For example, enter the search string {\tt *matrix*} into the input +area and click on {\bf Constructors}.\footnote{To get only +categories, domains, or packages, rather than all constructors, +you can click on the corresponding button to the right of {\bf +Constructors}.} +What you get is a table of all constructors whose names contain +the string ``{\tt matrix}.'' + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-consearch.ps} +\end{picture} +\caption{Table of exposed constructors matching {\tt *matrix*} .} +\end{figure} + +%% Following para replaced 1995oct30 MGR +%These are all the exposed constructors in +%Axiom. +%To see how to get all exposed and unexposed constructors in +%Axiom, skip to the section entitled {\bf Exposure} in +%\ref{ugBrowseOptions} on page~\pageref{ugBrowseOptions}. +All constructors containing the string are listed, whether +exposed or unexposed. +You can hide the names of the unexposed constructors by clicking +on the {\it *=}{\bf unexposed} button in the {\it Views} panel at +the bottom of the window. +(The button will change to {\bf exposed} {\it only}.) + +One of the names in this table is {\tt Matrix}. +Click on {\tt Matrix}. +What you get is again the constructor page for {\tt Matrix}. +As you see, Browse gives you a large network of +information in which there are many ways to reach the same +pages. +\index{Matrix} + +Again click on the \UpBitmap{} to return to the table of constructors +whose names contain {\tt matrix}. +%Below the table is a {\bf Views} panel. % here & globally MGR 1995oct30 +Below the table is a {\it Views} panel. +This panel contains buttons that let you view constructors in different +ways. +To learn about views of constructors, skip to +\ref{ugBrowseViewsOfConstructors} on +page~\pageref{ugBrowseViewsOfConstructors}. + +Click on \UpBitmap{} to return to the Browse front page. + +\subsubsection{Operations} + +Enter {\tt *matrix} into the input area and click on {\bf +Operations}. +This time you get a table of {\it operations} whose names end with {\tt +matrix} or {\tt Matrix}. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matrixops.ps} +\end{picture} +\caption{Table of operations matching {\tt *matrix} .} +\end{figure} + +If you select an operation name, you go to a page describing all +the operations in Axiom of that name. +At the bottom of an operation page is another kind of {\it Views} panel, +one for operation pages. +To learn more about these views, skip to +\ref{ugBrowseViewsOfOperations} on page~\pageref{ugBrowseViewsOfOperations}. + +Click on \UpBitmap{} to return to the Browse front page. + +\subsubsection{Attributes} + +This button gives you a table of attribute names that match the +search string. Enter the search string {\tt *} and click on +{\bf Attributes} to get a list +of all system attributes. + +Click on \UpBitmap{} to return to the Browse front page. + + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-atsearch.ps} +\end{picture} +\caption{Table of Axiom attributes.} +\end{figure} + +Again there is a {\it Views} panel at the bottom with buttons that let +you view the attributes in different ways. + +\subsubsection{General} + +This button does a general search for all constructor, operation, and +attribute names matching the search string. +Enter the search string \allowbreak +{\tt *matrix*} into the input area. +Click on {\bf General} to find all constructs that have {\tt +matrix} as a part of their name. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-gensearch.ps} +\end{picture} +\caption{Table of all constructs matching {\tt *matrix*} .} +\end{figure} + +The summary gives you all the names under a heading when the number of +entries is less than 10. % "less than 10." replaces the following: + % sufficiently small%\footnote{See +%\ref{ugBrowseOptions} to see how you can change this.}. +%% MGR 1995oct31 + +Click on \UpBitmap{} to return to the Browse front page. + +\subsubsection{Documentation} + +Again enter the search key {\tt *matrix*} and this time click on +{\bf Documentation}. +This search matches any constructor, operation, or attribute +name whose documentation contains a substring matching {\tt +matrix}. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-docsearch.ps} +\end{picture} +\caption{Table of constructs with documentation matching {\tt *matrix*} .} +\end{figure} + +Click on \UpBitmap{} to return to the Browse front page. + +\subsubsection{Complete} + +This search combines both {\bf General} and {\bf Documentation}. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-comsearch.ps} +\end{picture} +\caption{Table summarizing complete search for pattern {\tt *matrix*} .} +\end{figure} + +\section{The Constructor Page} +\label{ugBrowseDomain} + +In this section we look in detail at a constructor page for domain +{\tt Matrix}. +Enter {\tt matrix} into the input area on the main Browse page +and click on {\bf Constructors}. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matpage.ps} +\end{picture} +\caption{Constructor page for {\tt Matrix}.} +\end{figure} + +The header part tells you that {\tt Matrix} has abbreviation +{\tt MATRIX} and one argument called {\tt R} that must be a +domain of category {\tt Ring}. +Just what domains can be arguments of {\tt Matrix}? +To find this out, click on the {\tt R} on the second line of the +heading. +What you get is a table of all acceptable domain parameter values +of {\tt R}, or a table of {\it rings} in Axiom. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matargs.ps} +\end{picture} +\caption{Table of acceptable domain parameters to {\tt Matrix}.} +\end{figure} + +Click on \UpBitmap{} to return to the constructor page for +{\tt Matrix}. +\newpage + +If you have access to the source code of Axiom, the third +\index{source code} +line of the heading gives you the name of the source file +containing the definition of {\tt Matrix}. +Click on it to pop up an editor window containing the source code +of {\tt Matrix}. + +\begin{figure}[htbp] +\begin{picture}(324,168)%(-54,0) +\special{psfile=ps/h-matsource.ps} +\end{picture} +\caption{Source code for {\tt Matrix}.} +\end{figure} + +We recommend that you leave the editor window up while working +through this chapter as you occasionally may want to refer to it. +\newpage + +\subsection{Constructor Page Buttons} +\label{ugBrowseDomainButtons} + +We examine each button on this page in order. + +\subsubsection{Description} + +Click here to bring up a page with a brief description of +constructor {\tt Matrix}. +If you have access to system source code, note that these comments +can be found directly over the constructor definition. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matdesc.ps} +\end{picture} +\caption{Description page for {\tt Matrix}.} +\end{figure} + +\subsubsection{Operations} + +Click here to get a table of operations exported by +{\tt Matrix}. +You may wish to widen the window to have multiple columns as +below. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matops.ps} +\end{picture} +\caption{Table of operations from {\tt Matrix}.} +\end{figure} + +If you click on an operation name, you bring up a description +page for the operations. +For a detailed description of these pages, skip to +\ref{ugBrowseViewsOfOperations} on page~\pageref{ugBrowseViewsOfOperations}. + +\subsubsection{Attributes} + +Click here to get a table of the two attributes exported by +{\tt Matrix}: +\index{attribute} +{\bf fi\-nite\-Ag\-gre\-gate} and {\bf shallowlyMutable}. +These are two computational properties that result from +{\tt Matrix} being regarded as a data structure. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matats.ps} +\end{picture} +\caption{Attributes from {\tt Matrix}.} +\end{figure} + +\subsubsection{Examples} + +Click here to get an {\it examples page} with examples of operations to +create and manipulate matrices. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matexamp.ps} +\end{picture} +\caption{Example page for {\tt Matrix}.} +\end{figure} + +Read through this section. +Try selecting the various buttons. +Notice that if you click on an operation name, such as +\spadfunFrom{new}{Matrix}, you bring up a description page for that +operation from {\tt Matrix}. + +Example pages have several examples of Axiom commands. +Each example has an active button to its left. +Click on it! +A pre-computed answer is pasted into the page immediately following the +command. +If you click on the button a second time, the answer disappears. +This button thus acts as a toggle: +``now you see it; now you don't.'' + +Note also that the Axiom commands themselves are active. +If you want to see Axiom execute the command, then click on it! +A new Axiom window appears on your screen and the command is +executed. + +At the end of the page is generally a menu of buttons that lead +you to further sections. +Select one of these topics to explore its contents. + +\subsubsection{Exports} + +Click here to see a page describing the exports of {\tt Matrix} +exactly as described by the source code. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matexports.ps} +\end{picture} +\caption{Exports of {\tt Matrix}.} +\end{figure} + +As you see, {\tt Matrix} declares that it exports all the operations +and attributes exported by category +{\tt MatrixCategory(R, Row, Col)}. +In addition, two operations, {\bf diagonalMatrix} and +{\bf inverse}, are explicitly exported. + +To learn a little about the structure of Axiom, we suggest you do +the following exercise. + +Otherwise, go on to the next section. + +{\tt Matrix} explicitly exports only two operations. +The other operations are thus exports of {\tt MatrixCategory}. +In general, operations are usually not explicitly exported by a domain. +Typically they are inherited from several +different categories. +Let's find out from where the operations of {\tt Matrix} come. + +\begin{enumerate} +\item Click on {\tt MatrixCategory}, then on {\bf Exports}. +Here you see that {\tt MatrixCategory} explicitly exports many matrix +operations. +Also, it inherits its operations from +{\tt TwoDimensionalArrayCategory}. + +\item Click on {\tt TwoDimensionalArrayCategory}, then on {\bf Exports}. +Here you see explicit operations dealing with rows and columns. +In addition, it inherits operations from +{\tt HomogeneousAggregate}. + +%\item Click on {\tt HomogeneousAggregate}, then on {\bf Exports}. +%And so on. +%If you continue doing this, eventually you will + +\item Click on \UpBitmap{} and then +click on {\tt Object}, then on {\bf Exports}, where you see +there are no exports. + +\item Click on \UpBitmap{} repeatedly to return to the constructor page +for {\tt Matrix}. + +\end{enumerate} + +\subsubsection{Related Operations} + +Click here bringing up a table of operations that are exported by +packages but not by {\tt Matrix} itself. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matrelops.ps} +\end{picture} +\caption{Related operations of {\tt Matrix}.} +\end{figure} + +To see a table of such packages, use the {\bf Relatives} button on the +{\bf Cross Reference} page described next. + + +\subsection{Cross Reference} +\label{ugBrowseCrossReference} +Click on the {\bf Cross Reference} button on the main constructor page +for {\tt Matrix}. +This gives you a page having various cross reference information stored +under the respective buttons. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matxref.ps} +\end{picture} +\caption{Cross-reference page for {\tt Matrix}.} +\end{figure} + +\subsubsection{Parents} + +The parents of a domain are the same as the categories mentioned under +the {\bf Exports} button on the first page. +Domain {\tt Matrix} has only one parent but in general a domain can +have any number. + +\subsubsection{Ancestors} + +The ancestors of a constructor consist of its parents, the +parents of its parents, and so on. +Did you perform the exercise in the last section under {\bf Exports}? +If so, you see here all the categories you found while ascending the +{\bf Exports} chain for {\tt Matrix}. + +\subsubsection{Relatives} + +The relatives of a domain constructor are package +constructors that provide operations in addition to those +exported by the domain. + +Try this exercise. +\begin{enumerate} +\item Click on {\bf Relatives}, bringing up a list of +packages. + +\item Click on {\tt LinearSystemMatrixPackage} bringing up its +constructor page.\footnote{You may want to widen your HyperDoc +window to make what follows more legible.} + +\item Click on {\bf Operations}. +Here you see {\bf rank}, an operation also exported by +{\tt Matrix} itself. + +\item Click on {\bf rank}. +This \spadfunFrom{rank}{LinearSystemMatrixPackage} has two arguments and +thus is different from the \spadfunFrom{rank}{Matrix} from +{\tt Matrix}. + +\item Click on \UpBitmap{} to return to the list of operations for the +package {\bf LinearSystemMatrixPackage}. + +\item Click on {\bf solve} to bring up a +\spadfunFrom{solve}{LinearSystemMatrixPackage} for linear systems of +equations. + +\item Click on \UpBitmap{} several times to return to the cross +reference page for {\tt Matrix}. +\end{enumerate} + +\subsubsection{Dependents} + +The dependents of a constructor are those +domains or packages +that mention that +constructor either as an argument or in its exports. + +If you click on {\bf Dependents} two entries may surprise you: +{\tt RectangularMatrix} and {\tt SquareMatrix}. +This happens because {\tt Matrix}, as it turns out, appears in +signatures of operations exported by these domains. + +\subsubsection{Lineage} + +The term {\it lineage} refers to the {\it search order} for +functions. +If you are an expert user or curious about how the Axiom system +works, try the following exercise. +Otherwise, you best skip this button and go on to {\bf Clients}. + +Clicking on {\bf Lineage} gives you a +list of domain constructors: +{\tt InnerIndexedTwoDimensionalArray}, +\aliascon{MatrixCategory\&}{MATCAT-}, +\aliascon{TwoDimensionalArrayCategory\&}{ARR2CAT-}, +\aliascon{HomogeneousAggregate\&}{HOAGG-}, +\aliascon{Aggregate\&}{AGG-}. +What are these constructors and how are they used? + +We explain by an example. +Suppose you create a matrix using the interpreter, then ask for its +{\bf rank}. +Axiom must then find a function implementing the {\bf rank} +operation for matrices. +The first place Axiom looks for {\bf rank} is in the {\tt Matrix} +domain. + +If not there, the lineage of {\tt Matrix} tells Axiom where +else to look. +Associated with the matrix domain are five other lineage domains. +Their order is important. +Axiom first searches the first one, +{\tt InnerIndexedTwoDimensionalArray}. +If not there, it searches the second \aliascon{MatrixCategory\&}{MATCAT-}. +And so on. + +Where do these {\it lineage constructors} come from? +The source code for {\tt Matrix} contains this syntax for the +{\it function body} of +{\tt Matrix}:\footnote{{\tt InnerIndexedTwoDimensionalArray} +is a special domain implemented for matrix-like domains to provide +efficient implementations of two-di\-men\-sion\-al arrays. +For example, domains of category {\tt TwoDimensionalArrayCategory} +can have any integer as their $minIndex$. +Matrices and other members of this special ``inner'' array have their +$minIndex$ defined as $1$.} +\begin{verbatim} +InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col) + add ... +\end{verbatim} +where the ``{\tt ...}'' denotes all the code that follows. +In English, this means: +``The functions for matrices are defined as those from +{\tt InnerIndexedTwoDimensionalArray} domain augmented by those +defined in `{\tt ...}','' where the latter take precedence. + +This explains {\tt InnerIndexedTwoDimensionalArray}. +The other names, those with names ending with an ampersand {\tt \&} are +default packages +for categories to which {\tt Matrix} belongs. +Default packages are ordered by the notion of ``closest ancestor.'' + +\subsubsection{Clients} + +A client of {\tt Matrix} is any constructor that uses +{\tt Matrix} in its implementation. +For example, {\tt Complex} is a client of {\tt Matrix}; it +exports several operations that take matrices as arguments or return +matrices as values.\footnote{A constructor is a client of +{\tt Matrix} if it handles any matrix. +For example, a constructor having internal (unexported) operations +dealing with matrices is also a client.} + +\subsubsection{Benefactors} + +A {\it benefactor} of {\tt Matrix} is any constructor that +{\tt Matrix} uses in its implementation. +This information, like that for clients, is gathered from run-time +structures.\footnote{The benefactors exclude constructors such as +{\tt PrimitiveArray} whose operations macro-expand and so vanish +from sight!} + +Cross reference pages for categories have some different buttons on +them. +Starting with the constructor page of {\tt Matrix}, click on +{\tt Ring} producing its constructor page. +Click on {\bf Cross Reference}, +producing the cross-reference page for {\tt Ring}. +Here are buttons {\bf Parents} and {\bf Ancestors} similar to the notion +for domains, except for categories the relationship between parent and +child is defined through {\it category extension}. + +\subsubsection{Children} + +Category hierarchies go both ways. +There are children as well as parents. +A child can have any number of parents, but always at least one. +Every category is therefore a descendant of exactly one category: +{\tt Object}. + +\subsubsection{Descendants} + +These are children, children of children, and so on. + +Category hierarchies are complicated by the fact that categories take +parameters. +Where a parameterized category fits into a hierarchy {\it may} depend on +values of its parameters. +In general, the set of categories in Axiom forms a {\it directed +acyclic graph}, that is, a graph with directed arcs and no cycles. + +\subsubsection{Domains} + +This produces a table of all domain constructors that can possibly be +rings (members of category {\tt Ring}). +Some domains are unconditional rings. +Others are rings for some parameters and not for others. +To find out which, select the {\bf conditions} button in the views +panel. +For example, {\tt DirectProduct(n, R)} is a ring if {\tt R} is a +ring. + + + +\subsection{Views Of Constructors} +\label{ugBrowseViewsOfConstructors} + +Below every constructor table page is a {\it Views} panel. +As an example, click on {\bf Cross Reference} from +the constructor page of {\tt Matrix}, +then on {\bf Benefactors} to produce a +short table of constructor names. + +The {\it Views} panel is at the bottom of the page. +Two items, {\it names} and {\it conditions,} are in italics. +Others are active buttons. +The active buttons are those that give you useful alternative views +on this table of constructors. +Once you select a view, you notice that the button turns +off (becomes italicized) so that you cannot reselect it. + +\subsubsection{names} + +This view gives you a table of names. +Selecting any of these names brings up the constructor page for that +constructor. + +\subsubsection{abbrs} + +This view gives you a table of abbreviations, in the same order as the +original constructor names. +Abbreviations are in capitals and are limited to 7 characters. +They can be used interchangeably with constructor names in input areas. + +\subsubsection{kinds} + +This view organizes constructor names into +the three kinds: categories, domains and packages. + +\subsubsection{files} + +This view gives a table of file names for the source +code of the constructors in alphabetic order after removing +duplicates. + +\subsubsection{parameters} + +This view presents constructors with the arguments. +This view of the benefactors of {\tt Matrix} shows that +{\tt Matrix} uses as many as five different {\tt List} domains +in its implementation. + +\subsubsection{filter} + +This button is used to refine the list of names or abbreviations. +Starting with the {\it names} view, enter {\tt m*} into the input area +and click on {\tt filter}. +You then get a shorter table with only the names beginning with {\tt m}. + +\subsubsection{documentation} + +This gives you documentation for each of the constructors. + +\subsubsection{conditions} + +This page organizes the constructors according to predicates. +The view is not available for your example page since all constructors +are unconditional. +For a table with conditions, return to the {\bf Cross Reference} page +for {\tt Matrix}, click on {\bf Ancestors}, then on {\bf +conditions} in the view panel. +This page shows you that {\tt CoercibleTo(OutputForm)} and +{\tt SetCategory} are ancestors of {\tt Matrix(R)} only if {\tt R} +belongs to category {\tt SetCategory}. + +\subsection{Giving Parameters to Constructors} +\label{ugBrowseGivingParameters} + +Notice the input area at the bottom of the constructor page. +If you leave this blank, then the information you get is for the +domain constructor {\tt Matrix(R)}, that is, {\tt Matrix} for an +arbitrary underlying domain {\tt R}. + +In general, however, the exports and other information {\it do} usually +depend on the actual value of {\tt R}. +For example, {\tt Matrix} exports the {\bf inverse} operation +only if the domain {\tt R} is a {\tt Field}. +To see this, try this from the main constructor page: + +\begin{enumerate} +\item Enter {\tt Integer} into the input area at the bottom of the page. + +\item Click on {\bf Operations}, producing a table of operations. +Note the number of operation names that appear at the top of the +page. + +\item Click on \UpBitmap{} to return to the constructor page. + +\item Use the +\fbox{\bf Delete} +or +\fbox{\bf Backspace} +keys to erase {\tt Integer} from the input area. + +\item Click on {\bf Operations} to produce a new table of operations. +Look at the number of operations you get. +This number is greater than what you had before. +Find, for example, the operation {\bf inverse}. + +\item Click on {\bf inverse} to produce a page describing the operation +{\bf inverse}. +At the bottom of the description, you notice that the {\bf +Conditions} line says ``{\tt R} has {\tt Field}.'' +This operation is {\it not} exported by {\tt Matrix(Integer)} since +{\tt Integer} is not a {\it field}. + +Try putting the name of a domain such as {\tt Fraction Integer} +(which is a field) into the input area, then clicking on {\bf Operations}. +As you see, the operation {\bf inverse} is exported. +\end{enumerate} + +\section{Miscellaneous Features of Browse} +\label{ugBrowseMiscellaneousFeatures} + +\subsection{The Description Page for Operations} +\label{ugBrowseDescriptionPage} + +From the constructor page of {\tt Matrix}, +click on {\bf Operations} to bring up the table of operations +for {\tt Matrix}. + +Find the operation {\bf inverse} in the table and click on it. +This takes you to a page showing the documentation for this operation. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matinv.ps} +\end{picture} +\caption{Operation \spadfunFrom{inverse}{Matrix} from {\tt Matrix}.} +\end{figure} + +Here is the significance of the headings you see. + +\subsubsection{Arguments} + +This lists each of the arguments of the operation in turn, paraphrasing +the {\it signature} of the operation. +As for signatures, a {\tt \$} is used to designate {\em this domain}, +that is, {\tt Matrix(R)}. + +\subsubsection{Returns} + +This describes the return value for the operation, analogous to the {\bf +Arguments} part. + +\subsubsection{Origin} + +This tells you which domain or category explicitly exports the +operation. +In this example, the domain itself is the {\it Origin}. + + +\subsubsection{Conditions} + +This tells you that the operation is exported by {\tt Matrix(R)} only if +``{\tt R} has {\tt Field},'' that is, ``{\tt R} is a member of +category {\tt Field}.'' +When no {\bf Conditions} part is given, the operation is exported for +all values of {\tt R}. + +\subsubsection{Description} + +Here are the {\tt ++} comments +that appear in the source code of its {\it Origin}, here {\tt Matrix}. +You find these comments in the source code for {\tt Matrix}. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matmap.ps} +\end{picture} +\caption{Operations {\bf map} from {\tt Matrix}.} +\end{figure} + +Click on \UpBitmap{} to return to the table of operations. +Click on {\bf map}. +Here you find three different operations named {\bf map}. +This should not surprise you. +Operations are identified by name and {\it signature}. +There are three operations named {\bf map}, each with +different signatures. +What you see is the {\it descriptions} view of the operations. +If you like, select the button in the heading of one of these +descriptions to get {\it only} that operation. + +\subsubsection{Where} + +This part qualifies domain parameters mentioned in the arguments to the +operation. + +\subsection{Views of Operations} +\label{ugBrowseViewsOfOperations} + +We suggest that you go to the constructor page for {\tt Matrix} +and click on {\bf Operations} to bring up a table of operations +with a {\it Views} panel at the bottom. + +\subsubsection{names} + +This view lists the names of the operations. +Unlike constructors, however, there may be several operations with the +same name. +The heading for the page tells you the number of unique names and the +number of distinct operations when these numbers are different. + +\subsubsection{filter} + +As for constructors, you can use this button to cut down the list of +operations you are looking at. +Enter, for example, {\tt m*} into the input area to the right of {\bf +filter} then click on {\bf filter}. +As usual, any logical expression is permitted. +For example, use +\begin{verbatim} +*! or *? +\end{verbatim} +to get a list of destructive operations and predicates. + +\subsubsection{documentation} + +This gives you the most information: +a detailed description of all the operations in the form you have seen +before. +Every other button summarizes these operations in some form. + +\subsubsection{signatures} + +This views the operations by showing their signatures. + +\subsubsection{parameters} + +This views the operations by their distinct syntactic forms with +parameters. + +\subsubsection{origins} + +This organizes the operations according to the constructor that +explicitly exports them. + +\subsubsection{conditions} + +This view organizes the operations into conditional and unconditional +operations. + +\subsubsection{usage} + +This button is only available if your user-level is set to {\it +\index{user-level} +development}. +The {\bf usage} button produces a table of constructors that reference this +operation.\footnote{Axiom requires an especially long time to +produce this table, so anticipate this when requesting this +information.} + +\subsubsection{implementation} + +This button is only available if your user-level is set to {\it +development}. +\index{user-level} +If you enter values for all domain parameters on the constructor page, +then the {\bf implementation} button appears in place of the {\bf +conditions} button. +This button tells you what domains or packages actually implement the +various operations.\footnote{This button often takes a long time; expect +a delay while you wait for an answer.} + +With your user-level set to {\it development}, we suggest you try this +exercise. +Return to the main constructor page for {\tt Matrix}, then enter +{\tt Integer} into the input area at the bottom as the value of {\tt R}. +Then click on {\bf Operations} to produce a table of operations. +Note that the {\bf conditions} part of the {\it Views} table is +replaced by {\bf implementation}. +Click on {\bf implementation}. +After some delay, you get a page describing what implements each of +the matrix operations, organized by the various domains and packages. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-matimp.ps} +\end{picture} +\caption{Implementation domains for {\tt Matrix}.} +\end{figure} + +\subsubsection{generalize} + +This button only appears for an operation page of a constructor +involving a unique operation name. + +From an operations page for {\tt Matrix}, select any +operation name, say {\bf rank}. +In the views panel, the {\bf filter} button is replaced by +{\bf generalize}. +Click on it! +%% Replaced {\bf threshold} with 10 below. MGR 1995oct31 +What you get is a description of all Axiom operations +named {\bf rank}.\footnote{If there were more than 10 +operations of the name, you get instead a page +with a {\it Views} panel at the bottom and the message to {\bf +Select a view below}. +To get the descriptions of all these operations as mentioned +above, select the {\bf description} button.} +%See the discussion of {\bf threshold} in +%\ref{ugBrowseOptions}.} %% Removed MGR 1995oct31 + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-allrank.ps} +\end{picture} +\caption{All operations named {\bf rank} in Axiom.} +\end{figure} + +\subsubsection{all domains} + +This button only appears on an operation page resulting from a +search from the front page of Browse or from selecting +{\bf generalize} from an operation page for a constructor. + +Note that the {\bf filter} button in the {\it Views} panel is +replaced by {\bf all domains}. +Click on it to produce a table of {\it all} domains or packages that +export a {\bf rank} operation. + +\begin{figure}[htbp] +\begin{picture}(324,180)%(-54,0) +\special{psfile=ps/h-alldoms.ps} +\end{picture} +\caption{Table of all domains that export {\bf rank}.} +\end{figure} + +We note that this table specifically refers to all the {\bf rank} +operations shown in the preceding page. +Return to the descriptions of all the {\bf rank} operations and +select one of them by clicking on the button in its heading. +Select {\bf all domains}. +As you see, you have a smaller table of constructors. +When there is only one constructor, you get the +constructor page for that constructor. +\newpage + +\subsection{Capitalization Convention} +\label{ugBrowseCapitalizationConvention} + +When entering search keys for constructors, you can use capital +letters to search for abbreviations. +For example, enter {\tt UTS} into the input area and click on {\bf +Constructors}. +Up comes a page describing {\tt UnivariateTaylorSeries} +whose abbreviation is {\tt UTS}. + +Constructor abbreviations always have three or more capital +letters. +For short constructor names (six letters or less), abbreviations +are not generally helpful as their abbreviation is typically the +constructor name in capitals. +For example, the abbreviation for {\tt Matrix} is +{\tt MATRIX}. + +Abbreviations can also contain numbers. +For example, {\tt POLY2} is the abbreviation for constructor +{\tt PolynomialFunctions2}. +For default packages, the abbreviation is the same as the +abbreviation for the corresponding category with the ``\&'' +replaced by ``-''. +For example, for the category default package +\aliascon{MatrixCategory\&}{MATCAT-} the abbreviation is +{\tt MATCAT-} since the corresponding category +{\tt MatrixCategory} has abbreviation {\tt MATCAT}. + +%% ********************************************************************* +%\subsection{Browse Options} +%\label{ugBrowseOptions} +%% ********************************************************************* +% +%You can set two options for using Browse: exposure and threshold. +% +%% ********************************************************************* +%\subsubsection{Exposure} +%% ********************************************************************* +% +%By default, the only constructors, operations, and attributes +%shown by Browse are those from exposed constructors. +%To change this, you can issue +%\index{set hyperdoc browse exposure} +%\begin{verbatim} +%)set hyperdoc browse exposure on +%\end{verbatim} +%After you make this setting, you will see +%both exposed and unexposed constructs. +%By definition, an operation or attribute is exposed only if it is +%exported from an exposed constructor. +%Unexposed items are generally marked by Browse with an asterisk. +%For more information on exposure, see \ref{ugTypesExpose}. +% +%With this setting, try the following experiment. +%Starting with the main Browse page, enter {\tt *matrix*} into the +%input area and click on {\bf Constructors}. +%The result is the following table. %% This line should be texonly. MGR +% +%\begin{figure}[htbp] +%\begin{picture}(324,180)%(-54,0) +%\hspace*{\baseLeftSkip}\special{psfile=ps/h-consearch2.ps} +%\end{picture} +%\caption{Table of all constructors matching {\tt *matrix*} .} +%\end{figure} +% +% +%% ********************************************************************* +%\subsubsection{Threshold} +%% ********************************************************************* +% +%For General, Documentation or Complete searches, a summary is presented +%of all matches. +%When the number of items of a given kind is less than a number called +%{\bf threshold}, Axiom presents a table of names with the heading +%for that kind. +% +%Also, when an operation name is chosen and there are less than {\bf +%threshold} distinct operations, the operations are initially shown in +%{\bf description} mode. +% +%The default value of {\bf threshold} is 10. +%To change its value to say 5, issue +%\index{set hyperdoc browse threshold} +%\begin{verbatim} +%)set hyperdoc browse threshold 5 +%\end{verbatim} +%Notice that the headings in +%the summary are active. +%If you click on a heading, you bring up a separate page for those +%entries. +%% +%% Above section removed by MGR, 1995oct30, as these two options do +%% not exist. + + +\setcounter{chapter}{14} % Chapter 15 + +\chapter{What's New in Axiom Version 2.0} +\label{ugWhatsNew} + +Many things have changed in this new version of Axiom and +we describe many of the more important topics here. + +%------------------------------------------------------------------------ +\section{Important Things to Read First} +\index{ugWhatsNewImportant} +%------------------------------------------------------------------------ + +If you have any private {\tt .spad} files (that is, library files +which were not shipped with Axiom) you will need to +recompile them. For example, if you wrote the file {\tt +regress.spad} then you should issue {\tt )compile regress.spad} +before trying to use it. + +The internal representation of {\tt Union} has changed. +This means that \linebreak Axiom data saved +with Release 1.x may not +be readable by this Release. If you cannot recreate the saved data +by recomputing in Release 2.0, please contact NAG for assistance. + +%------------------------------------------------------------------------ +\section{The New Axiom Library Compiler} +\index{ugWhatsNewAsharp} +%------------------------------------------------------------------------ + +A new compiler is now available for Axiom. +The programming language is referred to as the Aldor, and +improves upon the old Axiom language in many ways. +The {\tt )compile} command has been upgraded to be able to +invoke the new or old compilers. +The language and the compiler are described in the hard-copy +documentation which came with your Axiom system. + +To ease the chore of upgrading your {\it .spad} files (old +compiler) to {\it .as} files (new compiler), the +{\tt )compile} command has been given a {\tt )translate} +option. This invokes a special version of the old compiler which +parses and analyzes your old code and produces augmented code +using the new syntax. +Please be aware that the translation is not necessarily one +hundred percent complete or correct. +You should attempt to compile the output with the Aldor compiler +and make any necessary corrections. + + +% ---------------------------------------------------------------------- +\section{The NAG Library Link} +\index{nagLinkIntro} +% ---------------------------------------------------------------------- + +The Nag Library link allows you to call NAG Fortran +routines from within Axiom, passing Axiom objects as parameters +and getting them back as results. + +The Nag Library and, consequently, the link are divided into {\em chapters}, +which cover different areas of numerical analysis. The statistical +and sorting {\em chapters} of the Library, however, are not included in the +link and various support and utility routines (mainly the F06 and X +{\em chapters}) have been omitted. + +Each {\em chapter} has a short (at most three-letter) name; +for example, the {\em chapter} devoted to the +solution of ordinary differential equations is called D02. When +using the link via the HyperDoc interface. +you will be presented with a complete menu of these {\em chapters}. The +names of individual routines within each {\em chapter} are formed by +adding three letters to the {\em chapter} name, so for example the routine +for solving ODEs by Adams method is called +\spadfunFrom{d02cjf}{NagOrdinaryDifferentialEquationsPackage}. + +% ---------------------------------------------------------------------- +\subsection{Interpreting NAG Documentation} +\index{nagDocumentation} +% ---------------------------------------------------------------------- + +Information about using the Nag Library in general, and about using +individual routines in particular, can be accessed via HyperDoc. +This documentation refers to the Fortran routines directly; the +purpose of this subsection is to explain how this corresponds to the +Axiom routines. + +For general information about the Nag Library users should consult +Essential Introduction to the NAG Foundation Library +\index{manpageXXintro}. +The documentation is in ASCII format, and a description of the conventions +used to represent mathematical symbols is given in +Introduction to NAG On-Line Documentation +\index{manpageXXonline}. +Advice about choosing a routine from a particular {\em chapter} can be found in +the Chapter Documents \index{FoundationLibraryDocPage}. + +% ---------------------------------------------------------------------- +\subsubsection{Correspondence Between Fortran and Axiom types} +% ---------------------------------------------------------------------- + +The NAG documentation refers to the Fortran types of objects; in +general, the correspondence to Axiom types is as follows. +\begin{itemize} +\item Fortran INTEGER corresponds to Axiom {\tt Integer}. +\item Fortran DOUBLE PRECISION corresponds to Axiom {\tt DoubleFloat}. +\item Fortran COMPLEX corresponds to Axiom {\tt Complex DoubleFloat}. +\item Fortran LOGICAL corresponds to Axiom {\tt Boolean}. +\item Fortran CHARACTER*(*) corresponds to Axiom {\tt String}. +\end{itemize} +(Exceptionally, for NAG EXTERNAL parameters -- ASPs in link parlance +-- REAL and COMPLEX correspond to {\tt MachineFloat} and {\tt MachineComplex}, +respectively; see \ref{aspSection} on page~\pageref{aspSection}.) + +The correspondence for aggregates is as follows. +\begin{itemize} +\item A one-dimensional Fortran array corresponds to an Axiom \linebreak + {\tt Matrix} with one column. +\item A two-dimensional Fortran ARRAY corresponds to an Axiom \linebreak + {\tt Matrix}. +\item A three-dimensional Fortran ARRAY corresponds to an Axiom \linebreak + {\tt ThreeDimensionalMatrix}. +\end{itemize} +Higher-dimensional arrays are not currently needed for the Nag Library. + +Arguments which are Fortran FUNCTIONs or SUBROUTINEs correspond +to special ASP domains in Axiom. See \ref{aspSection} on +page~\pageref{aspSection}. + +% ---------------------------------------------------------------------- +\subsubsection{Classification of NAG parameters} +% ---------------------------------------------------------------------- + +NAG parameters are classified as belonging to one (or more) +of the following categories: {\tt Input}, {\tt Output}, {\tt Workspace} or {\tt External} procedure. +Within {\tt External} procedures a similar classification is used, and parameters +may also be {\tt Dummies}, or {\tt User Workspace} (data structures not used by the +NAG routine but provided for the convenience of the user). + +When calling a NAG routine via the link the user only provides values +for {\tt Input} and {\tt External} parameters. + +The order of the parameters is, in general, different from the order +specified in the Nag Library documentation. The Browser description +for each routine helps in determining the correspondence. As a rule of +thumb, {\tt Input} parameters come first followed by {\tt Input/Output} +parameters. The {\tt External} parameters are always found at the end. + + +% ---------------------------------------------------------------------- +\subsubsection{IFAIL} +% ---------------------------------------------------------------------- + +NAG routines often return diagnostic information through a parameter called +$ifail$. With a few exceptions, the principle is that on input +$ifail$ takes +one of the values $-1,0,1$. This determines how the routine behaves when +it encounters an error: +\begin{itemize} +\item a value of 1 causes the NAG routine to return without printing an error +message; +\item a value of 0 causes the NAG routine to print an error message and abort; +\item a value of -1 causes the NAG routine to return and print an error message. +\end{itemize} + +The user is STRONGLY ADVISED to set $ifail$ to $-1$ when using the link. +If $ifail$ has been set to $1$ or $-1$ on input, then its value on output +will determine the possible cause of any error. A value of $0$ indicates +successful completion, otherwise it provides an index into a table of +diagnostics provided as part of the routine documentation (accessible via +Browse). + +% ---------------------------------------------------------------------- +\subsection{Using the Link} +\index{nagLinkUsage} +% ---------------------------------------------------------------------- + +The easiest way to use the link is via the +HyperDoc interface \index{htxl1}. +You will be presented with a set of fill-in forms where +you can specify the parameters for each call. Initially, the forms +contain example values, demonstrating the use of each routine (these, +in fact, correspond to the standard NAG example program for the +routine in question). For some parameters, these values can provide +reasonable defaults; others, of course, represent data. When you +change a parameter which controls the size of an array, the data in +that array are reset to a ``neutral'' value -- usually zero. + +When you are satisfied with the values entered, clicking on the +``Continue'' button will display the Axiom command needed to +run the chosen NAG routine with these values. Clicking on the +``Do It'' button will then cause Axiom to execute this command +and return the result in the parent Axiom session, as described +below. Note that, for some routines, multiple HyperDoc ``pages'' are +required, due to the structure of the data. For these, returning to +an earlier page causes HyperDoc to reset the later pages (this is a +general feature of HyperDoc); in such a case, the simplest way to +repeat a call, varying a parameter on an earlier page, is probably to +modify the call displayed in the parent session. + +An alternative approach is to call NAG routines directly in your +normal Axiom session (that is, using the Axiom +interpreter). Such calls return an +object of type {\bf Result}. As not +all parameters in the underlying NAG routine are required in the +AXIOM call (and the parameter ordering may be different), before +calling a NAG routine you should consult the description of the +Axiom operation in the Browser. (The quickest route to this +is to type the routine name, in lower case, into the Browser's +input area, then click on {\tt Operations}.) The parameter names +used coincide with NAG's, although they will appear here in lower +case. Of course, it is also possible to become familiar with the +Axiom form of a routine by first using it through the +HyperDoc interface \index{htxl1}. + +As an example of this mode of working, we can find a zero +of a function, lying between 3 and 4, as follows: + +\spadcommand{answer:=c05adf(3.0,4.0,1.0e-5,0.0,-1,sin(X)::ASP1(F)) } + +By default, {\bf Result} only displays the type of returned values, +since the amount of information returned can be quite large. Individual +components can be examined as follows: + +\spadcommand{answer . x} + +\spadcommand{answer . ifail} + +In order to avoid conflict with names defined in the workspace, you can also +get the values by using the {\tt String} type (the interpreter automatically +coerces them to {\tt Symbol}) + +\spadcommand{answer "x"} + +It is possible to have Axiom display the values of scalar or array +results automatically. For more details, see the commands +\spadfunFrom{showScalarValues}{Result} +and \spadfunFrom{showArrayValues}{Result}. + +There is also a {\bf .input} file for each NAG routine, containing +Axiom interpreter commands to set up and run the standard NAG +example for that routine. + +\spadcommand{)read c05adf.input} + +% ---------------------------------------------------------------------- +\subsection{Providing values for Argument Subprograms} +\index{aspSection} +% ---------------------------------------------------------------------- + +There are a number of ways in which users can provide values for argument +subprograms (ASPs). At the top level the user will see that NAG routines +require +an object from the {\tt Union} of a {\tt Filename} and an ASP. + +For example {\bf c05adf} requires an object of type \linebreak +{\tt Union}(fn: {\tt FileName},fp: {\tt Asp1 F}) + +\spadcommand{)display operation c05adf} + + +The user thus has a choice of providing the name of a file containing +Fortran source code, or of somehow generating the ASP within Axiom. +If a filename is specified, it is searched for in the {\it local} +machine, i.e., the machine that Axiom is running on. + +% ---------------------------------------------------------------------- +\subsubsection{Providing ASPs via {\tt FortranExpression}} +% ---------------------------------------------------------------------- + +The {\tt FortranExpression} domain is used to represent expressions +which can be translated into Fortran under certain circumstances. It is +very similar to {\tt Expression} except that only operators which exist +in Fortran can be used, and only certain variables can occur. +For +example the instantiation {\tt FortranExpression([X],[M],MachineFloat)} +is the domain of expressions containing the scalar $X$ and the array +$M$. + +This allows us to create expressions like: + +\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(X)+M[3,1]} + +but not + +\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(M)+Y} + +Those ASPs which represent expressions usually export a {\bf coerce} from +an appropriate instantiation of {\tt FortranExpression} (or perhaps +{\tt Vector FortranExpression} etc.). For convenience there are also +retractions from appropriate instantiations of {\tt Expression}, +{\tt Polynomial} and {\tt Fraction Polynomial}. + +% ---------------------------------------------------------------------- +\subsubsection{Providing ASPs via {\tt FortranCode}} +% ---------------------------------------------------------------------- + +\index{FortranCode} +{\tt FortranCode} allows us to build arbitrarily complex ASPs via a +kind of pseudo-code. It is described fully in +\ref{generalFortran} on page~\pageref{generalFortran}. + +Every ASP exports two {\bf coerce} functions: one from +{\tt FortranCode} and one from {\tt List FortranCode}. There +is also a {\bf coerce} from \linebreak +{\tt Record( localSymbols: SymbolTable, code: List FortranCode)} +which is used for passing extra symbol information about the ASP. + +So for example, to integrate the function abs(x) we could use the built-in +{\bf abs} function. But suppose we want to get back to basics and define +it directly, then we could do the following: + +\spadcommand{d01ajf(-1.0, 1.0, 0.0, 1.0e-5, 800, 200, -1, cond(LT(X,0), assign(F,-X), assign(F,X))) result } + +The \spadfunFrom{cond}{FortranCode} operation creates a conditional clause +and the \spadfunFrom{assign}{FortranCode} an assignment statement. + +% ---------------------------------------------------------------------- +\subsubsection{Providing ASPs via {\tt FileName}} +% ---------------------------------------------------------------------- + +Suppose we have created the file ``asp.f'' as follows: +\begin{verbatim} + DOUBLE PRECISION FUNCTION F(X) + DOUBLE PRECISION X + F=4.0D0/(X*X+1.0D0) + RETURN + END +\end{verbatim} +and wish to pass it to the NAG +routine {\bf d01ajf} which performs one-dimensional quadrature. +We can do this as follows: +\begin{verbatim} +d01ajf(0.0 ,1.0, 0.0, 1.0e-5, 800, 200, -1, "asp.f") +\end{verbatim} + +% ---------------------------------------------------------------------- +\subsection{General Fortran-generation utilities in Axiom} +\index{generalFortran} +% ---------------------------------------------------------------------- + +This section describes more advanced facilities which are available to users +who wish to generate Fortran code from within Axiom. There are +facilities to manipulate templates, store type information, and generate +code fragments or complete programs. + +% ---------------------------------------------------------------------- +\subsubsection{Template Manipulation} +% ---------------------------------------------------------------------- + +A template is a skeletal program which is ``fleshed out'' with data when +it is processed. It is a sequence of {\em active} and {\em passive} parts: +active parts are sequences of Axiom commands which are processed as if they +had been typed into the interpreter; passive parts are simply echoed +verbatim on the Fortran output stream. + +Suppose, for example, that we have the following template, stored in +the file ``test.tem'': +\begin{verbatim} +-- A simple template +beginVerbatim + DOUBLE PRECISION FUNCTION F(X) + DOUBLE PRECISION X +endVerbatim +outputAsFortran("F",f) +beginVerbatim + RETURN + END +endVerbatim +\end{verbatim} +The passive parts lie between the two +tokens {\tt beginVerbatim} and \linebreak {\tt endVerbatim}. There +are two active statements: one which is simply an Axiom ( +\verb+--+) +comment, and one which produces an assignment to the current value +of {\tt f}. We could use it as follows: +\begin{verbatim} +(4) ->f := 4.0/(1+X**2) + + 4 + (4) ------ + 2 + X + 1 + +(5) ->processTemplate "test.tem" + DOUBLE PRECISION FUNCTION F(X) + DOUBLE PRECISION X + F=4.0D0/(X*X+1.0D0) + RETURN + END + + (5) "CONSOLE" +\end{verbatim} + +(A more reliable method of specifying the filename will be introduced +below.) Note that the Fortran assignment {\tt F=4.0D0/(X*X+1.0D0)} +automatically converted 4.0 and 1 into DOUBLE PRECISION numbers; in +general, the Axiom Fortran generation facility will convert +anything which should be a floating point object into either +a Fortran REAL or DOUBLE PRECISION object. + +Which alternative is used is determined by the command + +\spadcommand{)set fortran precision} + +It is sometimes useful to end a template before the file itself ends (e.g. to +allow the template to be tested incrementally or so that a piece of text +describing how the template works can be included). It is of course possible +to ``comment-out'' the remainder of the file. Alternatively, the single token +{\tt endInput} as part of an active portion of the template will cause +processing to be ended prematurely at that point. + +The {\bf processTemplate} command comes in two flavours. In the first case, +illustrated above, it takes one argument of domain {\tt FileName}, +the name of the template to be processed, and writes its output on the +current Fortran output stream. In general, a filename can be generated +from {\em directory}, {\em name} and {\em extension} components, using +the operation {\bf filename}, as in +\begin{verbatim} +processTemplate filename("","test","tem") +\end{verbatim} +There is an alternative version of {\bf processTemplate}, which +takes two arguments (both of domain {\tt FileName}). In this case the +first argument is the name of the template to be processed, and the +second is the file in which to write the results. Both versions return +the location of the generated Fortran code as their result +(``{\tt CONSOLE}'' in the above example). + +It is sometimes useful to be able to mix active and passive parts of a +line or statement. For example you might want to generate a Fortran +Comment describing your data set. For this kind of application we +provide three functions as follows: +\begin{tabular}{p{1.8in}p{2.6in}} +{\bf fortranLiteral} & writes a string on the Fortran output stream \\ + & \\ +{\bf fortranCarriageReturn} & writes a carriage return on the Fortran output stream \\ +& \\ +{\bf fortranLiteralLine} & writes a string followed by a return +on the Fortran output stream \\ +\end{tabular} + +So we could create our comment as follows: +\spadcommand{m := matrix [ [1,2,3],[4,5,6] ]} + +\spadcommand{fortranLiteralLine concat ["C\ \ \ \ \ \ The\ Matrix\ has\ ", nrows(m)::String, "\ rows\ and\ ", ncols(m)::String, "\ columns"]} + +or, alternatively: +\spadcommand{fortranLiteral "C\ \ \ \ \ \ The\ Matrix\ has\ "} + +\spadcommand{fortranLiteral(nrows(m)::String)} + +\spadcommand{fortranLiteral "\ rows\ and\ "} + +\spadcommand{fortranLiteral(ncols(m)::String)} + +\spadcommand{fortranLiteral "\ columns"} + +\spadcommand{fortranCarriageReturn()} + +We should stress that these functions, together with the {\bf outputAsFortran} +function are the {\em only} sure ways +of getting output to appear on the Fortran output stream. Attempts to use +Axiom commands such as {\bf output} or {\bf writeline} may appear to give +the required result when displayed on the console, but will give the wrong +result when Fortran and algebraic output are sent to differing locations. On +the other hand, these functions can be used to send helpful messages to the +user, without interfering with the generated Fortran. + +% ---------------------------------------------------------------------- +\subsubsection{Manipulating the Fortran Output Stream} +% ---------------------------------------------------------------------- +\index{FortranOutputStackPackage} + +Sometimes it is useful to manipulate the Fortran output stream in a program, +possibly without being aware of its current value. The main use of this is +for gathering type declarations (see ``Fortran Types'' below) but it can be useful +in other contexts as well. Thus we provide a set of commands to manipulate +a stack of (open) output streams. Only one stream can be written to at +any given time. The stack is never empty---its initial value is the +console or the current value of the Fortran output stream, and can be +determined using + +\spadcommand{topFortranOutputStack()} + +(see below). +The commands available to manipulate the stack are: + +\begin{tabular}{ll} +{\bf clearFortranOutputStack} & resets the stack to the console \\ + & \\ +{\bf pushFortranOutputStack} & pushes a {\tt FileName} onto the stack \\ + & \\ +{\bf popFortranOutputStack} & pops the stack \\ + & \\ +{\bf showFortranOutputStack} & returns the current stack \\ + & \\ +{\bf topFortranOutputStack} & returns the top element of the stack \\ +\end{tabular} + +These commands are all part of {\tt FortranOutputStackPackage}. + +% ---------------------------------------------------------------------- +\subsubsection{Fortran Types} +% ---------------------------------------------------------------------- + +When generating code it is important to keep track of the Fortran types of +the objects which we are generating. This is useful for a number of reasons, +not least to ensure that we are actually generating legal Fortran code. The +current type system is built up in several layers, and we shall describe each +in turn. + +% ---------------------------------------------------------------------- +\subsubsection{FortranScalarType} +% ---------------------------------------------------------------------- +\index{FortranScalarType} + +This domain represents the simple Fortran datatypes: REAL, DOUBLE PRECISION, +COMPLEX, LOGICAL, INTEGER, and CHARACTER. +It is possible to {\bf coerce} a {\tt String} or {\tt Symbol} +into the domain, test whether two objects are equal, and also apply +the predicate functions \spadfunFrom{real?}{FortranScalarType} etc. + +% ---------------------------------------------------------------------- +\subsubsection{FortranType} +% ---------------------------------------------------------------------- +\index{FortranType} + +This domain represents ``full'' types: i.e., datatype plus array dimensions +(where appropriate) plus whether or not the parameter is an external +subprogram. It is possible to {\bf coerce} an object of +{\tt FortranScalarType} into the domain or {\bf construct} one +from an element of {\tt FortranScalarType}, a list of +{\tt Polynomial Integer}s (which can of course be simple integers or +symbols) representing its dimensions, and +a {\tt Boolean} declaring whether it is external or not. The list +of dimensions must be empty if the {\tt Boolean} is {\tt true}. +The functions {\bf scalarTypeOf}, {\bf dimensionsOf} and +{\bf external?} return the appropriate +parts, and it is possible to get the various basic Fortran Types via +functions like {\bf fortranReal}. + +For example: +\spadcommand{type:=construct(real,[i,10],false)\$FortranType} + +or +\spadcommand{type:=[real,[i,10],false]\$FortranType} + +\spadcommand{scalarTypeOf type} + +\spadcommand{dimensionsOf type} + +\spadcommand{external? type} + +\spadcommand{fortranLogical()} + +\spadcommand{construct(integer,[],true)\$FortranType} + +% ---------------------------------------------------------------------- +\subsubsection{SymbolTable} +% ---------------------------------------------------------------------- +\index{SymbolTable} + +This domain creates and manipulates a symbol table for generated Fortran code. +This is used by {\tt FortranProgram} to represent the types of objects in +a subprogram. The commands available are: + +\begin{tabular}{ll} +{\bf empty} & creates a new {\tt SymbolTable} \\ + & \\ +{\bf declare} & creates a new entry in a table \\ + & \\ +{\bf fortranTypeOf} & returns the type of an object in a table \\ + & \\ +{\bf parametersOf} & returns a list of all the symbols in the table \\ + & \\ +{\bf typeList} & returns a list of all objects of a given type \\ + & \\ +{\bf typeLists} & returns a list of lists of all objects sorted by type \\ + & \\ +{\bf externalList} & returns a list of all {\tt EXTERNAL} objects \\ + & \\ +{\bf printTypes} & produces Fortran type declarations from a table\\ +\end{tabular} + +\spadcommand{symbols := empty()\$SymbolTable} + +\spadcommand{declare!(X,fortranReal(),symbols)} + +\spadcommand{declare!(M,construct(real,[i,j],false)\$FortranType,symbols)} + +\spadcommand{declare!([i,j],fortranInteger(),symbols)} + +\spadcommand{symbols} + +\spadcommand{fortranTypeOf(i,symbols)} + +\spadcommand{typeList(real,symbols)} + +\spadcommand{printTypes symbols} + +% ---------------------------------------------------------------------- +\subsubsection{TheSymbolTable} +% ---------------------------------------------------------------------- +\index{TheSymbolTable} + +This domain creates and manipulates one global symbol table to be used, for +example, during template processing. It is +also used when +linking to external Fortran routines. The +information stored for each subprogram (and the main program segment, where +relevant) is: +\begin{itemize} +\item its name; +\item its return type; +\item its argument list; +\item and its argument types. +\end{itemize} +Initially, any information provided is deemed to be for the main program +segment. + +Issuing the following command indicates that from now on all information +refers to the subprogram $F$. + +\spadcommand{newSubProgram F} + +It is possible to return to processing the main program segment by issuing +the command: + +\spadcommand{endSubProgram()} + +The following commands exist: + +\begin{tabular}{p{1.6in}p{2.8in}} +{\bf returnType} & declares the return type of the current subprogram \\ + & \\ +{\bf returnTypeOf} & returns the return type of a subprogram \\ + & \\ +{\bf argumentList} & declares the argument list of the current subprogram \\ + & \\ +{\bf argumentListOf} & returns the argument list of a subprogram \\ + & \\ +{\bf declare} & provides type declarations for parameters of the current subprogram \\ + & \\ +{\bf symbolTableOf} & returns the symbol table of a subprogram \\ + & \\ +{\bf printHeader} & produces the Fortran header for the current subprogram \\ +\end{tabular} + +In addition there are versions of these commands which are parameterised by +the name of a subprogram, and others parameterised by both the name of a +subprogram and by an instance of {\tt TheSymbolTable}. + +\spadcommand{newSubProgram F} + +\spadcommand{argumentList!(F,[X])} + +\spadcommand{returnType!(F,real)} + +\spadcommand{declare!(X,fortranReal(),F)} + +\spadcommand{printHeader F} + +% ---------------------------------------------------------------------- +\subsubsection{Advanced Fortran Code Generation} +% ---------------------------------------------------------------------- + +This section describes facilities for representing Fortran statements, and +building up complete subprograms from them. + +% ---------------------------------------------------------------------- +\subsubsection{Switch} +% ---------------------------------------------------------------------- +\index{Switch} + +This domain is used to represent statements like {\tt x < y}. Although +these can be represented directly in Axiom, it is a little cumbersome, +since Axiom evaluates the last statement, for example, to {\tt true} +(since $x$ is lexicographically less than $y$). + +Instead we have a set of operations, such as {\bf LT} to represent $<$, +to let us build such statements. The available constructors are: + +\begin{center} +\begin{tabular}{ll} +{\bf LT} & $<$ \\ +{\bf GT} & $>$ \\ +{\bf LE} & $\leq$ \\ +{\bf GE} & $\geq$ \\ +{\bf EQ} & $=$ \\ +{\bf AND} & {\tt and}\\ +{\bf OR} & {\tt or} \\ +{\bf NOT} & {\tt not} \\ +\end{tabular} +\end{center} + +So for example: +\spadcommand{LT(x,y)} + +% ---------------------------------------------------------------------- +\subsubsection{FortranCode} +% ---------------------------------------------------------------------- + +This domain represents code segments or operations: currently assignments, +conditionals, blocks, comments, gotos, continues, various kinds of loops, +and return statements. + +For example we can create quite a complicated conditional statement using +assignments, and then turn it into Fortran code: + +\spadcommand{c := cond(LT(X,Y),assign(F,X),cond(GT(Y,Z),assign(F,Y),assign(F,Z)))} + +\spadcommand{printCode c} + +The Fortran code is printed +on the current Fortran output stream. + +% ---------------------------------------------------------------------- +\subsubsection{FortranProgram} +% ---------------------------------------------------------------------- +\index{FortranProgram} + +This domain is used to construct complete Fortran subprograms out of +elements of {\tt FortranCode}. It is parameterised by the name of the +target subprogram (a {\tt Symbol}), its return type (from +{\tt Union}({\tt FortranScalarType},``void'')), +its arguments (from {\tt List Symbol}), and +its symbol table (from {\tt SymbolTable}). One can +{\bf coerce} elements of either {\tt FortranCode} +or {\tt Expression} into it. + + +First of all we create a symbol table: + +\spadcommand{symbols := empty()\$SymbolTable} + +Now put some type declarations into it: + +\spadcommand{declare!([X,Y],fortranReal(),symbols)} + +Then (for convenience) +we set up the particular instantiation of {\tt FortranProgram} + +\spadcommand{FP := FortranProgram(F,real,[X,Y],symbols)} + +Create an object of type {\tt Expression(Integer)}: + +\spadcommand{asp := X*sin(Y)} + +Now {\bf coerce} it into {\tt FP}, and print its Fortran form: + +\spadcommand{outputAsFortran(asp::FP)} + +We can generate a {\tt FortranProgram} using $FortranCode$. For +example: + +Augment our symbol table: + +\spadcommand{declare!(Z,fortranReal(),symbols)} + +and transform the conditional expression we prepared earlier: + +\spadcommand{outputAsFortran([c,returns()]::FP)} + +%------------------------------------------------------------------------ +\subsection{Some technical information} +\index{nagTechnical} +%------------------------------------------------------------------------ + +The model adopted for the link is a server-client configuration +-- Axiom acting as a client via a local agent +(a process called {\tt nagman}). The server side is implemented +by the {\tt nagd} daemon process which may run on a different host. +The {\tt nagman} local agent is started by default whenever you +start Axiom. The {\tt nagd} server must be started separately. +Instructions for installing and running the server are supplied +in \ref{nugNagd} on page~\pageref{nugNagd}. +Use the {\tt )set naglink host} system command +to point your local agent to a server in your network. + + + +On the Axiom side, one sees a set of {\em packages} +(ask Browse for {\em Nag*}) for each chapter, each exporting +operations with the same name as a routine in the Nag Library. +The arguments and return value of each operation belong to +standard Axiom types. + +The {\tt man} pages for the Nag Library are accessible via the description +of each operation in Browse (among other places). + +In the implementation of each operation, the set of inputs is passed +to the local agent {\tt nagman}, which makes a +Remote Procedure Call (RPC) to the +remote {\tt nagd} daemon process. The local agent receives the RPC +results and forwards them to the Axiom workspace where they +are interpreted appropriately. + +How are Fortran subroutines turned into RPC calls? +For each Fortran routine in the Nag Library, a C main() routine +is supplied. +Its job is to assemble the RPC input (numeric) data stream into +the appropriate Fortran data structures for the routine, call the Fortran +routine from C and serialize the results into an RPC output data stream. + +Many Nag Library routines accept ASPs (Argument Subprogram Parameters). +These specify user-supplied Fortran routines (e.g. a routine to +supply values of a function is required for numerical integration). +How are they handled? There are new facilities in Axiom to help. +A set of Axiom domains has been provided to turn values in standard + Axiom types (such as Expression Integer) into the appropriate +piece of Fortran for each case (a filename pointing to Fortran source +for the ASP can always be supplied instead). +Ask Browse for {\em Asp*} to see these domains. The Fortran fragments +are included in the outgoing RPC stream, but {\tt nagd} intercepts them, +compiles them, and links them with the main() C program before executing +the resulting program on the numeric part of the RPC stream. + + +%------------------------------------------------------------------------ +\section{Interactive Front-end and Language} +\index{ugWhatsNewLanguage} +%------------------------------------------------------------------------ + +The {\tt leave} keyword has been replaced by the +{\tt break} keyword for compatibility with the new Axiom +extension language. +See section \ref{ugLangLoopsBreak} on page~\pageref{ugLangLoopsBreak} +for more information. + +Curly braces are no longer used to create sets. Instead, use +{\bf set} followed by a bracketed expression. For example, + +\spadcommand{set [1,2,3,4]} + +Curly braces are now used to enclose a block (see section +\ref{ugLangBlocks} on page~\pageref{ugLangBlocks} +for more information). For compatibility, a block can still be +enclosed by parentheses as well. + +``Free functions'' created by the Aldor compiler can now be +loaded and used within the Axiom interpreter. A {\it free +function} is a library function that is implemented outside a +domain or category constructor. + +New coercions to and from type {\tt Expression} have been +added. For example, it is now possible to map a polynomial +represented as an expression to an appropriate polynomial type. + +Various messages have been added or rewritten for clarity. + +%------------------------------------------------------------------------ +\section{Library} +\index{ugWhatsNewLibrary} +%------------------------------------------------------------------------ + +The {\tt FullPartialFractionExpansion} +domain has been added. This domain computes factor-free full +partial fraction expansions. +See section +FullPartialFractionExpansion +for examples. + +We have implemented the Bertrand/Cantor algorithm for integrals of +hyperelliptic functions. This brings a major speedup for some +classes of algebraic integrals. + +We have implemented a new (direct) algorithm for integrating trigonometric +functions. This brings a speedup and an improvement in the answer +quality. + +The {\sf SmallFloat} domain has been renamed +{\tt DoubleFloat} and {\sf SmallInteger} has been renamed +{\tt SingleInteger}. The new abbreviations as +{\tt DFLOAT} and {\tt SINT}, respectively. +We have defined the macro {\sf SF}, the old abbreviation for {\sf +SmallFloat}, to expand to {\tt DoubleFloat} and modified +the documentation and input file examples to use the new names +and abbreviations. You should do the same in any private Axiom +files you have. + +There are many new categories, domains and packages related to the +NAG Library Link facility. See the file + +src/algebra/exposed.lsp + +for a list of constructors in the {\bf naglink} Axiom exposure group. + +We have made improvements to the differential equation solvers +and there is a new facility for solving systems of first-order +linear differential equations. +In particular, an important fix was made to the solver for +inhomogeneous linear ordinary differential equations that +corrected the calculation of particular solutions. +We also made improvements to the polynomial +and transcendental equation solvers including the +ability to solve some classes of systems of transcendental +equations. + +The efficiency of power series have been improved and left and right +expansions of $tan(f(x))$ at $x =$ a pole of $f(x)$ +can now be computed. +A number of power series bugs were fixed and the {\tt GeneralUnivariatePowerSeries} +domain was added. +The power series variable can appear in the coefficients and when this +happens, you cannot differentiate or integrate the series. Differentiation +and integration with respect to other variables is supported. + +A domain was added for representing asymptotic expansions of a +function at an exponential singularity. + +For limits, the main new feature is the exponential expansion domain used +to treat certain exponential singularities. Previously, such singularities +were treated in an {\it ad hoc} way and only a few cases were covered. Now +Axiom can do things like + +\begin{verbatim} +limit( (x+1)**(x+1)/x**x - x**x/(x-1)**(x-1), x = %plusInfinity) +\end{verbatim} + +in a systematic way. It only does one level of nesting, though. In other +words, we can handle $exp(some function with a pole)$, but not +\linebreak $exp(exp(some function with a pole))$. + +The computation of integral bases has been improved through careful +use of Hermite row reduction. A P-adic algorithm +for function fields of algebraic curves in finite characteristic has also +been developed. + +Miscellaneous: +There is improved conversion of definite and indefinite integrals to +{\tt InputForm}; +binomial coefficients are displayed in a new way; +some new simplifications of radicals have been implemented; +the operation {\bf complexForm} for converting to rectangular coordinates +has been added; +symmetric product operations have been added to {\tt LinearOrdinaryDifferentialOperator}. + +%------------------------------------------------------------------------ +\section{HyperTex} +\index{ugWhatsNewHyperDoc} +%------------------------------------------------------------------------ + +The buttons on the titlebar and scrollbar have been replaced +with ones which have a 3D effect. You can change the foreground and +background colors of these ``controls'' by including and modifying +the following lines in your {\bf .Xdefaults} file. +\begin{verbatim} +Axiom.hyperdoc.ControlBackground: White +Axiom.hyperdoc.ControlForeground: Black +\end{verbatim} + +For various reasons, HyperDoc sometimes displays a +secondary window. You can control the size and placement of this +window by including and modifying +the following line in your {\bf .Xdefaults} file. +% +\begin{verbatim} +Axiom.hyperdoc.FormGeometry: =950x450+100+0 +\end{verbatim} +% +This setting is a standard X Window System geometry specification: +you are requesting a window 950 pixels wide by 450 deep and placed in +the upper left corner. + +Some key definitions have been changed to conform more closely +with the CUA guidelines. Press +F9 +to see the current definitions. + +Input boxes (for example, in the Browser) now accept paste-ins from +the X Window System. Use the second button to paste in something +you have previously copied or cut. An example of how you can use this +is that you can paste the type from an Axiom computation +into the main Browser input box. + + +%------------------------------------------------------------------------ +\section{Documentation} +\index{ugWhatsNewDocumentation} +%------------------------------------------------------------------------ + +We describe here a few additions to the on-line +version of the AXIOM book which you can read with +HyperDoc. + + +A section has been added to the graphics chapter, describing +how to build two-di\-men\-sion\-al graphs from lists of points. An example is +given showing how to read the points from a file. +See section \ref{ugGraphTwoDbuild} on page~\pageref{ugGraphTwoDbuild} +for details. + +A further section has been added to that same chapter, describing +how to add a two-di\-men\-sion\-al graph to a viewport which already +contains other graphs. +See section +\ref{ugGraphTwoDappend} on page~\pageref{ugGraphTwoDappend} +for details. + +Chapter 3 +and the on-line HyperDoc help have been unified. + +An explanation of operation names ending in ``?'' and ``!'' has +been added to the first chapter. +See the +end of the section +\ref{ugIntroCallFun} on page~\pageref{ugIntroCallFun} +for details. + +An expanded explanation of using predicates has +been added to the sixth chapter. See the +example involving {\bf evenRule} in the middle of the section +\ref{ugUserRules} on page~\pageref{ugUserRules} +for details. + +Documentation for the {\tt )compile}, {\tt )library} and +{\tt )load} commands has been greatly changed. This reflects +the ability of the {\tt )compile} to now invoke the Aldor +compiler, the impending deletion of the {\tt )load} command +and the new {\tt )library} command. +The {\tt )library} command replaces {\tt )load} and is +compatible with the compiled output from both the old and new +compilers. + + +\setcounter{chapter}{0} % Appendix A + +\newcommand{\lanb}{{\tt [}} +\newcommand{\ranb}{{\tt ]}} +\newcommand{\vertline}{$|$} + +\chapter{Axiom System Commands} +\label{ugSysCmd} + +This chapter describes system commands, the command-line +facilities used to control the Axiom environment. +The first section is an introduction and discusses the common +syntax of the commands available. + +\section{Introduction} +\label{ugSysCmdOverview} + +System commands are used to perform Axiom environment +management. +Among the commands are those that display what has been defined or +computed, set up multiple logical Axiom environments +(frames), clear definitions, read files of expressions and +commands, show what functions are available, and terminate +Axiom. + +Some commands are restricted: the commands +\index{set userlevel interpreter} +\index{set userlevel compiler} +\index{set userlevel development} +\begin{verbatim} +)set userlevel interpreter +)set userlevel compiler +)set userlevel development +\end{verbatim} +set the user-access level to the three possible choices. +All commands are available at {\tt development} level and the fewest +are available at {\tt interpreter} level. +The default user-level is {\tt interpreter}. +\index{user-level} +In addition to the {\tt )set} command (discussed in +\ref{ugSysCmdset} on page~\pageref{ugSysCmdset}) +you can use the HyperDoc settings facility to change the {\it user-level.} + + +Each command listing begins with one or more syntax pattern descriptions +plus examples of related commands. +The syntax descriptions are intended to be easy to read and do not +necessarily represent the most compact way of specifying all +possible arguments and options; the descriptions may occasionally +be redundant. + +All system commands begin with a right parenthesis which should be in +the first available column of the input line (that is, immediately +after the input prompt, if any). +System commands may be issued directly to Axiom or be +included in {\bf .input} files. +\index{file!input} + +A system command {\it argument} is a word that directly +follows the command name and is not followed or preceded by a +right parenthesis. +A system command {\it option} follows the system command and +is directly preceded by a right parenthesis. +Options may have arguments: they directly follow the option. +This example may make it easier to remember what is an option and +what is an argument: + +\begin{center} +{\tt )syscmd {\it arg1 arg2} )opt1 {\it opt1arg1 opt1arg2} )opt2 {\it opt2arg1} ...} +\end{center} + +In the system command descriptions, optional arguments and options are +enclosed in brackets (``\lanb'' and ``\ranb''). +If an argument or option name is in italics, it is +meant to be a variable and must have some actual value substituted +for it when the system command call is made. +For example, the syntax pattern description + +\noindent +{\tt )read} {\it fileName} {\tt \lanb{})quietly\ranb{}} + +\noindent +would imply that you must provide an actual file name for +{\it fileName} but need not use the {\tt )quietly} option. +Thus +\begin{verbatim} +)read matrix.input +\end{verbatim} +is a valid instance of the above pattern. + +System command names and options may be abbreviated and may be in +upper or lower case. +The case of actual arguments may be significant, depending on the +particular situation (such as in file names). +System command names and options may be abbreviated to the minimum +number of starting letters so that the name or option is unique. +Thus +\begin{verbatim} +)s Integer +\end{verbatim} +is not a valid abbreviation for the {\tt )set} command, +because both {\tt )set} and {\tt )show} +begin with the letter ``s''. +Typically, two or three letters are sufficient for disambiguating names. +In our descriptions of the commands, we have used no abbreviations for +either command names or options. + +In some syntax descriptions we use a vertical line ``\vertline'' +to indicate that you must specify one of the listed choices. +For example, in +\begin{verbatim} +)set output fortran on | off +\end{verbatim} +only {\tt on} and {\tt off} are acceptable words for following +{\tt boot}. +We also sometimes use ``...'' to indicate that additional arguments +or options of the listed form are allowed. +Finally, in the syntax descriptions we may also list the syntax of +related commands. + +\section{)abbreviation} + +\index{abbreviation} + + +\par\noindent{\bf User Level Required:} compiler + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )abbreviation query \lanb{}{\it nameOrAbbrev}\ranb{}} +\item {\tt )abbreviation category {\it abbrev fullname} \lanb{})quiet\ranb{}} +\item {\tt )abbreviation domain {\it abbrev fullname} \lanb{})quiet\ranb{}} +\item {\tt )abbreviation package {\it abbrev fullname} \lanb{})quiet\ranb{}} +\item {\tt )abbreviation remove {\it nameOrAbbrev}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to query, set and remove abbreviations for category, +domain and package constructors. +Every constructor must have a unique abbreviation. +This abbreviation is part of the name of the subdirectory +under which the components of the compiled constructor are +stored. +%% BEGIN OBSOLETE +% It is this abbreviation that is used to bring compiled code into +% Axiom with the {\tt )load} command. +%% END OBSOLETE +Furthermore, by issuing this command you +let the system know what file to load automatically if you use a new +constructor. +Abbreviations must start with a letter and then be followed by +up to seven letters or digits. +Any letters appearing in the abbreviation must be in uppercase. + +When used with the {\tt query} argument, +\index{abbreviation query} +this command may be used to list the name +associated with a particular abbreviation or the abbreviation for a +constructor. +If no abbreviation or name is given, the names and corresponding +abbreviations for {\it all} constructors are listed. + +The following shows the abbreviation for the constructor {\tt List}: +\begin{verbatim} +)abbreviation query List +\end{verbatim} +The following shows the constructor name corresponding to the +abbreviation {\tt NNI}: +\begin{verbatim} +)abbreviation query NNI +\end{verbatim} +The following lists all constructor names and their abbreviations. +\begin{verbatim} +)abbreviation query +\end{verbatim} + +To add an abbreviation for a constructor, use this command with +{\tt category}, {\tt domain} or {\tt package}. +\index{abbreviation package} +\index{abbreviation domain} +\index{abbreviation category} +The following add abbreviations to the system for a +category, domain and package, respectively: +\begin{verbatim} +)abbreviation domain SET Set +)abbreviation category COMPCAT ComplexCategory +)abbreviation package LIST2MAP ListToMap +\end{verbatim} +If the {\tt )quiet} option is used, +no output is displayed from this command. +You would normally only define an abbreviation in a library source file. +If this command is issued for a constructor that has already been loaded, the +constructor will be reloaded next time it is referenced. In particular, you +can use this command to force the automatic reloading of constructors. + +To remove an abbreviation, the {\tt remove} argument is used. +\index{abbreviation remove} +This is usually +only used to correct a previous command that set an abbreviation for a +constructor name. +If, in fact, the abbreviation does exist, you are prompted +for confirmation of the removal request. +Either of the following commands +will remove the abbreviation {\tt VECTOR2} and the +constructor name {\tt VectorFunctions2} from the system: +\begin{verbatim} +)abbreviation remove VECTOR2 +)abbreviation remove VectorFunctions2 +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )compile} \index{ugSysCmdcompile} + +\section{)boot} +\index{ugSysCmdboot} + +\index{boot} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )boot} {\it bootExpression} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to execute +expressions written in the BOOT language. +For example, +\begin{verbatim} +)boot times3(x) == 3*x +\end{verbatim} +creates and compiles the Common Lisp function ``times3'' +obtained by translating the BOOT code. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{ugSysCmdfin}, +{\tt )lisp} \index{ugSysCmdlisp}, +{\tt )set} \index{ugSysCmdset}, and +{\tt )system} \index{ugSysCmdsystem}. + + +\section{)cd} +\index{ugSysCmdcd} + +\index{cd} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )cd} {\it directory} +\end{list} + +\par\noindent{\bf Command Description:} + +This command sets the Axiom working current directory. +The current directory is used for looking for +input files (for {\tt )read}), +Axiom library source files (for {\tt )compile}), +saved history environment files (for {\tt )history )restore}), +compiled Axiom library files (for {\tt )library}), and +files to edit (for {\tt )edit}). +It is also used for writing +spool files (via {\tt )spool}), +writing history input files (via {\tt )history )write}) and +history environment files (via {\tt )history )save}),and +compiled Axiom library files (via {\tt )compile}). +\index{read} +\index{compile} +\index{history )restore} +\index{edit} +\index{spool} +\index{history )write} +\index{history )save} + +If issued with no argument, this command sets the Axiom +current directory to your home directory. +If an argument is used, it must be a valid directory name. +Except for the ``{\tt )}'' at the beginning of the command, +this has the same syntax as the operating system {\tt cd} command. + +\par\noindent{\bf Also See:} +{\tt )compile} \index{ugSysCmdcompile}, +{\tt )edit} \index{ugSysCmdedit}, +{\tt )history} \index{ugSysCmdhistory}, +{\tt )library} \index{ugSysCmdlibrary}, +{\tt )read} \index{ugSysCmdread}, and +{\tt )spool} \index{ugSysCmdspool}. + +\section{)close} +\index{ugSysCmdclose} + +\index{close} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )close} +\item{\tt )close )quietly} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to close down interpreter client processes. +Such processes are started by HyperDoc to run Axiom examples +when you click on their text. When you have finished examining or modifying the +example and you do not want the extra window around anymore, issue +\begin{verbatim} +)close +\end{verbatim} +to the Axiom prompt in the window. + +If you try to close down the last remaining interpreter client +process, Axiom will offer to close down the entire Axiom +session and return you to the operating system by displaying something +like +\begin{verbatim} + This is the last AXIOM session. Do you want to kill AXIOM? +\end{verbatim} +Type ``{\tt y}'' (followed by the Return key) if this is what you had in mind. +Type ``{\tt n}'' (followed by the Return key) to cancel the command. + +You can use the {\tt )quietly} option to force Axiom to +close down the interpreter client process without closing down +the entire Axiom session. + +\par\noindent{\bf Also See:} +{\tt )quit} \index{ugSysCmdquit} and +{\tt )pquit} \index{ugSysCmdpquit}. + + + +\section{)clear} +\index{ugSysCmdclear} + +\index{clear} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )clear all} +\item{\tt )clear completely} +\item{\tt )clear properties all} +\item{\tt )clear properties} {\it obj1 \lanb{}obj2 ...\ranb{}} +\item{\tt )clear value all} +\item{\tt )clear value} {\it obj1 \lanb{}obj2 ...\ranb{}} +\item{\tt )clear mode all} +\item{\tt )clear mode} {\it obj1 \lanb{}obj2 ...\ranb{}} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to remove function and variable declarations, definitions +and values from the workspace. +To empty the entire workspace and reset the +step counter to 1, issue +\begin{verbatim} +)clear all +\end{verbatim} +To remove everything in the workspace but not reset the step counter, issue +\begin{verbatim} +)clear properties all +\end{verbatim} +To remove everything about the object {\tt x}, issue +\begin{verbatim} +)clear properties x +\end{verbatim} +To remove everything about the objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear properties x y f +\end{verbatim} + +The word {\tt properties} may be abbreviated to the single letter +``{\tt p}''. +\begin{verbatim} +)clear p all +)clear p x +)clear p x y f +\end{verbatim} +All definitions of functions and values of variables may be removed by either +\begin{verbatim} +)clear value all +)clear v all +\end{verbatim} +This retains whatever declarations the objects had. To remove definitions and +values for the specific objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear value x y f +)clear v x y f +\end{verbatim} +To remove the declarations of everything while leaving the definitions and +values, issue +\begin{verbatim} +)clear mode all +)clear m all +\end{verbatim} +To remove declarations for the specific objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear mode x y f +)clear m x y f +\end{verbatim} +The {\tt )display names} and {\tt )display properties} commands may be used +to see what is currently in the workspace. + +The command +\begin{verbatim} +)clear completely +\end{verbatim} +does everything that {\tt )clear all} does, and also clears the internal +system function and constructor caches. + +\par\noindent{\bf Also See:} +{\tt )display} \index{ugSysCmddisplay}, +{\tt )history} \index{ugSysCmdhistory}, and +{\tt )undo} \index{ugSysCmdundo}. + + +\section{)compile} +\index{ugSysCmdcompile} + +\index{compile} + + +\par\noindent{\bf User Level Required:} compiler + +\par\noindent{\bf Command Syntax:} + +\begin{list}{} +\item {\tt )compile} +\item {\tt )compile {\it fileName}} +\item {\tt )compile {\it fileName}.as} +\item {\tt )compile {\it directory/fileName}.as} +\item {\tt )compile {\it fileName}.ao} +\item {\tt )compile {\it directory/fileName}.ao} +\item {\tt )compile {\it fileName}.al} +\item {\tt )compile {\it directory/fileName}.al} +\item {\tt )compile {\it fileName}.lsp} +\item {\tt )compile {\it directory/fileName}.lsp} +\item {\tt )compile {\it fileName}.spad} +\item {\tt )compile {\it directory/fileName}.spad} +\item {\tt )compile {\it fileName} )new} +\item {\tt )compile {\it fileName} )old} +\item {\tt )compile {\it fileName} )translate} +\item {\tt )compile {\it fileName} )quiet} +\item {\tt )compile {\it fileName} )noquiet} +\item {\tt )compile {\it fileName} )moreargs} +\item {\tt )compile {\it fileName} )onlyargs} +\item {\tt )compile {\it fileName} )break} +\item {\tt )compile {\it fileName} )nobreak} +\item {\tt )compile {\it fileName} )library} +\item {\tt )compile {\it fileName} )nolibrary} +\item {\tt )compile {\it fileName} )vartrace} +\item {\tt )compile {\it fileName} )constructor} {\it nameOrAbbrev} +\end{list} + +\par\noindent{\bf Command Description:} + +You use this command to invoke the new Axiom library compiler or +the old Axiom system compiler. +The {\tt )compile} system command is actually a combination of +Axiom processing and a call to the Aldor compiler. +It is performing double-duty, acting as a front-end to +both the Aldor compiler and the old Axiom system +compiler. +(The old Axiom system compiler was written in Lisp and was +an integral part of the Axiom environment. +The Aldor compiler is written in C and executed by the operating system +when called from within Axiom.) + +This command compiles files with file extensions {\it .as, .ao} +and {\it .al} with the +Aldor compiler and files with file extension {\it .spad} with the +old Axiom system compiler. +It also can compile files with file extension {\it .lsp}. These +are assumed to be Lisp files genererated by the Aldor +compiler. +If you omit the file extension, the command looks to see if you +have specified the {\tt )new} or {\tt )old} option. +If you have given one of these options, the corresponding compiler +is used. +Otherwise, the command first looks in the standard system +directories for files with extension {\it .as, .ao} and {\it +.al} and then files with extension {\it .spad}. +The first file found has the appropriate compiler invoked on it. +If the command cannot find a matching file, an error message is +displayed and the command terminates. + +The {\tt )translate} option is used to invoke a special version +of the old system compiler that will translate a {\it .spad} file +to a {\it .as} file. That is, the {\it .spad} file will be parsed and +analyzed and a file using the new syntax will be created. By default, +the {\it .as} file is created in the same directory as the +{\it .spad} file. If that directory is not writable, the current +directory is used. If the current directory is not writable, an +error message is given and the command terminates. +Note that {\tt )translate} implies the {\tt )old} option so the +file extension can safely be omitted. If {\tt )translate} is +given, all other options are ignored. +Please be aware that the translation is not necessarily one +hundred percent complete or correct. +You should attempt to compile the output with the Aldor compiler +and make any necessary corrections. + +We now describe the options for the new Aldor compiler. + +The first thing {\tt )compile} does is look for a source code +filename among its arguments. +Thus +\begin{verbatim} +)compile mycode.as +)compile /u/jones/as/mycode.as +)compile mycode +\end{verbatim} +all invoke {\tt )compiler} on the file {\tt +/u/jones/as/mycode.as} if the current Axiom working +directory is {\tt /u/jones/as.} (Recall that you can set the +working directory via the {\tt )cd} command. If you don't set it +explicitly, it is the directory from which you started +Axiom.) + +This is frequently all you need to compile your file. +This simple command: +\begin{enumerate} +\item Invokes the Aldor compiler and produces Lisp output. +\item Calls the Lisp compiler if the Aldor compilation was +successful. +\item Uses the {\tt )library} command to tell Axiom about +the contents of your compiled file and arrange to have those +contents loaded on demand. +\end{enumerate} + +Should you not want the {\tt )library} command automatically +invoked, call {\tt )compile} with the {\tt )nolibrary} option. +For example, +\begin{verbatim} +)compile mycode.as )nolibrary +\end{verbatim} + +The general description of Aldor command line arguments is in +the Aldor documentation. +The default options used by the {\tt )compile} command can be +viewed and set using the {\tt )set compiler args} Axiom +system command. +The current defaults are +\begin{verbatim} +-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom +\end{verbatim} +These options mean: +\begin{itemize} +\item {\tt -O}: perform all optimizations, +\item {\tt -Fasy}: generate a {\tt .asy} file, +\item {\tt -Fao}: generate a {\tt .ao} file, +\item {\tt -Flsp}: generate a {\tt .lsp} (Lisp) +file, +\index{Lisp!code generation} +\item {\tt -laxiom}: use the {\tt axiom} library {\tt libaxiom.al}, +\item {\tt -Mno-AXL\_W\_WillObsolete}: do not display messages +about older generated files becoming obsolete, and +\item {\tt -DAxiom}: define the global assertion {\tt Axiom} so that the +Aldor libraries for generating stand-alone code +are not accidentally used with Axiom. +\end{itemize} + +To supplement these default arguments, use the {\tt )moreargs} option on +{\tt )compile.} +For example, +\begin{verbatim} +)compile mycode.as )moreargs "-v" +\end{verbatim} +uses the default arguments and appends the {\tt -v} (verbose) +argument flag. +The additional argument specification {\bf must be enclosed in +double quotes.} + +To completely replace these default arguments for a particular +use of {\tt )compile}, use the {\tt )onlyargs} option. +For example, +\begin{verbatim} +)compile mycode.as )onlyargs "-v -O" +\end{verbatim} +only uses the {\tt -v} (verbose) and {\tt -O} (optimize) +arguments. +The argument specification {\bf must be enclosed in double quotes.} +In this example, Lisp code is not produced and so the compilation +output will not be available to Axiom. + +To completely replace the default arguments for all calls to {\tt +)compile} within your Axiom session, use {\tt )set compiler args.} +For example, to use the above arguments for all compilations, issue +\begin{verbatim} +)set compiler args "-v -O" +\end{verbatim} +Make sure you include the necessary {\tt -l} and {\tt -Y} +arguments along with those needed for Lisp file creation. +As above, {\bf the argument specification must be enclosed in double +quotes.} + +By default, the {\tt )library} system command {\it exposes} all +domains and categories it processes. +This means that the Axiom intepreter will consider those +domains and categories when it is trying to resolve a reference +to a function. +Sometimes domains and categories should not be exposed. +For example, a domain may just be used privately by another +domain and may not be meant for top-level use. +The {\tt )library} command should still be used, though, so that +the code will be loaded on demand. +In this case, you should use the {\tt )nolibrary} option on {\tt +)compile} and the {\tt )noexpose} option in the {\tt )library} +command. For example, +\begin{verbatim} +)compile mycode.as )nolibrary +)library mycode )noexpose +\end{verbatim} + +Once you have established your own collection of compiled code, +you may find it handy to use the {\tt )dir} option on the +{\tt )library} command. +This causes {\tt )library} to process all compiled code in the +specified directory. For example, +\begin{verbatim} +)library )dir /u/jones/as/quantum +\end{verbatim} +You must give an explicit directory after {\tt )dir}, even if you +want all compiled code in the current working directory +processed, e.g. +\begin{verbatim} +)library )dir . +\end{verbatim} + +The {\tt )compile} command works with several file extensions. We saw +above what happens when it is invoked on a file with extension {\tt +.as.} A {\tt .ao} file is a portable binary compiled version of a +{\tt .as} file, and {\tt )compile} simply passes the {\tt .ao} file +onto Aldor. The generated Lisp file is compiled and {\tt )library} +is automatically called, just as if you had specified a {\tt .as} file. + +A {\tt .al} file is an archive file containing {\tt .ao} files. The +archive is created (on Unix systems) with the {\tt ar} program. When +{\tt )compile} is given a {\tt .al} file, it creates a directory whose +name is based on that of the archive. For example, if you issue +\begin{verbatim} +)compile mylib.al +\end{verbatim} +the directory {\tt mylib.axldir} is created. All +members of the archive are unarchived into the +directory and {\tt )compile} is called on each {\tt .ao} file found. It +is your responsibility to remove the directory and its contents, if you +choose to do so. + +A {\tt .lsp} file is a Lisp source file, presumably, in our context, +generated by Aldor when called with the {\tt -Flsp} option. When +{\tt )compile} is used with a {\tt .lsp} file, the Lisp file is +compiled and {\tt )library} is called. You must also have present a +{\tt .asy} generated from the same source file. + +The following are descriptions of options for the old system compiler. + +You can compile category, domain, and package constructors +contained in files with file extension {\it .spad}. +You can compile individual constructors or every constructor +in a file. + +The full filename is remembered between invocations of this command and +{\tt )edit} commands. +The sequence of commands +\begin{verbatim} +)compile matrix.spad +)edit +)compile +\end{verbatim} +will call the compiler, edit, and then call the compiler again +on the file {\bf matrix.spad.} +If you do not specify a {\it directory,} the working current +directory (see +\ref{ugSysCmdcd} on page~\pageref{ugSysCmdcd}) +is searched for the file. +If the file is not found, the standard system directories are searched. + +If you do not give any options, all constructors within a file are +compiled. +Each constructor should have an {\tt )abbreviation} command in +the file in which it is defined. +We suggest that you place the {\tt )abbreviation} commands at the +top of the file in the order in which the constructors are +defined. +The list of commands serves as a table of contents for the file. +\index{abbreviation} + +The {\tt )library} option causes directories containing the +compiled code for each constructor +to be created in the working current directory. +The name of such a directory consists of the constructor +abbreviation and the {\bf .nrlib} file extension. +For example, the directory containing the compiled code for +the {\tt MATRIX} constructor is called {\bf MATRIX.nrlib.} +The {\tt )nolibrary} option says that such files should not +be created. +The default is {\tt )library.} +Note that the semantics of {\tt )library} and {\tt )nolibrary} +for the new Aldor compiler and for the old system compiler are +completely different. + +The {\tt )vartrace} option causes the compiler to generate +extra code for the constructor to support conditional tracing of +variable assignments. (see +\ref{ugSysCmdtrace} on page~\pageref{ugSysCmdtrace}). Without +this option, this code is suppressed and one cannot use +the {\tt )vars} option for the trace command. + +The {\tt )constructor} option is used to +specify a particular constructor to compile. +All other constructors in the file are ignored. +The constructor name or abbreviation follows {\tt )constructor.} +Thus either +\begin{verbatim} +)compile matrix.spad )constructor RectangularMatrix +\end{verbatim} +or +\begin{verbatim} +)compile matrix.spad )constructor RMATRIX +\end{verbatim} +compiles the {\tt RectangularMatrix} constructor +defined in {\bf matrix.spad.} + +The {\tt )break} and {\tt )nobreak} options determine what +the old system compiler does when it encounters an error. +{\tt )break} is the default and it indicates that processing +should stop at the first error. +The value of the {\tt )set break} variable then controls what happens. + + +%% BEGIN OBSOLTE +% It is important for you to realize that it does not suffice to compile a +% constructor to use the new code in the interpreter. +% After compilation, the {\tt )load} command with the +% {\tt )update} option should be used to bring in the new code +% and update internal system tables with information about the +% constructor. +%% END OBSOLTE + +\par\noindent{\bf Also See:} +{\tt )abbreviation} \index{ugSysCmdabbreviation}, +{\tt )edit} \index{ugSysCmdedit}, and +{\tt )library} \index{ugSysCmdlibrary}. + + +\section{)display} +\index{ugSysCmddisplay} + +\index{display} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )display all} +\item {\tt )display properties} +\item {\tt )display properties all} +\item {\tt )display properties} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display value all} +\item {\tt )display value} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display mode all} +\item {\tt )display mode} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display names} +\item {\tt )display operations} {\it opName} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to display the contents of the workspace and +signatures of functions with a given name.\footnote{A +{\it signature} gives the argument and return types of a +function.} + +The command +\begin{verbatim} +)display names +\end{verbatim} +lists the names of all user-defined objects in the workspace. This is useful +if you do not wish to see everything about the objects and need only be +reminded of their names. + +The commands +\begin{verbatim} +)display all +)display properties +)display properties all +\end{verbatim} +all do the same thing: show the values and types and declared modes of all +variables in the workspace. If you have defined functions, their signatures +and definitions will also be displayed. + +To show all information about a particular variable or user functions, +for example, something named {\tt d}, issue +\begin{verbatim} +)display properties d +\end{verbatim} +To just show the value (and the type) of {\tt d}, issue +\begin{verbatim} +)display value d +\end{verbatim} +To just show the declared mode of {\tt d}, issue +\begin{verbatim} +)display mode d +\end{verbatim} + +All modemaps for a given operation may be +displayed by using {\tt )display operations}. +A {\it modemap} is a collection of information about a particular +reference +to an operation. This includes the types of the arguments and the return +value, the location of the implementation and any conditions on the types. +The modemap may contain patterns. The following displays the modemaps for the +operation \spadfunFrom{complex}{ComplexCategory}: +\begin{verbatim} +)d op complex +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )clear} \index{ugSysCmdclear}, +{\tt )history} \index{ugSysCmdhistory}, +{\tt )set} \index{ugSysCmdset}, +{\tt )show} \index{ugSysCmdshow}, and +{\tt )what} \index{ugSysCmdwhat}. + + +\section{)edit} +\index{ugSysCmdedit} + +\index{edit} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )edit} \lanb{}{\it filename}\ranb{} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to edit files. +It works in conjunction with the {\tt )read} +and {\tt )compile} commands to remember the name +of the file on which you are working. +By specifying the name fully, you can edit any file you wish. +Thus +\begin{verbatim} +)edit /u/julius/matrix.input +\end{verbatim} +will place you in an editor looking at the file +{\tt /u/julius/matrix.input}. +\index{editing files} +By default, the editor is {\tt vi}, +\index{vi} +but if you have an EDITOR shell environment variable defined, that editor +will be used. +When Axiom is running under the X Window System, +it will try to open a separate {\tt xterm} running your editor if +it thinks one is necessary. +\index{Korn shell} +For example, under the Korn shell, if you issue +\begin{verbatim} +export EDITOR=emacs +\end{verbatim} +then the emacs +\index{emacs} +editor will be used by {\tt )edit}. + +If you do not specify a file name, the last file you edited, +read or compiled will be used. +If there is no ``last file'' you will be placed in the editor editing +an empty unnamed file. + +It is possible to use the {\tt )system} command to edit a file directly. +For example, +\begin{verbatim} +)system emacs /etc/rc.tcpip +\end{verbatim} +calls {\tt emacs} to edit the file. +\index{emacs} + +\par\noindent{\bf Also See:} +{\tt )system} \index{ugSysCmdsystem}, +{\tt )compile} \index{ugSysCmdcompile}, and +{\tt )read} \index{ugSysCmdread}. + + +\section{)fin} +\index{ugSysCmdfin} + +\index{fin} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )fin} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used by Axiom +developers to leave the Axiom system and return +to the underlying Common Lisp system. +To return to Axiom, issue the +``{\tt (\vertline{}spad\vertline{})}'' +function call to Common Lisp. + +\par\noindent{\bf Also See:} +{\tt )pquit} \index{ugSysCmdpquit} and +{\tt )quit} \index{ugSysCmdquit}. + + +\section{)frame} +\label{ugSysCmdframe} +\index{ugSysCmdframe} + +\index{frame} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )frame new {\it frameName}} +\item{\tt )frame drop {\it \lanb{}frameName\ranb{}}} +\item{\tt )frame next} +\item{\tt )frame last} +\item{\tt )frame names} +\item{\tt )frame import {\it frameName} {\it \lanb{}objectName1 \lanb{}objectName2 ...\ranb{}\ranb{}}} +\item{\tt )set message frame on \vertline{} off} +\item{\tt )set message prompt frame} +\end{list} + +\par\noindent{\bf Command Description:} + +A {\it frame} can be thought of as a logical session within the +physical session that you get when you start the system. You can +have as many frames as you want, within the limits of your computer's +storage, paging space, and so on. +Each frame has its own {\it step number}, {\it environment} and {\it history.} +You can have a variable named {\tt a} in one frame and it will +have nothing to do with anything that might be called {\tt a} in +any other frame. + +Some frames are created by the HyperDoc program and these can +have pretty strange names, since they are generated automatically. +\index{frame names} +To find out the names +of all frames, issue +\begin{verbatim} +)frame names +\end{verbatim} +It will indicate the name of the current frame. + +You create a new frame +\index{frame new} +``{\bf quark}'' by issuing +\begin{verbatim} +)frame new quark +\end{verbatim} +The history facility can be turned on by issuing either +{\tt )set history on} or {\tt )history )on}. +If the history facility is on and you are saving history information +in a file rather than in the Axiom environment +then a history file with filename {\bf quark.axh} will +be created as you enter commands. +If you wish to go back to what +you were doing in the +\index{frame next} +``{\bf initial}'' frame, use +\index{frame last} +\begin{verbatim} +)frame next +\end{verbatim} +or +\begin{verbatim} +)frame last +\end{verbatim} +to cycle through the ring of available frames to get back to +``{\bf initial}''. + +If you want to throw +away a frame (say ``{\bf quark}''), issue +\begin{verbatim} +)frame drop quark +\end{verbatim} +If you omit the name, the current frame is dropped. +\index{frame drop} + +If you do use frames with the history facility on and writing to a file, +you may want to delete some of the older history files. +\index{file!history} +These are directories, so you may want to issue a command like +{\tt rm -r quark.axh} to the operating system. + +You can bring things from another frame by using +\index{frame import} +{\tt )frame import}. +For example, to bring the {\tt f} and {\tt g} from the frame ``{\bf quark}'' +to the current frame, issue +\begin{verbatim} +)frame import quark f g +\end{verbatim} +If you want everything from the frame ``{\bf quark}'', issue +\begin{verbatim} +)frame import quark +\end{verbatim} +You will be asked to verify that you really want everything. + +There are two {\tt )set} flags +\index{set message frame} +to make it easier to tell where you are. +\begin{verbatim} +)set message frame on | off +\end{verbatim} +will print more messages about frames when it is set on. +By default, it is off. +\begin{verbatim} +)set message prompt frame +\end{verbatim} +will give a prompt +\index{set message prompt frame} +that looks like +\begin{verbatim} +initial (1) -> +\end{verbatim} +\index{prompt!with frame name} +when you start up. In this case, the frame name and step make up the +prompt. + +\par\noindent{\bf Also See:} +{\tt )history} \index{ugSysCmdhistory} and +{\tt )set} \index{ugSysCmdset}. + + +\section{)help} +\index{ugSysCmdhelp} + +\index{help} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )help} +\item{\tt )help} {\it commandName} +\end{list} + +\par\noindent{\bf Command Description:} + +This command displays help information about system commands. +If you issue +\begin{verbatim} +)help +\end{verbatim} +then this very text will be shown. +You can also give the name or abbreviation of a system command +to display information about it. +For example, +\begin{verbatim} +)help clear +\end{verbatim} +will display the description of the {\tt )clear} system command. + +All this material is available in the Axiom User Guide +and in HyperDoc. +In HyperDoc, choose the {\bf Commands} item from the +{\bf Reference} menu. + + + +\section{)history} +\index{ugSysCmdhistory} + +\index{history} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )history )on} +\item{\tt )history )off} +\item{\tt )history )write} {\it historyInputFileName} +\item{\tt )history )show \lanb{}{\it n}\ranb{} \lanb{}both\ranb{}} +\item{\tt )history )save} {\it savedHistoryName} +\item{\tt )history )restore} \lanb{}{\it savedHistoryName}\ranb{} +\item{\tt )history )reset} +\item{\tt )history )change} {\it n} +\item{\tt )history )memory} +\item{\tt )history )file} +\item{\tt \%} +\item{\tt \%\%({\it n})} +\item{\tt )set history on \vertline{} off} +\end{list} + +\par\noindent{\bf Command Description:} + +The {\it history} facility within Axiom allows you to restore your +environment to that of another session and recall previous +computational results. +Additional commands allow you to review previous +input lines and to create an {\bf .input} file of the lines typed to +\index{file!input} +Axiom. + +Axiom saves your input and output if the history facility is +turned on (which is the default). +This information is saved if either of +\begin{verbatim} +)set history on +)history )on +\end{verbatim} +has been issued. +Issuing either +\begin{verbatim} +)set history off +)history )off +\end{verbatim} +will discontinue the recording of information. +\index{history )on} +\index{set history on} +\index{set history off} +\index{history )off} + +Whether the facility is disabled or not, +the value of {\tt \%} in Axiom always +refers to the result of the last computation. +If you have not yet entered anything, +{\tt \%} evaluates to an object of type +{\tt Variable('\%)}. +The function {\tt \%\%} may be used to refer +to other previous results if the history facility is enabled. +In that case, +{\tt \%\%(n)} is the output from step {\tt n} if {\tt n > 0}. +If {\tt n < 0}, the step is computed relative to the current step. +Thus {\tt \%\%(-1)} is also the previous step, +{\tt \%\%(-2)}, is the step before that, and so on. +If an invalid step number is given, Axiom will signal an error. + +The {\it environment} information can either be saved in a file or entirely in +memory (the default). +Each frame +(\ref{ugSysCmdframe} on page~\pageref{ugSysCmdframe}) +has its own history database. +When it is kept in a file, some of it may also be kept in memory for +efficiency. +When the information is saved in a file, the name of the file is +of the form {\bf FRAME.axh} where ``{\bf FRAME}'' is the name of the +current frame. +The history file is placed in the current working directory +(see \ref{ugSysCmdcd} on page~\pageref{ugSysCmdcd}). +Note that these history database files are not text files (in fact, +they are directories themselves), and so are not in human-readable +format. + +The options to the {\tt )history} command are as follows: + +\begin{description} +\item[{\tt )change} {\it n}] +will set the number of steps that are saved in memory to {\it n}. +This option only has effect when the history data is maintained in a +file. +If you have issued {\tt )history )memory} (or not changed the default) +there is no need to use {\tt )history )change}. +\index{history )change} + +\item[{\tt )on}] +will start the recording of information. +If the workspace is not empty, you will be asked to confirm this +request. +If you do so, the workspace will be cleared and history data will begin +being saved. +You can also turn the facility on by issuing {\tt )set history on}. + +\item[{\tt )off}] +will stop the recording of information. +The {\tt )history )show} command will not work after issuing this +command. +Note that this command may be issued to save time, as there is some +performance penalty paid for saving the environment data. +You can also turn the facility off by issuing {\tt )set history off}. + +\item[{\tt )file}] +indicates that history data should be saved in an external file on disk. + +\item[{\tt )memory}] +indicates that all history data should be kept in memory rather than +saved in a file. +Note that if you are computing with very large objects it may not be +practical to kept this data in memory. + +\item[{\tt )reset}] +will flush the internal list of the most recent workspace calculations +so that the data structures may be garbage collected by the underlying +Common Lisp system. +Like {\tt )history )change}, this option only has real effect when +history data is being saved in a file. + +\item[{\tt )restore} \lanb{}{\it savedHistoryName}\ranb{}] +completely clears the environment and restores it to a saved session, if +possible. +The {\tt )save} option below allows you to save a session to a file +with a given name. If you had issued +{\tt )history )save jacobi} +the command +{\tt )history )restore jacobi} +would clear the current workspace and load the contents of the named +saved session. If no saved session name is specified, the system looks +for a file called {\bf last.axh}. + +\item[{\tt )save} {\it savedHistoryName}] +is used to save a snapshot of the environment in a file. +This file is placed in the current working directory +(see \ref{ugSysCmdcd} on page~\pageref{ugSysCmdcd}). +Use {\tt )history )restore} to restore the environment to the state +preserved in the file. +This option also creates an input file containing all the lines of input +since you created the workspace frame (for example, by starting your +Axiom session) or last did a {\tt )clear all} or +{\tt )clear completely}. + +\item[{\tt )show} \lanb{}{\it n}\ranb{} \lanb{}{\tt both}\ranb{}] +can show previous input lines and output results. +{\tt )show} will display up to twenty of the last input lines +(fewer if you haven't typed in twenty lines). +{\tt )show} {\it n} will display up to {\it n} of the last input lines. +{\tt )show both} will display up to five of the last input lines and +output results. +{\tt )show} {\it n} {\tt both} will display up to {\it n} of the last +input lines and output results. + +\item[{\tt )write} {\it historyInputFile}] +creates an {\bf .input} file with the input lines typed since the start +of the session/frame or the last {\tt )clear all} or {\tt )clear +completely}. +If {\it historyInputFileName} does not contain a period (``.'') in the filename, +{\bf .input} is appended to it. +For example, +{\tt )history )write chaos} +and +{\tt )history )write chaos.input} +both write the input lines to a file called {\bf chaos.input} in your +current working directory. +If you issued one or more {\tt )undo} commands, +{\tt )history )write} +eliminates all +input lines backtracked over as a result of {\tt )undo}. +You can edit this file and then use {\tt )read} to have Axiom process +the contents. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )frame} \index{ugSysCmdframe}, +{\tt )read} \index{ugSysCmdread}, +{\tt )set} \index{ugSysCmdset}, and +{\tt )undo} \index{ugSysCmdundo}. + + +\section{)library} +\index{ugSysCmdlibrary} + +\index{library} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )library {\it libName1 \lanb{}libName2 ...\ranb{}}} +\item{\tt )library )dir {\it dirName}} +\item{\tt )library )only {\it objName1 \lanb{}objlib2 ...\ranb{}}} +\item{\tt )library )noexpose} +\end{list} + +\par\noindent{\bf Command Description:} + +This command replaces the {\tt )load} system command that +was available in Axiom releases before version 2.0. +The {\tt )library} command makes available to Axiom the compiled +objects in the libraries listed. + +For example, if you {\tt )compile dopler.as} in your home +directory, issue {\tt )library dopler} to have Axiom look +at the library, determine the category and domain constructors present, +update the internal database with various properties of the +constructors, and arrange for the constructors to be +automatically loaded when needed. +If the {\tt )noexpose} option has not been given, the +constructors will be exposed (that is, available) in the current +frame. + +If you compiled a file with the old system compiler, you will +have an {\it nrlib} present, for example, {\it DOPLER.nrlib,} +where {\tt DOPLER} is a constructor abbreviation. +The command {\tt )library DOPLER} will then do the analysis and +database updates as above. + +To tell the system about all libraries in a directory, use +{\tt )library )dir dirName} where {\tt dirName} is an explicit +directory. +You may specify ``.'' as the directory, which means the current +directory from which you started the system or the one you set +via the {\tt )cd} command. The directory name is required. + +You may only want to tell the system about particular +constructors within a library. In this case, use the {\tt )only} +option. The command {\tt )library dopler )only Test1} will only +cause the {\sf Test1} constructor to be analyzed, autoloaded, +etc.. + +Finally, each constructor in a library are usually automatically exposed when the +{\tt )library} command is used. Use the {\tt )noexpose} +option if you not want them exposed. At a later time you can use +{\tt )set expose add constructor} to expose any hidden +constructors. + +{\bf Note for Axiom beta testers:} At various times this +command was called {\tt )local} and {\tt )with} before the name +{\tt )library} became the official name. + +\par\noindent{\bf Also See:} +{\tt )cd} \index{ugSysCmdcd}, +{\tt )compile} \index{ugSysCmdcompile}, +{\tt )frame} \index{ugSysCmdframe}, and +{\tt )set} \index{ugSysCmdset}. + +\section{)lisp} +\index{ugSysCmdlisp} + +\index{lisp} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )lisp} {\it\lanb{}lispExpression\ranb{}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to have single +expressions evaluated by the Common Lisp system on which +Axiom is built. +The {\it lispExpression} is read by the Common Lisp reader and +evaluated. +If this expression is not complete (unbalanced parentheses, say), the reader +will wait until a complete expression is entered. + +Since this command is only useful for evaluating single expressions, the +{\tt )fin} +command may be used to drop out of Axiom into Common Lisp. + +\par\noindent{\bf Also See:} +{\tt )system} \index{ugSysCmdsystem}, +{\tt )boot} \index{ugSysCmdboot}, and +{\tt )fin} \index{ugSysCmdfin}. + + + +\section{)load} +\index{ugSysCmdload} + +\index{load} + + +\par\noindent{\bf User Level Required:} interpreter + +%% BEGIN OBSOLETE +% \par\noindent{\bf Command Syntax:} +% \begin{list}{} +% \item{\tt )load {\it libName1 \lanb{}libName2 ...\ranb{}} \lanb{})update\ranb{}} +% \item{\tt )load {\it libName1 \lanb{}libName2 ...\ranb{}} )cond \lanb{})update\ranb{}} +% \item{\tt )load {\it libName1 \lanb{}libName2 ...\ranb{}} )query} +% \item{\tt )load {\it libName1 \lanb{}libName2 ...\ranb{}} )noexpose} +% \end{list} +%% END OBSOLETE + +\par\noindent{\bf Command Description:} + +This command is obsolete. Use {\tt )library} instead. + +%% BEGIN OBSOLETE + +% The {\tt )load} command is used to bring in the compiled library code +% for constructors and update internal system tables with information +% about the constructors. +% This command is usually only used by Axiom library developers. +% +% The abbreviation of a constructor serves as part of the name of the +% directory in which the compiled code is stored (see +% \ref{ugSysCmdabbreviation} for a discussion of defining and querying +% abbreviations). +% The abbreviation is used in the {\tt )load} command. +% For example, to load the constructors {\tt Integer}, +% {\tt NonNegativeInteger} and {\tt List} which have +% abbreviations {\tt INT}, {\tt NNI} and {\tt LIST}, +% respectively, issue the command +% \begin{verbatim} +% )load INT NNI LIST +% \end{verbatim} +% To load constructors only if they have not already been +% loaded (that is., load {\it conditionally}), use the {\tt )cond} +% option: +% \begin{verbatim} +% )load INT NNI LIST )cond +% \end{verbatim} +% To query whether particular constructors have been loaded, use the +% {\tt )query} option: +% \begin{verbatim} +% )load I NNI L )query +% \end{verbatim} +% When constructors are loaded from Axiom system directories, some +% checks and updates are not performed because it is assumed that the system +% knows about these constructors. +% To force these checks and updates to occur, add the {\tt )update} +% option to the command: +% \begin{verbatim} +% )load INT NNI LIST )update +% )load INT NNI LIST )cond )update +% \end{verbatim} +% The only time it is really necessary to use the {\tt )load} command is +% when a new constructor has been compiled or an existing constructor has +% been modified and then compiled. +% If an {\tt )abbreviate} command has been issued for a constructor, it +% will be automatically loaded when needed. +% In particular, any constructor that comes with the Axiom system +% will be automatically loaded. +% +% If you write several interdependent constructors it is important that +% they all get loaded when needed. +% To accomplish this, either load them manually or issue +% {\tt )abbreviate} commands for each of the constructors so that they +% will be automatically loaded when needed. +% +% Constructors are automatically exposed in the frame in which you load +% them unless you use the {\tt )noexpose} option. +% \begin{verbatim} +% )load MATCAT- )noexpose +% \end{verbatim} +% See \ref{ugTypesExpose} +% for more information about constructor exposure. +% +% \par\noindent{\bf Also See:} +% {\tt )abbreviation} \index{ugSysCmdabbreviation} and +% {\tt )compile} \index{ugSysCmdcompile}. + +%% END OBSOLETE + + +\section{)trace} +\index{ugSysCmdltrace} + +\index{ltrace} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} + +This command has the same arguments as options as the +{\tt )trace} command. + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to trace +Common Lisp or +BOOT functions. +It is not supported for general use. + +\par\noindent{\bf Also See:} +{\tt )boot} \index{ugSysCmdboot}, +{\tt )lisp} \index{ugSysCmdlisp}, and +{\tt )trace} \index{ugSysCmdtrace}. + + +\section{)pquit} +\index{ugSysCmdpquit} + +\index{pquit} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )pquit} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to terminate Axiom and return to the +operating system. +Other than by redoing all your computations or by +using the {\tt )history )restore} +command to try to restore your working environment, +you cannot return to Axiom in the same state. + +{\tt )pquit} differs from the {\tt )quit} in that it always asks for +confirmation that you want to terminate Axiom (the ``p'' is for +``protected''). +\index{quit} +When you enter the {\tt )pquit} command, Axiom responds +% +\begin{center} +Please enter {\bf y} or {\bf yes} if you really want to leave the interactive \\ +environment and return to the operating system: +\end{center} +% +If you respond with {\tt y} or {\tt yes}, you will see the message +% +\begin{center} +You are now leaving the Axiom interactive environment. \\ +Issue the command {\bf axiom} to the operating system to start a new session. +\end{center} +% +and Axiom will terminate and return you to the operating +system (or the environment from which you invoked the system). +If you responded with something other than {\tt y} or {\tt yes}, then +the message +% +\begin{center} +You have chosen to remain in the Axiom interactive environment. +\end{center} +% +will be displayed and, indeed, Axiom would still be running. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{ugSysCmdfin}, +{\tt )history} \index{ugSysCmdhistory}, +{\tt )close} \index{ugSysCmdclose}, +{\tt )quit} \index{ugSysCmdquit}, and +{\tt )system} \index{ugSysCmdsystem}. + + +\section{)quit} +\index{ugSysCmdquit} + +\index{quit} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )quit} +\item{\tt )set quit protected \vertline{} unprotected} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to terminate Axiom and return to the +operating system. +Other than by redoing all your computations or by +using the {\tt )history )restore} +command to try to restore your working environment, +you cannot return to Axiom in the same state. + +{\tt )quit} differs from the {\tt )pquit} in that it asks for +\index{pquit} +confirmation only if the command +\begin{verbatim} +)set quit protected +\end{verbatim} +has been issued. +\index{set quit protected} +Otherwise, {\tt )quit} will make Axiom terminate and return you +to the operating system (or the environment from which you invoked the +system). + +The default setting is {\tt )set quit protected} so that {\tt )quit} +and {\tt )pquit} behave in the same way. +If you do issue +\begin{verbatim} +)set quit unprotected +\end{verbatim} +we +\index{set quit unprotected} +suggest that you do not (somehow) assign {\tt )quit} to be +executed when you press, say, a function key. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{ugSysCmdfin}, +{\tt )history} \index{ugSysCmdhistory}, +{\tt )close} \index{ugSysCmdclose}, +{\tt )pquit} \index{ugSysCmdpquit}, and +{\tt )system} \index{ugSysCmdsystem}. + + +\section{)read} +\index{ugSysCmdread} + +\index{read} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )read} {\it \lanb{}fileName\ranb{}} +\item {\tt )read} {\it \lanb{}fileName\ranb{}} \lanb{}{\tt )quiet}\ranb{} \lanb{}{\tt )ifthere}\ranb{} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to read {\bf .input} files into Axiom. +\index{file!input} +The command +\begin{verbatim} +)read matrix.input +\end{verbatim} +will read the contents of the file {\bf matrix.input} into +Axiom. +The ``.input'' file extension is optional. +See \ref{ugInOutIn} on page~\pageref{ugInOutIn} +for more information about {\bf .input} files. + +This command remembers the previous file you edited, read or compiled. +If you do not specify a file name, the previous file will be read. + +The {\tt )ifthere} option checks to see whether the {\bf .input} file +exists. +If it does not, the {\tt )read} command does nothing. +If you do not use this option and the file does not exist, +you are asked to give the name of an existing {\bf .input} file. + +The {\tt )quiet} option suppresses output while the file is being read. + +\par\noindent{\bf Also See:} +{\tt )compile} \index{ugSysCmdcompile}, +{\tt )edit} \index{ugSysCmdedit}, and +{\tt )history} \index{ugSysCmdhistory}. + + +\section{)set} +\index{ugSysCmdset} + +\index{set} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )set} +\item {\tt )set} {\it label1 \lanb{}... labelN\ranb{}} +\item {\tt )set} {\it label1 \lanb{}... labelN\ranb{} newValue} +\end{list} +\par\noindent{\bf Command Description:} + +The {\tt )set} command is used to view or set system variables that +control what messages are displayed, the type of output desired, the +status of the history facility, the way Axiom user functions are +cached, and so on. +Since this collection is very large, we will not discuss them here. +Rather, we will show how the facility is used. +We urge you to explore the {\tt )set} options to familiarize yourself +with how you can modify your Axiom working environment. +There is a HyperDoc version of this same facility available from the +main HyperDoc menu. + + +The {\tt )set} command is command-driven with a menu display. +It is tree-structured. +To see all top-level nodes, issue {\tt )set} by itself. +\begin{verbatim} +)set +\end{verbatim} +Variables with values have them displayed near the right margin. +Subtrees of selections have ``{\tt ...}'' +displayed in the value field. +For example, there are many kinds of messages, so issue +{\tt )set message} to see the choices. +\begin{verbatim} +)set message +\end{verbatim} +The current setting for the variable that displays +\index{computation timings!displaying} +whether computation times +\index{timings!displaying} +are displayed is visible in the menu displayed by the last command. +To see more information, issue +\begin{verbatim} +)set message time +\end{verbatim} +This shows that time printing is on now. +To turn it off, issue +\begin{verbatim} +)set message time off +\end{verbatim} +\index{set message time} + +As noted above, not all settings have so many qualifiers. +For example, to change the {\tt )quit} command to being unprotected +(that is, you will not be prompted for verification), you need only issue +\begin{verbatim} +)set quit unprotected +\end{verbatim} +\index{set quit unprotected} + +\par\noindent{\bf Also See:} +{\tt )quit} \index{ugSysCmdquit}. + + +\section{)show} +\index{ugSysCmdshow} + +\index{show} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )show {\it nameOrAbbrev}} +\item {\tt )show {\it nameOrAbbrev} )operations} +\item {\tt )show {\it nameOrAbbrev} )attributes} +\end{list} + +\par\noindent{\bf Command Description:} +This command displays information about Axiom +domain, package and category {\it constructors}. +If no options are given, the {\tt )operations} option is assumed. +For example, +\begin{verbatim} +)show POLY +)show POLY )operations +)show Polynomial +)show Polynomial )operations +\end{verbatim} +each display basic information about the +{\tt Polynomial} domain constructor and then provide a +listing of operations. +Since {\tt Polynomial} requires a {\tt Ring} (for example, +{\tt Integer}) as argument, the above commands all refer +to a unspecified ring {\tt R}. +In the list of operations, {\tt \$} means +{\tt Polynomial(R)}. + +The basic information displayed includes the {\it signature} +of the constructor (the name and arguments), the constructor +{\it abbreviation}, the {\it exposure status} of the constructor, and the +name of the {\it library source file} for the constructor. + +If operation information about a specific domain is wanted, +the full or abbreviated domain name may be used. +For example, +\begin{verbatim} +)show POLY INT +)show POLY INT )operations +)show Polynomial Integer +)show Polynomial Integer )operations +\end{verbatim} +are among the combinations that will +display the operations exported by the +domain {\tt Polynomial(Integer)} (as opposed to the general +{\it domain constructor} {\tt Polynomial}). +Attributes may be listed by using the {\tt )attributes} option. + +\par\noindent{\bf Also See:} +{\tt )display} \index{ugSysCmddisplay}, +{\tt )set} \index{ugSysCmdset}, and +{\tt )what} \index{ugSysCmdwhat}. + + +\section{)spool} +\index{ugSysCmdspool} + +\index{spool} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )spool} \lanb{}{\it fileName}\ranb{} +\item{\tt )spool} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to save {\it (spool)} all Axiom input and output +\index{file!spool} +into a file, called a {\it spool file.} +You can only have one spool file active at a time. +To start spool, issue this command with a filename. For example, +\begin{verbatim} +)spool integrate.out +\end{verbatim} +To stop spooling, issue {\tt )spool} with no filename. + +If the filename is qualified with a directory, then the output will +be placed in that directory. +If no directory information is given, the spool file will be placed in the +\index{directory!for spool files} +{\it current directory.} +The current directory is the directory from which you started +Axiom or is the directory you specified using the +{\tt )cd} command. +\index{cd} + +\par\noindent{\bf Also See:} +{\tt )cd} \index{ugSysCmdcd}. + + +\section{)synonym} +\index{ugSysCmdsynonym} + +\index{synonym} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )synonym} +\item{\tt )synonym} {\it synonym fullCommand} +\item{\tt )what synonyms} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to create short synonyms for system command expressions. +For example, the following synonyms might simplify commands you often +use. +\begin{verbatim} +)synonym save history )save +)synonym restore history )restore +)synonym mail system mail +)synonym ls system ls +)synonym fortran set output fortran +\end{verbatim} +Once defined, synonyms can be +used in place of the longer command expressions. +Thus +\begin{verbatim} +)fortran on +\end{verbatim} +is the same as the longer +\begin{verbatim} +)set fortran output on +\end{verbatim} +To list all defined synonyms, issue either of +\begin{verbatim} +)synonyms +)what synonyms +\end{verbatim} +To list, say, all synonyms that contain the substring +``{\tt ap}'', issue +\begin{verbatim} +)what synonyms ap +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )set} \index{ugSysCmdset} and +{\tt )what} \index{ugSysCmdwhat}. + + +\section{)system} +\index{ugSysCmdsystem} + +\index{system} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )system} {\it cmdExpression} +\end{list} + +\par\noindent{\bf Command Description:} + +This command may be used to issue commands to the operating system while +remaining in Axiom. +The {\it cmdExpression} is passed to the operating system for +execution. + +To get an operating system shell, issue, for example, +{\tt )system sh}. +When you enter the key combination, +\fbox{\bf Ctrl}--\fbox{\bf D} +(pressing and holding the +\fbox{\bf Ctrl} key and then pressing the +\fbox{\bf D} key) +the shell will terminate and you will return to Axiom. +We do not recommend this way of creating a shell because +Common Lisp may field some interrupts instead of the shell. +If possible, use a shell running in another window. + +If you execute programs that misbehave you may not be able to return to +Axiom. +If this happens, you may have no other choice than to restart +Axiom and restore the environment via {\tt )history )restore}, if +possible. + +\par\noindent{\bf Also See:} +{\tt )boot} \index{ugSysCmdboot}, +{\tt )fin} \index{ugSysCmdfin}, +{\tt )lisp} \index{ugSysCmdlisp}, +{\tt )pquit} \index{ugSysCmdpquit}, and +{\tt )quit} \index{ugSysCmdquit}. + + +\section{)trace} +\index{ugSysCmdtrace} + +\index{trace} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )trace} +\item{\tt )trace )off} + +\item{\tt )trace} {\it function \lanb{}options\ranb{}} +\item{\tt )trace} {\it constructor \lanb{}options\ranb{}} +\item{\tt )trace} {\it domainOrPackage \lanb{}options\ranb{}} +\end{list} +% +where options can be one or more of +% +\begin{list}{} +\item{\tt )after} {\it S-expression} +\item{\tt )before} {\it S-expression} +\item{\tt )break after} +\item{\tt )break before} +\item{\tt )cond} {\it S-expression} +\item{\tt )count} +\item{\tt )count} {\it n} +\item{\tt )depth} {\it n} +\item{\tt )local} {\it op1 \lanb{}... opN\ranb{}} +\item{\tt )nonquietly} +\item{\tt )nt} +\item{\tt )off} +\item{\tt )only} {\it listOfDataToDisplay} +\item{\tt )ops} +\item{\tt )ops} {\it op1 \lanb{}... opN \ranb{}} +\item{\tt )restore} +\item{\tt )stats} +\item{\tt )stats reset} +\item{\tt )timer} +\item{\tt )varbreak} +\item{\tt )varbreak} {\it var1 \lanb{}... varN \ranb{}} +\item{\tt )vars} +\item{\tt )vars} {\it var1 \lanb{}... varN \ranb{}} +\item{\tt )within} {\it executingFunction} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to trace the execution of functions that make +up the Axiom system, functions defined by users, +and functions from the system library. +Almost all options are available for each type of function but +exceptions will be noted below. + +To list all functions, constructors, domains and packages that are +traced, simply issue +\begin{verbatim} +)trace +\end{verbatim} +To untrace everything that is traced, issue +\begin{verbatim} +)trace )off +\end{verbatim} +When a function is traced, the default system action is to display +the arguments to the function and the return value when the +function is exited. +Note that if a function is left via an action such as a {\tt THROW}, no +return value will be displayed. +Also, optimization of tail recursion may decrease the number of +times a function is actually invoked and so may cause less trace +information to be displayed. +Other information can be displayed or collected when a function is +traced and this is controlled by the various options. +Most options will be of interest only to Axiom system +developers. +If a domain or package is traced, the default action is to trace +all functions exported. + +Individual interpreter, lisp or boot +functions can be traced by listing their names after +{\tt )trace}. +Any options that are present must follow the functions to be +traced. +\begin{verbatim} +)trace f +\end{verbatim} +traces the function {\tt f}. +To untrace {\tt f}, issue +\begin{verbatim} +)trace f )off +\end{verbatim} +Note that if a function name contains a special character, it will +be necessary to escape the character with an underscore +% +\begin{verbatim} +)trace _/D_,1 +\end{verbatim} +% +To trace all domains or packages that are or will be created from a particular +constructor, give the constructor name or abbreviation after +{\tt )trace}. +% +\begin{verbatim} +)trace MATRIX +)trace List Integer +\end{verbatim} +% +The first command traces all domains currently instantiated with +{\tt Matrix}. +If additional domains are instantiated with this constructor +(for example, if you have used {\tt Matrix(Integer)} and +{\tt Matrix(Float)}), they will be automatically traced. +The second command traces {\tt List(Integer)}. +It is possible to trace individual functions in a domain or +package. +See the {\tt )ops} option below. + +The following are the general options for the {\tt )trace} +command. + +%!! system command parser doesn't treat general s-expressions correctly, +%!! I recommand not documenting )after )before and )cond +\begin{description} +%\item[{\tt )after} {\it S-expression}] +%causes the given Common Lisp {\it S-expression} to be +%executed after exiting the traced function. + +%\item[{\tt )before} {\it S-expression}] +%causes the given Common Lisp {\it S-expression} to be +%executed before entering the traced function. + +\item[{\tt )break after}] +causes a Common Lisp break loop to be entered after +exiting the traced function. + +\item[{\tt )break before}] +causes a Common Lisp break loop to be entered before +entering the traced function. + +\item[{\tt )break}] +is the same as {\tt )break before}. + +%\item[{\tt )cond} {\it S-expression}] +%causes trace information to be shown only if the given +%Common Lisp {\it S-expression} evaluates to non-NIL. For +%example, the following command causes the system function +%{\tt resolveTT} to be traced but to have the information +%displayed only if the value of the variable +%{\tt \$reportBottomUpFlag} is non-NIL. +%\begin{verbatim} +%)trace resolveTT )cond \_\$reportBottomUpFlag} +%\end{verbatim} + +\item[{\tt )count}] +causes the system to keep a count of the number of times the +traced function is entered. The total can be displayed with +{\tt )trace )stats} and cleared with {\tt )trace )stats reset}. + +\item[{\tt )count} {\it n}] +causes information about the traced function to be displayed for +the first {\it n} executions. After the \it n-th execution, the +function is untraced. + +\item[{\tt )depth} {\it n}] +causes trace information to be shown for only {\it n} levels of +recursion of the traced function. The command +\begin{verbatim} +)trace fib )depth 10 +\end{verbatim} +will cause the display of only 10 levels of trace information for +the recursive execution of a user function {\bf fib}. + +\item[{\tt )math}] +causes the function arguments and return value to be displayed in the +Axiom monospace two-dimensional math format. + +\item[{\tt )nonquietly}] +causes the display of additional messages when a function is +traced. + +\item[{\tt )nt}] +This suppresses all normal trace information. This option is +useful if the {\tt )count} or {\tt )timer} options are used and +you are interested in the statistics but not the function calling +information. + +\item[{\tt )off}] +causes untracing of all or specific functions. Without an +argument, all functions, constructors, domains and packages are +untraced. Otherwise, the given functions and other objects +are untraced. To +immediately retrace the untraced functions, issue {\tt )trace +)restore}. + +\item[{\tt )only} {\it listOfDataToDisplay}] +causes only specific trace information to be shown. The items are +listed by using the following abbreviations: +\begin{description} +\item[a] display all arguments +\item[v] display return value +\item[1] display first argument +\item[2] display second argument +\item[15] display the 15th argument, and so on +\end{description} +\end{description} +\begin{description} + +\item[{\tt )restore}] +causes the last untraced functions to be retraced. If additional +options are present, they are added to those previously in effect. + +\item[{\tt )stats}] +causes the display of statistics collected by the use of the +{\tt )count} and {\tt )timer} options. + +\item[{\tt )stats reset}] +resets to 0 the statistics collected by the use of the +{\tt )count} and {\tt )timer} options. + +\item[{\tt )timer}] +causes the system to keep a count of execution times for the +traced function. The total can be displayed with {\tt )trace +)stats} and cleared with {\tt )trace )stats reset}. + +%!! only for lisp, boot, may not work in any case, recommend removing +%\item[{\tt )varbreak}] +%causes a Common Lisp break loop to be entered after +%the assignment to any variable in the traced function. + +\item[{\tt )varbreak} {\it var1 \lanb{}... varN\ranb{}}] +causes a Common Lisp break loop to be entered after +the assignment to any of the listed variables in the traced +function. + +\item[{\tt )vars}] +causes the display of the value of any variable after it is +assigned in the traced function. +Note that library code must +have been compiled (see \ref{ugSysCmdcompile} on +page~\pageref{ugSysCmdcompile}) +using the {\tt )vartrace} option in order +to support this option. + +\item[{\tt )vars} {\it var1 \lanb{}... varN\ranb{}}] +causes the display of the value of any of the specified variables +after they are assigned in the traced function. +Note that library code must +have been compiled (see \ref{ugSysCmdcompile} on +page~\pageref{ugSysCmdcompile}) +using the {\tt )vartrace} option in order +to support this option. + +\item[{\tt )within} {\it executingFunction}] +causes the display of trace information only if the traced +function is called when the given {\it executingFunction} is running. +\end{description} + +The following are the options for tracing constructors, domains +and packages. + +\begin{description} +\item[{\tt )local} {\it \lanb{}op1 \lanb{}... opN\ranb{}\ranb{}}] +causes local functions of the constructor to be traced. Note that +to untrace an individual local function, you must use the fully +qualified internal name, using the escape character +{\tt \_} before the semicolon. +\begin{verbatim} +)trace FRAC )local +)trace FRAC_;cancelGcd )off +\end{verbatim} + +\item[{\tt )ops} {\it op1 \lanb{}... opN\ranb{}}] +By default, all operations from a domain or package are traced +when the domain or package is traced. This option allows you to +specify that only particular operations should be traced. The +command +% +\begin{verbatim} +)trace Integer )ops min max _+ _- +\end{verbatim} +% +traces four operations from the domain {\tt Integer}. Since +{\tt +} and {\tt -} are special +characters, it is necessary +to escape them with an underscore. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )boot} \index{ugSysCmdboot}, +{\tt )lisp} \index{ugSysCmdlisp}, and +{\tt )ltrace} \index{ugSysCmdltrace}. + +\section{)undo} +\index{ugSysCmdundo} + +\index{undo} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )undo} +\item{\tt )undo} {\it integer} +\item{\tt )undo} {\it integer \lanb{}option\ranb{}} +\item{\tt )undo} {\tt )redo} +\end{list} +% +where {\it option} is one of +% +\begin{list}{} +\item{\tt )after} +\item{\tt )before} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to +restore the state of the user environment to an earlier +point in the interactive session. +The argument of an {\tt )undo} is an integer which must designate some +step number in the interactive session. + +\begin{verbatim} +)undo n +)undo n )after +\end{verbatim} +These commands return the state of the interactive +environment to that immediately after step {\tt n}. +If {\tt n} is a positive number, then {\tt n} refers to step nummber +{\tt n}. If {\tt n} is a negative number, it refers to the {\tt n}-th +previous command (that is, undoes the effects of the last $-n$ +commands). + +A {\tt )clear all} resets the {\tt )undo} facility. +Otherwise, an {\tt )undo} undoes the effect of {\tt )clear} with +options {\tt properties}, {\tt value}, and {\tt mode}, and +that of a previous {\tt undo}. +If any such system commands are given between steps $n$ and +$n + 1$ ($n > 0$), their effect is undone +for {\tt )undo m} for any $0 < m \leq n$.. + +The command {\tt )undo} is equivalent to {\tt )undo -1} (it undoes +the effect of the previous user expression). +The command {\tt )undo 0} undoes any of the above system commands +issued since the last user expression. + +\begin{verbatim} +)undo n )before +\end{verbatim} +This command returns the state of the interactive +environment to that immediately before step {\tt n}. +Any {\tt )undo} or {\tt )clear} system commands +given before step {\tt n} will not be undone. + +\begin{verbatim} +)undo )redo +\end{verbatim} +This command reads the file {\tt redo.input}. +created by the last {\tt )undo} command. +This file consists of all user input lines, excluding those +backtracked over due to a previous {\tt )undo}. + +\par\noindent{\bf Also See:} +{\tt )history} \index{ugSysCmdhistory}. +The command {\tt )history )write} will eliminate the ``undone'' command +lines of your program. + +\section{)what} +\index{ugSysCmdwhat} + +\index{what} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )what categories} {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what commands } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what domains } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what operations} {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what packages } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what synonym } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )what things } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\item{\tt )apropos } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to display lists of things in the system. The +patterns are all strings and, if present, restrict the contents of the +lists. Only those items that contain one or more of the strings as +substrings are displayed. For example, +\begin{verbatim} +)what synonym +\end{verbatim} +displays all command synonyms, +\begin{verbatim} +)what synonym ver +\end{verbatim} +displays all command synonyms containing the substring ``{\tt ver}'', +\begin{verbatim} +)what synonym ver pr +\end{verbatim} +displays all command synonyms +containing the substring ``{\tt ver}'' or the substring +``{\tt pr}''. +Output similar to the following will be displayed +\begin{verbatim} +---------------- System Command Synonyms ----------------- + +user-defined synonyms satisfying patterns: + ver pr + + )apr ........................... )what things + )apropos ....................... )what things + )prompt ........................ )set message prompt + )version ....................... )lisp *yearweek* +\end{verbatim} + +Several other things can be listed with the {\tt )what} command: + +\begin{description} +\item[{\tt categories}] displays a list of category constructors. +\index{what categories} +\item[{\tt commands}] displays a list of system commands available at your +user-level. +\index{what commands} +Your user-level +\index{user-level} +is set via the {\tt )set userlevel} command. +\index{set userlevel} +To get a description of a particular command, such as ``{\tt )what}'', issue +{\tt )help what}. +\item[{\tt domains}] displays a list of domain constructors. +\index{what domains} +\item[{\tt operations}] displays a list of operations in the system library. +\index{what operations} +It is recommended that you qualify this command with one or +more patterns, as there are thousands of operations available. For +example, say you are looking for functions that involve computation of +eigenvalues. To find their names, try {\tt )what operations eig}. +A rather large list of operations is loaded into the workspace when +this command is first issued. This list will be deleted when you +clear the workspace via {\tt )clear all} or {\tt )clear completely}. +It will be re-created if it is needed again. +\item[{\tt packages}] displays a list of package constructors. +\index{what packages} +\item[{\tt synonym}] lists system command synonyms. +\index{what synonym} +\item[{\tt things}] displays all of the above types for items containing +\index{what things} +the pattern strings as substrings. +The command synonym {\tt )apropos} is equivalent to +\index{apropos} +{\tt )what things}. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )display} \index{ugSysCmddisplay}, +{\tt )set} \index{ugSysCmdset}, and +{\tt )show} \index{ugSysCmdshow}. + +\setcounter{chapter}{1} % Appendix B + +%\twocolumn[% +\chapter{Categories} +\label{ugAppCategories} + +This is a listing of all categories in the Axiom library at the +time this book was produced. +Use the Browse facility (described in Chapter +\ref{ugBrowse} on page~\pageref{ugBrowse}) +to get more information about these constructors. + +This sample entry will help you read the following table: + +CategoryName{CategoryAbbreviation}:{$\hbox{{\sf Category}}_{1}$% +\ldots$\hbox{{\sf Category}}_{N}$}{\sl with }% +{$\hbox{{\rm op}}_{1}$\ldots$\hbox{{\rm op}}_{M}$} + +where + +\begin{tabular}{@{\quad}ll} +CategoryName & is the full category name, e.g., {\sf CommutativeRing}. \\ +CategoryAbbreviation & is the category abbreviation, e.g., {\sf COMRING}. \\ +$\hbox{{\sf Category}}_{i}$ & is a category to which the category belongs. \\ +$\hbox{{\rm op}}_{j}$ & is an operation exported by the category. +\end{tabular} + +\appendix{Categories} +%tpdclip2 + +% ---------------------------------------------------------------------- +\chapter{constructorListing} +% ---------------------------------------------------------------------- +% +\condata{ABELGRP}{AbelianGroup}{CancellationAbelianMonoid}{{\tt *} {\tt -}} +% +\condata{AMR}{AbelianMonoidRing}{Algebra BiModule CharacteristicNonZero CharacteristicZero CommutativeRing + IntegralDomain Ring}{{\tt /} coefficient degree leadingCoefficient leadingMonomial map monomial monomial? reductum} +% +\condata{ABELMON}{AbelianMonoid}{AbelianSemiGroup}{{\tt *} Zero zero?} +% +\condata{ABELSG}{AbelianSemiGroup}{SetCategory}{{\tt *} {\tt +}} +% +\condata{AGG}{Aggregate}{Object}{{\tt \#} copy empty empty? eq? less? more? size?} +% +\condata{ACF}{AlgebraicallyClosedField}{Field RadicalCategory}{rootOf rootsOf zeroOf zerosOf} +% +\condata{ACFS}{AlgebraicallyClosedFunctionSpace}{AlgebraicallyClosedField FunctionSpace}{rootOf rootsOf zeroOf zerosOf} +% +\condata{ALGEBRA}{Algebra}{Module Ring}{coerce} +% +\condata{AHYP}{ArcHyperbolicFunctionCategory}{}{acosh acoth acsch asech asinh atanh} +% +\condata{ATRIG}{ArcTrigonometricFunctionCategory}{}{acos acot acsc asec asin atan} +% +\condata{ALAGG}{AssociationListAggregate}{ListAggregate TableAggregate}{assoc} +% +\condata{ATTREG}{AttributeRegistry}{}{} +% +\condata{BGAGG}{BagAggregate}{HomogeneousAggregate}{bag extract! insert! inspect} +% +\condata{BMODULE}{BiModule}{LeftModule RightModule}{} +% +\condata{BRAGG}{BinaryRecursiveAggregate}{RecursiveAggregate}{elt left right setelt setleft! setright!} +% +\condata{BTCAT}{BinaryTreeCategory}{BinaryRecursiveAggregate}{node} +% +\condata{BTAGG}{BitAggregate}{OneDimensionalArrayAggregate OrderedSet}{{\tt \^{}} and nand nor not or xor} +% +\condata{CACHSET}{CachableSet}{OrderedSet}{position setPosition} +% +\condata{CABMON}{CancellationAbelianMonoid}{AbelianMonoid}{{\tt -}} +% +\condata{CHARNZ}{CharacteristicNonZero}{Ring}{charthRoot} +% +\condata{CHARZ}{CharacteristicZero}{Ring}{} +% +\condata{KOERCE}{CoercibleTo}{}{coerce} +% +\condata{CLAGG}{Collection}{ConvertibleTo HomogeneousAggregate}{construct find reduce remove removeDuplicates select} +% +\condata{CFCAT}{CombinatorialFunctionCategory}{}{binomial factorial permutation} +% +\condata{COMBOPC}{CombinatorialOpsCategory}{CombinatorialFunctionCategory}{factorials product summation} +% +\condata{COMRING}{CommutativeRing}{BiModule Ring}{} +% +\condata{COMPCAT}{ComplexCategory}{CharacteristicNonZero CharacteristicZero CommutativeRing ConvertibleTo + DifferentialExtension EuclideanDomain Field FullyEvalableOver FullyLinearlyExplicitRingOver FullyRetractableTo + IntegralDomain MonogenicAlgebra OrderedSet PolynomialFactorizationExplicit RadicalCategory + TranscendentalFunctionCategory}{abs argument complex conjugate exquo imag imaginary norm polarCoordinates rational + rational? rationalIfCan real} +% +\condata{KONVERT}{ConvertibleTo}{}{convert} +% +\condata{DQAGG}{DequeueAggregate}{QueueAggregate StackAggregate}{bottom! dequeue extractBottom! extractTop! height + insertBottom! insertTop! reverse! top!} +% +\condata{DIOPS}{DictionaryOperations}{BagAggregate Collection}{dictionary remove! select!} +% +\condata{DIAGG}{Dictionary}{DictionaryOperations}{} +% +\condata{DIFEXT}{DifferentialExtension}{DifferentialRing PartialDifferentialRing Ring}{D differentiate} +% +\condata{DPOLCAT}{DifferentialPolynomialCategory}{DifferentialExtension Evalable InnerEvalable PolynomialCategory + RetractableTo}{degree differentialVariables initial isobaric? leader makeVariable order separant weight weights} +% +\condata{DIFRING}{DifferentialRing}{Ring}{D differentiate} +% +\condata{DVARCAT}{DifferentialVariableCategory}{OrderedSet RetractableTo}{D coerce differentiate makeVariable order + variable weight} +% +\condata{DIRPCAT}{DirectProductCategory}{AbelianSemiGroup Algebra BiModule CancellationAbelianMonoid CoercibleTo + CommutativeRing DifferentialExtension Finite FullyLinearlyExplicitRingOver FullyRetractableTo IndexedAggregate + OrderedAbelianMonoidSup OrderedRing VectorSpace}{{\tt *} directProduct dot unitVector} +% +\condata{DIVRING}{DivisionRing}{Algebra EntireRing}{{\tt **} inv} +% +\condata{DLAGG}{DoublyLinkedAggregate}{RecursiveAggregate}{concat! head last next previous setnext! setprevious! tail} +% +\condata{ELEMFUN}{ElementaryFunctionCategory}{}{{\tt **} exp log} +% +\condata{ELTAGG}{EltableAggregate}{Eltable}{elt qelt qsetelt! setelt} +% +\condata{ELTAB}{Eltable}{}{elt} +% +\condata{ENTIRER}{EntireRing}{BiModule Ring}{} +% +\condata{EUCDOM}{EuclideanDomain}{PrincipalIdealDomain}{divide euclideanSize extendedEuclidean multiEuclidean quo rem + sizeLess?} +% +\condata{EVALAB}{Evalable}{}{eval} +% +\condata{ES}{ExpressionSpace}{Evalable InnerEvalable OrderedSet RetractableTo}{belong? box definingPolynomial + distribute elt eval freeOf? height is? kernel kernels mainKernel map minPoly operator operators paren subst tower} +% +\condata{ELAGG}{ExtensibleLinearAggregate}{LinearAggregate}{concat! delete! insert! merge! remove! removeDuplicates! + select!} +% +\condata{XF}{ExtensionField}{CharacteristicZero Field FieldOfPrimeCharacteristic RetractableTo VectorSpace}{Frobenius + algebraic? degree extensionDegree inGroundField? transcendenceDegree transcendent?} +% +\condata{FPC}{FieldOfPrimeCharacteristic}{CharacteristicNonZero Field}{discreteLog order primeFrobenius} +% +\condata{FIELD}{Field}{DivisionRing EuclideanDomain UniqueFactorizationDomain}{{\tt /}} +% +\condata{FILECAT}{FileCategory}{SetCategory}{close! iomode name open read! reopen! write!} +% +\condata{FNCAT}{FileNameCategory}{SetCategory}{coerce directory exists? extension filename name new readable? writable?} +% +\condata{FAMR}{FiniteAbelianMonoidRing}{AbelianMonoidRing FullyRetractableTo}{coefficients content exquo ground ground? + mapExponents minimumDegree numberOfMonomials primitivePart} +% +\condata{FAXF}{FiniteAlgebraicExtensionField}{ExtensionField FiniteFieldCategory RetractableTo}{basis coordinates + createNormalElement definingPolynomial degree extensionDegree generator minimalPolynomial norm normal? normalElement + represents trace} +% +\condata{FFIELDC}{FiniteFieldCategory}{FieldOfPrimeCharacteristic Finite StepThrough}{charthRoot conditionP + createPrimitiveElement discreteLog factorsOfCyclicGroupSize order primitive? primitiveElement representationType + tableForDiscreteLogarithm} +% +\condata{FLAGG}{FiniteLinearAggregate}{LinearAggregate OrderedSet}{copyInto! merge position reverse reverse! sort sort! + sorted?} +% +\condata{FINRALG}{FiniteRankAlgebra}{Algebra CharacteristicNonZero CharacteristicZero}{characteristicPolynomial + coordinates discriminant minimalPolynomial norm rank regularRepresentation represents trace traceMatrix} +% +\condata{FINAALG}{FiniteRankNonAssociativeAlgebra}{NonAssociativeAlgebra}{JacobiIdentity? JordanAlgebra? alternative? + antiAssociative? antiCommutative? associative? associatorDependence commutative? conditionsForIdempotents coordinates + flexible? jordanAdmissible? leftAlternative? leftCharacteristicPolynomial leftDiscriminant leftMinimalPolynomial + leftNorm leftRecip leftRegularRepresentation leftTrace leftTraceMatrix leftUnit leftUnits lieAdmissible? lieAlgebra? + noncommutativeJordanAlgebra? powerAssociative? rank recip represents rightAlternative? rightCharacteristicPolynomial + rightDiscriminant rightMinimalPolynomial rightNorm rightRecip rightRegularRepresentation rightTrace rightTraceMatrix + rightUnit rightUnits someBasis structuralConstants unit} +% +\condata{FSAGG}{FiniteSetAggregate}{Dictionary Finite SetAggregate}{cardinality complement max min universe} +% +\condata{FINITE}{Finite}{SetCategory}{index lookup random size} +% +\condata{FPS}{FloatingPointSystem}{RealNumberSystem}{base bits decreasePrecision digits exponent float + increasePrecision mantissa max order precision} +% +\condata{FRAMALG}{FramedAlgebra}{FiniteRankAlgebra}{basis convert coordinates discriminant regularRepresentation + represents traceMatrix} +% +\condata{FRNAALG}{FramedNonAssociativeAlgebra}{FiniteRankNonAssociativeAlgebra}{apply basis conditionsForIdempotents + convert coordinates elt leftDiscriminant leftRankPolynomial leftRegularRepresentation leftTraceMatrix represents + rightDiscriminant rightRankPolynomial rightRegularRepresentation rightTraceMatrix structuralConstants} +% +\condata{FAMONC}{FreeAbelianMonoidCategory}{CancellationAbelianMonoid RetractableTo}{{\tt *} {\tt +} coefficient + highCommonTerms mapCoef mapGen nthCoef nthFactor size terms} +% +\condata{FEVALAB}{FullyEvalableOver}{Eltable Evalable InnerEvalable}{map} +% +\condata{FLINEXP}{FullyLinearlyExplicitRingOver}{LinearlyExplicitRingOver}{} +% +\condata{FPATMAB}{FullyPatternMatchable}{Object PatternMatchable}{} +% +\condata{FRETRCT}{FullyRetractableTo}{RetractableTo}{} +% +\condata{FFCAT}{FunctionFieldCategory}{MonogenicAlgebra}{D absolutelyIrreducible? branchPoint? branchPointAtInfinity? + complementaryBasis differentiate elt genus integral? integralAtInfinity? integralBasis integralBasisAtInfinity + integralCoordinates integralDerivationMatrix integralMatrix integralMatrixAtInfinity integralRepresents + inverseIntegralMatrix inverseIntegralMatrixAtInfinity nonSingularModel normalizeAtInfinity numberOfComponents + primitivePart ramified? ramifiedAtInfinity? rationalPoint? rationalPoints reduceBasisAtInfinity represents singular? + singularAtInfinity? yCoordinates} +% +\condata{FS}{FunctionSpace}{AbelianGroup AbelianMonoid Algebra CharacteristicNonZero CharacteristicZero ConvertibleTo + ExpressionSpace Field FullyLinearlyExplicitRingOver FullyPatternMatchable FullyRetractableTo Group Monoid + PartialDifferentialRing Patternable RetractableTo Ring}{{\tt **} {\tt /} applyQuote coerce convert denom denominator + eval ground ground? isExpt isMult isPlus isPower isTimes numer numerator univariate variables} +% +\condata{GCDDOM}{GcdDomain}{IntegralDomain}{gcd lcm} +% +\condata{GRALG}{GradedAlgebra}{GradedModule}{One product} +% +\condata{GRMOD}{GradedModule}{RetractableTo SetCategory}{{\tt *} {\tt +} {\tt -} Zero degree} +% +\condata{GROUP}{Group}{Monoid}{{\tt **} {\tt /} commutator conjugate inv} +% +\condata{HOAGG}{HomogeneousAggregate}{Aggregate SetCategory}{any? count every? map map! member? members parts} +% +\condata{HYPCAT}{HyperbolicFunctionCategory}{}{cosh coth csch sech sinh tanh} +% +\condata{IXAGG}{IndexedAggregate}{EltableAggregate HomogeneousAggregate}{entries entry? fill! first index? indices + maxIndex minIndex swap!} +% +\condata{IDPC}{IndexedDirectProductCategory}{SetCategory}{leadingCoefficient leadingSupport map monomial reductum} +% +\condata{IEVALAB}{InnerEvalable}{}{eval} +% +\condata{INS}{IntegerNumberSystem}{CharacteristicZero CombinatorialFunctionCategory ConvertibleTo DifferentialRing + EuclideanDomain LinearlyExplicitRingOver OrderedRing PatternMatchable RealConstant RetractableTo StepThrough + UniqueFactorizationDomain}{addmod base bit? copy dec even? hash inc invmod length mask mulmod odd? positiveRemainder + powmod random rational rational? rationalIfCan shift submod symmetricRemainder} +% +\condata{INTDOM}{IntegralDomain}{Algebra CommutativeRing EntireRing}{associates? exquo unit? unitCanonical unitNormal} +% +\condata{KDAGG}{KeyedDictionary}{Dictionary}{key? keys remove! search} +% +\condata{LZSTAGG}{LazyStreamAggregate}{StreamAggregate}{complete explicitEntries? explicitlyEmpty? extend frst lazy? + lazyEvaluate numberOfComputedEntries remove rst select} +% +\condata{LALG}{LeftAlgebra}{LeftModule Ring}{coerce} +% +\condata{LMODULE}{LeftModule}{AbelianGroup}{{\tt *}} +% +\condata{LNAGG}{LinearAggregate}{Collection IndexedAggregate}{concat delete elt insert map new setelt} +% +\condata{LINEXP}{LinearlyExplicitRingOver}{Ring}{reducedSystem} +% +\condata{LFCAT}{LiouvillianFunctionCategory}{PrimitiveFunctionCategory TranscendentalFunctionCategory}{Ci Ei Si dilog + erf li} +% +\condata{LSAGG}{ListAggregate}{ExtensibleLinearAggregate FiniteLinearAggregate StreamAggregate}{list} +% +\condata{MATCAT}{MatrixCategory}{TwoDimensionalArrayCategory}{{\tt *} {\tt **} {\tt +} {\tt -} {\tt /} antisymmetric? + coerce determinant diagonal? diagonalMatrix elt exquo horizConcat inverse listOfLists matrix minordet nullSpace nullity + rank rowEchelon scalarMatrix setelt setsubMatrix! square? squareTop subMatrix swapColumns! swapRows! symmetric? + transpose vertConcat zero} +% +\condata{MODULE}{Module}{BiModule}{} +% +\condata{MONADWU}{MonadWithUnit}{Monad}{{\tt **} One leftPower leftRecip one? recip rightPower rightRecip} +% +\condata{MONAD}{Monad}{SetCategory}{{\tt *} {\tt **} leftPower rightPower} +% +\condata{MONOGEN}{MonogenicAlgebra}{CommutativeRing ConvertibleTo DifferentialExtension Field Finite + FiniteFieldCategory FramedAlgebra FullyLinearlyExplicitRingOver FullyRetractableTo}{convert definingPolynomial + derivationCoordinates generator lift reduce} +% +\condata{MLO}{MonogenicLinearOperator}{Algebra BiModule Ring}{coefficient degree leadingCoefficient minimumDegree + monomial reductum} +% +\condata{MONOID}{Monoid}{SemiGroup}{{\tt **} One one? recip} +% +\condata{MDAGG}{MultiDictionary}{DictionaryOperations}{duplicates insert! removeDuplicates!} +% +\condata{MSAGG}{MultisetAggregate}{MultiDictionary SetAggregate}{} +% +\condata{MTSCAT}{MultivariateTaylorSeriesCategory}{Evalable InnerEvalable PartialDifferentialRing PowerSeriesCategory + RadicalCategory TranscendentalFunctionCategory}{coefficient extend integrate monomial order polynomial} +% +\condata{NAALG}{NonAssociativeAlgebra}{Module NonAssociativeRng}{plenaryPower} +% +\condata{NASRING}{NonAssociativeRing}{MonadWithUnit NonAssociativeRng}{characteristic coerce} +% +\condata{NARNG}{NonAssociativeRng}{AbelianGroup Monad}{antiCommutator associator commutator} +% +\condata{OBJECT}{Object}{}{} +% +\condata{OC}{OctonionCategory}{Algebra CharacteristicNonZero CharacteristicZero ConvertibleTo Finite FullyEvalableOver + FullyRetractableTo OrderedSet}{abs conjugate imagE imagI imagJ imagK imagi imagj imagk inv norm octon rational + rational? rationalIfCan real} +% +\condata{A1AGG}{OneDimensionalArrayAggregate}{FiniteLinearAggregate}{} +% +\condata{OAGROUP}{OrderedAbelianGroup}{AbelianGroup OrderedCancellationAbelianMonoid}{} +% +\condata{OAMONS}{OrderedAbelianMonoidSup}{OrderedCancellationAbelianMonoid}{sup} +% +\condata{OAMON}{OrderedAbelianMonoid}{AbelianMonoid OrderedAbelianSemiGroup}{} +% +\condata{OASGP}{OrderedAbelianSemiGroup}{AbelianMonoid OrderedSet}{} +% +\condata{OCAMON}{OrderedCancellationAbelianMonoid}{CancellationAbelianMonoid OrderedAbelianMonoid}{} +% +\condata{ORDFIN}{OrderedFinite}{Finite OrderedSet}{} +% +\condata{ORDMON}{OrderedMonoid}{Monoid OrderedSet}{} +% +\condata{OMAGG}{OrderedMultisetAggregate}{MultisetAggregate PriorityQueueAggregate}{min} +% +\condata{ORDRING}{OrderedRing}{OrderedAbelianGroup OrderedMonoid Ring}{abs negative? positive? sign} +% +\condata{ORDSET}{OrderedSet}{SetCategory}{{\tt <} max min} +% +\condata{PADICCT}{PAdicIntegerCategory}{CharacteristicZero EuclideanDomain}{approximate complete digits extend moduloP + modulus order quotientByP sqrt} +% +\condata{PDRING}{PartialDifferentialRing}{Ring}{D differentiate} +% +\condata{PTRANFN}{PartialTranscendentalFunctions}{}{acosIfCan acoshIfCan acotIfCan acothIfCan acscIfCan acschIfCan + asecIfCan asechIfCan asinIfCan asinhIfCan atanIfCan atanhIfCan cosIfCan coshIfCan cotIfCan cothIfCan cscIfCan cschIfCan + expIfCan logIfCan nthRootIfCan secIfCan sechIfCan sinIfCan sinhIfCan tanIfCan tanhIfCan} +% +\condata{PATAB}{Patternable}{ConvertibleTo Object}{} +% +\condata{PATMAB}{PatternMatchable}{SetCategory}{patternMatch} +% +\condata{PERMCAT}{PermutationCategory}{Group OrderedSet}{{\tt <} cycle cycles elt eval orbit} +% +\condata{PPCURVE}{PlottablePlaneCurveCategory}{CoercibleTo}{listBranches xRange yRange} +% +\condata{PSCURVE}{PlottableSpaceCurveCategory}{CoercibleTo}{listBranches xRange yRange zRange} +% +\condata{PTCAT}{PointCategory}{VectorCategory}{convert cross dimension extend length point} +% +\condata{POLYCAT}{PolynomialCategory}{ConvertibleTo Evalable FiniteAbelianMonoidRing FullyLinearlyExplicitRingOver + GcdDomain InnerEvalable OrderedSet PartialDifferentialRing PatternMatchable PolynomialFactorizationExplicit + RetractableTo}{coefficient content degree discriminant isExpt isPlus isTimes mainVariable minimumDegree monicDivide + monomial monomials multivariate primitiveMonomials primitivePart resultant squareFree squareFreePart totalDegree + univariate variables} +% +\condata{PFECAT}{PolynomialFactorizationExplicit}{UniqueFactorizationDomain}{charthRoot conditionP factorPolynomial + factorSquareFreePolynomial gcdPolynomial solveLinearPolynomialEquation squareFreePolynomial} +% +\condata{PSCAT}{PowerSeriesCategory}{AbelianMonoidRing}{complete monomial pole? variables} +% +\condata{PRIMCAT}{PrimitiveFunctionCategory}{}{integral} +% +\condata{PID}{PrincipalIdealDomain}{GcdDomain}{expressIdealMember principalIdeal} +% +\condata{PRQAGG}{PriorityQueueAggregate}{BagAggregate}{max merge merge!} +% +\condata{QUATCAT}{QuaternionCategory}{Algebra CharacteristicNonZero CharacteristicZero ConvertibleTo + DifferentialExtension DivisionRing EntireRing FullyEvalableOver FullyLinearlyExplicitRingOver FullyRetractableTo + OrderedSet}{abs conjugate imagI imagJ imagK norm quatern rational rational? rationalIfCan real} +% +\condata{QUAGG}{QueueAggregate}{BagAggregate}{back dequeue! enqueue! front length rotate!} +% +\condata{QFCAT}{QuotientFieldCategory}{Algebra CharacteristicNonZero CharacteristicZero ConvertibleTo + DifferentialExtension Field FullyEvalableOver FullyLinearlyExplicitRingOver FullyPatternMatchable OrderedRing + OrderedSet Patternable PolynomialFactorizationExplicit RealConstant RetractableTo StepThrough}{{\tt /} ceiling denom + denominator floor fractionPart numer numerator random wholePart} +% +\condata{RADCAT}{RadicalCategory}{}{{\tt **} nthRoot sqrt} +% +\condata{REAL}{RealConstant}{ConvertibleTo}{} +% +\condata{RNS}{RealNumberSystem}{CharacteristicZero ConvertibleTo Field OrderedRing PatternMatchable RadicalCategory + RealConstant RetractableTo}{abs ceiling floor fractionPart norm round truncate wholePart} +% +\condata{RMATCAT}{RectangularMatrixCategory}{BiModule HomogeneousAggregate Module}{{\tt /} antisymmetric? column + diagonal? elt exquo listOfLists map matrix maxColIndex maxRowIndex minColIndex minRowIndex ncols nrows nullSpace + nullity qelt rank row rowEchelon square? symmetric?} +% +\condata{RCAGG}{RecursiveAggregate}{HomogeneousAggregate}{children cyclic? elt leaf? leaves node? nodes setchildren! + setelt setvalue! value} +% +\condata{RETRACT}{RetractableTo}{}{coerce retract retractIfCan} +% +\condata{RMODULE}{RightModule}{AbelianGroup}{{\tt *}} +% +\condata{RING}{Ring}{LeftModule Monoid Rng}{characteristic coerce} +% +\condata{RNG}{Rng}{AbelianGroup SemiGroup}{} +% +\condata{SEGCAT}{SegmentCategory}{SetCategory}{BY SEGMENT convert hi high incr lo low segment} +% +\condata{SEGXCAT}{SegmentExpansionCategory}{SegmentCategory}{expand map} +% +\condata{SGROUP}{SemiGroup}{SetCategory}{{\tt *} {\tt **}} +% +\condata{SETAGG}{SetAggregate}{Collection SetCategory}{{\tt <} brace difference intersect subset? symmetricDifference + union} +% +\condata{SETCAT}{SetCategory}{CoercibleTo Object}{{\tt =}} +% +\condata{SEXCAT}{SExpressionCategory}{SetCategory}{{\tt \#} atom? car cdr convert destruct elt eq expr float float? + integer integer? list? null? pair? string string? symbol symbol? uequal} +% +\condata{SPFCAT}{SpecialFunctionCategory}{}{Beta Gamma abs airyAi airyBi besselI besselJ besselK besselY digamma + polygamma} +% +\condata{SMATCAT}{SquareMatrixCategory}{Algebra BiModule DifferentialExtension FullyLinearlyExplicitRingOver + FullyRetractableTo Module RectangularMatrixCategory}{{\tt *} {\tt **} determinant diagonal diagonalMatrix + diagonalProduct inverse minordet scalarMatrix trace} +% +\condata{SKAGG}{StackAggregate}{BagAggregate}{depth pop! push! top} +% +\condata{STEP}{StepThrough}{SetCategory}{init nextItem} +% +\condata{STAGG}{StreamAggregate}{LinearAggregate UnaryRecursiveAggregate}{explicitlyFinite? possiblyInfinite?} +% +\condata{SRAGG}{StringAggregate}{OneDimensionalArrayAggregate}{coerce elt leftTrim lowerCase lowerCase! match match? + position prefix? replace rightTrim split substring? suffix? trim upperCase upperCase!} +% +\condata{STRICAT}{StringCategory}{StringAggregate}{string} +% +\condata{TBAGG}{TableAggregate}{IndexedAggregate KeyedDictionary}{map setelt table} +% +\condata{SPACEC}{ThreeSpaceCategory}{SetCategory}{check closedCurve closedCurve? coerce components composite composites + copy create3Space curve curve? enterPointData lllip lllp llprop lp lprop merge mesh mesh? modifyPointData + numberOfComponents numberOfComposites objects point point? polygon polygon? subspace} +% +\condata{TRANFUN}{TranscendentalFunctionCategory}{ArcHyperbolicFunctionCategory ArcTrigonometricFunctionCategory + ElementaryFunctionCategory HyperbolicFunctionCategory TrigonometricFunctionCategory}{pi} +% +\condata{TRIGCAT}{TrigonometricFunctionCategory}{}{cos cot csc sec sin tan} +% +\condata{ARR2CAT}{TwoDimensionalArrayCategory}{HomogeneousAggregate}{column elt fill! map map! maxColIndex maxRowIndex + minColIndex minRowIndex ncols new nrows parts qelt qsetelt! row setColumn! setRow! setelt} +% +\condata{URAGG}{UnaryRecursiveAggregate}{RecursiveAggregate}{concat concat! cycleEntry cycleLength cycleSplit! + cycleTail elt first last rest second setelt setfirst! setlast! setrest! split! tail third} +% +\condata{UFD}{UniqueFactorizationDomain}{GcdDomain}{factor prime? squareFree squareFreePart} +% +\condata{ULSCAT}{UnivariateLaurentSeriesCategory}{Field RadicalCategory TranscendentalFunctionCategory + UnivariatePowerSeriesCategory}{integrate multiplyCoefficients rationalFunction} +% +\condata{ULSCCAT}{UnivariateLaurentSeriesConstructorCategory}{QuotientFieldCategory RetractableTo + UnivariateLaurentSeriesCategory}{coerce degree laurent removeZeroes taylor taylorIfCan taylorRep} +% +\condata{UPOLYC}{UnivariatePolynomialCategory}{DifferentialExtension DifferentialRing Eltable EuclideanDomain + PolynomialCategory StepThrough}{D composite differentiate discriminant divideExponents elt integrate makeSUP + monicDivide multiplyExponents order pseudoDivide pseudoQuotient pseudoRemainder resultant separate subResultantGcd + unmakeSUP vectorise} +% +\condata{UPSCAT}{UnivariatePowerSeriesCategory}{DifferentialRing Eltable PowerSeriesCategory}{approximate center elt + eval extend multiplyExponents order series terms truncate variable} +% +\condata{UPXSCAT}{UnivariatePuiseuxSeriesCategory}{Field RadicalCategory TranscendentalFunctionCategory + UnivariatePowerSeriesCategory}{integrate multiplyExponents} +% +\condata{UPXSCCA}{UnivariatePuiseuxSeriesConstructorCategory}{RetractableTo UnivariatePuiseuxSeriesCategory}{coerce + degree laurent laurentIfCan laurentRep puiseux rationalPower} +% +\condata{UTSCAT}{UnivariateTaylorSeriesCategory}{RadicalCategory TranscendentalFunctionCategory + UnivariatePowerSeriesCategory}{{\tt **} coefficients integrate multiplyCoefficients polynomial quoByVar series} +% +\condata{VECTCAT}{VectorCategory}{OneDimensionalArrayAggregate}{{\tt *} {\tt +} {\tt -} dot zero} +% +\condata{VSPACE}{VectorSpace}{Module}{{\tt /} dimension} +% +% +% ---------------------------------------------------------------------- + + + +\setcounter{chapter}{2} % Appendix C + +%\twocolumn[% + +\chapter{Domains} +\label{ugAppDomains} + +This is a listing of all domains in the Axiom library at the +time this book was produced. +Use the Browse facility (described in Chapter +\ref{ugBrowse} on page~\pageref{ugBrowse}) +to get more information about these constructors. + +This sample entry will help you read the following table: + +DomainName{DomainAbbreviation}:{$\hbox{{\sf Category}}_{1}$% +\ldots$\hbox{{\sf Category}}_{N}$}{\sl with }% +{$\hbox{{\rm op}}_{1}$\ldots$\hbox{{\rm op}}_{M}$} + +where + +\begin{tabular}{@{\quad}ll} +DomainName & is the full domain name, e.g., {\sf Integer}. \\ +DomainAbbreviation & is the domain abbreviation, e.g., {\sf INT}. \\ +$\hbox{{\sf Category}}_{i}$ & is a category to which the domain belongs. \\ +$\hbox{{\rm op}}_{j}$ & is an operation exported by the domain. +\end{tabular} + +\appendix{Domains} + +% ---------------------------------------------------------------------- +%\begin{constructorListing} +% ---------------------------------------------------------------------- +\condata{ALGSC}{AlgebraGivenByStructuralConstants}{FramedNonAssociativeAlgebra LeftModule}{0 {\tt *} {\tt **} {\tt +} + {\tt -} {\tt =} JacobiIdentity? JordanAlgebra? alternative? antiAssociative? antiCommutative? antiCommutator apply + associative? associator associatorDependence basis coerce commutative? commutator conditionsForIdempotents convert + coordinates elt flexible? jordanAdmissible? leftAlternative? leftCharacteristicPolynomial leftDiscriminant + leftMinimalPolynomial leftNorm leftPower leftRankPolynomial leftRecip leftRegularRepresentation leftTrace + leftTraceMatrix leftUnit leftUnits lieAdmissible? lieAlgebra? noncommutativeJordanAlgebra? plenaryPower + powerAssociative? rank recip represents rightAlternative? rightCharacteristicPolynomial rightDiscriminant + rightMinimalPolynomial rightNorm rightPower rightRankPolynomial rightRecip rightRegularRepresentation rightTrace + rightTraceMatrix rightUnit rightUnits someBasis structuralConstants unit zero?} +% +\condata{ALGFF}{AlgebraicFunctionField}{FunctionFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + absolutelyIrreducible? associates? basis branchPoint? branchPointAtInfinity? characteristic characteristicPolynomial + charthRoot coerce complementaryBasis convert coordinates definingPolynomial derivationCoordinates differentiate + discriminant divide elt euclideanSize expressIdealMember exquo extendedEuclidean factor gcd generator genus integral? + integralAtInfinity? integralBasis integralBasisAtInfinity integralCoordinates integralDerivationMatrix integralMatrix + integralMatrixAtInfinity integralRepresents inv inverseIntegralMatrix inverseIntegralMatrixAtInfinity knownInfBasis lcm + lift minimalPolynomial multiEuclidean nonSingularModel norm normalizeAtInfinity numberOfComponents one? prime? + primitivePart principalIdeal quo ramified? ramifiedAtInfinity? rank rationalPoint? rationalPoints recip reduce + reduceBasisAtInfinity reducedSystem regularRepresentation rem represents retract retractIfCan singular? + singularAtInfinity? sizeLess? squareFree squareFreePart trace traceMatrix unit? unitCanonical unitNormal yCoordinates + zero?} +% +\condata{AN}{AlgebraicNumber}{AlgebraicallyClosedField CharacteristicZero ConvertibleTo DifferentialRing + ExpressionSpace LinearlyExplicitRingOver RealConstant RetractableTo}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + <} {\tt =} D associates? belong? box characteristic coerce convert definingPolynomial denom differentiate distribute + divide elt euclideanSize eval expressIdealMember exquo extendedEuclidean factor freeOf? gcd height inv is? kernel + kernels lcm mainKernel map max min minPoly multiEuclidean nthRoot numer one? operator operators paren prime? + principalIdeal quo recip reduce reducedSystem rem retract retractIfCan rootOf rootsOf sizeLess? sqrt squareFree + squareFreePart subst tower unit? unitCanonical unitNormal zero? zeroOf zerosOf} +% +\condata{ANON}{AnonymousFunction}{SetCategory}{{\tt =} coerce} +% +\condata{ANTISYM}{AntiSymm}{LeftAlgebra RetractableTo}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} characteristic + coefficient coerce degree exp generator homogeneous? leadingBasisTerm leadingCoefficient map one? recip reductum + retract retractIfCan retractable? zero?} +% +\condata{ANY}{Any}{SetCategory}{{\tt =} any coerce domain domainOf obj objectOf showTypeInOutput} +% +\condata{ASTACK}{ArrayStack}{StackAggregate}{{\tt \#} {\tt =} any? arrayStack bag coerce copy count depth empty empty? + eq? every? extract! insert! inspect less? map map! member? members more? parts pop! push! size? top} +% +\condata{JORDAN}{AssociatedJordanAlgebra}{CoercibleTo FiniteRankNonAssociativeAlgebra FramedNonAssociativeAlgebra + NonAssociativeAlgebra}{0 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} JacobiIdentity? JordanAlgebra? alternative? + antiAssociative? antiCommutative? antiCommutator apply associative? associator associatorDependence basis coerce + commutative? commutator conditionsForIdempotents convert coordinates elt flexible? jordanAdmissible? leftAlternative? + leftCharacteristicPolynomial leftDiscriminant leftMinimalPolynomial leftNorm leftPower leftRankPolynomial leftRecip + leftRegularRepresentation leftTrace leftTraceMatrix leftUnit leftUnits lieAdmissible? lieAlgebra? + noncommutativeJordanAlgebra? plenaryPower powerAssociative? rank recip represents rightAlternative? + rightCharacteristicPolynomial rightDiscriminant rightMinimalPolynomial rightNorm rightPower rightRankPolynomial + rightRecip rightRegularRepresentation rightTrace rightTraceMatrix rightUnit rightUnits someBasis structuralConstants + unit zero?} +% +\condata{LIE}{AssociatedLieAlgebra}{CoercibleTo FiniteRankNonAssociativeAlgebra FramedNonAssociativeAlgebra + NonAssociativeAlgebra}{0 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} JacobiIdentity? JordanAlgebra? alternative? + antiAssociative? antiCommutative? antiCommutator apply associative? associator associatorDependence basis coerce + commutative? commutator conditionsForIdempotents convert coordinates elt flexible? jordanAdmissible? leftAlternative? + leftCharacteristicPolynomial leftDiscriminant leftMinimalPolynomial leftNorm leftPower leftRankPolynomial leftRecip + leftRegularRepresentation leftTrace leftTraceMatrix leftUnit leftUnits lieAdmissible? lieAlgebra? + noncommutativeJordanAlgebra? plenaryPower powerAssociative? rank recip represents rightAlternative? + rightCharacteristicPolynomial rightDiscriminant rightMinimalPolynomial rightNorm rightPower rightRankPolynomial + rightRecip rightRegularRepresentation rightTrace rightTraceMatrix rightUnit rightUnits someBasis structuralConstants + unit zero?} +% +\condata{ALIST}{AssociationList}{AssociationListAggregate}{{\tt \#} {\tt =} any? assoc bag child? children coerce + concat concat! construct copy copyInto! count cycleEntry cycleLength cycleSplit! cycleTail cyclic? delete delete! + dictionary distance elt empty empty? entries entry? eq? every? explicitlyFinite? extract! fill! find first index? + indices insert insert! inspect key? keys last leaf? less? list map map! maxIndex member? members merge merge! minIndex + more? new node? nodes parts position possiblyInfinite? qelt qsetelt! reduce remove remove! removeDuplicates + removeDuplicates! rest reverse reverse! search second select select! setchildren! setelt setfirst! setlast! setrest! + setvalue! size? sort sort! sorted? split! swap! table tail third value} +% +\condata{BBTREE}{BalancedBinaryTree}{BinaryTreeCategory}{{\tt \#} {\tt =} any? balancedBinaryTree children coerce copy + count cyclic? elt empty empty? eq? every? leaf? leaves left less? map map! mapDown! mapUp! member? members more? node + node? nodes parts right setchildren! setelt setleaves! setleft! setright! setvalue! size? value} +% +\condata{BPADIC}{BalancedPAdicInteger}{PAdicIntegerCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} approximate + associates? characteristic coerce complete digits divide euclideanSize expressIdealMember exquo extend + extendedEuclidean gcd lcm moduloP modulus multiEuclidean one? order principalIdeal quo quotientByP recip rem sizeLess? + sqrt unit? unitCanonical unitNormal zero?} +% +\condata{BPADICRT}{BalancedPAdicRational}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + approximate associates? characteristic coerce continuedFraction denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor fractionPart gcd inv lcm map multiEuclidean numer numerator one? + prime? principalIdeal quo recip reducedSystem rem removeZeroes retract retractIfCan sizeLess? squareFree squareFreePart + unit? unitCanonical unitNormal wholePart zero?} +% +\condata{BOP}{BasicOperator}{OrderedSet}{{\tt <} {\tt =} arity assert coerce comparison copy deleteProperty! display + equality has? input is? max min name nary? nullary? operator properties property setProperties setProperty unary? + weight} +% +\condata{BINARY}{BinaryExpansion}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D + abs associates? binary ceiling characteristic coerce convert denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor floor fractionPart gcd init inv lcm map max min multiEuclidean + negative? nextItem numer numerator one? patternMatch positive? prime? principalIdeal quo random recip reducedSystem rem + retract retractIfCan sign sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal wholePart zero?} +% +\condata{BSTREE}{BinarySearchTree}{BinaryTreeCategory}{{\tt \#} {\tt =} any? binarySearchTree children coerce copy + count cyclic? elt empty empty? eq? every? insert! insertRoot! leaf? leaves left less? map map! member? members more? + node node? nodes parts right setchildren! setelt setleft! setright! setvalue! size? split value} +% +\condata{BTOURN}{BinaryTournament}{BinaryTreeCategory}{{\tt \#} {\tt =} any? binaryTournament children coerce copy + count cyclic? elt empty empty? eq? every? insert! leaf? leaves left less? map map! member? members more? node node? + nodes parts right setchildren! setelt setleft! setright! setvalue! size? value} +% +\condata{BTREE}{BinaryTree}{BinaryTreeCategory}{{\tt \#} {\tt =} any? binaryTree children coerce copy count cyclic? elt + empty empty? eq? every? leaf? leaves left less? map map! member? members more? node node? nodes parts right + setchildren! setelt setleft! setright! setvalue! size? value} +% +\condata{BITS}{Bits}{BitAggregate}{{\tt \#} {\tt <} {\tt =} {\tt \^{}} and any? bits coerce concat construct convert + copy copyInto! count delete elt empty empty? entries entry? eq? every? fill! find first index? indices insert less? map + map! max maxIndex member? members merge min minIndex more? nand new nor not or parts position qelt qsetelt! reduce + remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap! xor} +% +\condata{BOOLEAN}{Boolean}{ConvertibleTo Finite OrderedSet}{{\tt <} {\tt =} {\tt \^{}} and coerce convert false implies + index lookup max min nand nor not or random size true xor} +% +\condata{CARD}{CardinalNumber}{CancellationAbelianMonoid Monoid OrderedSet RetractableTo}{0 1 {\tt *} {\tt **} {\tt +} + {\tt -} {\tt <} {\tt =} Aleph coerce countable? finite? generalizedContinuumHypothesisAssumed + generalizedContinuumHypothesisAssumed? max min one? recip retract retractIfCan zero?} +% +\condata{CARTEN}{CartesianTensor}{GradedAlgebra}{0 1 {\tt *} {\tt +} {\tt -} {\tt =} coerce contract degree elt + kroneckerDelta leviCivitaSymbol product rank ravel reindex retract retractIfCan transpose unravel} +% +\condata{CCLASS}{CharacterClass}{ConvertibleTo FiniteSetAggregate SetCategory}{{\tt \#} {\tt <} {\tt =} alphabetic + alphanumeric any? bag brace cardinality charClass coerce complement construct convert copy count dictionary difference + digit empty empty? eq? every? extract! find hexDigit index insert! inspect intersect less? lookup lowerCase map map! + max member? members min more? parts random reduce remove remove! removeDuplicates select select! size size? subset? + symmetricDifference union universe upperCase} +% +\condata{CHAR}{Character}{OrderedFinite}{{\tt <} {\tt =} alphabetic? alphanumeric? char coerce digit? escape hexDigit? + index lookup lowerCase lowerCase? max min ord quote random size space upperCase upperCase?} +% +\condata{CLIF}{CliffordAlgebra}{Algebra Ring VectorSpace}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + characteristic coefficient coerce dimension e monomial one? recip zero?} +% +\condata{COLOR}{Color}{AbelianSemiGroup}{{\tt *} {\tt +} {\tt =} blue coerce color green hue numberOfHues red yellow} +% +\condata{COMM}{Commutator}{SetCategory}{{\tt =} coerce mkcomm} +% +\condata{COMPLEX}{Complex}{ComplexCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D abs acos + acosh acot acoth acsc acsch argument asec asech asin asinh associates? atan atanh basis characteristic + characteristicPolynomial charthRoot coerce complex conditionP conjugate convert coordinates cos cosh cot coth + createPrimitiveElement csc csch definingPolynomial derivationCoordinates differentiate discreteLog discriminant divide + elt euclideanSize eval exp expressIdealMember exquo extendedEuclidean factor factorPolynomial + factorSquareFreePolynomial factorsOfCyclicGroupSize gcd gcdPolynomial generator imag imaginary index init inv lcm lift + log lookup map max min minimalPolynomial multiEuclidean nextItem norm nthRoot one? order pi polarCoordinates prime? + primeFrobenius primitive? primitiveElement principalIdeal quo random rank rational rational? rationalIfCan real recip + reduce reducedSystem regularRepresentation rem representationType represents retract retractIfCan sec sech sin sinh + size sizeLess? solveLinearPolynomialEquation sqrt squareFree squareFreePart squareFreePolynomial + tableForDiscreteLogarithm tan tanh trace traceMatrix unit? unitCanonical unitNormal zero?} +% +\condata{CONTFRAC}{ContinuedFraction}{Algebra Field}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} approximants + associates? characteristic coerce complete continuedFraction convergents denominators divide euclideanSize + expressIdealMember exquo extend extendedEuclidean factor gcd inv lcm multiEuclidean numerators one? partialDenominators + partialNumerators partialQuotients prime? principalIdeal quo recip reducedContinuedFraction reducedForm rem sizeLess? + squareFree squareFreePart unit? unitCanonical unitNormal wholePart zero?} +% +\condata{DBASE}{Database}{SetCategory}{{\tt +} {\tt -} {\tt =} coerce display elt fullDisplay} +% +\condata{DFLOAT}{DoubleFloat}{ConvertibleTo DifferentialRing FloatingPointSystem TranscendentalFunctionCategory}{0 1 {\tt *} + {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D abs acos acosh acot acoth acsc acsch asec asech asin asinh + associates? atan atanh base bits ceiling characteristic coerce convert cos cosh cot coth csc csch decreasePrecision + differentiate digits divide euclideanSize exp exp1 exponent expressIdealMember exquo extendedEuclidean factor float + floor fractionPart gcd hash increasePrecision inv lcm log log10 log2 mantissa max min multiEuclidean negative? norm + nthRoot one? order patternMatch pi positive? precision prime? principalIdeal quo rationalApproximation recip rem + retract retractIfCan round sec sech sign sin sinh sizeLess? sqrt squareFree squareFreePart tan tanh truncate unit? + unitCanonical unitNormal wholePart zero?} +% +\condata{DLIST}{DataList}{ListAggregate}{{\tt \#} {\tt <} {\tt =} any? children coerce concat concat! construct convert + copy copyInto! count cycleEntry cycleLength cycleSplit! cycleTail cyclic? datalist delete delete! elt empty empty? + entries entry? eq? every? explicitlyFinite? fill! find first index? indices insert insert! last leaf? leaves less? list + map map! max maxIndex member? members merge merge! min minIndex more? new node? nodes parts position possiblyInfinite? + qelt qsetelt! reduce remove remove! removeDuplicates removeDuplicates! rest reverse reverse! second select select! + setchildren! setelt setfirst! setlast! setrest! setvalue! size? sort sort! sorted? split! swap! tail third value} +% +\condata{DECIMAL}{DecimalExpansion}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} + D abs associates? ceiling characteristic coerce convert decimal denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor floor fractionPart gcd init inv lcm map max min multiEuclidean + negative? nextItem numer numerator one? patternMatch positive? prime? principalIdeal quo random recip reducedSystem rem + retract retractIfCan sign sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal wholePart zero?} +% +\condata{DHMATRIX}{DenavitHartenbergMatrix}{MatrixCategory}{{\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + antisymmetric? any? coerce column copy count determinant diagonal? diagonalMatrix elt empty empty? eq? every? exquo + fill! horizConcat identity inverse less? listOfLists map map! matrix maxColIndex maxRowIndex member? members + minColIndex minRowIndex minordet more? ncols new nrows nullSpace nullity parts qelt qsetelt! rank rotatex rotatey + rotatez row rowEchelon scalarMatrix scale setColumn! setRow! setelt setsubMatrix! size? square? squareTop subMatrix + swapColumns! swapRows! symmetric? translate transpose vertConcat zero} +% +\condata{DEQUEUE}{Dequeue}{DequeueAggregate}{{\tt \#} {\tt =} any? back bag bottom! coerce copy count depth dequeue + dequeue! empty empty? enqueue! eq? every? extract! extractBottom! extractTop! front height insert! insertBottom! + insertTop! inspect length less? map map! member? members more? parts pop! push! reverse! rotate! size? top top!} +% +\condata{DERHAM}{DeRhamComplex}{LeftAlgebra RetractableTo}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} characteristic + coefficient coerce degree exteriorDifferential generator homogeneous? leadingBasisTerm leadingCoefficient map one? + recip reductum retract retractIfCan retractable? totalDifferential zero?} +% +\condata{DSMP}{DifferentialSparseMultivariatePolynomial}{DifferentialPolynomialCategory RetractableTo}{0 1 {\tt *} {\tt + **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce + conditionP content convert degree differentialVariables differentiate discriminant eval exquo factor factorPolynomial + factorSquareFreePolynomial gcd gcdPolynomial ground ground? initial isExpt isPlus isTimes isobaric? lcm leader + leadingCoefficient leadingMonomial mainVariable makeVariable map mapExponents max min minimumDegree monicDivide + monomial monomial? monomials multivariate numberOfMonomials one? order patternMatch prime? primitiveMonomials + primitivePart recip reducedSystem reductum resultant retract retractIfCan separant solveLinearPolynomialEquation + squareFree squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables weight + weights zero?} +% +\condata{DPMM}{DirectProductMatrixModule}{DirectProductCategory LeftModule}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt + -} {\tt /} {\tt <} {\tt =} D abs any? characteristic coerce copy count differentiate dimension directProduct dot elt + empty empty? entries entry? eq? every? fill! first index index? indices less? lookup map map! max maxIndex member? + members min minIndex more? negative? one? parts positive? qelt qsetelt! random recip reducedSystem retract retractIfCan + setelt sign size size? sup swap! unitVector zero?} +% +\condata{DPMO}{DirectProductModule}{DirectProductCategory LeftModule}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt <} {\tt =} D abs any? characteristic coerce copy count differentiate dimension directProduct dot elt empty + empty? entries entry? eq? every? fill! first index index? indices less? lookup map map! max maxIndex member? members + min minIndex more? negative? one? parts positive? qelt qsetelt! random recip reducedSystem retract retractIfCan setelt + sign size size? sup swap! unitVector zero?} +% +\condata{DIRPROD}{DirectProduct}{DirectProductCategory}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D abs any? characteristic coerce copy count differentiate dimension directProduct dot elt empty empty? entries + entry? eq? every? fill! first index index? indices less? lookup map map! max maxIndex member? members min minIndex + more? negative? one? parts positive? qelt qsetelt! random recip reducedSystem retract retractIfCan setelt sign size + size? sup swap! unitVector zero?} +% +\condata{DMP}{DistributedMultivariatePolynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP const content convert + degree differentiate discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial + ground ground? isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents max min + minimumDegree monicDivide monomial monomial? monomials multivariate numberOfMonomials one? prime? primitiveMonomials + primitivePart recip reducedSystem reductum reorder resultant retract retractIfCan solveLinearPolynomialEquation + squareFree squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables zero?} +% +\condata{DROPT}{DrawOption}{SetCategory}{{\tt =} adaptive clip coerce colorFunction coordinate coordinates curveColor + option option? pointColor range ranges space style title toScale tubePoints tubeRadius unit var1Steps var2Steps} +% +\condata{EFULS}{ElementaryFunctionsUnivariateLaurentSeries}{PartialTranscendentalFunctions}{{\tt **} acos acosIfCan + acosh acoshIfCan acot acotIfCan acoth acothIfCan acsc acscIfCan acsch acschIfCan asec asecIfCan asech asechIfCan asin + asinIfCan asinh asinhIfCan atan atanIfCan atanh atanhIfCan cos cosIfCan cosh coshIfCan cot cotIfCan coth cothIfCan csc + cscIfCan csch cschIfCan exp expIfCan log logIfCan nthRootIfCan sec secIfCan sech sechIfCan sin sinIfCan sinh sinhIfCan + tan tanIfCan tanh tanhIfCan} +% +\condata{EFUPXS}{ElementaryFunctionsUnivariatePuiseuxSeries}{PartialTranscendentalFunctions}{{\tt **} acos acosIfCan + acosh acoshIfCan acot acotIfCan acoth acothIfCan acsc acscIfCan acsch acschIfCan asec asecIfCan asech asechIfCan asin + asinIfCan asinh asinhIfCan atan atanIfCan atanh atanhIfCan cos cosIfCan cosh coshIfCan cot cotIfCan coth cothIfCan csc + cscIfCan csch cschIfCan exp expIfCan log logIfCan nthRootIfCan sec secIfCan sech sechIfCan sin sinIfCan sinh sinhIfCan + tan tanIfCan tanh tanhIfCan} +% +\condata{EQTBL}{EqTable}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt empty + empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map map! + maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{EQ}{Equation}{CoercibleTo InnerEvalable Object SetCategory}{{\tt *} {\tt **} {\tt +} {\tt -} {\tt =} coerce + equation eval lhs map rhs} +% +\condata{EMR}{EuclideanModularRing}{EuclideanDomain}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} associates? + characteristic coerce divide euclideanSize exQuo expressIdealMember exquo extendedEuclidean gcd inv lcm modulus + multiEuclidean one? principalIdeal quo recip reduce rem sizeLess? unit? unitCanonical unitNormal zero?} +% +\condata{EXIT}{Exit}{SetCategory}{{\tt =} coerce} +% +\condata{EXPR}{Expression}{AlgebraicallyClosedFunctionSpace CombinatorialOpsCategory FunctionSpace + LiouvillianFunctionCategory RetractableTo SpecialFunctionCategory TranscendentalFunctionCategory}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} Beta Ci D Ei Gamma Si abs acos acosh acot acoth acsc acsch airyAi airyBi + applyQuote asec asech asin asinh associates? atan atanh belong? besselI besselJ besselK besselY binomial box + characteristic charthRoot coerce commutator conjugate convert cos cosh cot coth csc csch definingPolynomial denom + denominator differentiate digamma dilog distribute divide elt erf euclideanSize eval exp expressIdealMember exquo + extendedEuclidean factor factorial factorials freeOf? gcd ground ground? height integral inv is? isExpt isMult isPlus + isPower isTimes kernel kernels lcm li log mainKernel map max min minPoly multiEuclidean nthRoot numer numerator one? + operator operators paren patternMatch permutation pi polygamma prime? principalIdeal product quo recip reduce + reducedSystem rem retract retractIfCan rootOf rootsOf sec sech sin sinh sizeLess? sqrt squareFree squareFreePart subst + summation tan tanh tower unit? unitCanonical unitNormal univariate variables zero? zeroOf zerosOf} +% +\condata{EAB}{ExtAlgBasis}{OrderedSet}{{\tt <} {\tt =} Nul coerce degree exponents max min} +% +\condata{FR}{Factored}{Algebra DifferentialExtension Eltable Evalable FullyEvalableOver FullyRetractableTo GcdDomain + InnerEvalable IntegralDomain RealConstant UniqueFactorizationDomain}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} D + associates? characteristic coerce convert differentiate elt eval expand exponent exquo factor factorList factors + flagFactor gcd irreducibleFactor lcm makeFR map nilFactor nthExponent nthFactor nthFlag numberOfFactors one? prime? + primeFactor rational rational? rationalIfCan recip retract retractIfCan sqfrFactor squareFree squareFreePart unit unit? + unitCanonical unitNormal unitNormalize zero?} +% +\condata{FNAME}{FileName}{FileNameCategory}{{\tt =} coerce directory exists? extension filename name new readable? + writable?} +% +\condata{FILE}{File}{FileCategory}{{\tt =} close! coerce iomode name open read! readIfCan! reopen! write!} +% +\condata{FDIV}{FiniteDivisor}{AbelianGroup}{0 {\tt *} {\tt +} {\tt -} {\tt =} algsplit coerce divisor finiteBasis + generator ideal lSpaceBasis mkBasicDiv principal? reduce zero?} +% +\condata{FFCGP}{FiniteFieldCyclicGroupExtensionByPolynomial}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt + +} {\tt -} {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP + coordinates createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide + euclideanSize expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator + getZechTable inGroundField? index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? + normalElement one? order prime? primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem + representationType represents retract retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm + trace transcendenceDegree transcendent? unit? unitCanonical unitNormal zero?} +% +\condata{FFCGX}{FiniteFieldCyclicGroupExtension}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates + createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize + expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator getZechTable + inGroundField? index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? + order prime? primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem representationType + represents retract retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace + transcendenceDegree transcendent? unit? unitCanonical unitNormal zero?} +% +\condata{FFCG}{FiniteFieldCyclicGroup}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates createNormalElement + createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize expressIdealMember exquo + extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator getZechTable inGroundField? index init + inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius + primitive? primitiveElement principalIdeal quo random recip rem representationType represents retract retractIfCan size + sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? + unitCanonical unitNormal zero?} +% +\condata{FFP}{FiniteFieldExtensionByPolynomial}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates + createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize + expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator inGroundField? + index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? order prime? + primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem representationType represents retract + retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? + unit? unitCanonical unitNormal zero?} +% +\condata{FFX}{FiniteFieldExtension}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates createNormalElement + createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize expressIdealMember exquo + extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator inGroundField? index init inv lcm + lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius primitive? + primitiveElement principalIdeal quo random recip rem representationType represents retract retractIfCan size sizeLess? + squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? unitCanonical + unitNormal zero?} +% +\condata{FFNBP}{FiniteFieldNormalBasisExtensionByPolynomial}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt + +} {\tt -} {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP + coordinates createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide + euclideanSize expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator + getMultiplicationMatrix getMultiplicationTable inGroundField? index init inv lcm lookup minimalPolynomial + multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius primitive? primitiveElement + principalIdeal quo random recip rem representationType represents retract retractIfCan size sizeLess? + sizeMultiplication squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? + unitCanonical unitNormal zero?} +% +\condata{FFNBX}{FiniteFieldNormalBasisExtension}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates + createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize + expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator + getMultiplicationMatrix getMultiplicationTable inGroundField? index init inv lcm lookup minimalPolynomial + multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius primitive? primitiveElement + principalIdeal quo random recip rem representationType represents retract retractIfCan size sizeLess? + sizeMultiplication squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? + unitCanonical unitNormal zero?} +% +\condata{FFNB}{FiniteFieldNormalBasis}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates createNormalElement + createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize expressIdealMember exquo + extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator getMultiplicationMatrix + getMultiplicationTable inGroundField? index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? + normalElement one? order prime? primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem + representationType represents retract retractIfCan size sizeLess? sizeMultiplication squareFree squareFreePart + tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? unitCanonical unitNormal zero?} +% +\condata{FF}{FiniteField}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} Frobenius + algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates createNormalElement + createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize expressIdealMember exquo + extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator inGroundField? index init inv lcm + lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius primitive? + primitiveElement principalIdeal quo random recip rem representationType represents retract retractIfCan size sizeLess? + squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? unitCanonical + unitNormal zero?} +% +\condata{FARRAY}{FlexibleArray}{ExtensibleLinearAggregate OneDimensionalArrayAggregate}{{\tt \#} {\tt <} {\tt =} any? + coerce concat concat! construct convert copy copyInto! count delete delete! elt empty empty? entries entry? eq? every? + fill! find first flexibleArray index? indices insert insert! less? map map! max maxIndex member? members merge merge! + min minIndex more? new parts physicalLength physicalLength! position qelt qsetelt! reduce remove remove! + removeDuplicates removeDuplicates! reverse reverse! select select! setelt shrinkable size? sort sort! sorted? swap!} +% +\condata{FLOAT}{Float}{CoercibleTo ConvertibleTo DifferentialRing FloatingPointSystem TranscendentalFunctionCategory}{0 + 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D abs acos acosh acot acoth acsc acsch asec asech asin asinh + associates? atan atanh base bits ceiling characteristic coerce convert cos cosh cot coth csc csch decreasePrecision + differentiate digits divide euclideanSize exp exp1 exponent expressIdealMember exquo extendedEuclidean factor float + floor fractionPart gcd increasePrecision inv lcm log log10 log2 mantissa max min multiEuclidean negative? norm + normalize nthRoot one? order outputFixed outputFloating outputGeneral outputSpacing patternMatch pi positive? precision + prime? principalIdeal quo rationalApproximation recip relerror rem retract retractIfCan round sec sech shift sign sin + sinh sizeLess? sqrt squareFree squareFreePart tan tanh truncate unit? unitCanonical unitNormal wholePart zero?} +% +\condata{FRIDEAL}{FractionalIdeal}{Group}{1 {\tt *} {\tt **} {\tt /} {\tt =} basis coerce commutator conjugate denom + ideal inv minimize norm numer one? randomLC recip} +% +\condata{FRAC}{Fraction}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D abs + associates? ceiling characteristic charthRoot coerce conditionP convert denom denominator differentiate divide elt + euclideanSize eval expressIdealMember exquo extendedEuclidean factor factorPolynomial factorSquareFreePolynomial floor + fractionPart gcd gcdPolynomial init inv lcm map max min multiEuclidean negative? nextItem numer numerator one? + patternMatch positive? prime? principalIdeal quo random recip reducedSystem rem retract retractIfCan sign sizeLess? + solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial unit? unitCanonical unitNormal wholePart + zero?} +% +\condata{FRMOD}{FramedModule}{Monoid}{1 {\tt *} {\tt **} {\tt =} basis coerce module norm one? recip} +% +\condata{FAGROUP}{FreeAbelianGroup}{AbelianGroup FreeAbelianMonoidCategory Module OrderedSet}{0 {\tt *} {\tt +} {\tt -} + {\tt <} {\tt =} coefficient coerce highCommonTerms mapCoef mapGen max min nthCoef nthFactor retract retractIfCan size + terms zero?} +% +\condata{FAMONOID}{FreeAbelianMonoid}{FreeAbelianMonoidCategory}{0 {\tt *} {\tt +} {\tt -} {\tt =} coefficient coerce + highCommonTerms mapCoef mapGen nthCoef nthFactor retract retractIfCan size terms zero?} +% +\condata{FGROUP}{FreeGroup}{Group RetractableTo}{1 {\tt *} {\tt **} {\tt /} {\tt =} coerce commutator conjugate factors + inv mapExpon mapGen nthExpon nthFactor one? recip retract retractIfCan size} +% +\condata{FM}{FreeModule}{BiModule IndexedDirectProductCategory Module}{0 {\tt *} {\tt +} {\tt -} {\tt =} coerce + leadingCoefficient leadingSupport map monomial reductum zero?} +% +\condata{FMONOID}{FreeMonoid}{Monoid OrderedSet RetractableTo}{1 {\tt *} {\tt **} {\tt <} {\tt =} coerce divide factors + hclf hcrf lquo mapExpon mapGen max min nthExpon nthFactor one? overlap recip retract retractIfCan rquo size} +% +\condata{FNLA}{FreeNilpotentLie}{NonAssociativeAlgebra}{0 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} antiCommutator + associator coerce commutator deepExpand dimension generator leftPower rightPower shallowExpand zero?} +% +\condata{FUNCTION}{FunctionCalled}{SetCategory}{{\tt =} coerce name} +% +\condata{GDMP}{GeneralDistributedMultivariatePolynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt + /} {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP const content + convert degree differentiate discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd + gcdPolynomial ground ground? isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents + max min minimumDegree monicDivide monomial monomial? monomials multivariate numberOfMonomials one? prime? + primitiveMonomials primitivePart recip reducedSystem reductum reorder resultant retract retractIfCan + solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal + univariate variables zero?} +% +\condata{GSTBL}{GeneralSparseTable}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary + elt empty empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map + map! maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{GCNAALG}{GenericNonAssociativeAlgebra}{FramedNonAssociativeAlgebra LeftModule}{0 {\tt *} {\tt **} {\tt +} {\tt + -} {\tt =} JacobiIdentity? JordanAlgebra? alternative? antiAssociative? antiCommutative? antiCommutator apply + associative? associator associatorDependence basis coerce commutative? commutator conditionsForIdempotents convert + coordinates elt flexible? generic genericLeftDiscriminant genericLeftMinimalPolynomial genericLeftNorm genericLeftTrace + genericLeftTraceForm genericRightDiscriminant genericRightMinimalPolynomial genericRightNorm genericRightTrace + genericRightTraceForm jordanAdmissible? leftAlternative? leftCharacteristicPolynomial leftDiscriminant + leftMinimalPolynomial leftNorm leftPower leftRankPolynomial leftRecip leftRegularRepresentation leftTrace + leftTraceMatrix leftUnit leftUnits lieAdmissible? lieAlgebra? noncommutativeJordanAlgebra? plenaryPower + powerAssociative? rank recip represents rightAlternative? rightCharacteristicPolynomial rightDiscriminant + rightMinimalPolynomial rightNorm rightPower rightRankPolynomial rightRecip rightRegularRepresentation rightTrace + rightTraceMatrix rightUnit rightUnits someBasis structuralConstants unit zero?} +% +\condata{GRIMAGE}{GraphImage}{SetCategory}{{\tt =} appendPoint coerce component graphImage key makeGraphImage point + pointLists putColorInfo ranges units} +% +\condata{HASHTBL}{HashTable}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt empty + empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map map! + maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{HEAP}{Heap}{PriorityQueueAggregate}{{\tt \#} {\tt =} any? bag coerce copy count empty empty? eq? every? + extract! heap insert! inspect less? map map! max member? members merge merge! more? parts size?} +% +\condata{HEXADEC}{HexadecimalExpansion}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D abs associates? ceiling characteristic coerce convert denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor floor fractionPart gcd hex init inv lcm map max min multiEuclidean + negative? nextItem numer numerator one? patternMatch positive? prime? principalIdeal quo random recip reducedSystem rem + retract retractIfCan sign sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal wholePart zero?} +% +\condata{ICARD}{IndexCard}{OrderedSet}{{\tt <} {\tt =} coerce display elt fullDisplay max min} +% +\condata{IBITS}{IndexedBits}{BitAggregate}{{\tt \#} {\tt <} {\tt =} And Not Or {\tt \^{}} and any? coerce concat + construct convert copy copyInto! count delete elt empty empty? entries entry? eq? every? fill! find first index? + indices insert less? map map! max maxIndex member? members merge min minIndex more? nand new nor not or parts position + qelt qsetelt! reduce remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap! xor} +% +\condata{IDPAG}{IndexedDirectProductAbelianGroup}{AbelianGroup IndexedDirectProductCategory}{0 {\tt *} {\tt +} {\tt -} + {\tt =} coerce leadingCoefficient leadingSupport map monomial reductum zero?} +% +\condata{IDPAM}{IndexedDirectProductAbelianMonoid}{AbelianMonoid IndexedDirectProductCategory}{0 {\tt *} {\tt +} {\tt + =} coerce leadingCoefficient leadingSupport map monomial reductum zero?} +% +\condata{IDPO}{IndexedDirectProductObject}{IndexedDirectProductCategory}{{\tt =} coerce leadingCoefficient + leadingSupport map monomial reductum} +% +\condata{IDPOAMS}{IndexedDirectProductOrderedAbelianMonoidSup}{IndexedDirectProductCategory OrderedAbelianMonoidSup}{0 + {\tt *} {\tt +} {\tt -} {\tt <} {\tt =} coerce leadingCoefficient leadingSupport map max min monomial reductum sup + zero?} +% +\condata{IDPOAM}{IndexedDirectProductOrderedAbelianMonoid}{IndexedDirectProductCategory OrderedAbelianMonoid}{0 {\tt *} + {\tt +} {\tt <} {\tt =} coerce leadingCoefficient leadingSupport map max min monomial reductum zero?} +% +\condata{INDE}{IndexedExponents}{IndexedDirectProductCategory OrderedAbelianMonoidSup}{0 {\tt *} {\tt +} {\tt -} {\tt + <} {\tt =} coerce leadingCoefficient leadingSupport map max min monomial reductum sup zero?} +% +\condata{IFARRAY}{IndexedFlexibleArray}{ExtensibleLinearAggregate OneDimensionalArrayAggregate}{{\tt \#} {\tt <} {\tt + =} any? coerce concat concat! construct convert copy copyInto! count delete delete! elt empty empty? entries entry? eq? + every? fill! find first flexibleArray index? indices insert insert! less? map map! max maxIndex member? members merge + merge! min minIndex more? new parts physicalLength physicalLength! position qelt qsetelt! reduce remove remove! + removeDuplicates removeDuplicates! reverse reverse! select select! setelt shrinkable size? sort sort! sorted? swap!} +% +\condata{ILIST}{IndexedList}{ListAggregate}{{\tt \#} {\tt <} {\tt =} any? child? children coerce concat concat! + construct convert copy copyInto! count cycleEntry cycleLength cycleSplit! cycleTail cyclic? delete delete! distance elt + empty empty? entries entry? eq? every? explicitlyFinite? fill! find first index? indices insert insert! last leaf? + less? list map map! max maxIndex member? members merge merge! min minIndex more? new node? nodes parts position + possiblyInfinite? qelt qsetelt! reduce remove remove! removeDuplicates removeDuplicates! rest reverse reverse! second + select select! setchildren! setelt setfirst! setlast! setrest! setvalue! size? sort sort! sorted? split! swap! tail + third value} +% +\condata{IMATRIX}{IndexedMatrix}{MatrixCategory}{{\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + antisymmetric? any? coerce column copy count determinant diagonal? diagonalMatrix elt empty empty? eq? every? exquo + fill! horizConcat inverse less? listOfLists map map! matrix maxColIndex maxRowIndex member? members minColIndex + minRowIndex minordet more? ncols new nrows nullSpace nullity parts qelt qsetelt! rank row rowEchelon scalarMatrix + setColumn! setRow! setelt setsubMatrix! size? square? squareTop subMatrix swapColumns! swapRows! symmetric? transpose + vertConcat zero} +% +\condata{IARRAY1}{IndexedOneDimensionalArray}{OneDimensionalArrayAggregate}{{\tt \#} {\tt <} {\tt =} any? coerce concat + construct convert copy copyInto! count delete elt empty empty? entries entry? eq? every? fill! find first index? + indices insert less? map map! max maxIndex member? members merge min minIndex more? new parts position qelt qsetelt! + reduce remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap!} +% +\condata{ISTRING}{IndexedString}{StringAggregate}{{\tt \#} {\tt <} {\tt =} any? coerce concat construct copy copyInto! + count delete elt empty empty? entries entry? eq? every? fill! find first hash index? indices insert leftTrim less? + lowerCase lowerCase! map map! match? max maxIndex member? members merge min minIndex more? new parts position prefix? + qelt qsetelt! reduce remove removeDuplicates replace reverse reverse! rightTrim select setelt size? sort sort! sorted? + split substring? suffix? swap! trim upperCase upperCase!} +% +\condata{IARRAY2}{IndexedTwoDimensionalArray}{TwoDimensionalArrayCategory}{{\tt \#} {\tt =} any? coerce column copy + count elt empty empty? eq? every? fill! less? map map! maxColIndex maxRowIndex member? members minColIndex minRowIndex + more? ncols new nrows parts qelt qsetelt! row setColumn! setRow! setelt size?} +% +\condata{IVECTOR}{IndexedVector}{VectorCategory}{{\tt \#} {\tt *} {\tt +} {\tt -} {\tt <} {\tt =} any? coerce concat + construct convert copy copyInto! count delete dot elt empty empty? entries entry? eq? every? fill! find first index? + indices insert less? map map! max maxIndex member? members merge min minIndex more? new parts position qelt qsetelt! + reduce remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap! zero} +% +\condata{ITUPLE}{InfiniteTuple}{CoercibleTo}{coerce construct filterUntil filterWhile generate map select} +% +\condata{IFF}{InnerFiniteField}{FiniteAlgebraicExtensionField}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP coordinates createNormalElement + createPrimitiveElement definingPolynomial degree dimension discreteLog divide euclideanSize expressIdealMember exquo + extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator inGroundField? index init inv lcm + lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? order prime? primeFrobenius primitive? + primitiveElement principalIdeal quo random recip rem representationType represents retract retractIfCan size sizeLess? + squareFree squareFreePart tableForDiscreteLogarithm trace transcendenceDegree transcendent? unit? unitCanonical + unitNormal zero?} +% +\condata{IFAMON}{InnerFreeAbelianMonoid}{FreeAbelianMonoidCategory}{0 {\tt *} {\tt +} {\tt -} {\tt =} coefficient + coerce highCommonTerms mapCoef mapGen nthCoef nthFactor retract retractIfCan size terms zero?} +% +\condata{IIARRAY2}{InnerIndexedTwoDimensionalArray}{TwoDimensionalArrayCategory}{{\tt \#} {\tt =} any? coerce column + copy count elt empty empty? eq? every? fill! less? map map! maxColIndex maxRowIndex member? members minColIndex + minRowIndex more? ncols new nrows parts qelt qsetelt! row setColumn! setRow! setelt size?} +% +\condata{IPADIC}{InnerPAdicInteger}{PAdicIntegerCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} approximate + associates? characteristic coerce complete digits divide euclideanSize expressIdealMember exquo extend + extendedEuclidean gcd lcm moduloP modulus multiEuclidean one? order principalIdeal quo quotientByP recip rem sizeLess? + sqrt unit? unitCanonical unitNormal zero?} +% +\condata{IPF}{InnerPrimeField}{ConvertibleTo FiniteAlgebraicExtensionField FiniteFieldCategory}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP + convert coordinates createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide + euclideanSize expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator + inGroundField? index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? + order prime? primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem representationType + represents retract retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace + transcendenceDegree transcendent? unit? unitCanonical unitNormal zero?} +% +\condata{ITAYLOR}{InnerTaylorSeries}{IntegralDomain Ring}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} associates? + characteristic coefficients coerce exquo one? order pole? recip series unit? unitCanonical unitNormal zero?} +% +\condata{INFORM}{InputForm}{ConvertibleTo SExpressionCategory}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt /} {\tt =} + atom? binary car cdr coerce compile convert declare destruct elt eq expr flatten float float? function integer integer? + interpret lambda list? null? pair? string string? symbol symbol? uequal unparse} +% +\condata{ZMOD}{IntegerMod}{CommutativeRing ConvertibleTo Finite StepThrough}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt + =} characteristic coerce convert index init lookup nextItem one? random recip size zero?} +% +\condata{INT}{Integer}{ConvertibleTo IntegerNumberSystem}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} D abs + addmod associates? base binomial bit? characteristic coerce convert copy dec differentiate divide euclideanSize even? + expressIdealMember exquo extendedEuclidean factor factorial gcd hash inc init invmod lcm length mask max min mulmod + multiEuclidean negative? nextItem odd? one? patternMatch permutation positive? positiveRemainder powmod prime? + principalIdeal quo random rational rational? rationalIfCan recip reducedSystem rem retract retractIfCan shift sign + sizeLess? squareFree squareFreePart submod symmetricRemainder unit? unitCanonical unitNormal zero?} +% +\condata{IR}{IntegrationResult}{Module RetractableTo}{0 {\tt *} {\tt +} {\tt -} {\tt =} D coerce differentiate elem? + integral logpart mkAnswer notelem ratpart retract retractIfCan zero?} +% +\condata{KERNEL}{Kernel}{CachableSet ConvertibleTo Patternable}{{\tt <} {\tt =} argument coerce convert height is? + kernel max min name operator position setPosition symbolIfCan} +% +\condata{KAFILE}{KeyedAccessFile}{FileCategory TableAggregate}{{\tt \#} {\tt =} any? bag close! coerce construct copy + count dictionary elt empty empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect + iomode key? keys less? map map! maxIndex member? members minIndex more? name open pack! parts qelt qsetelt! read! + reduce remove remove! removeDuplicates reopen! search select select! setelt size? swap! table write!} +% +\condata{LAUPOL}{LaurentPolynomial}{CharacteristicNonZero CharacteristicZero ConvertibleTo DifferentialExtension + EuclideanDomain FullyRetractableTo IntegralDomain RetractableTo}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} D + associates? characteristic charthRoot coefficient coerce convert degree differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean gcd lcm leadingCoefficient monomial monomial? multiEuclidean one? order + principalIdeal quo recip reductum rem retract retractIfCan separate sizeLess? trailingCoefficient unit? unitCanonical + unitNormal zero?} +% +\condata{LIB}{Library}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt empty + empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? library map + map! maxIndex member? members minIndex more? pack! parts qelt qsetelt! reduce remove remove! removeDuplicates search + select select! setelt size? swap! table} +% +\condata{LSQM}{LieSquareMatrix}{CoercibleTo FramedNonAssociativeAlgebra SquareMatrixCategory}{0 1 {\tt \#} {\tt *} {\tt + **} {\tt +} {\tt -} {\tt /} {\tt =} D JacobiIdentity? JordanAlgebra? alternative? antiAssociative? antiCommutative? + antiCommutator antisymmetric? any? apply associative? associator associatorDependence basis characteristic coerce + column commutative? commutator conditionsForIdempotents convert coordinates copy count determinant diagonal diagonal? + diagonalMatrix diagonalProduct differentiate elt empty empty? eq? every? exquo flexible? inverse jordanAdmissible? + leftAlternative? leftCharacteristicPolynomial leftDiscriminant leftMinimalPolynomial leftNorm leftPower + leftRankPolynomial leftRecip leftRegularRepresentation leftTrace leftTraceMatrix leftUnit leftUnits less? + lieAdmissible? lieAlgebra? listOfLists map map! matrix maxColIndex maxRowIndex member? members minColIndex minRowIndex + minordet more? ncols noncommutativeJordanAlgebra? nrows nullSpace nullity one? parts plenaryPower powerAssociative? + qelt rank recip reducedSystem represents retract retractIfCan rightAlternative? rightCharacteristicPolynomial + rightDiscriminant rightMinimalPolynomial rightNorm rightPower rightRankPolynomial rightRecip rightRegularRepresentation + rightTrace rightTraceMatrix rightUnit rightUnits row rowEchelon scalarMatrix size? someBasis square? + structuralConstants symmetric? trace unit zero?} +% +\condata{LODO}{LinearOrdinaryDifferentialOperator}{MonogenicLinearOperator}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt + =} D characteristic coefficient coerce degree elt leadingCoefficient leftDivide leftExactQuotient leftGcd leftLcm + leftQuotient leftRemainder minimumDegree monomial one? recip reductum rightDivide rightExactQuotient rightGcd rightLcm + rightQuotient rightRemainder zero?} +% +\condata{LMOPS}{ListMonoidOps}{RetractableTo SetCategory}{{\tt =} coerce leftMult listOfMonoms makeMulti makeTerm + makeUnit mapExpon mapGen nthExpon nthFactor outputForm plus retract retractIfCan reverse reverse! rightMult size} +% +\condata{LMDICT}{ListMultiDictionary}{MultiDictionary}{{\tt \#} {\tt =} any? bag coerce construct convert copy count + dictionary duplicates duplicates? empty empty? eq? every? extract! find insert! inspect less? map map! member? members + more? parts reduce remove remove! removeDuplicates removeDuplicates! select select! size? substitute} +% +\condata{LIST}{List}{ListAggregate}{{\tt \#} {\tt <} {\tt =} any? append child? children coerce concat concat! cons + construct convert copy copyInto! count cycleEntry cycleLength cycleSplit! cycleTail cyclic? delete delete! distance elt + empty empty? entries entry? eq? every? explicitlyFinite? fill! find first index? indices insert insert! last leaf? + less? list map map! max maxIndex member? members merge merge! min minIndex more? new nil node? nodes null parts + position possiblyInfinite? qelt qsetelt! reduce remove remove! removeDuplicates removeDuplicates! rest reverse reverse! + second select select! setDifference setIntersection setUnion setchildren! setelt setfirst! setlast! setrest! setvalue! + size? sort sort! sorted? split! swap! tail third value} +% +\condata{LA}{LocalAlgebra}{Algebra OrderedRing}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} abs + characteristic coerce denom max min negative? numer one? positive? recip sign zero?} +% +\condata{LO}{Localize}{Module OrderedAbelianGroup}{0 {\tt *} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} coerce denom max + min numer zero?} +% +\condata{MKCHSET}{MakeCachableSet}{CachableSet CoercibleTo}{{\tt <} {\tt =} coerce max min position setPosition} +% +\condata{MKODRING}{MakeOrdinaryDifferentialRing}{CoercibleTo DifferentialRing}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt =} D characteristic coerce differentiate one? recip zero?} +% +\condata{MATRIX}{Matrix}{MatrixCategory}{{\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} antisymmetric? any? + coerce column copy count determinant diagonal? diagonalMatrix elt empty empty? eq? every? exquo fill! horizConcat + inverse less? listOfLists map map! matrix maxColIndex maxRowIndex member? members minColIndex minRowIndex minordet + more? ncols new nrows nullSpace nullity parts qelt qsetelt! rank row rowEchelon scalarMatrix setColumn! setRow! setelt + setsubMatrix! size? square? squareTop subMatrix swapColumns! swapRows! symmetric? transpose vertConcat zero} +% +\condata{MODMON}{ModMonic}{Finite UnivariatePolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} An D UnVectorise Vectorise associates? characteristic charthRoot coefficient coefficients coerce composite + computePowers conditionP content degree differentiate discriminant divide divideExponents elt euclideanSize eval + expressIdealMember exquo extendedEuclidean factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground + ground? index init integrate isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial lift lookup mainVariable + makeSUP map mapExponents max min minimumDegree modulus monicDivide monomial monomial? monomials multiEuclidean + multiplyExponents multivariate nextItem numberOfMonomials one? order pow prime? primitiveMonomials primitivePart + principalIdeal pseudoDivide pseudoQuotient pseudoRemainder quo random recip reduce reducedSystem reductum rem resultant + retract retractIfCan separate setPoly size sizeLess? solveLinearPolynomialEquation squareFree squareFreePart + squareFreePolynomial subResultantGcd totalDegree unit? unitCanonical unitNormal univariate unmakeSUP variables + vectorise zero?} +% +\condata{MODFIELD}{ModularField}{Field}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} associates? characteristic + coerce divide euclideanSize exQuo expressIdealMember exquo extendedEuclidean factor gcd inv lcm modulus multiEuclidean + one? prime? principalIdeal quo recip reduce rem sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal + zero?} +% +\condata{MODRING}{ModularRing}{Ring}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} characteristic coerce exQuo inv + modulus one? recip reduce zero?} +% +\condata{MOEBIUS}{MoebiusTransform}{Group}{1 {\tt *} {\tt **} {\tt /} {\tt =} coerce commutator conjugate eval inv + moebius one? recip scale shift} +% +\condata{MRING}{MonoidRing}{Algebra CharacteristicNonZero CharacteristicZero Finite RetractableTo Ring}{0 1 {\tt *} + {\tt **} {\tt +} {\tt -} {\tt =} characteristic charthRoot coefficient coefficients coerce index leadingCoefficient + leadingMonomial lookup map monomial monomial? monomials numberOfMonomials one? random recip reductum retract + retractIfCan size terms zero?} +% +\condata{MSET}{Multiset}{MultisetAggregate}{{\tt \#} {\tt <} {\tt =} any? bag brace coerce construct convert copy count + dictionary difference duplicates empty empty? eq? every? extract! find insert! inspect intersect less? map map! member? + members more? multiset parts reduce remove remove! removeDuplicates removeDuplicates! select select! size? subset? + symmetricDifference union} +% +\condata{MPOLY}{MultivariatePolynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt + =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP content convert degree + differentiate discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground + ground? isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents max min + minimumDegree monicDivide monomial monomial? monomials multivariate numberOfMonomials one? prime? primitiveMonomials + primitivePart recip reducedSystem reductum resultant retract retractIfCan solveLinearPolynomialEquation squareFree + squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables zero?} +% +\condata{NDP}{NewDirectProduct}{DirectProductCategory}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D abs any? characteristic coerce copy count differentiate dimension directProduct dot elt empty empty? entries + entry? eq? every? fill! first index index? indices less? lookup map map! max maxIndex member? members min minIndex + more? negative? one? parts positive? qelt qsetelt! random recip reducedSystem retract retractIfCan setelt sign size + size? sup swap! unitVector zero?} +% +\condata{NDMP}{NewDistributedMultivariatePolynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} + {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP const content + convert degree differentiate discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd + gcdPolynomial ground ground? isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents + max min minimumDegree monicDivide monomial monomial? monomials multivariate numberOfMonomials one? prime? + primitiveMonomials primitivePart recip reducedSystem reductum reorder resultant retract retractIfCan + solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal + univariate variables zero?} +% +\condata{NONE}{None}{SetCategory}{{\tt =} coerce} +% +\condata{NNI}{NonNegativeInteger}{Monoid OrderedAbelianMonoidSup}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} + coerce divide exquo gcd max min one? quo recip rem sup zero?} +% +\condata{OCT}{Octonion}{FullyRetractableTo OctonionCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} abs + characteristic charthRoot coerce conjugate convert elt eval imagE imagI imagJ imagK imagi imagj imagk index inv lookup + map max min norm octon one? random rational rational? rationalIfCan real recip retract retractIfCan size zero?} +% +\condata{ARRAY1}{OneDimensionalArray}{OneDimensionalArrayAggregate}{{\tt \#} {\tt <} {\tt =} any? coerce concat + construct convert copy copyInto! count delete elt empty empty? entries entry? eq? every? fill! find first index? + indices insert less? map map! max maxIndex member? members merge min minIndex more? new oneDimensionalArray parts + position qelt qsetelt! reduce remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap!} +% +\condata{ONECOMP}{OnePointCompletion}{AbelianGroup FullyRetractableTo OrderedRing SetCategory}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt <} {\tt =} abs characteristic coerce finite? infinite? infinity max min negative? one? positive? + rational rational? rationalIfCan recip retract retractIfCan sign zero?} +% +\condata{OP}{Operator}{Algebra CharacteristicNonZero CharacteristicZero Eltable RetractableTo Ring}{0 1 {\tt *} {\tt + **} {\tt +} {\tt -} {\tt =} characteristic charthRoot coerce elt evaluate one? opeval recip retract retractIfCan zero?} +% +\condata{OMLO}{OppositeMonogenicLinearOperator}{DifferentialRing MonogenicLinearOperator}{0 1 {\tt *} {\tt **} {\tt +} + {\tt -} {\tt =} D characteristic coefficient coerce degree differentiate leadingCoefficient minimumDegree monomial one? + op po recip reductum zero?} +% +\condata{ORDCOMP}{OrderedCompletion}{AbelianGroup FullyRetractableTo OrderedRing SetCategory}{0 1 {\tt *} {\tt **} {\tt + +} {\tt -} {\tt <} {\tt =} abs characteristic coerce finite? infinite? max min minusInfinity negative? one? + plusInfinity positive? rational rational? rationalIfCan recip retract retractIfCan sign whatInfinity zero?} +% +\condata{ODP}{OrderedDirectProduct}{DirectProductCategory}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + <} {\tt =} D abs any? characteristic coerce copy count differentiate dimension directProduct dot elt empty empty? + entries entry? eq? every? fill! first index index? indices less? lookup map map! max maxIndex member? members min + minIndex more? negative? one? parts positive? qelt qsetelt! random recip reducedSystem retract retractIfCan setelt sign + size size? sup swap! unitVector zero?} +% +\condata{OVAR}{OrderedVariableList}{ConvertibleTo OrderedFinite}{{\tt <} {\tt =} coerce convert index lookup max min + random size variable} +% +\condata{ODPOL}{OrderlyDifferentialPolynomial}{DifferentialPolynomialCategory RetractableTo}{0 1 {\tt *} {\tt **} {\tt + +} {\tt -} {\tt /} {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP + content degree differentialVariables differentiate discriminant eval exquo factor factorPolynomial + factorSquareFreePolynomial gcd gcdPolynomial ground ground? initial isExpt isPlus isTimes isobaric? lcm leader + leadingCoefficient leadingMonomial mainVariable makeVariable map mapExponents max min minimumDegree monicDivide + monomial monomial? monomials multivariate numberOfMonomials one? order prime? primitiveMonomials primitivePart recip + reducedSystem reductum resultant retract retractIfCan separant solveLinearPolynomialEquation squareFree squareFreePart + squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables weight weights zero?} +% +\condata{ODVAR}{OrderlyDifferentialVariable}{DifferentialVariableCategory}{{\tt <} {\tt =} D coerce differentiate + makeVariable max min order retract retractIfCan variable weight} +% +\condata{ODR}{OrdinaryDifferentialRing}{Algebra DifferentialRing Field}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} + {\tt =} D associates? characteristic coerce differentiate divide euclideanSize expressIdealMember exquo + extendedEuclidean factor gcd inv lcm multiEuclidean one? prime? principalIdeal quo recip rem sizeLess? squareFree + squareFreePart unit? unitCanonical unitNormal zero?} +% +\condata{OSI}{OrdSetInts}{OrderedSet}{{\tt <} {\tt =} coerce max min value} +% +\condata{OUTFORM}{OutputForm}{SetCategory}{{\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt <=} {\tt =} {\tt >} + {\tt >=} D SEGMENT {\tt \^{}=} and assign blankSeparate box brace bracket center coerce commaSeparate differentiate div + dot elt empty exquo hconcat height hspace infix infix? int label left matrix message messagePrint not or outputForm + over overbar paren pile postfix prefix presub presuper prime print prod quo quote rarrow rem right root rspace scripts + semicolonSeparate slash string sub subHeight sum super superHeight supersub vconcat vspace width zag} +% +\condata{PADIC}{PAdicInteger}{PAdicIntegerCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt =} approximate + associates? characteristic coerce complete digits divide euclideanSize expressIdealMember exquo extend + extendedEuclidean gcd lcm moduloP modulus multiEuclidean one? order principalIdeal quo quotientByP recip rem sizeLess? + sqrt unit? unitCanonical unitNormal zero?} +% +\condata{PADICRC}{PAdicRationalConstructor}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D abs approximate associates? ceiling characteristic charthRoot coerce conditionP continuedFraction convert + denom denominator differentiate divide elt euclideanSize eval expressIdealMember exquo extendedEuclidean factor + factorPolynomial factorSquareFreePolynomial floor fractionPart gcd gcdPolynomial init inv lcm map max min + multiEuclidean negative? nextItem numer numerator one? patternMatch positive? prime? principalIdeal quo random recip + reducedSystem rem removeZeroes retract retractIfCan sign sizeLess? solveLinearPolynomialEquation squareFree + squareFreePart squareFreePolynomial unit? unitCanonical unitNormal wholePart zero?} +% +\condata{PADICRAT}{PAdicRational}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + approximate associates? characteristic coerce continuedFraction denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor fractionPart gcd inv lcm map multiEuclidean numer numerator one? + prime? principalIdeal quo recip reducedSystem rem removeZeroes retract retractIfCan sizeLess? squareFree squareFreePart + unit? unitCanonical unitNormal wholePart zero?} +% +\condata{PALETTE}{Palette}{SetCategory}{{\tt =} bright coerce dark dim hue light pastel shade} +% +\condata{PARPCURV}{ParametricPlaneCurve}{}{coordinate curve} +% +\condata{PARSCURV}{ParametricSpaceCurve}{}{coordinate curve} +% +\condata{PARSURF}{ParametricSurface}{}{coordinate surface} +% +\condata{PFR}{PartialFraction}{Algebra Field}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} associates? + characteristic coerce compactFraction divide euclideanSize expressIdealMember exquo extendedEuclidean factor firstDenom + firstNumer gcd inv lcm multiEuclidean nthFractionalTerm numberOfFractionalTerms one? padicFraction padicallyExpand + partialFraction prime? principalIdeal quo recip rem sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal + wholePart zero?} +% +\condata{PRTITION}{Partition}{ConvertibleTo OrderedCancellationAbelianMonoid}{0 {\tt *} {\tt +} {\tt -} {\tt <} {\tt =} + coerce conjugate convert max min partition pdct powers zero?} +% +\condata{PATLRES}{PatternMatchListResult}{SetCategory}{{\tt =} atoms coerce failed failed? lists makeResult new} +% +\condata{PATRES}{PatternMatchResult}{SetCategory}{{\tt =} addMatch addMatchRestricted coerce construct destruct failed + failed? getMatch insertMatch new satisfy? union} +% +\condata{PATTERN}{Pattern}{RetractableTo SetCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt /} {\tt =} addBadValue coerce + constant? convert copy depth elt generic? getBadValues hasPredicate? hasTopPredicate? inR? isExpt isList isOp isPlus + isPower isQuotient isTimes multiple? optional? optpair patternVariable predicates quoted? resetBadValues retract + retractIfCan setPredicates setTopPredicate symbol? topPredicate variables withPredicates} +% +\condata{PENDTREE}{PendantTree}{BinaryRecursiveAggregate}{{\tt \#} {\tt =} any? children coerce copy count cyclic? elt + empty empty? eq? every? leaf? leaves left less? map map! member? members more? node? nodes parts ptree right + setchildren! setelt setleft! setright! setvalue! size? value} +% +\condata{PERMGRP}{PermutationGroup}{SetCategory}{{\tt <} {\tt <=} {\tt =} base coerce degree elt generators + initializeGroupForWordProblem member? movedPoints orbit orbits order permutationGroup random strongGenerators + wordInGenerators wordInStrongGenerators wordsForStrongGenerators} +% +\condata{PERM}{Permutation}{PermutationCategory}{1 {\tt *} {\tt **} {\tt /} {\tt <} {\tt =} coerce coerceImages + coerceListOfPairs coercePreimagesImages commutator conjugate cycle cyclePartition cycles degree elt eval even? + fixedPoints inv listRepresentation max min movedPoints numberOfCycles odd? one? orbit order recip sign sort} +% +\condata{HACKPI}{Pi}{CharacteristicZero CoercibleTo ConvertibleTo Field RealConstant RetractableTo}{0 1 {\tt *} {\tt + **} {\tt +} {\tt -} {\tt /} {\tt =} associates? characteristic coerce convert divide euclideanSize expressIdealMember + exquo extendedEuclidean factor gcd inv lcm multiEuclidean one? pi prime? principalIdeal quo recip rem retract + retractIfCan sizeLess? squareFree squareFreePart unit? unitCanonical unitNormal zero?} +% +\condata{ACPLOT}{PlaneAlgebraicCurvePlot}{PlottablePlaneCurveCategory}{coerce listBranches makeSketch refine xRange + yRange} +% +\condata{PLOT3D}{Plot3D}{PlottableSpaceCurveCategory}{adaptive3D? coerce debug3D listBranches maxPoints3D minPoints3D + numFunEvals3D plot pointPlot refine screenResolution3D setAdaptive3D setMaxPoints3D setMinPoints3D + setScreenResolution3D tRange tValues xRange yRange zRange zoom} +% +\condata{PLOT}{Plot}{PlottablePlaneCurveCategory}{adaptive? coerce debug listBranches maxPoints minPoints numFunEvals + parametric? plot plotPolar pointPlot refine screenResolution setAdaptive setMaxPoints setMinPoints setScreenResolution + tRange xRange yRange zoom} +% +\condata{POINT}{Point}{PointCategory}{{\tt \#} {\tt *} {\tt +} {\tt -} {\tt <} {\tt =} any? coerce concat construct + convert copy copyInto! count cross delete dimension dot elt empty empty? entries entry? eq? every? extend fill! find + first index? indices insert length less? map map! max maxIndex member? members merge min minIndex more? new parts point + position qelt qsetelt! reduce remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap! + zero} +% +\condata{IDEAL}{PolynomialIdeals}{SetCategory}{{\tt *} {\tt **} {\tt +} {\tt =} backOldPos coerce contract dimension + element? generalPosition generators groebner groebner? groebnerIdeal ideal in? inRadical? intersect leadingIdeal + quotient relationsIdeal saturate zeroDim?} +% +\condata{PR}{PolynomialRing}{FiniteAbelianMonoidRing}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} associates? + characteristic charthRoot coefficient coefficients coerce content degree exquo ground ground? leadingCoefficient + leadingMonomial map mapExponents minimumDegree monomial monomial? numberOfMonomials one? primitivePart recip reductum + retract retractIfCan unit? unitCanonical unitNormal zero?} +% +\condata{POLY}{Polynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D + associates? characteristic charthRoot coefficient coefficients coerce conditionP content convert degree differentiate + discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground ground? integrate + isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents max min minimumDegree + monicDivide monomial monomial? monomials multivariate numberOfMonomials one? patternMatch prime? primitiveMonomials + primitivePart recip reducedSystem reductum resultant retract retractIfCan solveLinearPolynomialEquation squareFree + squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables zero?} +% +\condata{PI}{PositiveInteger}{AbelianSemiGroup Monoid OrderedSet}{1 {\tt *} {\tt **} {\tt +} {\tt <} {\tt =} coerce gcd + max min one? recip} +% +\condata{PF}{PrimeField}{ConvertibleTo FiniteAlgebraicExtensionField FiniteFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} + {\tt -} {\tt /} {\tt =} Frobenius algebraic? associates? basis characteristic charthRoot coerce conditionP convert + coordinates createNormalElement createPrimitiveElement definingPolynomial degree dimension discreteLog divide + euclideanSize expressIdealMember exquo extendedEuclidean extensionDegree factor factorsOfCyclicGroupSize gcd generator + inGroundField? index init inv lcm lookup minimalPolynomial multiEuclidean nextItem norm normal? normalElement one? + order prime? primeFrobenius primitive? primitiveElement principalIdeal quo random recip rem representationType + represents retract retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace + transcendenceDegree transcendent? unit? unitCanonical unitNormal zero?} +% +\condata{PRIMARR}{PrimitiveArray}{OneDimensionalArrayAggregate}{{\tt \#} {\tt <} {\tt =} any? coerce concat construct + convert copy copyInto! count delete elt empty empty? entries entry? eq? every? fill! find first index? indices insert + less? map map! max maxIndex member? members merge min minIndex more? new parts position qelt qsetelt! reduce remove + removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap!} +% +\condata{PRODUCT}{Product}{AbelianGroup AbelianMonoid CancellationAbelianMonoid Finite Group Monoid + OrderedAbelianMonoidSup OrderedSet SetCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} coerce + commutator conjugate index inv lookup makeprod max min one? random recip selectfirst selectsecond size sup zero?} +% +\condata{QFORM}{QuadraticForm}{AbelianGroup}{0 {\tt *} {\tt +} {\tt -} {\tt =} coerce elt matrix quadraticForm zero?} +% +\condata{QALGSET}{QuasiAlgebraicSet}{CoercibleTo SetCategory}{{\tt =} coerce definingEquations definingInequation + empty? idealSimplify quasiAlgebraicSet setStatus simplify} +% +\condata{QUAT}{Quaternion}{QuaternionCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} D abs + characteristic charthRoot coerce conjugate convert differentiate elt eval imagI imagJ imagK inv map max min norm one? + quatern rational rational? rationalIfCan real recip reducedSystem retract retractIfCan zero?} +% +\condata{QEQUAT}{QueryEquation}{}{equation value variable} +% +\condata{QUEUE}{Queue}{QueueAggregate}{{\tt \#} {\tt =} any? back bag coerce copy count dequeue! empty empty? enqueue! + eq? every? extract! front insert! inspect length less? map map! member? members more? parts queue rotate! size?} +% +\condata{RADFF}{RadicalFunctionField}{FunctionFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + absolutelyIrreducible? associates? basis branchPoint? branchPointAtInfinity? characteristic characteristicPolynomial + charthRoot coerce complementaryBasis convert coordinates definingPolynomial derivationCoordinates differentiate + discriminant divide elt euclideanSize expressIdealMember exquo extendedEuclidean factor gcd generator genus integral? + integralAtInfinity? integralBasis integralBasisAtInfinity integralCoordinates integralDerivationMatrix integralMatrix + integralMatrixAtInfinity integralRepresents inv inverseIntegralMatrix inverseIntegralMatrixAtInfinity lcm lift + minimalPolynomial multiEuclidean nonSingularModel norm normalizeAtInfinity numberOfComponents one? prime? primitivePart + principalIdeal quo ramified? ramifiedAtInfinity? rank rationalPoint? rationalPoints recip reduce reduceBasisAtInfinity + reducedSystem regularRepresentation rem represents retract retractIfCan singular? singularAtInfinity? sizeLess? + squareFree squareFreePart trace traceMatrix unit? unitCanonical unitNormal yCoordinates zero?} +% +\condata{RADIX}{RadixExpansion}{QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D + abs associates? ceiling characteristic coerce convert cycleRagits denom denominator differentiate divide euclideanSize + expressIdealMember exquo extendedEuclidean factor floor fractRadix fractRagits fractionPart gcd init inv lcm map max + min multiEuclidean negative? nextItem numer numerator one? patternMatch positive? prefixRagits prime? principalIdeal + quo random recip reducedSystem rem retract retractIfCan sign sizeLess? squareFree squareFreePart unit? unitCanonical + unitNormal wholePart wholeRadix wholeRagits zero?} +% +\condata{RMATRIX}{RectangularMatrix}{CoercibleTo RectangularMatrixCategory VectorSpace}{0 {\tt \#} {\tt *} {\tt +} {\tt + -} {\tt /} {\tt =} antisymmetric? any? coerce column copy count diagonal? dimension elt empty empty? eq? every? exquo + less? listOfLists map map! matrix maxColIndex maxRowIndex member? members minColIndex minRowIndex more? ncols nrows + nullSpace nullity parts qelt rank rectangularMatrix row rowEchelon size? square? symmetric? zero?} +% +\condata{REF}{Reference}{Object SetCategory}{{\tt =} coerce deref elt ref setelt setref} +% +\condata{RULE}{RewriteRule}{Eltable RetractableTo SetCategory}{{\tt =} coerce elt lhs pattern quotedOperators retract + retractIfCan rhs rule suchThat} +% +\condata{ROMAN}{RomanNumeral}{IntegerNumberSystem}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} D abs addmod + associates? base binomial bit? characteristic coerce convert copy dec differentiate divide euclideanSize even? + expressIdealMember exquo extendedEuclidean factor factorial gcd hash inc init invmod lcm length mask max min mulmod + multiEuclidean negative? nextItem odd? one? patternMatch permutation positive? positiveRemainder powmod prime? + principalIdeal quo random rational rational? rationalIfCan recip reducedSystem rem retract retractIfCan roman shift + sign sizeLess? squareFree squareFreePart submod symmetricRemainder unit? unitCanonical unitNormal zero?} +% +\condata{RULECOLD}{RuleCalled}{SetCategory}{{\tt =} coerce name} +% +\condata{RULESET}{Ruleset}{Eltable SetCategory}{{\tt =} coerce elt rules ruleset} +% +\condata{FORMULA1}{ScriptFormulaFormat1}{Object}{coerce} +% +\condata{FORMULA}{ScriptFormulaFormat}{SetCategory}{{\tt =} coerce convert display epilogue formula new prologue + setEpilogue! setFormula! setPrologue!} +% +\condata{SEGBIND}{SegmentBinding}{SetCategory}{{\tt =} coerce equation segment variable} +% +\condata{SEG}{Segment}{SegmentCategory SegmentExpansionCategory}{{\tt =} BY SEGMENT coerce convert expand hi high incr + lo low map segment} +% +\condata{SCFRAC}{SemiCancelledFraction}{ConvertibleTo QuotientFieldCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt + /} {\tt <} {\tt =} D abs associates? ceiling characteristic charthRoot coerce conditionP convert denom denominator + differentiate divide elt euclideanSize eval expressIdealMember exquo extendedEuclidean factor factorPolynomial + factorSquareFreePolynomial floor fractionPart gcd gcdPolynomial init inv lcm map max min multiEuclidean negative? + nextItem normalize numer numerator one? patternMatch positive? prime? principalIdeal quo random recip reducedSystem rem + retract retractIfCan sign sizeLess? solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial unit? + unitCanonical unitNormal wholePart zero?} +% +\condata{SDPOL}{SequentialDifferentialPolynomial}{DifferentialPolynomialCategory RetractableTo}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce + conditionP content degree differentialVariables differentiate discriminant eval exquo factor factorPolynomial + factorSquareFreePolynomial gcd gcdPolynomial ground ground? initial isExpt isPlus isTimes isobaric? lcm leader + leadingCoefficient leadingMonomial mainVariable makeVariable map mapExponents max min minimumDegree monicDivide + monomial monomial? monomials multivariate numberOfMonomials one? order prime? primitiveMonomials primitivePart recip + reducedSystem reductum resultant retract retractIfCan separant solveLinearPolynomialEquation squareFree squareFreePart + squareFreePolynomial totalDegree unit? unitCanonical unitNormal univariate variables weight weights zero?} +% +\condata{SDVAR}{SequentialDifferentialVariable}{DifferentialVariableCategory}{{\tt <} {\tt =} D coerce differentiate + makeVariable max min order retract retractIfCan variable weight} +% +\condata{SET}{Set}{FiniteSetAggregate}{{\tt \#} {\tt <} {\tt =} any? bag brace cardinality coerce complement construct + convert copy count dictionary difference empty empty? eq? every? extract! find index insert! inspect intersect less? + lookup map map! max member? members min more? parts random reduce remove remove! removeDuplicates select select! size + size? subset? symmetricDifference union universe} +% +\condata{SEXOF}{SExpressionOf}{SExpressionCategory}{{\tt \#} {\tt =} atom? car cdr coerce convert destruct elt eq expr + float float? integer integer? list? null? pair? string string? symbol symbol? uequal} +% +\condata{SEX}{SExpression}{SExpressionCategory}{{\tt \#} {\tt =} atom? car cdr coerce convert destruct elt eq expr + float float? integer integer? list? null? pair? string string? symbol symbol? uequal} +% +\condata{SAE}{SimpleAlgebraicExtension}{MonogenicAlgebra}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + associates? basis characteristic characteristicPolynomial charthRoot coerce conditionP convert coordinates + createPrimitiveElement definingPolynomial derivationCoordinates differentiate discreteLog discriminant divide + euclideanSize expressIdealMember exquo extendedEuclidean factor factorsOfCyclicGroupSize gcd generator index init inv + lcm lift lookup minimalPolynomial multiEuclidean nextItem norm one? order prime? primeFrobenius primitive? + primitiveElement principalIdeal quo random rank recip reduce reducedSystem regularRepresentation rem representationType + represents retract retractIfCan size sizeLess? squareFree squareFreePart tableForDiscreteLogarithm trace traceMatrix + unit? unitCanonical unitNormal zero?} +% +\condata{SAOS}{SingletonAsOrderedSet}{OrderedSet}{{\tt <} {\tt =} coerce create max min} +% +\condata{SINT}{SingleInteger}{IntegerNumberSystem}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt <} {\tt =} And D Not Or {\tt + \^{}} abs addmod and associates? base binomial bit? characteristic coerce convert copy dec differentiate divide + euclideanSize even? expressIdealMember exquo extendedEuclidean factor factorial gcd hash inc init invmod lcm length + mask max min mulmod multiEuclidean negative? nextItem not odd? one? or patternMatch permutation positive? + positiveRemainder powmod prime? principalIdeal quo random rational rational? rationalIfCan recip reducedSystem rem + retract retractIfCan shift sign sizeLess? squareFree squareFreePart submod symmetricRemainder unit? unitCanonical + unitNormal xor zero?} +% +\condata{SMP}{SparseMultivariatePolynomial}{PolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce conditionP content convert degree + differentiate discriminant eval exquo factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground + ground? isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable map mapExponents max min + minimumDegree monicDivide monomial monomial? monomials multivariate numberOfMonomials one? patternMatch prime? + primitiveMonomials primitivePart recip reducedSystem reductum resultant retract retractIfCan + solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial totalDegree unit? unitCanonical unitNormal + univariate variables zero?} +% +\condata{SMTS}{SparseMultivariateTaylorSeries}{MultivariateTaylorSeriesCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt =} D acos acosh acot acoth acsc acsch asec asech asin asinh associates? atan atanh characteristic + charthRoot coefficient coerce complete cos cosh cot coth csc csch csubst degree differentiate eval exp exquo extend + fintegrate integrate leadingCoefficient leadingMonomial log map monomial monomial? nthRoot one? order pi pole? + polynomial recip reductum sec sech sin sinh sqrt tan tanh unit? unitCanonical unitNormal variables zero?} +% +\condata{STBL}{SparseTable}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt empty + empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map map! + maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{SUP}{SparseUnivariatePolynomial}{UnivariatePolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} + {\tt <} {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce composite conditionP content + degree differentiate discriminant divide divideExponents elt euclideanSize eval expressIdealMember exquo + extendedEuclidean factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground ground? init integrate + isExpt isPlus isTimes lcm leadingCoefficient leadingMonomial mainVariable makeSUP map mapExponents max min + minimumDegree monicDivide monomial monomial? monomials multiEuclidean multiplyExponents multivariate nextItem + numberOfMonomials one? order outputForm prime? primitiveMonomials primitivePart principalIdeal pseudoDivide + pseudoQuotient pseudoRemainder quo recip reducedSystem reductum rem resultant retract retractIfCan separate sizeLess? + solveLinearPolynomialEquation squareFree squareFreePart squareFreePolynomial subResultantGcd totalDegree unit? + unitCanonical unitNormal univariate unmakeSUP variables vectorise zero?} +% +\condata{SUTS}{SparseUnivariateTaylorSeries}{UnivariateTaylorSeriesCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt + /} {\tt =} D acos acosh acot acoth acsc acsch approximate asec asech asin asinh associates? atan atanh center + characteristic charthRoot coefficient coefficients coerce complete cos cosh cot coth csc csch degree differentiate elt + eval exp exquo extend integrate leadingCoefficient leadingMonomial log map monomial monomial? multiplyCoefficients + multiplyExponents nthRoot one? order pi pole? polynomial quoByVar recip reductum sec sech series sin sinh sqrt tan tanh + terms truncate unit? unitCanonical unitNormal variable variables zero?} +% +\condata{SQMATRIX}{SquareMatrix}{CoercibleTo SquareMatrixCategory}{0 1 {\tt \#} {\tt *} {\tt **} {\tt +} {\tt -} {\tt + /} {\tt =} D antisymmetric? any? characteristic coerce column copy count determinant diagonal diagonal? diagonalMatrix + diagonalProduct differentiate elt empty empty? eq? every? exquo inverse less? listOfLists map map! matrix maxColIndex + maxRowIndex member? members minColIndex minRowIndex minordet more? ncols nrows nullSpace nullity one? parts qelt rank + recip reducedSystem retract retractIfCan row rowEchelon scalarMatrix size? square? squareMatrix symmetric? trace + transpose zero?} +% +\condata{STACK}{Stack}{StackAggregate}{{\tt \#} {\tt =} any? bag coerce copy count depth empty empty? eq? every? + extract! insert! inspect less? map map! member? members more? parts pop! push! size? stack top} +% +\condata{STREAM}{Stream}{LazyStreamAggregate}{{\tt \#} {\tt =} any? child? children coerce complete concat concat! cons + construct convert copy count cycleEntry cycleLength cycleSplit! cycleTail cyclic? delay delete distance elt empty + empty? entries entry? eq? every? explicitEntries? explicitlyEmpty? explicitlyFinite? extend fill! filterUntil + filterWhile find findCycle first frst generate index? indices insert last lazy? lazyEvaluate leaf? less? map map! + maxIndex member? members minIndex more? new node? nodes numberOfComputedEntries output parts possiblyInfinite? qelt + qsetelt! reduce remove removeDuplicates repeating repeating? rest rst second select setchildren! setelt setfirst! + setlast! setrest! setvalue! showAll? showAllElements size? split! swap! tail third value} +% +\condata{STRTBL}{StringTable}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt + empty empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map + map! maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{STRING}{String}{StringCategory}{{\tt \#} {\tt <} {\tt =} any? coerce concat construct copy copyInto! count + delete elt empty empty? entries entry? eq? every? fill! find first index? indices insert leftTrim less? lowerCase + lowerCase! map map! match? max maxIndex member? members merge min minIndex more? new parts position prefix? qelt + qsetelt! reduce remove removeDuplicates replace reverse reverse! rightTrim select setelt size? sort sort! sorted? split + string substring? suffix? swap! trim upperCase upperCase!} +% +\condata{COMPPROP}{SubSpaceComponentProperty}{SetCategory}{{\tt =} close closed? coerce copy new solid solid?} +% +\condata{SUBSPACE}{SubSpace}{SetCategory}{{\tt =} addPoint addPoint2 addPointLast birth child children closeComponent + coerce deepCopy defineProperty extractClosed extractIndex extractPoint extractProperty internal? leaf? level merge + modifyPoint new numberOfChildren parent pointData root? separate shallowCopy subspace traverse} +% +\condata{SUCH}{SuchThat}{SetCategory}{{\tt =} coerce construct lhs rhs} +% +\condata{SYMBOL}{Symbol}{ConvertibleTo OrderedSet PatternMatchable}{{\tt <} {\tt =} argscript coerce convert elt list + max min name new patternMatch resetNew script scripted? scripts string subscript superscript} +% +\condata{SYMPOLY}{SymmetricPolynomial}{FiniteAbelianMonoidRing}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} + associates? characteristic charthRoot coefficient coefficients coerce content degree exquo ground ground? + leadingCoefficient leadingMonomial map mapExponents minimumDegree monomial monomial? numberOfMonomials one? + primitivePart recip reductum retract retractIfCan unit? unitCanonical unitNormal zero?} +% +\condata{TABLEAU}{Tableau}{Object}{coerce listOfLists tableau} +% +\condata{TABLE}{Table}{TableAggregate}{{\tt \#} {\tt =} any? bag coerce construct copy count dictionary elt empty + empty? entries entry? eq? every? extract! fill! find first index? indices insert! inspect key? keys less? map map! + maxIndex member? members minIndex more? parts qelt qsetelt! reduce remove remove! removeDuplicates search select + select! setelt size? swap! table} +% +\condata{TS}{TaylorSeries}{MultivariateTaylorSeriesCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt =} D + acos acosh acot acoth acsc acsch asec asech asin asinh associates? atan atanh characteristic charthRoot coefficient + coerce complete cos cosh cot coth csc csch degree differentiate eval exp exquo extend fintegrate integrate + leadingCoefficient leadingMonomial log map monomial monomial? nthRoot one? order pi pole? polynomial recip reductum sec + sech sin sinh sqrt tan tanh unit? unitCanonical unitNormal variables zero?} +% +\condata{TEX1}{TexFormat1}{Object}{coerce} +% +\condata{TEX}{TexFormat}{SetCategory}{{\tt =} coerce convert display epilogue new prologue setEpilogue! setPrologue! + setTex! tex} +% +\condata{TEXTFILE}{TextFile}{FileCategory}{{\tt =} close! coerce endOfFile? iomode name open read! readIfCan! readLine! + readLineIfCan! reopen! write! writeLine!} +% +\condata{VIEW3D}{ThreeDimensionalViewport}{SetCategory}{{\tt =} axes clipSurface close coerce colorDef controlPanel + diagonals dimensions drawStyle eyeDistance hitherPlane intensity key lighting makeViewport3D modifyPointData move + options outlineRender perspective reset resize rotate showClipRegion showRegion subspace title translate + viewDeltaXDefault viewDeltaYDefault viewPhiDefault viewThetaDefault viewZoomDefault viewpoint viewport3D write zoom} +% +\condata{SPACE3}{ThreeSpace}{ThreeSpaceCategory}{{\tt =} check closedCurve closedCurve? coerce components composite + composites copy create3Space curve curve? enterPointData lllip lllp llprop lp lprop merge mesh mesh? modifyPointData + numberOfComponents numberOfComposites objects point point? polygon polygon? subspace} +% +\condata{TREE}{Tree}{RecursiveAggregate}{{\tt \#} {\tt =} any? children coerce copy count cyclic? elt empty empty? eq? + every? leaf? leaves less? map map! member? members more? node? nodes parts setchildren! setelt setvalue! size? tree + value} +% +\condata{TUBE}{TubePlot}{}{closed? getCurve listLoops open? setClosed tube} +% +\condata{TUPLE}{Tuple}{CoercibleTo SetCategory}{{\tt =} coerce length select} +% +\condata{ARRAY2}{TwoDimensionalArray}{TwoDimensionalArrayCategory}{{\tt \#} {\tt =} any? coerce column copy count elt + empty empty? eq? every? fill! less? map map! maxColIndex maxRowIndex member? members minColIndex minRowIndex more? + ncols new nrows parts qelt qsetelt! row setColumn! setRow! setelt size?} +% +\condata{VIEW2D}{TwoDimensionalViewport}{SetCategory}{{\tt =} axes close coerce connect controlPanel dimensions + getGraph graphState graphStates graphs key makeViewport2D move options points putGraph region reset resize scale show + title translate units viewport2D write} +% +\condata{ULSCONS}{UnivariateLaurentSeriesConstructor}{UnivariateLaurentSeriesConstructorCategory}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt /} {\tt <} {\tt =} D abs acos acosh acot acoth acsc acsch approximate asec asech asin asinh + associates? atan atanh ceiling center characteristic charthRoot coefficient coerce complete conditionP convert cos cosh + cot coth csc csch degree denom denominator differentiate divide elt euclideanSize eval exp expressIdealMember exquo + extend extendedEuclidean factor factorPolynomial factorSquareFreePolynomial floor fractionPart gcd gcdPolynomial init + integrate inv laurent lcm leadingCoefficient leadingMonomial log map max min monomial monomial? multiEuclidean + multiplyCoefficients multiplyExponents negative? nextItem nthRoot numer numerator one? order patternMatch pi pole? + positive? prime? principalIdeal quo random rationalFunction recip reducedSystem reductum rem removeZeroes retract + retractIfCan sec sech series sign sin sinh sizeLess? solveLinearPolynomialEquation sqrt squareFree squareFreePart + squareFreePolynomial tan tanh taylor taylorIfCan taylorRep terms truncate unit? unitCanonical unitNormal variable + variables wholePart zero?} +% +\condata{ULS}{UnivariateLaurentSeries}{UnivariateLaurentSeriesConstructorCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} + {\tt /} {\tt =} D acos acosh acot acoth acsc acsch approximate asec asech asin asinh associates? atan atanh center + characteristic charthRoot coefficient coerce complete cos cosh cot coth csc csch degree denom denominator differentiate + divide elt euclideanSize eval exp expressIdealMember exquo extend extendedEuclidean factor gcd integrate inv laurent + lcm leadingCoefficient leadingMonomial log map monomial monomial? multiEuclidean multiplyCoefficients multiplyExponents + nthRoot numer numerator one? order pi pole? prime? principalIdeal quo rationalFunction recip reducedSystem reductum rem + removeZeroes retract retractIfCan sec sech series sin sinh sizeLess? sqrt squareFree squareFreePart tan tanh taylor + taylorIfCan taylorRep terms truncate unit? unitCanonical unitNormal variable variables zero?} +% +\condata{UP}{UnivariatePolynomial}{UnivariatePolynomialCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt <} + {\tt =} D associates? characteristic charthRoot coefficient coefficients coerce composite conditionP content degree + differentiate discriminant divide divideExponents elt euclideanSize eval expressIdealMember exquo extendedEuclidean + factor factorPolynomial factorSquareFreePolynomial gcd gcdPolynomial ground ground? init integrate isExpt isPlus + isTimes lcm leadingCoefficient leadingMonomial mainVariable makeSUP map mapExponents max min minimumDegree monicDivide + monomial monomial? monomials multiEuclidean multiplyExponents multivariate nextItem numberOfMonomials one? order prime? + primitiveMonomials primitivePart principalIdeal pseudoDivide pseudoQuotient pseudoRemainder quo recip reducedSystem + reductum rem resultant retract retractIfCan separate sizeLess? solveLinearPolynomialEquation squareFree squareFreePart + squareFreePolynomial subResultantGcd totalDegree unit? unitCanonical unitNormal univariate unmakeSUP variables + vectorise zero?} +% +\condata{UPXSCONS}{UnivariatePuiseuxSeriesConstructor}{UnivariatePuiseuxSeriesConstructorCategory}{0 1 {\tt *} {\tt **} + {\tt +} {\tt -} {\tt /} {\tt =} D acos acosh acot acoth acsc acsch approximate asec asech asin asinh associates? atan + atanh center characteristic charthRoot coefficient coerce complete cos cosh cot coth csc csch degree differentiate + divide elt euclideanSize eval exp expressIdealMember exquo extend extendedEuclidean factor gcd integrate inv laurent + laurentIfCan laurentRep lcm leadingCoefficient leadingMonomial log map monomial monomial? multiEuclidean + multiplyExponents nthRoot one? order pi pole? prime? principalIdeal puiseux quo rationalPower recip reductum rem + retract retractIfCan sec sech series sin sinh sizeLess? sqrt squareFree squareFreePart tan tanh terms truncate unit? + unitCanonical unitNormal variable variables zero?} +% +\condata{UPXS}{UnivariatePuiseuxSeries}{UnivariatePuiseuxSeriesConstructorCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt + -} {\tt /} {\tt =} D acos acosh acot acoth acsc acsch approximate asec asech asin asinh associates? atan atanh center + characteristic charthRoot coefficient coerce complete cos cosh cot coth csc csch degree differentiate divide elt + euclideanSize eval exp expressIdealMember exquo extend extendedEuclidean factor gcd integrate inv laurent laurentIfCan + laurentRep lcm leadingCoefficient leadingMonomial log map monomial monomial? multiEuclidean multiplyExponents nthRoot + one? order pi pole? prime? principalIdeal puiseux quo rationalPower recip reductum rem retract retractIfCan sec sech + series sin sinh sizeLess? sqrt squareFree squareFreePart tan tanh terms truncate unit? unitCanonical unitNormal + variable variables zero?} +% +\condata{UTS}{UnivariateTaylorSeries}{UnivariateTaylorSeriesCategory}{0 1 {\tt *} {\tt **} {\tt +} {\tt -} {\tt /} {\tt + =} D acos acosh acot acoth acsc acsch approximate asec asech asin asinh associates? atan atanh center characteristic + charthRoot coefficient coefficients coerce complete cos cosh cot coth csc csch degree differentiate elt eval + evenlambert exp exquo extend generalLambert integrate invmultisect lagrange lambert leadingCoefficient leadingMonomial + log map monomial monomial? multiplyCoefficients multiplyExponents multisect nthRoot oddlambert one? order pi pole? + polynomial quoByVar recip reductum revert sec sech series sin sinh sqrt tan tanh terms truncate unit? unitCanonical + unitNormal univariatePolynomial variable variables zero?} +% +\condata{UNISEG}{UniversalSegment}{SegmentCategory SegmentExpansionCategory}{{\tt =} BY SEGMENT coerce convert expand + hasHi hi high incr lo low map segment} +% +\condata{VARIABLE}{Variable}{CoercibleTo SetCategory}{{\tt =} coerce variable} +% +\condata{VECTOR}{Vector}{VectorCategory}{{\tt \#} {\tt *} {\tt +} {\tt -} {\tt <} {\tt =} any? coerce concat construct + convert copy copyInto! count delete dot elt empty empty? entries entry? eq? every? fill! find first index? indices + insert less? map map! max maxIndex member? members merge min minIndex more? new parts position qelt qsetelt! reduce + remove removeDuplicates reverse reverse! select setelt size? sort sort! sorted? swap! vector zero} +% +\condata{VOID}{Void}{}{coerce void} +% +% +% ---------------------------------------------------------------------- +%\end{constructorListing} +% ---------------------------------------------------------------------- + + + +\setcounter{chapter}{3} % Appendix D + +%\twocolumn[% +\chapter{Packages} +\label{ugAppPackages} + +This is a listing of all packages in the Axiom library at the +time this book was produced. +Use the Browse facility (described in Chapter +\ref{ugBrowse} on page~\pageref{ugBrowse}) +to get more information about these constructors. + +This sample entry will help you read the following table: + +PackageName{PackageAbbreviation}:{$\hbox{{\sf Category}}_{1}$% +\ldots$\hbox{{\sf Category}}_{N}$}{\sl with }% +{$\hbox{{\rm op}}_{1}$\ldots$\hbox{{\rm op}}_{M}$} + +where + +\begin{tabular}{@{\quad}ll} +PackageName & is the full package name, e.g., {\sf PadeApproximantPackage}. \\ +PackageAbbreviation & is the package abbreviation, e.g., {\sf PADEPAC}.\\ +$\hbox{{\sf Category}}_{i}$ & is a category to which the package belongs. \\ +$\hbox{{\rm op}}_{j}$ & is an operation exported by the package. +\end{tabular} + +\appendix{Packages} + +% ---------------------------------------------------------------------- +%\begin{constructorListing} +% ---------------------------------------------------------------------- +\condata{AF}{AlgebraicFunction}{}{{\tt **} belong? definingPolynomial inrootof iroot minPoly operator rootOf} +% +\condata{INTHERAL}{AlgebraicHermiteIntegration}{}{HermiteIntegrate} +% +\condata{INTALG}{AlgebraicIntegrate}{}{algintegrate palginfieldint palgintegrate} +% +\condata{INTAF}{AlgebraicIntegration}{}{algint} +% +\condata{ALGMANIP}{AlgebraicManipulations}{}{ratDenom ratPoly rootKerSimp rootSimp rootSplit} +% +\condata{ALGMFACT}{AlgebraicMultFact}{}{factor} +% +\condata{ALGPKG}{AlgebraPackage}{}{basisOfCenter basisOfCentroid basisOfCommutingElements basisOfLeftAnnihilator + basisOfLeftNucleus basisOfLeftNucloid basisOfMiddleNucleus basisOfNucleus basisOfRightAnnihilator basisOfRightNucleus + basisOfRightNucloid biRank doubleRank leftRank radicalOfLeftTraceForm rightRank weakBiRank} +% +\condata{ALGFACT}{AlgFactor}{}{doublyTransitive? factor split} +% +\condata{ANY1}{AnyFunctions1}{}{coerce retract retractIfCan retractable?} +% +\condata{APPRULE}{ApplyRules}{}{applyRules localUnquote} +% +\condata{PMPRED}{AttachPredicates}{}{suchThat} +% +\condata{BALFACT}{BalancedFactorisation}{}{balancedFactorisation} +% +\condata{BOP1}{BasicOperatorFunctions1}{}{constantOpIfCan constantOperator derivative evaluate} +% +\condata{BEZOUT}{BezoutMatrix}{}{bezoutDiscriminant bezoutMatrix bezoutResultant} +% +\condata{BOUNDZRO}{BoundIntegerRoots}{}{integerBound} +% +\condata{CARTEN2}{CartesianTensorFunctions2}{}{map reshape} +% +\condata{CHVAR}{ChangeOfVariable}{}{chvar eval goodPoint mkIntegral radPoly rootPoly} +% +\condata{CHARPOL}{CharacteristicPolynomialPackage}{}{characteristicPolynomial} +% +\condata{CVMP}{CoerceVectorMatrixPackage}{}{coerce coerceP} +% +\condata{COMBF}{CombinatorialFunction}{}{{\tt **} belong? binomial factorial factorials iibinom iidprod iidsum iifact + iiperm iipow ipow operator permutation product summation} +% +\condata{CDEN}{CommonDenominator}{}{clearDenominator commonDenominator splitDenominator} +% +\condata{COMMONOP}{CommonOperators}{}{operator} +% +\condata{COMMUPC}{CommuteUnivariatePolynomialCategory}{}{swap} +% +\condata{COMPFACT}{ComplexFactorization}{}{factor} +% +\condata{COMPLEX2}{ComplexFunctions2}{}{map} +% +\condata{CINTSLPE}{ComplexIntegerSolveLinearPolynomialEquation}{}{solveLinearPolynomialEquation} +% +\condata{CRFP}{ComplexRootFindingPackage}{}{complexZeros divisorCascade factor graeffe norm pleskenSplit + reciprocalPolynomial rootRadius schwerpunkt setErrorBound startPolynomial} +% +\condata{CMPLXRT}{ComplexRootPackage}{}{complexZeros} +% +\condata{ODECONST}{ConstantLODE}{}{constDsolve} +% +\condata{COORDSYS}{CoordinateSystems}{}{bipolar bipolarCylindrical cartesian conical cylindrical elliptic + ellipticCylindrical oblateSpheroidal parabolic parabolicCylindrical paraboloidal polar prolateSpheroidal spherical + toroidal} +% +\condata{CRAPACK}{CRApackage}{}{chineseRemainder modTree multiEuclideanTree} +% +\condata{CYCLES}{CycleIndicators}{}{SFunction alternating cap complete cup cyclic dihedral elementary eval graphs + powerSum skewSFunction wreath} +% +\condata{CSTTOOLS}{CyclicStreamTools}{}{computeCycleEntry computeCycleLength cycleElt} +% +\condata{CYCLOTOM}{CyclotomicPolynomialPackage}{}{cyclotomic cyclotomicDecomposition cyclotomicFactorization} +% +\condata{DEGRED}{DegreeReductionPackage}{}{expand reduce} +% +\condata{DIOSP}{DiophantineSolutionPackage}{}{dioSolve} +% +\condata{DIRPROD2}{DirectProductFunctions2}{}{map reduce scan} +% +\condata{DLP}{DiscreteLogarithmPackage}{}{shanksDiscLogAlgorithm} +% +\condata{DISPLAY}{DisplayPackage}{}{bright center copies newLine say sayLength} +% +\condata{DDFACT}{DistinctDegreeFactorize}{}{distdfact exptMod factor irreducible? separateDegrees separateFactors + tracePowMod} +% +\condata{DBLRESP}{DoubleResultantPackage}{}{doubleResultant} +% +\condata{DRAWHACK}{DrawNumericHack}{}{coerce} +% +\condata{DROPT0}{DrawOptionFunctions0}{}{adaptive clipBoolean coordinate curveColorPalette pointColorPalette ranges + space style title toScale tubePoints tubeRadius units var1Steps var2Steps} +% +\condata{DROPT1}{DrawOptionFunctions1}{}{option} +% +\condata{EP}{EigenPackage}{}{characteristicPolynomial eigenvalues eigenvector eigenvectors inteigen} +% +\condata{ODEEF}{ElementaryFunctionODESolver}{}{solve} +% +\condata{SIGNEF}{ElementaryFunctionSign}{}{sign} +% +\condata{EFSTRUC}{ElementaryFunctionStructurePackage}{}{normalize realElementary rischNormalize validExponential} +% +\condata{EFUTS}{ElementaryFunctionsUnivariateTaylorSeries}{}{{\tt **} acos acosh acot acoth acsc acsch asec asech asin + asinh atan atanh cos cosh cot coth csc csch exp log sec sech sin sincos sinh sinhcosh tan tanh} +% +\condata{EF}{ElementaryFunction}{}{acos acosh acot acoth acsc acsch asec asech asin asinh atan atanh belong? cos cosh + cot coth csc csch exp iiacos iiacosh iiacot iiacoth iiacsc iiacsch iiasec iiasech iiasin iiasinh iiatan iiatanh iicos + iicosh iicot iicoth iicsc iicsch iiexp iilog iisec iisech iisin iisinh iitan iitanh log operator pi sec sech sin sinh + specialTrigs tan tanh} +% +\condata{INTEF}{ElementaryIntegration}{}{lfextendedint lfextlimint lfinfieldint lfintegrate lflimitedint} +% +\condata{RDEEF}{ElementaryRischDE}{}{rischDE} +% +\condata{ELFUTS}{EllipticFunctionsUnivariateTaylorSeries}{}{cn dn sn sncndn} +% +\condata{EQ2}{EquationFunctions2}{}{map} +% +\condata{ERROR}{ErrorFunctions}{}{error} +% +\condata{GBEUCLID}{EuclideanGroebnerBasisPackage}{}{euclideanGroebner euclideanNormalForm} +% +\condata{EVALCYC}{EvaluateCycleIndicators}{}{eval} +% +\condata{EXPR2}{ExpressionFunctions2}{}{map} +% +\condata{ES1}{ExpressionSpaceFunctions1}{}{map} +% +\condata{ES2}{ExpressionSpaceFunctions2}{}{map} +% +\condata{EXPRODE}{ExpressionSpaceODESolver}{}{seriesSolve} +% +\condata{EXPR2UPS}{ExpressionToUnivariatePowerSeries}{}{laurent puiseux series taylor} +% +\condata{EXPRTUBE}{ExpressionTubePlot}{}{constantToUnaryFunction tubePlot} +% +\condata{FR2}{FactoredFunctions2}{}{map} +% +\condata{FACTFUNC}{FactoredFunctions}{}{log nthRoot} +% +\condata{FRUTIL}{FactoredFunctionUtilities}{}{mergeFactors refine} +% +\condata{FACUTIL}{FactoringUtilities}{}{completeEval degree lowerPolynomial normalDeriv raisePolynomial ran variables} +% +\condata{FORDER}{FindOrderFinite}{}{order} +% +\condata{FDIV2}{FiniteDivisorFunctions2}{}{map} +% +\condata{FFF}{FiniteFieldFunctions}{}{createMultiplicationMatrix createMultiplicationTable createZechTable + sizeMultiplication} +% +\condata{FFHOM}{FiniteFieldHomomorphisms}{}{coerce} +% +\condata{FFPOLY2}{FiniteFieldPolynomialPackage2}{}{rootOfIrreduciblePoly} +% +\condata{FFPOLY}{FiniteFieldPolynomialPackage}{}{createIrreduciblePoly createNormalPoly createNormalPrimitivePoly + createPrimitiveNormalPoly createPrimitivePoly leastAffineMultiple nextIrreduciblePoly nextNormalPoly + nextNormalPrimitivePoly nextPrimitiveNormalPoly nextPrimitivePoly normal? numberOfIrreduciblePoly numberOfNormalPoly + numberOfPrimitivePoly primitive? random reducedQPowers} +% +\condata{FFSLPE}{FiniteFieldSolveLinearPolynomialEquation}{}{solveLinearPolynomialEquation} +% +\condata{FLAGG2}{FiniteLinearAggregateFunctions2}{}{map reduce scan} +% +\condata{FLASORT}{FiniteLinearAggregateSort}{}{heapSort quickSort shellSort} +% +\condata{FSAGG2}{FiniteSetAggregateFunctions2}{}{map reduce scan} +% +\condata{FLOATCP}{FloatingComplexPackage}{}{complexRoots complexSolve} +% +\condata{FLOATRP}{FloatingRealPackage}{}{realRoots solve} +% +\condata{FRIDEAL2}{FractionalIdealFunctions2}{}{map} +% +\condata{FRAC2}{FractionFunctions2}{}{map} +% +\condata{FSPECF}{FunctionalSpecialFunction}{}{Beta Gamma abs airyAi airyBi belong? besselI besselJ besselK besselY + digamma iiGamma iiabs operator polygamma} +% +\condata{FFCAT2}{FunctionFieldCategoryFunctions2}{}{map} +% +\condata{FFINTBAS}{FunctionFieldIntegralBasis}{}{integralBasis} +% +\condata{PMASSFS}{FunctionSpaceAssertions}{}{assert constant multiple optional} +% +\condata{PMPREDFS}{FunctionSpaceAttachPredicates}{}{suchThat} +% +\condata{FSCINT}{FunctionSpaceComplexIntegration}{}{complexIntegrate internalIntegrate} +% +\condata{FS2}{FunctionSpaceFunctions2}{}{map} +% +\condata{FSINT}{FunctionSpaceIntegration}{}{integrate} +% +\condata{FSPRMELT}{FunctionSpacePrimitiveElement}{}{primitiveElement} +% +\condata{FSRED}{FunctionSpaceReduce}{}{bringDown newReduc} +% +\condata{SUMFS}{FunctionSpaceSum}{}{sum} +% +\condata{FS2UPS}{FunctionSpaceToUnivariatePowerSeries}{}{exprToGenUPS exprToUPS} +% +\condata{FSUPFACT}{FunctionSpaceUnivariatePolynomialFactor}{}{ffactor qfactor} +% +\condata{GAUSSFAC}{GaussianFactorizationPackage}{}{factor prime? sumSquares} +% +\condata{GHENSEL}{GeneralHenselPackage}{}{HenselLift completeHensel} +% +\condata{GENPGCD}{GeneralPolynomialGcdPackage}{}{gcdPolynomial randomR} +% +\condata{GENUPS}{GenerateUnivariatePowerSeries}{}{laurent puiseux series taylor} +% +\condata{GENEEZ}{GenExEuclid}{}{compBound reduction solveid tablePow testModulus} +% +\condata{GENUFACT}{GenUFactorize}{}{factor} +% +\condata{INTG0}{GenusZeroIntegration}{}{palgLODE0 palgRDE0 palgextint0 palgint0 palglimint0} +% +\condata{GOSPER}{GosperSummationMethod}{}{GospersMethod} +% +\condata{GRDEF}{GraphicsDefaults}{}{adaptive clipPointsDefault drawToScale maxPoints minPoints screenResolution} +% +\condata{GRAY}{GrayCode}{}{firstSubsetGray nextSubsetGray} +% +\condata{GBF}{GroebnerFactorizationPackage}{}{factorGroebnerBasis groebnerFactorize} +% +\condata{GBINTERN}{GroebnerInternalPackage}{}{credPol critB critBonD critM critMTonD1 critMonD1 critT critpOrder + fprindINFO gbasis hMonic lepol makeCrit minGbasis prinb prindINFO prinpolINFO prinshINFO redPo redPol sPol updatD + updatF virtualDegree} +% +\condata{GB}{GroebnerPackage}{}{groebner normalForm} +% +\condata{GROEBSOL}{GroebnerSolve}{}{genericPosition groebSolve testDim} +% +\condata{HB}{HallBasis}{}{generate inHallBasis? lfunc} +% +\condata{HEUGCD}{HeuGcd}{}{content contprim gcd gcdcofact gcdcofactprim gcdprim lintgcd} +% +\condata{IDECOMP}{IdealDecompositionPackage}{}{primaryDecomp prime? radical zeroDimPrimary? zeroDimPrime?} +% +\condata{INCRMAPS}{IncrementingMaps}{}{increment incrementBy} +% +\condata{ITFUN2}{InfiniteTupleFunctions2}{}{map} +% +\condata{ITFUN3}{InfiniteTupleFunctions3}{}{map} +% +\condata{INFINITY}{Infinity}{}{infinity minusInfinity plusInfinity} +% +\condata{IALGFACT}{InnerAlgFactor}{}{factor} +% +\condata{ICDEN}{InnerCommonDenominator}{}{clearDenominator commonDenominator splitDenominator} +% +\condata{IMATLIN}{InnerMatrixLinearAlgebraFunctions}{}{determinant inverse nullSpace nullity rank rowEchelon} +% +\condata{IMATQF}{InnerMatrixQuotientFieldFunctions}{}{inverse nullSpace nullity rank rowEchelon} +% +\condata{INMODGCD}{InnerModularGcd}{}{modularGcd reduction} +% +\condata{INNMFACT}{InnerMultFact}{}{factor} +% +\condata{INBFF}{InnerNormalBasisFieldFunctions}{}{{\tt *} {\tt **} {\tt /} basis dAndcExp expPot index inv lookup + minimalPolynomial norm normal? normalElement pol qPot random repSq setFieldInfo trace xn} +% +\condata{INEP}{InnerNumericEigenPackage}{}{charpol innerEigenvectors} +% +\condata{INFSP}{InnerNumericFloatSolvePackage}{}{innerSolve innerSolve1 makeEq} +% +\condata{INPSIGN}{InnerPolySign}{}{signAround} +% +\condata{ISUMP}{InnerPolySum}{}{sum} +% +\condata{ITRIGMNP}{InnerTrigonometricManipulations}{}{F2FG FG2F GF2FG explogs2trigs trigs2explogs} +% +\condata{INFORM1}{InputFormFunctions1}{}{interpret packageCall} +% +\condata{COMBINAT}{IntegerCombinatoricFunctions}{}{binomial factorial multinomial partition permutation stirling1 + stirling2} +% +\condata{INTFACT}{IntegerFactorizationPackage}{}{BasicMethod PollardSmallFactor factor squareFree} +% +\condata{ZLINDEP}{IntegerLinearDependence}{}{linearDependenceOverZ linearlyDependentOverZ? solveLinearlyOverQ} +% +\condata{INTHEORY}{IntegerNumberTheoryFunctions}{}{bernoulli chineseRemainder divisors euler eulerPhi fibonacci + harmonic jacobi legendre moebiusMu numberOfDivisors sumOfDivisors sumOfKthPowerDivisors} +% +\condata{PRIMES}{IntegerPrimesPackage}{}{nextPrime prevPrime prime? primes} +% +\condata{INTRET}{IntegerRetractions}{}{integer integer? integerIfCan} +% +\condata{IROOT}{IntegerRoots}{}{approxNthRoot approxSqrt perfectNthPower? perfectNthRoot perfectSqrt perfectSquare?} +% +\condata{IBATOOL}{IntegralBasisTools}{}{diagonalProduct idealiser leastPower} +% +\condata{IR2}{IntegrationResultFunctions2}{}{map} +% +\condata{IRRF2F}{IntegrationResultRFToFunction}{}{complexExpand complexIntegrate expand integrate split} +% +\condata{IR2F}{IntegrationResultToFunction}{}{complexExpand expand split} +% +\condata{INTTOOLS}{IntegrationTools}{}{kmax ksec mkPrim union vark varselect} +% +\condata{INVLAPLA}{InverseLaplaceTransform}{}{inverseLaplace} +% +\condata{IRREDFFX}{IrredPolyOverFiniteField}{}{generateIrredPoly} +% +\condata{IRSN}{IrrRepSymNatPackage}{}{dimensionOfIrreducibleRepresentation irreducibleRepresentation} +% +\condata{KERNEL2}{KernelFunctions2}{}{constantIfCan constantKernel} +% +\condata{KOVACIC}{Kovacic}{}{kovacic} +% +\condata{LAPLACE}{LaplaceTransform}{}{laplace} +% +\condata{LEADCDET}{LeadingCoefDetermination}{}{distFact polCase} +% +\condata{LINDEP}{LinearDependence}{}{linearDependence linearlyDependent? solveLinear} +% +\condata{LPEFRAC}{LinearPolynomialEquationByFractions}{}{solveLinearPolynomialEquationByFractions} +% +\condata{LSMP}{LinearSystemMatrixPackage}{}{aSolution hasSolution? rank solve} +% +\condata{LSPP}{LinearSystemPolynomialPackage}{}{linSolve} +% +\condata{LGROBP}{LinGrobnerPackage}{}{anticoord choosemon computeBasis coordinate groebgen intcompBasis linGenPos + minPol totolex transform} +% +\condata{LF}{LiouvillianFunction}{}{Ci Ei Si belong? dilog erf integral li operator} +% +\condata{LIST2}{ListFunctions2}{}{map reduce scan} +% +\condata{LIST3}{ListFunctions3}{}{map} +% +\condata{LIST2MAP}{ListToMap}{}{match} +% +\condata{MKBCFUNC}{MakeBinaryCompiledFunction}{}{binaryFunction compiledFunction} +% +\condata{MKFLCFN}{MakeFloatCompiledFunction}{}{makeFloatFunction} +% +\condata{MKFUNC}{MakeFunction}{}{function} +% +\condata{MKRECORD}{MakeRecord}{}{makeRecord} +% +\condata{MKUCFUNC}{MakeUnaryCompiledFunction}{}{compiledFunction unaryFunction} +% +\condata{MAPPKG1}{MappingPackage1}{}{{\tt **} coerce fixedPoint id nullary recur} +% +\condata{MAPPKG2}{MappingPackage2}{}{const constant curry diag} +% +\condata{MAPPKG3}{MappingPackage3}{}{{\tt *} constantLeft constantRight curryLeft curryRight twist} +% +\condata{MAPHACK1}{MappingPackageInternalHacks1}{}{iter recur} +% +\condata{MAPHACK2}{MappingPackageInternalHacks2}{}{arg1 arg2} +% +\condata{MAPHACK3}{MappingPackageInternalHacks3}{}{comp} +% +\condata{MATCAT2}{MatrixCategoryFunctions2}{}{map reduce} +% +\condata{MCDEN}{MatrixCommonDenominator}{}{clearDenominator commonDenominator splitDenominator} +% +\condata{MATLIN}{MatrixLinearAlgebraFunctions}{}{determinant inverse minordet nullSpace nullity rank rowEchelon} +% +\condata{MTHING}{MergeThing}{}{mergeDifference} +% +\condata{MESH}{MeshCreationRoutinesForThreeDimensions}{}{meshFun2Var meshPar1Var meshPar2Var ptFunc} +% +\condata{MDDFACT}{ModularDistinctDegreeFactorizer}{}{ddFact exptMod factor gcd separateFactors} +% +\condata{MHROWRED}{ModularHermitianRowReduction}{}{rowEch rowEchelon} +% +\condata{MRF2}{MonoidRingFunctions2}{}{map} +% +\condata{MSYSCMD}{MoreSystemCommands}{}{systemCommand} +% +\condata{MPC2}{MPolyCatFunctions2}{}{map reshape} +% +\condata{MPC3}{MPolyCatFunctions3}{}{map} +% +\condata{MPRFF}{MPolyCatRationalFunctionFactorizer}{}{factor pushdown pushdterm pushucoef pushuconst pushup totalfract} +% +\condata{MRATFAC}{MRationalFactorize}{}{factor} +% +\condata{MFINFACT}{MultFiniteFactorize}{}{factor} +% +\condata{MMAP}{MultipleMap}{}{map} +% +\condata{MULTFACT}{MultivariateFactorize}{}{factor} +% +\condata{MLIFT}{MultivariateLifting}{}{corrPoly lifting lifting1} +% +\condata{MULTSQFR}{MultivariateSquareFree}{}{squareFree squareFreePrim} +% +\condata{NCODIV}{NonCommutativeOperatorDivision}{}{leftDivide leftExactQuotient leftGcd leftLcm leftQuotient + leftRemainder} +% +\condata{NONE1}{NoneFunctions1}{}{coerce} +% +\condata{NODE1}{NonLinearFirstOrderODESolver}{}{solve} +% +\condata{NLINSOL}{NonLinearSolvePackage}{}{solve solveInField} +% +\condata{NPCOEF}{NPCoef}{}{listexp npcoef} +% +\condata{NFINTBAS}{NumberFieldIntegralBasis}{}{discriminant integralBasis} +% +\condata{NUMFMT}{NumberFormats}{}{FormatArabic FormatRoman ScanArabic ScanRoman} +% +\condata{NTPOLFN}{NumberTheoreticPolynomialFunctions}{}{bernoulliB cyclotomic eulerE} +% +\condata{NUMODE}{NumericalOrdinaryDifferentialEquations}{}{rk4 rk4a rk4f rk4qc} +% +\condata{NUMQUAD}{NumericalQuadrature}{}{aromberg asimpson atrapezoidal romberg rombergo simpson simpsono trapezoidal + trapezoidalo} +% +\condata{NCEP}{NumericComplexEigenPackage}{}{characteristicPolynomial complexEigenvalues complexEigenvectors} +% +\condata{NCNTFRAC}{NumericContinuedFraction}{}{continuedFraction} +% +\condata{NREP}{NumericRealEigenPackage}{}{characteristicPolynomial realEigenvalues realEigenvectors} +% +\condata{NUMTUBE}{NumericTubePlot}{}{tube} +% +\condata{NUMERIC}{Numeric}{}{complexNumeric numeric} +% +\condata{OCTCT2}{OctonionCategoryFunctions2}{}{map} +% +\condata{ODEINT}{ODEIntegration}{}{expint int} +% +\condata{ODETOOLS}{ODETools}{}{particularSolution variationOfParameters wronskianMatrix} +% +\condata{ARRAY12}{OneDimensionalArrayFunctions2}{}{map reduce scan} +% +\condata{ONECOMP2}{OnePointCompletionFunctions2}{}{map} +% +\condata{OPQUERY}{OperationsQuery}{}{getDatabase} +% +\condata{ORDCOMP2}{OrderedCompletionFunctions2}{}{map} +% +\condata{ORDFUNS}{OrderingFunctions}{}{pureLex reverseLex totalLex} +% +\condata{ORTHPOL}{OrthogonalPolynomialFunctions}{}{ChebyshevU chebyshevT hermiteH laguerreL legendreP} +% +\condata{OUT}{OutputPackage}{}{output} +% +\condata{PADEPAC}{PadeApproximantPackage}{}{pade} +% +\condata{PADE}{PadeApproximants}{}{pade padecf} +% +\condata{YSTREAM}{ParadoxicalCombinatorsForStreams}{}{Y} +% +\condata{PARTPERM}{PartitionsAndPermutations}{}{conjugate conjugates partitions permutations sequences shuffle + shufflein} +% +\condata{PATTERN1}{PatternFunctions1}{}{addBadValue badValues predicate satisfy? suchThat} +% +\condata{PATTERN2}{PatternFunctions2}{}{map} +% +\condata{PMASS}{PatternMatchAssertions}{}{assert constant multiple optional} +% +\condata{PMFS}{PatternMatchFunctionSpace}{}{patternMatch} +% +\condata{PMINS}{PatternMatchIntegerNumberSystem}{}{patternMatch} +% +\condata{PMKERNEL}{PatternMatchKernel}{}{patternMatch} +% +\condata{PMLSAGG}{PatternMatchListAggregate}{}{patternMatch} +% +\condata{PMPLCAT}{PatternMatchPolynomialCategory}{}{patternMatch} +% +\condata{PMDOWN}{PatternMatchPushDown}{}{fixPredicate patternMatch} +% +\condata{PMQFCAT}{PatternMatchQuotientFieldCategory}{}{patternMatch} +% +\condata{PATRES2}{PatternMatchResultFunctions2}{}{map} +% +\condata{PMSYM}{PatternMatchSymbol}{}{patternMatch} +% +\condata{PMTOOLS}{PatternMatchTools}{}{patternMatch patternMatchTimes} +% +\condata{PATMATCH}{PatternMatch}{}{Is is?} +% +\condata{PERMAN}{Permanent}{}{permanent} +% +\condata{PGE}{PermutationGroupExamples}{}{abelianGroup alternatingGroup cyclicGroup dihedralGroup janko2 mathieu11 + mathieu12 mathieu22 mathieu23 mathieu24 rubiksGroup symmetricGroup youngGroup} +% +\condata{PICOERCE}{PiCoercions}{}{coerce} +% +\condata{PLOT1}{PlotFunctions1}{}{plot plotPolar} +% +\condata{PLOTTOOL}{PlotTools}{}{calcRanges} +% +\condata{PTFUNC2}{PointFunctions2}{}{map} +% +\condata{PTPACK}{PointPackage}{}{color hue phiCoord rCoord shade thetaCoord xCoord yCoord zCoord} +% +\condata{PFOQ}{PointsOfFiniteOrderRational}{}{order torsion? torsionIfCan} +% +\condata{PFOTOOLS}{PointsOfFiniteOrderTools}{}{badNum doubleDisc getGoodPrime mix polyred} +% +\condata{PFO}{PointsOfFiniteOrder}{}{order torsion? torsionIfCan} +% +\condata{POLTOPOL}{PolToPol}{}{dmpToNdmp dmpToP ndmpToDmp ndmpToP pToDmp pToNdmp} +% +\condata{PGROEB}{PolyGroebner}{}{lexGroebner totalGroebner} +% +\condata{PAN2EXPR}{PolynomialAN2Expression}{}{coerce} +% +\condata{POLYLIFT}{PolynomialCategoryLifting}{}{map} +% +\condata{POLYCATQ}{PolynomialCategoryQuotientFunctions}{}{isExpt isPlus isPower isTimes mainVariable multivariate + univariate variables} +% +\condata{PFBRU}{PolynomialFactorizationByRecursionUnivariate}{}{bivariateSLPEBR factorByRecursion factorSFBRlcUnit + factorSquareFreeByRecursion randomR solveLinearPolynomialEquationByRecursion} +% +\condata{PFBR}{PolynomialFactorizationByRecursion}{}{bivariateSLPEBR factorByRecursion factorSFBRlcUnit + factorSquareFreeByRecursion randomR solveLinearPolynomialEquationByRecursion} +% +\condata{POLY2}{PolynomialFunctions2}{}{map} +% +\condata{PGCD}{PolynomialGcdPackage}{}{gcd gcdPrimitive} +% +\condata{PINTERPA}{PolynomialInterpolationAlgorithms}{}{LagrangeInterpolation} +% +\condata{PINTERP}{PolynomialInterpolation}{}{interpolate} +% +\condata{PNTHEORY}{PolynomialNumberTheoryFunctions}{}{bernoulli chebyshevT chebyshevU cyclotomic euler fixedDivisor + hermite laguerre legendre} +% +\condata{POLYROOT}{PolynomialRoots}{}{froot qroot rroot} +% +\condata{SOLVEFOR}{PolynomialSolveByFormulas}{}{aCubic aLinear aQuadratic aQuartic aSolution cubic linear mapSolve + quadratic quartic solve} +% +\condata{PSQFR}{PolynomialSquareFree}{}{squareFree} +% +\condata{POLY2UP}{PolynomialToUnivariatePolynomial}{}{univariate} +% +\condata{LIMITPS}{PowerSeriesLimitPackage}{}{complexLimit limit} +% +\condata{PRIMARR2}{PrimitiveArrayFunctions2}{}{map reduce scan} +% +\condata{PRIMELT}{PrimitiveElement}{}{primitiveElement} +% +\condata{ODEPRIM}{PrimitiveRatDE}{}{denomLODE} +% +\condata{ODEPRRIC}{PrimitiveRatRicDE}{}{changevar constantCoefficientRicDE denomRicDE leadingCoefficientRicDE polyRicDE + singRicDE} +% +\condata{PRINT}{PrintPackage}{}{print} +% +\condata{INTPAF}{PureAlgebraicIntegration}{}{palgLODE palgRDE palgextint palgint palglimint} +% +\condata{ODEPAL}{PureAlgebraicLODE}{}{algDsolve} +% +\condata{QALGSET2}{QuasiAlgebraicSet2}{}{radicalSimplify} +% +\condata{QUATCT2}{QuaternionCategoryFunctions2}{}{map} +% +\condata{QFCAT2}{QuotientFieldCategoryFunctions2}{}{map} +% +\condata{REP}{RadicalEigenPackage}{}{eigenMatrix gramschmidt normalise orthonormalBasis radicalEigenvalues + radicalEigenvector radicalEigenvectors} +% +\condata{SOLVERAD}{RadicalSolvePackage}{}{contractSolve radicalRoots radicalSolve} +% +\condata{RADUTIL}{RadixUtilities}{}{radix} +% +\condata{RANDSRC}{RandomNumberSource}{}{randnum reseed size} +% +\condata{RATFACT}{RationalFactorize}{}{factor} +% +\condata{DEFINTRF}{RationalFunctionDefiniteIntegration}{}{integrate} +% +\condata{RFFACTOR}{RationalFunctionFactorizer}{}{factorFraction} +% +\condata{RFFACT}{RationalFunctionFactor}{}{factor} +% +\condata{INTRF}{RationalFunctionIntegration}{}{extendedIntegrate infieldIntegrate internalIntegrate limitedIntegrate} +% +\condata{LIMITRF}{RationalFunctionLimitPackage}{}{complexLimit limit} +% +\condata{SIGNRF}{RationalFunctionSign}{}{sign} +% +\condata{SUMRF}{RationalFunctionSum}{}{sum} +% +\condata{RF}{RationalFunction}{}{coerce eval mainVariable multivariate univariate variables} +% +\condata{INTRAT}{RationalIntegration}{}{extendedint infieldint integrate limitedint} +% +\condata{ODERAT}{RationalLODE}{}{ratDsolve} +% +\condata{RATRET}{RationalRetractions}{}{rational rational? rationalIfCan} +% +\condata{ODERTRIC}{RationalRicDE}{}{changevar constantCoefficientRicDE polyRicDE ricDsolve singRicDE} +% +\condata{RTODETLS}{RatODETools}{}{genericPolynomial} +% +\condata{REALSOLV}{RealSolvePackage}{}{realSolve solve} +% +\condata{REAL0Q}{RealZeroPackageQ}{}{realZeros refine} +% +\condata{REAL0}{RealZeroPackage}{}{midpoint midpoints realZeros refine} +% +\condata{RMCAT2}{RectangularMatrixCategoryFunctions2}{}{map reduce} +% +\condata{RDIV}{ReducedDivisor}{}{order} +% +\condata{ODERED}{ReduceLODE}{}{reduceLODE} +% +\condata{REDORDER}{ReductionOfOrder}{}{ReduceOrder} +% +\condata{REPDB}{RepeatedDoubling}{}{double} +% +\condata{REPSQ}{RepeatedSquaring}{}{expt} +% +\condata{REP1}{RepresentationPackage1}{}{antisymmetricTensors createGenericMatrix permutationRepresentation + symmetricTensors tensorProduct} +% +\condata{REP2}{RepresentationPackage2}{}{areEquivalent? completeEchelonBasis createRandomElement cyclicSubmodule + isAbsolutelyIrreducible? meatAxe scanOneDimSubspaces split standardBasisOfCyclicSubmodule} +% +\condata{RESLATC}{ResolveLatticeCompletion}{}{coerce} +% +\condata{RETSOL}{RetractSolvePackage}{}{solveRetract} +% +\condata{SAERFFC}{SAERationalFunctionAlgFactor}{}{factor} +% +\condata{SEGBIND2}{SegmentBindingFunctions2}{}{map} +% +\condata{SEG2}{SegmentFunctions2}{}{map} +% +\condata{SAEFACT}{SimpleAlgebraicExtensionAlgFactor}{}{factor} +% +\condata{DFLOATSFUN}{DoubleFloatSpecialFunctions}{}{Beta Gamma airyAi airyBi besselI besselJ besselK besselY digamma + hypergeometric0F1 logGamma polygamma} +% +\condata{SCACHE}{SortedCache}{}{cache clearCache enterInCache} +% +\condata{SUP2}{SparseUnivariatePolynomialFunctions2}{}{map} +% +\condata{SPECOUT}{SpecialOutputPackage}{}{outputAsFortran outputAsScript outputAsTex} +% +\condata{MATSTOR}{StorageEfficientMatrixOperations}{}{{\tt **} copy! leftScalarTimes! minus! plus! power! + rightScalarTimes! times!} +% +\condata{STREAM1}{StreamFunctions1}{}{concat} +% +\condata{STREAM2}{StreamFunctions2}{}{map reduce scan} +% +\condata{STREAM3}{StreamFunctions3}{}{map} +% +\condata{STTAYLOR}{StreamTaylorSeriesOperations}{}{{\tt *} {\tt +} {\tt -} {\tt /} addiag coerce compose deriv eval + evenlambert gderiv generalLambert int integers integrate invmultisect lagrange lambert lazyGintegrate lazyIntegrate + mapdiv mapmult monom multisect nlde oddintegers oddlambert power powern recip revert} +% +\condata{STTF}{StreamTranscendentalFunctions}{}{{\tt **} acos acosh acot acoth acsc acsch asec asech asin asinh atan + atanh cos cosh cot coth csc csch exp log sec sech sin sincos sinh sinhcosh tan tanh} +% +\condata{SUBRESP}{SubResultantPackage}{}{primitivePart subresultantVector} +% +\condata{SYMFUNC}{SymmetricFunctions}{}{symFunc} +% +\condata{SGCF}{SymmetricGroupCombinatoricFunctions}{}{coleman inverseColeman listYoungTableaus makeYoungTableau + nextColeman nextLatticePermutation nextPartition numberOfImproperPartitions subSet unrankImproperPartitions0 + unrankImproperPartitions1} +% +\condata{ODESYS}{SystemODESolver}{}{solveInField triangulate} +% +\condata{SYSSOLP}{SystemSolvePackage}{}{solve triangularSystems} +% +\condata{TABLBUMP}{TableauxBumpers}{}{bat bat1 bumprow bumptab bumptab1 inverse lex maxrow mr slex tab tab1 untab} +% +\condata{TANEXP}{TangentExpansions}{}{tanAn tanNa tanSum} +% +\condata{TOOLSIGN}{ToolsForSign}{}{direction nonQsign sign} +% +\condata{DRAWCURV}{TopLevelDrawFunctionsForAlgebraicCurves}{}{draw} +% +\condata{DRAWCFUN}{TopLevelDrawFunctionsForCompiledFunctions}{}{draw makeObject recolor} +% +\condata{DRAW}{TopLevelDrawFunctions}{}{draw makeObject} +% +\condata{TOPSP}{TopLevelThreeSpace}{}{createThreeSpace} +% +\condata{INTHERTR}{TranscendentalHermiteIntegration}{}{HermiteIntegrate} +% +\condata{INTTR}{TranscendentalIntegration}{}{expextendedint expintegrate expintfldpoly explimitedint primextendedint + primextintfrac primintegrate primintegratefrac primintfldpoly primlimintfrac primlimitedint} +% +\condata{TRMANIP}{TranscendentalManipulations}{}{cos2sec cosh2sech cot2tan cot2trig coth2tanh coth2trigh csc2sin + csch2sinh expand expandLog expandPower htrigs removeCosSq removeCoshSq removeSinSq removeSinhSq sec2cos sech2cosh + simplify simplifyExp sin2csc sinh2csch tan2cot tan2trig tanh2coth tanh2trigh} +% +\condata{RDETR}{TranscendentalRischDE}{}{DSPDE SPDE baseRDE expRDE primRDE} +% +\condata{SOLVESER}{TransSolvePackageService}{}{decomposeFunc unvectorise} +% +\condata{SOLVETRA}{TransSolvePackage}{}{solve} +% +\condata{TRIMAT}{TriangularMatrixOperations}{}{LowTriBddDenomInv UpTriBddDenomInv} +% +\condata{TRIGMNIP}{TrigonometricManipulations}{}{complexElementary complexNormalize imag real real? trigs} +% +\condata{TUBETOOL}{TubePlotTools}{}{{\tt *} {\tt +} {\tt -} cosSinInfo cross dot loopPoints point unitVector} +% +\condata{CLIP}{TwoDimensionalPlotClipping}{}{clip clipParametric clipWithRanges} +% +\condata{TWOFACT}{TwoFactorize}{}{generalSqFr generalTwoFactor twoFactor} +% +\condata{UNIFACT}{UnivariateFactorize}{}{factor factorSquareFree genFact henselFact henselfact quadratic sqroot + trueFactors} +% +\condata{ULS2}{UnivariateLaurentSeriesFunctions2}{}{map} +% +\condata{UPOLYC2}{UnivariatePolynomialCategoryFunctions2}{}{map} +% +\condata{UPCDEN}{UnivariatePolynomialCommonDenominator}{}{clearDenominator commonDenominator splitDenominator} +% +\condata{UP2}{UnivariatePolynomialFunctions2}{}{map} +% +\condata{UPSQFREE}{UnivariatePolynomialSquareFree}{}{BumInSepFFE squareFree squareFreePart} +% +\condata{UPXS2}{UnivariatePuiseuxSeriesFunctions2}{}{map} +% +\condata{UTS2}{UnivariateTaylorSeriesFunctions2}{}{map} +% +\condata{UTSODE}{UnivariateTaylorSeriesODESolver}{}{mpsode ode ode1 ode2 stFunc1 stFunc2 stFuncN} +% +\condata{UNISEG2}{UniversalSegmentFunctions2}{}{map} +% +\condata{UDPO}{UserDefinedPartialOrdering}{}{getOrder largest less? more? setOrder userOrdered?} +% +\condata{UDVO}{UserDefinedVariableOrdering}{}{getVariableOrder resetVariableOrder setVariableOrder} +% +\condata{VECTOR2}{VectorFunctions2}{}{map reduce scan} +% +\condata{VIEWDEF}{ViewDefaultsPackage}{}{axesColorDefault lineColorDefault pointColorDefault pointSizeDefault + tubePointsDefault tubeRadiusDefault unitsColorDefault var1StepsDefault var2StepsDefault viewDefaults viewPosDefault + viewSizeDefault viewWriteAvailable viewWriteDefault} +% +\condata{VIEW}{ViewportPackage}{}{coerce drawCurves graphCurves} +% +\condata{WEIER}{WeierstrassPreparation}{}{cfirst clikeUniv crest qqq sts2stst weierstrass} +% +\condata{WFFINTBS}{WildFunctionFieldIntegralBasis}{}{integralBasis listSquaredFactors} +% +% +% ---------------------------------------------------------------------- +%\end{constructorListing} +% ---------------------------------------------------------------------- + + +\setcounter{chapter}{4} % Appendix E +% +{ +%\twocolumn[% +\chapter{Operations} +\label{ugAppOperations} + +This appendix contains a partial list of Axiom operations +with brief descriptions. +For more details, use the Browse facility of HyperDoc: +enter the name of the operation for which you want more information +in the input area on the main Browse menu and then click on +{\bf Operations.} + +\vskip \baselineskip +%] +\appendix{Operations} +\def\alt#1#2{{$\lbrace$#1$\mid$#2$\rbrace$}} +\def\altx#1#2#3{{$\lbrace$#1$\mid$#2$\mid$#3$\rbrace$}} +\def\opt#1{{$\,\lbrack$#1$\rbrack$}} +\def\bigLeftBrack{{\tt \[}} +\def\bigRightBrack{{\tt \]}} +\def\smallLeftBrack{{\tt \[}} +\def\smallRightBrack{{\tt \]}} +\def\optinit#1{{$\lbrack$#1$\rbrack$}} +\def\optfirst#1{{\lbrack#1\rbrack}} +\def\from#1{{From {\bf #1}.}} +\def\consultType#1{{Consult {\bf #1} using Browse for details.}} +\def\colx#1#2#3{{\lbrack#1,#2,\ldots,#3\rbrack}} +\def\col#1#2{{\lbrack#1,\ldots,\rbrack}} +\def\code#1{{\tt #1}} +\def\Unix{{\bf Unix}} +\def\Script{IBM SCRIPT Formula Formatter} + +\def\smallbreak{{\hfill{\break}}} +\def\newitem{{\smallbreak}} +\def\bigitem{{\medbreak}} +\def\largerbreak{{\hfill{\smallskip\break}}} +\def\medbreak{{\hfill{\medskip\break}}} +\def\bigbreak{{\hfill{\bigskip\break}}} +\long\def\bigopkey#1{{#1}} +\long\def\opkey#1{{#1}} + +\def\and{{\ {\bf and}\ }} +\def\or{{\ {\bf or}\ }} +\def\mod{{\ {\bf mod}\ }} +\def\quo{{\ {\bf quo}\ }} +\def\rem{{\ {\bf rem}\ }} +\def\opLeftPren{\nobreak\,{\tt (}} +\def\opRightPren{\nobreak{\tt )}\allowbreak} + +\def\seeType#1{{See {\bf #1} using Browse.}} +\def\seeAlso#1{{See also #1}.} +\def\seeOther#1{{For additional information on $#1$, consult Browse.}} +\def\sayOption#1#2{{This command may be given as a draw option: $#1 == #2$.}} +\def\seeDetails#1{{Consult {\bf #1} using Browse for details.}} + +\long\def\opdataQual#1#2#3#4#5#6{{ + \opdata{#1}{#2}{#3}{#4}{#5\newitem#6} +}} + +\long\def\opdata#1#2#3#4#5{{ + %#1 name #2 number of args #3 sig #4 con #5 documentation + \hyphenpenalty=1000 + \exhyphenpenalty=1000 + \par\vskip 4pt\optitle{#1}\nopagebreak\par\vskip -\parskip\vskip 2pt\nopagebreak\noindent% + {\def\TYsize{\SMTYfont}#5}\par}} + +\def\keydata#1#2#3#4#5{{\opdata{#1}{#2}{#3}{#4}{#5}}} +\def\keyop#1{{{\large{$#1$}}}} +\def\optitle#1{{\bf #1}} +\def\opand{\optand} +\def\optand{{\par\vskip -\parskip}} +\def\optinner#1{{\lbrack#1\rbrack}} +\def\opname#1{{\tt #1}} +\def\opoption#1#2{{{\tt #1}$== #2$}} + +\long\def\xxdata#1#2#3#4#5{{}} + +\def\indented#1{{#1}} +\def\spadsyscom#1{{#1}} + +\sloppy\raggedright +%\input{oplist} + +\fussy +}\onecolumn + + +\setcounter{chapter}{5} % Appendix F + +\chapter{Programs for AXIOM Images} +\label{ugAppGraphics} + +% +This appendix contains the Axiom programs used to generate +the images in the gallery color insert of this book. +All these input files are included +with the Axiom system. +To produce the images +on page 6 of the gallery insert, for example, issue the command: +\begin{verbatim} +)read images6 +\end{verbatim} + +These images were produced on an IBM RS/6000 model 530 with a +standard color graphics adapter. The smooth shaded images +were made from X Window System screen dumps. +The remaining images were produced with Axiom-generated +PostScript output. The images were reproduced from slides made on an Agfa +ChromaScript PostScript interpreter with a Matrix Instruments QCR camera. + + +\section{images1.input} +\label{ugFimagesOne} + + +\begin{verbatim} +)read tknot Read torus knot program + +torusKnot(15,17, 0.1, 6, 700) A (15,17) torus knot +\end{verbatim} +\index{torus knot} + +\newpage + +\section{images2.input} +\label{ugFimagesTwo} + +These images illustrate how Newton's method converges when computing the +\index{Newton iteration} +complex cube roots of 2. Each point in the $(x,y)$-plane represents the +complex number $x + iy,$ which is given as a starting point for Newton's +method. The poles in these images represent bad starting values. +The flat areas are the regions of convergence to the three roots. + +\begin{verbatim} +)read newton Read the programs from +)read vectors Chapter 10 +f := newtonStep(x**3 - 2) Create a Newton's iteration + function for $x^3 = 2$ +\end{verbatim} + +The function $f^n$ computes $n$ steps of Newton's method. + +\begin{verbatim} +clipValue := 4 Clip values with magnitude > 4 +drawComplexVectorField(f**3, -3..3, -3..3) The vector field for $f^3$ +drawComplex(f**3, -3..3, -3..3) The surface for $f^3$ +drawComplex(f**4, -3..3, -3..3) The surface for $f^4$ +\end{verbatim} + +\section{images3.input} +\label{ugFimagesThree} + + +\begin{verbatim} +)r tknot +for i in 0..4 repeat torusKnot(2, 2 + i/4, 0.5, 25, 250) +\end{verbatim} + +\section{images5.input} +\label{ugFimagesFive} + + +The parameterization of the Etruscan Venus is due to George Frances. +\index{Etruscan Venus} + +\begin{verbatim} +venus(a,r,steps) == + surf := (u:DFLOAT, v:DFLOAT): Point DFLOAT +-> + cv := cos(v) + sv := sin(v) + cu := cos(u) + su := sin(u) + x := r * cos(2*u) * cv + sv * cu + y := r * sin(2*u) * cv - sv * su + z := a * cv + point [x,y,z] + draw(surf, 0..\%pi, -\%pi..\%pi, var1Steps==steps, + var2Steps==steps, title == "Etruscan Venus") + +venus(5/2, 13/10, 50) The Etruscan Venus +\end{verbatim} + +The Figure-8 Klein Bottle +\index{Klein bottle} +parameterization is from +``Differential Geometry and Computer Graphics'' by Thomas Banchoff, +in {\it Perspectives in Mathematics,} Anniversary of Oberwolfasch 1984, +Birkh\"{a}user-Verlag, Basel, pp. 43-60. + +\begin{verbatim} +klein(x,y) == + cx := cos(x) + cy := cos(y) + sx := sin(x) + sy := sin(y) + sx2 := sin(x/2) + cx2 := cos(x/2) + sq2 := sqrt(2.0@DFLOAT) + point [cx * (cx2 * (sq2 + cy) + (sx2 * sy * cy)), _ + sx * (cx2 * (sq2 + cy) + (sx2 * sy * cy)), _ + -sx2 * (sq2 + cy) + cx2 * sy * cy] + +draw(klein, 0..4*\%pi, 0..2*\%pi, var1Steps==50, Figure-8 Klein bottle + var2Steps==50,title=="Figure Eight Klein Bottle") +\end{verbatim} + +The next two images are examples of generalized tubes. + +\begin{verbatim} +)read ntube +rotateBy(p, theta) == Rotate a point $p$ by + c := cos(theta) $\theta$ around the origin + s := sin(theta) + point [p.1*c - p.2*s, p.1*s + p.2*c] + +bcircle t == A circle in three-space + point [3*cos t, 3*sin t, 0] + +twist(u, t) == An ellipse that twists + theta := 4*t around four times as + p := point [sin u, cos(u)/2] $t$ revolves once + rotateBy(p, theta) + +ntubeDrawOpt(bcircle, twist, 0..2*\%pi, 0..2*\%pi, Twisted Torus + var1Steps == 70, var2Steps == 250) + +twist2(u, t) == Create a twisting circle + theta := t + p := point [sin u, cos(u)] + rotateBy(p, theta) + +cf(u,v) == sin(21*u) Color function with $21$ stripes + +ntubeDrawOpt(bcircle, twist2, 0..2*\%pi, 0..2*\%pi, Striped Torus + colorFunction == cf, var1Steps == 168, + var2Steps == 126) +\end{verbatim} + +\section{images6.input} +\label{ugFimagesSix} + +\begin{verbatim} +gam(x,y) == The height and color are the + g := Gamma complex(x,y) real and argument parts + point [x,y,max(min(real g, 4), -4), argument g] of the Gamma function, + respectively. + +draw(gam, -\%pi..\%pi, -\%pi..\%pi, The Gamma Function + title == "Gamma(x + \%i*y)", _ + var1Steps == 100, var2Steps == 100) + +b(x,y) == Beta(x,y) + +draw(b, -3.1..3, -3.1 .. 3, title == "Beta(x,y)") The Beta Function + +atf(x,y) == + a := atan complex(x,y) + point [x,y,real a, argument a] + +draw(atf, -3.0..\%pi, -3.0..\%pi) The Arctangent function +\end{verbatim} +\index{function!Gamma} +\index{function!Euler Beta} +\index{Euler!Beta function} + + +\section{images7.input} +\label{ugFimagesSeven} + +First we look at the conformal +\index{conformal map} +map $z \mapsto z + 1/z$. +\begin{verbatim} +)read conformal Read program for drawing + conformal maps + +f z == z The coordinate grid for the + complex plane +conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") Mapping 1: Source + +f z == z + 1/z The map $z \mapsto z + 1/z$ + +conformalDraw(f, -2..2, -2..2, 9, 9, "cartesian") Mapping 1: Target +\end{verbatim} + +The map $z \mapsto -(z+1)/(z-1)$ maps +the unit disk to the right half-plane, as shown +\index{Riemann!sphere} +on the Riemann sphere. + +\begin{verbatim} +f z == z The unit disk + +riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") Mapping 2: Source + +f z == -(z+1)/(z-1) The map $x \mapsto -(z+1)/(z-1)$ + +riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar") Mapping 2: Target + +riemannSphereDraw(-4..4, -4..4, 7, 7, "cartesian") Riemann Sphere Mapping +\end{verbatim} + +\section{images8.input} +\label{ugFimagesEight} + +\begin{verbatim} +)read dhtri +)read tetra +drawPyramid 4 Sierpinsky's Tetrahedron + +Sierpinsky's Tetrahedron +)read antoine +drawRings 2 Antoine's Necklace + +Aintoine's Necklace +)read scherk +drawScherk(3,3) Scherk's Minimal Surface + +)read ribbonsnew +drawRibbons([x**i for i in 1..5], x=-1..1, y=0..2) Ribbon Plot +\end{verbatim} +\index{Scherk's minimal surface} + + +%\input{gallery/conformal.htex} +\section{conformal.input} +\label{ugFconformal} +% +The functions in this section draw conformal maps both on the +\index{conformal map} +plane and on the Riemann sphere. +\index{Riemann!sphere} + +%-- Compile, don't interpret functions. +%\xmpLine{)set fun comp on}{} +\begin{verbatim} +C := Complex DoubleFloat Complex Numbers +S := Segment DoubleFloat Draw ranges +R3 := Point DFLOAT Points in 3-space +\end{verbatim} + +{\bf conformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)} +draws the image of the coordinate grid under {\it f} in the complex plane. +The grid may be given in either polar or Cartesian coordinates. +Argument {\it f} is the function to draw; +{\it rRange} is the range of the radius (in polar) or real (in Cartesian); +{\it tRange} is the range of $\theta$ (in polar) or imaginary (in Cartesian); +{\it tSteps, rSteps}, are the number of intervals in the {\it r} and +$\theta$ directions; and +{\it coord} is the coordinate system to use (either {\tt "polar"} or +{\tt "cartesian"}). + +\begin{verbatim} +conformalDraw: (C -> C, S, S, PI, PI, String) -> VIEW3D +conformalDraw(f,rRange,tRange,rSteps,tSteps,coord) == + transformC := Function for changing an $(x,y)$ + coord = "polar" => polar2Complex pair into a complex number + cartesian2Complex + cm := makeConformalMap(f, transformC) + sp := createThreeSpace() Create a fresh space + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Plot the coordinate lines + makeViewport3D(sp, "Conformal Map") Draw the image +\end{verbatim} + +{\bf riemannConformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)} +draws the image of the coordinate grid under {\it f} on the Riemann sphere. +The grid may be given in either polar or Cartesian coordinates. +Its arguments are the same as those for {\bf conformalDraw}. + +\begin{verbatim} +riemannConformalDraw:(C->C,S,S,PI,PI,String)->VIEW3D +riemannConformalDraw(f, rRange, tRange, + rSteps, tSteps, coord) == + transformC := Function for changing an $(x,y)$ + coord = "polar" => polar2Complex pair into a complex number + cartesian2Complex + sp := createThreeSpace() Create a fresh space + cm := makeRiemannConformalMap(f, transformC) + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Plot the coordinate lines + curve(sp,[point [0,0,2.0@DFLOAT,0],point [0,0,2.0@DFLOAT,0] ]) + Add an invisible point at + makeViewport3D(sp,"Map on the Riemann Sphere") the north pole for scaling + +adaptGrid(sp, f, uRange, vRange, uSteps, vSteps) == Plot the coordinate grid + delU := (hi(uRange) - lo(uRange))/uSteps using adaptive plotting for + delV := (hi(vRange) - lo(vRange))/vSteps coordinate lines, and draw + uSteps := uSteps + 1; vSteps := vSteps + 1 tubes around the lines + u := lo uRange + for i in 1..uSteps repeat Draw coordinate lines in the $v$ + c := curryLeft(f,u) direction; curve $c$ fixes the + cf := (t:DFLOAT):DFLOAT +-> 0 current value of $u$ + makeObject(c,vRange::SEG Float,colorFunction==cf, + Draw the $v$ coordinate line + space == sp, tubeRadius == .02, tubePoints == 6) + u := u + delU + v := lo vRange + for i in 1..vSteps repeat Draw coodinate lines in the $u$ + c := curryRight(f,v) direction; curve $c$ fixes the + cf := (t:DFLOAT):DFLOAT +-> 1 current value of $v$ + makeObject(c,uRange::SEG Float,colorFunction==cf, + Draw the $u$ coordinate line + space == sp, tubeRadius == .02, tubePoints == 6) + v := v + delV + void() + +riemannTransform(z) == Map a point in the complex + r := sqrt norm z plane to the Riemann sphere + cosTheta := (real z)/r + sinTheta := (imag z)/r + cp := 4*r/(4+r**2) + sp := sqrt(1-cp*cp) + if r>2 then sp := -sp + point [cosTheta*cp, sinTheta*cp, -sp + 1] + +cartesian2Complex(r:DFLOAT, i:DFLOAT):C == Convert Cartesian coordinates to + complex(r, i) complex Cartesian form + +polar2Complex(r:DFLOAT, th:DFLOAT):C == Convert polar coordinates to + complex(r*cos(th), r*sin(th)) complex Cartesian form + +makeConformalMap(f, transformC) == Convert complex function $f$ + (u:DFLOAT,v:DFLOAT):R3 +-> to a mapping: + (DFLOAT,DFLOAT) $mapsto$ R3 + z := f transformC(u, v) in the complex plane + point [real z, imag z, 0.0@DFLOAT] + +makeRiemannConformalMap(f, transformC) == Convert a complex function $f$ + (u:DFLOAT, v:DFLOAT):R3 +-> to a mapping: + (DFLOAT,DFLOAT) $\mapsto$ R3 + riemannTransform f transformC(u, v) on the Riemann sphere + +riemannSphereDraw: (S, S, PI, PI, String) -> VIEW3D + Draw a picture of the mapping + of the complex plane to + the Riemann sphere +riemannSphereDraw(rRange,tRange,rSteps,tSteps,coord) == + transformC := + coord = "polar" => polar2Complex + cartesian2Complex + grid := (u:DFLOAT, v:DFLOAT): R3 +-> Coordinate grid function + z1 := transformC(u, v) + point [real z1, imag z1, 0] + sp := createThreeSpace() Create a fresh space + adaptGrid(sp, grid, rRange, tRange, rSteps, tSteps) Draw the flat grid + connectingLines(sp,grid,rRange,tRange,rSteps,tSteps) + makeObject(riemannSphere,0..2*\%pi,0..\%pi,space==sp) Draw the sphere + f := (z:C):C +-> z + cm := makeRiemannConformalMap(f, transformC) + adaptGrid(sp, cm, rRange, tRange, rSteps, tSteps) Draw the sphere grid + makeViewport3D(sp, "Riemann Sphere") + +connectingLines(sp,f,uRange,vRange,uSteps,vSteps) == + Draw the lines that connect + delU := (hi(uRange) - lo(uRange))/uSteps the points in the complex + delV := (hi(vRange) - lo(vRange))/vSteps plane to the north pole + uSteps := uSteps + 1; vSteps := vSteps + 1 of the Riemann sphere + u := lo uRange + for i in 1..uSteps repeat For each u + v := lo vRange + for j in 1..vSteps repeat For each v + p1 := f(u,v) + p2 := riemannTransform complex(p1.1, p1.2) Project p1 onto the sphere + fun := lineFromTo(p1,p2) Create a line function + cf := (t:DFLOAT):DFLOAT +-> 3 + makeObject(fun, 0..1,space==sp,tubePoints==4, Draw the connecting line + tubeRadius==0.01,colorFunction==cf) + v := v + delV + u := u + delU + void() + +riemannSphere(u,v) == A sphere sitting on the + sv := sin(v) complex plane, with radius 1 + 0.99@DFLOAT*(point [cos(u)*sv,sin(u)*sv,cos(v),0.0@DFLOAT])+ + point [0.0@DFLOAT, 0.0@DFLOAT, 1.0@DFLOAT, 4.0@DFLOAT] + +lineFromTo(p1, p2) == Create a line function + d := p2 - p1 that goes from p1 to p2 + (t:DFLOAT):Point DFLOAT +-> + p1 + t*d +\end{verbatim} + +%\input{gallery/tknot.htex} +\section{tknot.input} +\label{ugFtknot} +% +Create a $(p,q)$ torus-knot with radius $r$ around the curve. +The formula was derived by Larry Lambe. + +\begin{verbatim} +)read ntube +torusKnot: (DFLOAT, DFLOAT, DFLOAT, PI, PI) -> VIEW3D +torusKnot(p, q ,r, uSteps, tSteps) == + knot := (t:DFLOAT):Point DFLOAT +-> Function for the torus knot + fac := 4/(2.2@DFLOAT-sin(q*t)) + fac * point [cos(p*t), sin(p*t), cos(q*t)] + circle := (u:DFLOAT, t:DFLOAT): Point DFLOAT +-> The cross section + r * point [cos u, sin u] + ntubeDrawOpt(knot, circle, 0..2*\%pi, 0..2*\%pi, + Draw the circle around the knot + var1Steps == uSteps, var2Steps == tSteps) + +\end{verbatim} + +%\input{gallery/ntube.htex} +\section{ntube.input} +\label{ugFntube} +% +The functions in this file create generalized tubes (also known as generalized +cylinders). +These functions draw a 2-d curve in the normal +planes around a 3-d curve. + +\begin{verbatim} +R3 := Point DFLOAT Points in 3-Space +R2 := Point DFLOAT Points in 2-Space +S := Segment Float Draw ranges + Introduce types for functions for: +ThreeCurve := DFLOAT -> R3 --the space curve function +TwoCurve := (DFLOAT, DFLOAT) -> R2 --the plane curve function +Surface := (DFLOAT, DFLOAT) -> R3 --the surface function + Frenet frames define a +FrenetFrame := coordinate system around a + Record(value:R3,tangent:R3,normal:R3,binormal:R3) + point on a space curve +frame: FrenetFrame The current Frenet frame + for a point on a curve +\end{verbatim} + +{\bf ntubeDraw}{\it (spaceCurve, planeCurve,} +$u_0 .. u_1,$ $t_0 .. t_1)$ +draws {\it planeCurve} in the normal planes of {\it spaceCurve.} +The parameter $u_0 .. u_1$ specifies +the parameter range for {\it planeCurve} +and $t_0 .. t_1$ specifies the parameter range for {\it spaceCurve}. +Additionally, the plane curve function takes +a second parameter: the current parameter of {\it spaceCurve}. +This allows the plane curve to change shape +as it goes around the space curve. +See \ref{ugFimagesFive} for an example of this. +% +\begin{verbatim} +ntubeDraw: (ThreeCurve,TwoCurve,S,S) -> VIEW3D +ntubeDraw(spaceCurve,planeCurve,uRange,tRange) == + ntubeDrawOpt(spaceCurve, planeCurve, uRange, _ + tRange, []$List DROPT) + +ntubeDrawOpt: (ThreeCurve,TwoCurve,S,S,List DROPT) + -> VIEW3D +ntubeDrawOpt(spaceCurve,planeCurve,uRange,tRange,l) == + This function is similar + to ntubeDraw, but takes + delT:DFLOAT := (hi(tRange) - lo(tRange))/10000 optional parameters that it + oldT:DFLOAT := lo(tRange) - 1 passes to the draw command + fun := ngeneralTube(spaceCurve,planeCurve,delT,oldT) + draw(fun, uRange, tRange, l) + +\end{verbatim} + +{\bf nfrenetFrame}{\it (c, t, delT)} +numerically computes the Frenet frame +about the curve {\it c} at {\it t}. +Parameter {\it delT} is a small number used to +compute derivatives. +\begin{verbatim} +nfrenetFrame(c, t, delT) == + f0 := c(t) + f1 := c(t+delT) + t0 := f1 - f0 The tangent + n0 := f1 + f0 + b := cross(t0, n0) The binormal + n := cross(b,t0) The normal + ln := length n + lb := length b + ln = 0 or lb = 0 => + error "Frenet Frame not well defined" + n := (1/ln)*n Make into unit length vectors + b := (1/lb)*b + [f0, t0, n, b]$FrenetFrame +\end{verbatim} + +{\bf ngeneralTube}{\it (spaceCurve, planeCurve,}{\it delT, oltT)} +creates a function that can be passed to the system axiomFun{draw} command. +The function is a parameterized surface for the general tube +around {\it spaceCurve}. {\it delT} is a small number used to compute +derivatives. {\it oldT} is used to hold the current value of the +{\it t} parameter for {\it spaceCurve.} This is an efficiency measure +to ensure that frames are only computed once for each value of {\it t}. +\begin{verbatim} +ngeneralTube: (ThreeCurve, TwoCurve, DFLOAT, DFLOAT) -> Surface +ngeneralTube(spaceCurve, planeCurve, delT, oldT) == + free frame Indicate that $frame$ is global + (v:DFLOAT, t: DFLOAT): R3 +-> + if (t $\sim$= oldT) then If not already computed + frame := nfrenetFrame(spaceCurve, t, delT) compute new frame + oldT := t + p := planeCurve(v, t) + frame.value + p.1*frame.normal + p.2*frame.binormal + Project $p$ into the normal plane +\end{verbatim} + +%\input{gallery/dhtri.htex} +\section{dhtri.input} +\label{ugFdhtri} +% +Create affine transformations (DH matrices) that transform +a given triangle into another. + +\begin{verbatim} +tri2tri: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT) + Compute a DHMATRIX that +tri2tri(t1, t2) == transforms $t1$ to $t2,$ where + n1 := triangleNormal(t1) $t1$ and $t2$ are the vertices + n2 := triangleNormal(t2) of two triangles in 3-space + tet2tet(concat(t1, n1), concat(t2, n2)) + +tet2tet: (List Point DFLOAT, List Point DFLOAT) -> DHMATRIX(DFLOAT) + Compute a DHMATRIX that +tet2tet(t1, t2) == transforms $t1$ to $t2,$ + m1 := makeColumnMatrix t1 where $t1$ and $t2$ are the + m2 := makeColumnMatrix t2 vertices of two tetrahedrons + m2 * inverse(m1) in 3-space + +makeColumnMatrix(t) == Put the vertices of a tetra- + m := new(4,4,0)$DHMATRIX(DFLOAT) hedron into matrix form + for x in t for i in 1..repeat + for j in 1..3 repeat + m(j,i) := x.j + m(4,i) := 1 + m + +triangleNormal(t) == Compute a vector normal to + a := triangleArea t the given triangle, whose + p1 := t.2 - t.1 length is the square root + p2 := t.3 - t.2 of the area of the triangle + c := cross(p1, p2) + len := length(c) + len = 0 => error "degenerate triangle!" + c := (1/len)*c + t.1 + sqrt(a) * c + +triangleArea t == Compute the area of a + a := length(t.2 - t.1) triangle using Heron's + b := length(t.3 - t.2) formula + c := length(t.1 - t.3) + s := (a+b+c)/2 + sqrt(s*(s-a)*(s-b)*(s-c)) +\end{verbatim} + +\section{tetra.input} +\label{ugFtetra} +% +%\input{gallery/tetra.htex} +%\outdent{Sierpinsky's Tetrahedron} + +\begin{verbatim} +)set expose add con DenavitHartenbergMatrix Bring DH matrices into the + environment +x1:DFLOAT := sqrt(2.0@DFLOAT/3.0@DFLOAT) Set up the coordinates of the +x2:DFLOAT := sqrt(3.0@DFLOAT)/6 corners of the tetrahedron. + +p1 := point [-0.5@DFLOAT, -x2, 0.0@DFLOAT] Some needed points +p2 := point [0.5@DFLOAT, -x2, 0.0@DFLOAT] +p3 := point [0.0@DFLOAT, 2*x2, 0.0@DFLOAT] +p4 := point [0.0@DFLOAT, 0.0@DFLOAT, x1] + +baseTriangle := [p2, p1, p3] The base of the tetrahedron + +mt := [0.5@DFLOAT*(p2+p1), 0.5@DFLOAT*(p1+p3), 0.5@DFLOAT*(p3+p2)] + The middle triangle inscribed + in the base of the tetrahedron +bt1 := [mt.1, p1, mt.2] The bases of the triangles of +bt2 := [p2, mt.1, mt.3] the subdivided tetrahedron +bt3 := [mt.2, p3, mt.3] +bt4 := [0.5@DFLOAT*(p2+p4), 0.5@DFLOAT*(p1+p4), 0.5@DFLOAT*(p3+p4)] + +tt1 := tri2tri(baseTriangle, bt1) Create the transformations +tt2 := tri2tri(baseTriangle, bt2) that bring the base of the +tt3 := tri2tri(baseTriangle, bt3) tetrahedron to the bases of +tt4 := tri2tri(baseTriangle, bt4) the subdivided tetrahedron + +drawPyramid(n) == Draw a Sierpinsky tetrahedron + s := createThreeSpace() with $n$ levels of recursive + dh := rotatex(0.0@DFLOAT) subdivision + drawPyramidInner(s, n, dh) + makeViewport3D(s, "Sierpinsky Tetrahedron") + +drawPyramidInner(s, n, dh) == Recursively draw a Sierpinsky + n = 0 => makeTetrahedron(s, dh, n) tetrahedron + drawPyramidInner(s, n-1, dh * tt1) Draw the 4 recursive pyramids + drawPyramidInner(s, n-1, dh * tt2) + drawPyramidInner(s, n-1, dh * tt3) + drawPyramidInner(s, n-1, dh * tt4) + +makeTetrahedron(sp, dh, color) == Draw a tetrahedron into the + w1 := dh*p1 given space with the given + w2 := dh*p2 color, transforming it by + w3 := dh*p3 the given DH matrix + w4 := dh*p4 + polygon(sp, [w1, w2, w4]) + polygon(sp, [w1, w3, w4]) + polygon(sp, [w2, w3, w4]) + void() +\end{verbatim} +\index{Sierpinsky's Tetrahedron} + + +%\input{gallery/antoine.htex} +\section{antoine.input} +\label{ugFantoine} +% +Draw Antoine's Necklace. +\index{Antoine's Necklace} +Thank you to Matthew Grayson at IBM's T.J Watson Research Center for the idea. + +\begin{verbatim} +)set expose add con DenavitHartenbergMatrix Bring DH matrices into + the environment +torusRot: DHMATRIX(DFLOAT) The transformation for + drawing a sub ring + +drawRings(n) == Draw Antoine's Necklace with $n$ + s := createThreeSpace() levels of recursive subdivision + dh:DHMATRIX(DFLOAT) := identity() The number of subrings is $10^n$ + drawRingsInner(s, n, dh) Do the real work + makeViewport3D(s, "Antoine's Necklace") + +\end{verbatim} + +In order to draw Antoine rings, we take one ring, scale it down to +a smaller size, rotate it around its central axis, translate it +to the edge of the larger ring and rotate it around the edge to +a point corresponding to its count (there are 10 positions around +the edge of the larger ring). For each of these new rings we +recursively perform the operations, each ring becoming 10 smaller +rings. Notice how the {\bf DHMATRIX} operations are used to build up +the proper matrix composing all these transformations. + +\begin{verbatim} +drawRingsInner(s, n, dh) == Recursively draw Antoine's + n = 0 => Necklace + drawRing(s, dh) + void() + t := 0.0@DFLOAT Angle around ring + p := 0.0@DFLOAT Angle of subring from plane + tr := 1.0@DFLOAT Amount to translate subring + inc := 0.1@DFLOAT The translation increment + for i in 1..10 repeat Subdivide into 10 linked rings + tr := tr + inc + inc := -inc + dh' := dh*rotatez(t)*translate(tr,0.0@DFLOAT,0.0@DFLOAT)* + Transform ring in center + to a link + rotatey(p)*scale(0.35@DFLOAT, 0.48@DFLOAT, 0.4@DFLOAT) + drawRingsInner(s, n-1, dh') + t := t + 36.0@DFLOAT + p := p + 90.0@DFLOAT + void() + +drawRing(s, dh) == Draw a single ring into + free torusRot the given subspace, + torusRot := dh transformed by the given + DHMATRIX + makeObject(torus, 0..2*\%pi, 0..2*\%pi, var1Steps == 6, + space == s, var2Steps == 15) + +torus(u ,v) == Parameterization of a torus, + cu := cos(u)/6 transformed by the + DHMATRIX in $torusRot.$ + torusRot*point [(1+cu)*cos(v),(1+cu)*sin(v),(sin u)/6] +\end{verbatim} + +%\input{gallery/scherk.htex} +\section{scherk.input} +\label{ugFscherk} +% + +Scherk's minimal surface, defined by: +\index{Scherk's minimal surface} +$e^z \cos(x) = \cos(y)$. +See: {\it A Comprehensive Introduction to Differential Geometry,} Vol. 3, +by Michael Spivak, Publish Or Perish, Berkeley, 1979, pp. 249-252. + +\begin{verbatim} +(xOffset, yOffset):DFLOAT Offsets for a single piece + of Scherk's minimal surface + +drawScherk(m,n) == Draw Scherk's minimal surface + free xOffset, yOffset on an $m$ by $n$ patch + space := createThreeSpace() + for i in 0..m-1 repeat + xOffset := i*\%pi + for j in 0 .. n-1 repeat + rem(i+j, 2) = 0 => 'iter Draw only odd patches + yOffset := j*\%pi + drawOneScherk(space) Draw a patch + makeViewport3D(space, "Scherk's Minimal Surface") + +scherk1(u,v) == The first patch that makes + x := cos(u)/exp(v) up a single piece of + point [xOffset + acos(x), yOffset + u, v, abs(v)] Scherk's minimal surface + +scherk2(u,v) == The second patch + x := cos(u)/exp(v) + point [xOffset - acos(x), yOffset + u, v, abs(v)] + +scherk3(u,v) == The third patch + x := exp(v) * cos(u) + point [xOffset + u, yOffset + acos(x), v, abs(v)] + +scherk4(u,v) == The fourth patch + x := exp(v) * cos(u) + point [xOffset + u, yOffset - acos(x), v, abs(v)] + +drawOneScherk(s) == Draw the surface by + breaking it into four + patches and then drawing + the patches + makeObject(scherk1,-\%pi/2..\%pi/2,0..\%pi/2,space==s, + var1Steps == 28, var2Steps == 28) + makeObject(scherk2,-\%pi/2..\%pi/2,0..\%pi/2,space==s, + var1Steps == 28, var2Steps == 28) + makeObject(scherk3,-\%pi/2..\%pi/2,-\%pi/2..0,space==s, + var1Steps == 28, var2Steps == 28) + makeObject(scherk4,-\%pi/2..\%pi/2,-\%pi/2..0,space==s, + var1Steps == 28, var2Steps == 28) + void() +\end{verbatim} + + +{ + +\setcounter{chapter}{6} % Appendix G +\chapter{Glossary} +\label{ugGlossary} + +\appendix{Glossary} + +\sloppy + +\ourGloss{\glossarySyntaxTerm{!}}{% +{\it (syntax)} +Suffix character +\index{operation!destructive} +for destructive operations. +} + +\ourGloss{\glossarySyntaxTerm{,}}{% +{\it (syntax)} +a separator for items in a {\it tuple}, +for example, to separate arguments of a function $f(x,y)$. +} + + +\ourGloss{\glossarySyntaxTerm{\tt =>}}{% +{\it (syntax)} +the expression $a => b$ is equivalent to $if a +then$ {\it exit} $b$. +} + + +\ourGloss{\glossarySyntaxTerm{?}}{% +1. +{\it (syntax)} a suffix character for Boolean-valued {\bf function} +names, for example, {\bf odd?}. +2. +Prefix character for ``optional'' pattern variables. For example, +the pattern $f(x + y)$ does not match the expression $f(7)$, +but $f(?x + y)$ does, with $x$ matching 0 and $y$ matching 7. +3. +The special type {\bf ?} means {\it don't care}. +For example, the declaration: $x : Polynomial ?$ means +that values assigned to $x$ must be polynomials over an arbitrary +{\it underlying domain}. +} + +\ourGloss{\glossaryTerm{abstract datatype}}{% +a programming language principle used in Axiom where a datatype definition has +defined in two parts: (1) a {\it public} part describing a set of +exports, +principally operations that apply to objects of that +type, and (2) a {\it private} part describing the implementation of the +datatype usually in terms of a {\it representation} for objects of +the type. +Programs that create and otherwise manipulate objects of the type may +only do so through its exports. +The representation and other implementation information is specifically +hidden. +} + +\ourGloss{\glossaryTerm{abstraction}}{% +described functionally or conceptually without regard to implementation. +} + + +\ourGloss{\glossaryTerm{accuracy}}{% +the degree of exactness of an approximation or measurement. +In computer algebra systems, computations are typically carried out with +complete accuracy using integers or rational numbers of indefinite size. +Domain {\bf Float} provides a function +\spadfunFrom{precision}{Float} to change the precision for floating-point +computations. Computations using {\bf DoubleFloat} have a fixed +precision but uncertain accuracy. +} + + +\ourGloss{\glossaryTerm{add-chain}}{% +a hierarchy formed by domain extensions. +If domain $A$ extends domain $B$ and domain $B$ extends +domain $C$, then $A$ has {\it add-chain} $B$-$C$. +} + + +\ourGloss{\glossaryTerm{aggregate}}{% +a data structure designed to hold multiple values. +Examples of aggregates are {\bf List}, {\bf Set}, +{\bf Matrix} and {\bf Bits}. +} + + +\ourGloss{\glossaryTerm{AKCL}}{% +Austin Kyoto Common LISP, a version of {\it KCL} produced +by William Schelter, Austin, Texas. +} + + +\ourGloss{\glossaryTerm{algorithm}}{% +a step-by-step procedure for a solution of a problem; a program +} + + +\ourGloss{\glossaryTerm{ancestor}}{% +(of a domain or category) a category that is a {\it parent}, or +a {\it parent} of a {\it parent}, and so on. +See a {\bf Cross Reference} page of a constructor in Browse. +} + + +\ourGloss{\glossaryTerm{application}}{% +{\it (syntax)} an expression denoting ``application'' of a function to a +set of {\it argument} parameters. +Applications are written as a {\it parameterized form}. +For example, the form $f(x,y)$ indicates the ``application of the +function $f$ to the tuple of arguments $x$ and $y$.'' +See also {\it evaluation} and {\it invocation}. +} + + +\ourGloss{\glossaryTerm{apply}}{% +See {\it application}. +} + + +\ourGloss{\glossaryTerm{argument}}{% +1. +(actual argument) a value passed to a function at the time of a +function call; also called an {\it actual +parameter}. 2. +(formal argument) a variable used in the definition of a function to +denote the actual argument passed when the function is called. +} + + +\ourGloss{\glossaryTerm{arity}}{% +1. +(function) the number of arguments. +2. +(operator or operation) corresponds to the arity of a function +implementing the operator or operation. +} + + +\ourGloss{\glossaryTerm{assignment}}{% +{\it (syntax)} an expression of the form $x := e$, meaning +``assign the value of $e$ to $x$.'' +After {\it evaluation}, the {\it variable} $x$ +points to an object obtained by evaluating +the expression $e$. +If $x$ has a {\it type} as a result of a previous +{\it declaration}, the object assigned to $x$ must have +that type. +The interpreter must often coerce the +value of $e$ to make that happen. +For example, the expression $x : Float := 11$ +first declares $x$ to be a float, then +forces the interpreter to coerce the integer $11$ to +$11.0$ in order to assign a floating-point value to $x$. +} + + +\ourGloss{\glossaryTerm{attribute}}{% +a name or functional form denoting {\it any} useful computational or mathematical +property. For example, {\bf commutative($"*"$)} asserts that +$*$ is commutative. +Also, {\bf finiteAggregate} is used to assert that an aggregate has +a finite number of immediate components. +} + + +\ourGloss{\glossaryTerm{basis}}{% +{\it (algebra)} $S$ is a basis of a module $M$ over a +{\it ring} if $S$ generates $M$, and $S$ is linearly +independent. +} + + +\ourGloss{\glossaryTerm{benefactor}}{% +(of a given domain) a domain or package that the given domain explicitly +references (for example, calls functions from) in its implementation. +See a {\bf Cross Reference} page of a constructor in Browse. +} + + +\ourGloss{\glossaryTerm{binary}}{% +operation or function with {\it arity} 2. +} + + +\ourGloss{\glossaryTerm{binding}}{% +the association of a variable with properties such as {\it value} +and {\it type}. +The top-level {\it environment} in the interpreter consists of +bindings for all user variables and functions. +When a {\it function} is applied to arguments, a local environment +of bindings is created, one for +each formal {\it argument} and {\it local variable}. +} + + +\ourGloss{\glossaryTerm{block}}{% +{\it (syntax)} a control structure where expressions are sequentially +evaluated. +} + + +\ourGloss{\glossaryTerm{body}}{% +a {\it function body} or {\it loop body}. +} + + +\ourGloss{\glossaryTerm{boolean}}{% +objects denoted by the literals {\tt true} and +{\tt false}; elements of domain {\bf Boolean}. +\index{Boolean} +See also {\bf Bits}. +} + + +\ourGloss{\glossaryTerm{built-in function}}{% +a {\it function} in the standard Axiom library. +\index{function!built-in} +Contrast {\it user function}. +} + +v +\ourGloss{\glossaryTerm{cache}}{% +1. +(noun) a mechanism for immediate retrieval of previously computed data. +For example, a function that does a lengthy computation might store its +values in a {\it hash table} using the function argument as +the key. +The hash table then serves as a cache for the function (see also +{\tt )set function cache}). +Also, when recurrence relations that depend upon $n$ +previous values are compiled, the previous $n$ values are normally +cached (use {\tt )set functions recurrence} to change this). +2. +(verb) to save values in a cache. +} + + +\ourGloss{\glossaryTerm{capsule}}{% +the part of the body of a +{\it domain constructor} that defines the functions implemented by +the constructor. +} + + +\ourGloss{\glossaryTerm{case}}{% +{\it (syntax)} an operator used to evaluate code conditionally based on +the branch of a {\bf Union}. +\index{Union} +For example, if value $u$ is $Union(Integer, "failed")$, the +conditional expression $if u case Integer then A else B$ evaluates +$A$ if $u$ is an integer and $B$ otherwise. +} + + +\ourGloss{\glossaryTerm{Category}}{% +the distinguished object denoting the type of a category; the class of +all categories. +} + + +\ourGloss{\glossaryTerm{category}}{% +{\it (basic concept)} types denoting +classes of domains. +Examples of categories are {\bf Ring} (``the class of all +rings'') and {\bf Aggregate} (``the class of all aggregates''). +Categories form a hierarchy (formally, a directed acyclic graph) +with the distinquished category {\bf Type} +at the top. +Each category inherits the properties of all its ancestors. +Categories optionally provide ``default definitions'' for +operations they export. +Categories are defined in Axiom by functions called +category constructors. +Technically, a category designates a class of domains with common +operations and +attributes but usually with different +functions +and representations for its +constituent objects. +Categories are always defined using the Axiom library +language (see also {\it category extension}). +See also file {\bf catdef.spad} for definitions of basic algebraic +\index{file!catdef.spad @{\bf catdef.spad}} +categories in Axiom, {\bf aggcat.spad} for data structure +\index{file!aggcat.spad @{\bf aggcat.spad}} +} + + +\ourGloss{\glossaryTerm{category constructor}}{% +a function that creates categories, described by an abstract datatype in +\index{constructor!category} +the Axiom programming language. +For example, the category constructor {\bf Module} is a function +that takes a domain parameter $R$ and creates the category ``modules +over $R$.'' +} + + +\ourGloss{\glossaryTerm{category extension}}{% +A category $A$ {\it directly extends} a category $B$ if +its definition has the form $A == B with ...$ or +$A == Join(...,B,...)$. +In this case, we also say that $B$ is the {\it parent} of $A$. +We say that a category $A$ extends $B$ if $B$ is +an {\it ancestor} of $A$. +A category $A$ may also directly extend $B$ if +$B$ appears in a conditional expression within the {\tt Exports} part +of the definition to the right of a {\tt with}. +See, for example, file {\bf catdef.spad} for definitions of the +\index{file!catdef.spad @{\bf catdef.spad}} +algebra categories in Axiom, {\bf aggcat.spad} for data structure +\index{file!aggcat.spad @{\bf aggcat.spad}} +categories. +} + +\ourGloss{\glossaryTerm{category hierarchy}}{% +hierarchy formed by category extensions. +The root category is {\bf Type}. +\index{Type} +A category can be defined as a {\it Join} of two or more +categories so as to have multiple parents. +Categories may also be parameterized so as to allow conditional +inheritance. +} + +\ourGloss{\glossaryTerm{character}}{% +1. +an element of a character set, as represented by a keyboard key. +2. +a component of a string. +For example, the $1$st element of the string $"hello there"$ is the +character {\it h}. +} + + +\ourGloss{\glossaryTerm{client}}{% +(of a given domain) any domain or package that explicitly calls +functions from the given domain. +See a {\bf Cross Reference} page of a constructor in Browse. +} + + +\ourGloss{\glossaryTerm{coercion}}{% +an automatic transformation of an object of one {\it type} to an +object of a similar or desired target type. +In the interpreter, coercions and retractions +are done +automatically by the interpreter when a type mismatch occurs. +Compare {\it conversion}. +} + + +\ourGloss{\glossaryTerm{comment}}{% +textual remarks imbedded in code. +Comments are preceded by a double dash ({\tt --}). +For Axiom library code, stylized comments for on-line +documentation are preceded by two plus signs ({\tt ++}). +} + + +\ourGloss{\glossaryTerm{Common LISP}}{% +A version of {\it LISP} adopted as an informal standard by major +users and suppliers of LISP. +} + + +\ourGloss{\glossaryTerm{compile-time}}{% +the time when category or domain constructors are compiled. +Contrast {\it run-time}. +} + + +\ourGloss{\glossaryTerm{compiler}}{% +a program that generates low-level code from a higher-level source +language. Axiom has three compilers. +A {\it graphics compiler} converts graphical formulas to a compiled +subroutine so that points can be rapidly produced for graphics commands. +An {\it interpreter compiler} optionally compiles +user functions +when first invoked (use +{\tt )set functions compile} to turn this feature on). +A {\it library compiler} compiles all constructors (available on an ``as-is'' +basis for Release 1). +} + + +\ourGloss{\glossaryTerm{computational object}}{% +In Axiom, domains are objects. +This term is used to distinguish the objects that are members of +domains rather than the domains themselves. +} + + +\ourGloss{\glossaryTerm{conditional}}{% +a {\it control structure} of the form $if A then B else C$. +\index{if} +The {\it evaluation} of $A$ produces {\tt true} or +{\tt false}. If {\tt true}, $B$ evaluates to produce a value; +otherwise $C$ evaluates to produce a value. +When the value is not required, the $else C$ part can be omitted. +} + + +\ourGloss{\glossaryTerm{constant}}{% +{\it (syntax)} a reserved word used in signatures in +Axiom programming language to signify that an operation always +returns the same value. +For example, the signature $0: constant -> \$$ in the source code +of {\bf AbelianMonoid} tells the Axiom compiler that $0$ +is a constant so that suitable optimizations might be performed. +} + + +\ourGloss{\glossaryTerm{constructor}}{% +a {\it function} that creates a {\it category}, +{\it domain}, or {\it package}. +} + + +\ourGloss{\glossaryTerm{continuation}}{% +when a line of a program is so long that it must be broken into several +lines, then all but the first line are called {\it continuation lines}. +If such a line is given interactively, then each incomplete line must +end with an underscore. +} + + +\ourGloss{\glossaryTerm{control structure}}{% +program structures that can specify a departure from normal sequential +execution. Axiom has four kinds of control structures: +blocks, {\it case} statements, +conditionals, and loops. +} + + +\ourGloss{\glossaryTerm{conversion}}{% +the transformation of an object of one {\it type} to one of +another type. +Conversions that can be performed automatically by the +interpreter are called +coercions. These happen when the +interpreter encounters a type mismatch and a similar or declared +target type is needed. +In general, the user must use the infix operation {\tt ::} +to cause this transformation. +} + + +\ourGloss{\glossaryTerm{copying semantics}}{% +the programming language semantics used in PASCAL +\index{PASCAL} +but {\it not} in +\index{semantics!copying} +Axiom. See also {\it pointer semantics} for details. +\index{semantics!pointer} +} + + +\ourGloss{\glossaryTerm{data structure}}{% +a structure for storing data in the computer. +Examples are lists +and hash tables. +} + + +\ourGloss{\glossaryTerm{datatype}}{% +equivalent to {\it domain} in Axiom. +} + + +\ourGloss{\glossaryTerm{declaration}}{% +{\it (syntax)} an expression of the form $x : T$ where $T$ is +some {\it type}. +A declaration forces all values assigned to +$x$ to be of that type. +If a value is of a different type, the interpreter will try to +coerce the value to type $T$. +Declarations are necessary in case of ambiguity or when a user wants to +introduce an unexposed domain. +} + + +\ourGloss{\glossaryTerm{default definition}}{% +a function defined by a {\it category}. +Such definitions appear in category definitions of the form \newline +$C: Category == T add I$ \newline +in an optional implementation part $I$ to the +right of the keyword $add$. +} + + +\ourGloss{\glossaryTerm{default package}}{% +an optional {\it package} +of functions associated with a category. +Such functions are necessarily defined in terms of other operations +exported by the category. +} + + +\ourGloss{\glossaryTerm{definition}}{% +{\it (syntax)} 1. +An expression of the form $f(a) == b$ defining function $f$ +with formal arguments +$a$ and {\it body} $b$; +equivalent to the statement $f == (a) +-> b$. +2. +An expression of the form $a == b$ where $a$ is a +{\it symbol}, equivalent to $a() == b$. +See also {\it macro} where a similar substitution is done at +{\it parse} time. +} + + +\ourGloss{\glossaryTerm{delimiter}}{% +a {\it character} that marks the beginning or end of some +syntactically correct unit in the language, for example, +{\tt "} for strings, blanks for identifiers. +} + +\ourGloss{\glossaryTerm{dependent}}{% +(of a given constructor) another constructor that mentions the given +constructor as an argument or among the types of an exported operation. +See a {\bf Cross Reference} page of a constructor in Browse. +} + + +\ourGloss{\glossaryTerm{destructive operation}}{% +An operation that changes a component or structure of a value. +\index{operation!destructive} +In Axiom, destructive operations have names ending +with an exclamation mark ({\tt !}). +For example, domain {\bf List} has two operations to reverse +the elements of a list, one named \spadfunFrom{reverse}{List} +that returns a copy of the original list with the elements +reversed, another named \spadfunFrom{reverse}{List} that +reverses the elements {\it in place,} thus destructively changing +the original list. +} + +\ourGloss{\glossaryTerm{documentation}}{% +1. +on-line or hard-copy descriptions of Axiom; 2. +text in library code preceded by {\tt ++} comments as opposed to general +comments preceded by {\tt --}. +} + + +\ourGloss{\glossaryTerm{domain}}{% +{\it (basic concept)} a domain corresponds to the usual notion of +datatypes. +Examples of domains are +{\bf List Float} (``lists of floats''), +{\bf Fraction Polynomial Integer} (``fractions of polynomials of integers''), +and {\bf Matrix Stream CardinalNumber} +(``matrices of infinite streams of cardinal numbers''). +The term {\it domain} actually abbreviates {\it domain of +computation}. +Technically, a domain denotes a class of objects, a class of +operations for creating and otherwise +manipulating these objects, and a class of +attributes describing computationally +useful properties. +Domains may also define functions for its exported operations, +often in terms of some {\it representation} for the objects. +A domain itself is an {\it object} created by a +{\it function} called a {\it domain constructor}. +The types of the exported operations of a domain are arbitary; this gives +rise to a special class of domains called packages. +} + + +\ourGloss{\glossaryTerm{domain constructor}}{% +a function that creates domains, described by an abstract datatype in +\index{constructor!domain} +the Axiom programming language. +Simple domains like {\bf Integer} and {\bf Boolean} are +created by domain constructors with no arguments. +Most domain constructors take one or more parameters, one usually +denoting an {\it underlying domain}. +For example, the domain {\bf Matrix(R)} denotes ``matrices over +$R$.'' +\index{Mapping} +Domains {\bf Mapping}, +\index{Record} +{\bf Record}, and {\bf Union} are +\index{Union} +primitive domains. +All other domains are written in the Axiom programming language +and can be modified by users with access to the library source code +and the library compiler. +} + + +\ourGloss{\glossaryTerm{domain extension}}{% +a domain constructor $A$ is said to {\it extend} a domain +constructor $B$ if $A$'s definition has the form $A +== B add ...$. +This intuitively means ``functions not defined by $A$ are assumed to +come from $B$.'' +Successive domain extensions form add-chains +affecting the +search order for functions not implemented +directly by the domain during {\it dynamic lookup}. +} + + +\ourGloss{\glossaryTerm{dot notation}}{% +using an infix dot ({\tt .}) for the +operation {\bf elt}. +If $u$ is the list $[7,4,-11]$ then both $u(2)$ and +$u.2$ return $4$. +Dot notation nests to the left: +$f.g.h$ is equivalent to $(f.g).h$. +} + + +\ourGloss{\glossaryTerm{dynamic}}{% +that which is done at {\it run-time} as opposed to +{\it compile-time}. +For example, the interpreter may build a domain ``matrices over +integers'' dynamically in response to user input. +However, the compilation of all functions for matrices and +integers is done during {\it compile-time}. +Constrast {\it static}. +} + + +\ourGloss{\glossaryTerm{dynamic lookup}}{% +In Axiom, a {\it domain} may or may not explicitly provide +{\it function} definitions for all its exported +operations. These definitions may instead come from domains +in the {\it add-chain} or from default packages. +When a function call is made for an +operation in the domain, up to five steps are carried out. +\begin{enumerate} +\item If the domain itself implements a function for the operation, +that function is returned. +\item Each of the domains in the {\it add-chain} are searched; +if one of these domains implements the function, that function is +returned. +\item Each of the default packages +for the +domain are searched in order of the {\it lineage}. +If any of the default packages implements the function, the first one +found is returned. +\item Each of the default packages +for each of the +domains in the {\it add-chain} are searched in the order of their +{\it lineage}. If any of the default packages implements the +function, the first one found is returned. +\item If all of the above steps fail, an error message is reported. +\end{enumerate} +} + + +\ourGloss{\glossaryTerm{empty}}{% +the unique value of objects with type {\bf Void}. +} + + +\ourGloss{\glossaryTerm{environment}}{% +a set of bindings. +} + + +\ourGloss{\glossaryTerm{evaluation}}{% +a systematic process that transforms an {\it expression} into an +object called the {\it value} of the expression. +Evaluation may produce side effects. +} + +%tpdclip1 + + +\ourGloss{\glossaryTerm{exit}}{% +{\it (reserved word)} an {\it operator} that forces an exit from +the current {\it block}. +For example, the block $(a := 1; if i > 0 then exit a; +a := 2)$ will prematurely exit at the second statement with value 1 if +the value of $i$ is greater than zero. +See {\tt =>} for an alternate syntax. +} + +%tpdhere +%tpdhere +%tpdhere +%tpdhere + +\ourGloss{\glossaryTerm{explicit export}}{% +1. +(of a domain $D$) any {\it attribute}, {\it operation}, +\index{export!explicit} +or {\it category} explicitly mentioned in the {\it type} +exports part $E$ for the domain constructor definition +$D: E == I$ 2. +(of a category $C$) any {\it attribute}, +{\it operation}, or {\it category} explicitly mentioned in +the {\it type} specification part $E$ for the category +constructor definition $C: {\it Category} == E$ +} + + +\ourGloss{\glossaryTerm{export}}{% +{\it explicit export} or {\it implicit export} of a domain +or category +} + + +\ourGloss{\glossaryTerm{expose}}{% +some constructors are {\it exposed}, others {\it unexposed}. +Exposed domains and packages are recognized by the interpreter. +Use {\tt )set expose} to control what is exposed. +Unexposed constructors will appear in Browse prefixed by a star (``{\tt *}''). +} + + +\ourGloss{\glossaryTerm{expression}}{% +1. +any syntactically correct program fragment. +2. +an element of domain {\bf Expression}. +} + + +\ourGloss{\glossaryTerm{extend}}{% +see {\it category extension} or {\it domain extension}. +} + + +\ourGloss{\glossaryTerm{field}}{% +{\it (algebra)} a {\it domain} that is a {\it ring} where +every non-zero element is invertible and where $xy=yx$; a member of +category {\bf Field}. +For a complete list of fields, click on {\bf Domains} under {\bf Cross +Reference} for {\bf Field} in Browse. +} + + +\ourGloss{\glossaryTerm{file}}{% +1. a program or collection of data stored on disk, tape or other medium. +2. an object of a {\bf File} domain. +} + + +\ourGloss{\glossaryTerm{float}}{% +a floating-point number with user-specified precision; an element of +domain {\bf Float}. +\index{Float} +Floats are literals written either without an +exponent (for example, $3.1416$), or with an exponent +(for example, $3.12E-12$). +Use function {\it precision} to change the precision of the +mantissa ($20$ digits by default). +See also {\it small float}. +} + + +\ourGloss{\glossaryTerm{formal parameter}}{% +(of a function) an identifier bound to the value +of an actual {\it argument} on {\it invocation}. +In the function definition $f(x,y) == u$, for example, $x$ and +$y$ are the formal parameters. +} + + +\ourGloss{\glossaryTerm{frame}}{% +the basic unit of an interactive session; each frame has its own +{\it step number}, {\it environment}, and +{\it history}. In one interactive session, users can create and +drop frames, and have several active frames simultaneously. +} + + +\ourGloss{\glossaryTermNoIndex{free}}{% +{\it (syntax)} +A keyword used in user-defined functions to declare that +a variable is a {\it free variable} of that function. +\index{free} +For example, the statement $free x$ declares the variable $x$ +within the body of a function $f$ to be a free variable in $f$. +Without such a declaration, any variable $x$ that appears on the +left-hand side of an assignment before it is referenced +is regarded as a {\it local variable} of that function. +If the intention of the assignment is to give a value to a +{\it global variable} $x$, the body of that function must +contain the statement $free x$. +A variable that is a parameter to the function is always local. +%This should be a reported bug---> +%, even +%if it has a $free$ declaration. +} + + +\ourGloss{\glossaryTerm{free variable}}{% +(of a function) a variable that appears in a body of a function but is +not bound by that function. +Contrast with {\it local variable}. +} + +\ourGloss{\glossaryTerm{function}}{% +implementation of {\it operation}. +A function takes zero or more {\it argument} parameters and +produces a single return value. +Functions are objects that can be passed as parameters to +functions and can be returned as values of functions. +Functions can also create other functions (see also +{\bf InputForm}). +See also {\it application} and {\it invocation}. +The terms {\it operation} and {\it function} are distinct notions +in Axiom. +An operation is an abstraction of a function, described by +a {\it name} and a {\it signature}. +A function is created by providing an implementation of that +operation by Axiom code. +Consider the example of defining a user-function $fact$ to +compute the {\bf factorial} of a nonnegative integer. +The Axiom statement $fact: Integer -> Integer$ +describes the operation, whereas the statement $fact(n) = +reduce(*,[1..n])$ defines the function. +See also {\it generic function}. +} + + +\ourGloss{\glossaryTerm{function body}}{% +the part of a {\it function}'s definition that is +evaluated when the function is called at {\it run-time}; the part +of the function definition to the right of the {\tt ==}. +} + + +\ourGloss{\glossaryTerm{garbage collection}}{% +a system function that automatically recycles memory cells from the +{\it heap}. Axiom is built upon {\it Common LISP} that +provides this facility. +} + + +\ourGloss{\glossaryTerm{garbage collector}}{% +a mechanism for reclaiming storage in the {\it heap}. +} + + +\ourGloss{\glossaryTerm{Gaussian}}{% +a complex-valued expression, for example, one with both a real and +imaginary part; a member of a {\bf Complex} domain. +} + + +\ourGloss{\glossaryTerm{generic function}}{% +the use of one function to operate on objects of different types. +One might regard Axiom as supporting generic +operations but not generic functions. +One operation $+: (D, D) -> D$ exists for adding elements in a +ring; each ring however provides its own type-specific function +for implementing this operation. +} + + +\ourGloss{\glossaryTerm{global variable}}{% +A variable that can be referenced freely by functions. +\index{variable!global} +In Axiom, all top-level user-defined variables defined during an +interactive user session are global variables. +Axiom does not allow {\it fluid variables}, that is, variables +\index{variable!fluid} +bound by a function $f$ that can be referenced by +functions that $f$ calls. +} + + +\ourGloss{\glossaryTermNoIndex{Gr\protect\"{o}bner basis}}{% +{\it (algebra)} a special basis for a polynomial ideal that allows a +\index{Groebner basis @{Gr\protect\"{o}bner basis}} +simple test for membership. +\index{basis!Groebner@{Gr\protect\"{o}bner}} +It is useful in solving systems of polynomial equations. +} + + + +\ourGloss{\glossaryTerm{group}}{% +{\it (algebra)} a monoid where every element has a multiplicative +inverse. +} + + +\ourGloss{\glossaryTerm{hash table}}{% +a data structure designed for fast lookup of information stored under ``keys''. +A hash table consists of a set of {\it entries}, each of which +associates a {\it key} with a {\it value}. +Finding the object stored under a key can be fast for +a large number of entries since keys are {\it hashed} into numerical +codes for fast lookup. +} + + +\ourGloss{\glossaryTerm{heap}}{% +1. an area of storage used by data in programs. +For example, Axiom will use the heap to hold the partial results of +symbolic computations. +When cancellations occur, these results remain in the heap until +garbage collected. +2. an object of a {\bf Heap} domain. +} + + +\ourGloss{\glossaryTerm{history}}{% +a mechanism that records input and output data for an interactive session. +Using the history facility, users can save computations, review previous +steps of a computation, and restore a previous interactive session at +some later time. +For details, issue the system command {\it )history ?} to the +interpreter. See also {\it frame}. +} + + +\ourGloss{\glossaryTerm{ideal}}{% +{\it (algebra)} a subset of a ring that is closed under addition and +multiplication by arbitrary ring elements; thus an ideal is +a module over the ring. +} + + +\ourGloss{\glossaryTerm{identifier}}{% +{\it (syntax)} an Axiom name; a {\it literal} of type +{\bf Symbol}. An identifier begins with an alphabetical character, +\%, ?, or !, and may be followed by any of these or digits. +Certain distinguished reserved words are not allowed as +identifiers but have special meaning in Axiom. +} + +\ourGloss{\glossaryTerm{immutable}}{% +an object is immutable if it cannot be changed by an +{\it operation}; it is not a mutable object. +Algebraic objects are generally immutable: changing an algebraic expression +involves copying parts of the original object. +One exception is an object of type {\bf Matrix}. +Examples of mutable objects are data structures such as those of type +\index{semantics!pointer} +{\bf List}. See also {\it pointer semantics}. +} + + +\ourGloss{\glossaryTerm{implicit export}}{% +(of a domain or category) +\index{export!implicit} +any exported {\it attribute} or {\it operation} or {\it category} +that is not an {\it explicit export}. +For example, {\bf Monoid} and {\bf *} are implicit exports of +{\bf Ring}. +} + + +\ourGloss{\glossaryTerm{index}}{% +1. +a variable that counts the number of times a {\it loop} is +repeated. +2. the ``address'' of an element in a data structure (see also category +{\bf LinearAggregate}). +} + + +\ourGloss{\glossaryTerm{infix}}{% +{\it (syntax)} an {\it operator} placed between two +operands; also called a {\it binary +operator}. +For example, in the expression $a + b$, $+$ is the +infix operator. +An infix operator may also be used as a {\it prefix}. +Thus $+(a,b)$ is also permissible in the Axiom +language. +Infix operators have a {\it precedence} relative to one another. +% relative to what? +} + + +\ourGloss{\glossaryTerm{input area}}{% +a rectangular area on a HyperDoc screen into which users can enter +text. +} + + +\ourGloss{\glossaryTerm{instantiate}}{% +to build a {\it category}, {\it domain}, or +{\it package} at run-time. +} + + +\ourGloss{\glossaryTerm{integer}}{% +a {\it literal} object of domain {\bf Integer}, the class of +integers with an unbounded number of digits. +Integer literals consist of one or more consecutive digits (0-9) with no +embedded blanks. +Underscores can be used to separate digits in long integers if +desirable. +} + + +\ourGloss{\glossaryTerm{interactive}}{% +a system where the user interacts with the computer step-by-step. +} + + +\ourGloss{\glossaryTerm{interpreter}}{% +the part of Axiom responsible for handling user input +during an interactive session. +%The following is a somewhat simplified description of the typical action of +%the interpreter. +The interpreter parses the user's input expression to create an +expression tree, then does a bottom-up traversal of the tree. +Each subtree encountered that is not a value consists of a root node +denoting an operation name and one or more leaf nodes denoting +operands. +The interpreter resolves type mismatches and uses +type-inferencing and a library database to determine appropriate types for +the operands and the result, and an operation to be performed. +The interpreter next builds a domain to perform the indicated operation, +and invokes a function from the domain to compute a value. +The subtree is then replaced by that value and the process continues. +Once the entire tree has been processed, the value replacing the top +node of the tree is displayed back to the user as the value of the +expression. +} + +\ourGloss{\glossaryTerm{invocation}}{% +(of a function) the run-time process involved in +evaluating a {\it function} +{\it application}. This process has two steps. +First, a local {\it environment} is created where +formal arguments +are locally bound by +{\it assignment} to their respective actual {\it argument}. +Second, the {\it function body} is evaluated in that local +environment. The evaluation of a function is terminated either by +completely evaluating the function body or by the evaluation of a +{\tt return} expression. +\index{return} +} + +\ourGloss{\glossaryTerm{iteration}}{% +repeated evaluation of an expression or a sequence of expressions. +Iterations use the reserved words +{\tt for}, +\index{for} +{\tt while}, +\index{while} +and {\tt repeat}. +\index{repeat} +} + +\ourGloss{\glossaryTerm{Join}}{% +a primitive Axiom function taking two or more categories as +arguments and producing a category containing all of the operations and +attributes from the respective categories. +} + + +\ourGloss{\glossaryTerm{KCL}}{% +Kyoto Common LISP, a version of {\it Common LISP} that features +compilation of LISP into the $C$ Programming +Language. +} + + +\ourGloss{\glossaryTerm{library}}{% +In Axiom, a collection of compiled modules respresenting +{\it category} or {\it domain} constructors. +} + + +\ourGloss{\glossaryTerm{lineage}}{% +the sequence of +default packages +for a given domain to be +searched during {\it dynamic lookup}. +This sequence is computed first by ordering the category +ancestors of the domain according to their {\it level +number}, an integer equal to the minimum distance of the domain from +the category. +Parents have level 1, parents of parents have level 2, and so on. +Among categories with equal level numbers, ones that appear in the +left-most branches of {\tt Join}s in the source code come first. +See a {\bf Cross Reference} page of a constructor in Browse. +See also {\it dynamic lookup}. +} + +\ourGloss{\glossaryTerm{LISP}}{% +acronym for List Processing Language, a language designed for the +manipulation of non-numerical data. +The Axiom library is translated into LISP then compiled into +machine code by an underlying LISP system. +} + + +\ourGloss{\glossaryTerm{list}}{% +an object of a {\bf List} domain. +} + + +\ourGloss{\glossaryTerm{literal}}{% +an object with a special syntax in the language. +In Axiom, there are five types of literals: +booleans, +integers, +floats, +strings, and +symbols. +} + + +\ourGloss{\glossaryTerm{local}}{% +{\it (syntax)} +A keyword used in user-defined functions to declare that +a variable is a {\it local variable} of that function. +Because of default assumptions on variables, such a declaration is +often not necessary but is available to the user for clarity when appropriate. +} + +\ourGloss{\glossaryTerm{local variable}}{% +(of a function) a variable bound by that +function and such that its binding is invisible to any function +that function calls. +Also called a {\it lexical} variable. +By default in the interpreter: +\begin{enumerate} +\item +any variable $x$ that appears on the left-hand side of an +assignment +is normally regarded a local variable of that function. +%The right-hand side of an assignment is looked at before the +%left-hand side. +If the intention of an assignment is to change the value of a +{\it global variable} $x$, the body of the function must then +contain the statement $free x$. +\item +any other variable is regarded as a {\it free variable}. +\end{enumerate} +An optional declaration $local x$ is available to declare +explicitly a variable to be a local variable. +All formal parameters +are local variables to the function. +} + +\ourGloss{\glossaryTerm{loop}}{% +1. +an expression containing a {\tt repeat}. +\index{repeat} +2. +a collection expression having a +{\tt for} or a +\index{for} +{\tt while}, +\index{while} +for example, $[f(i) for i in S]$. +\index{while} +} + + +\ourGloss{\glossaryTerm{loop body}}{% +the part of a loop following the {\tt repeat} +\index{repeat} +that tells what to do each iteration. +For example, the body of the loop $for x in S repeat B$ is +$B$. For a collection expression, the body of the loop precedes the +initial {\tt for} +\index{for} +or {\tt while}. +\index{while} +} + + +\ourGloss{\glossaryTerm{macro}}{% +1. {\it (interactive syntax)} +An expression of the form $macro a == b$ where $a$ is a +{\it symbol} causes $a$ to be textually replaced by the +expression $b$ at {\it parse} time. +2. +An expression of the form $macro f(a) == b$ defines a parameterized +macro expansion for a parameterized form $f$. This macro causes a +form $f$($x$) to be textually replaced by the expression +$c$ at parse time, where $c$ is the expression obtained by +replacing $a$ by $x$ everywhere in $b$. +See also {\it definition} where a similar substitution is done +during {\it evaluation}. +3. {\it (programming language syntax)} +An expression of the form $a ==> b$ where $a$ is a symbol. +} + + +\ourGloss{\glossaryTerm{mode}}{% +a type expression containing a question-mark ({\tt ?}). +For example, the mode {\sf POLY ?} designates {\it the class of all +polynomials over an arbitrary ring}. +} + + +\ourGloss{\glossaryTerm{mutable}}{% +objects that contain pointers +to other objects and that +\index{semantics!pointer} +have operations defined on them that alter these pointers. +Contrast {\it immutable}. +Axiom uses {\it pointer semantics} as does {\it LISP} +\index{semantics!copying} +in contrast with many other languages such as PASCAL +\index{PASCAL} +that use +{\it copying semantics}. +See {\it pointer semantics} for details. +} + +\ourGloss{\glossaryTerm{name}}{% +1. +a {\it symbol} denoting a {\it variable}, such as +the variable $x$. +2. +a {\it symbol} denoting an {\it operation}, +that is, the operation $divide: (Integer, Integer) -> Integer$. +} + + + +\ourGloss{\glossaryTerm{nullary}}{% +a function with no arguments, for example, +{\bf characteristic}; operation or function with {\it arity} zero. +} + + + +\ourGloss{\glossaryTerm{object}}{% +a data entity created or manipulated by programs. +Elements of domains, functions, and domains themselves are objects. +The most basic objects are literals; all other objects must +be created by functions. +Objects can refer to other objects using pointers +and can be {\it mutable}. +} + +\ourGloss{\glossaryTerm{object code}}{% +code that can be directly executed by hardware; also known as {\it +machine language}. +} + + +\ourGloss{\glossaryTerm{operand}}{% +an argument of an {\it operator} (regarding an operator as a +{\it function}). +} + + +\ourGloss{\glossaryTerm{operation}}{% +an abstraction of a {\it function}, described by a +{\it signature}. For example, $fact: +NonNegativeInteger -> NonNegativeInteger$ describes an operation for +``the factorial of a (non-negative) integer.'' +} + + +\ourGloss{\glossaryTerm{operator}}{% +special reserved words in the language such as $+$ and +$*$; operators can be either {\it prefix} or +{\it infix} and have a relative {\it precedence}. +} + + +\ourGloss{\glossaryTerm{overloading}}{% +the use of the same name to denote distinct operations; an +operation is +identified by a {\it signature} identifying its name, the number +and types of its arguments, and its return types. +If two functions can have identical signatures, a {\it package +call} must be made to distinguish the two. +} + + +\ourGloss{\glossaryTerm{package}}{% +a special case of a domain, one for which the exported operations +depend solely on the parameters +and other explicit domains (contain no \$). +Intuitively, packages are collections of +({\it polymorphic}) functions. +Facilities for integration, differential equations, solution of +linear or polynomial equations, and group theory are provided by +packages. +} + + +\ourGloss{\glossaryTerm{package call}}{% +{\it (syntax)} an expression of the form $e \$ P$ where $e$ is +an {\it application} and $P$ denotes some {\it package} +(or {\it domain}). +} + + +\ourGloss{\glossaryTerm{package constructor}}{% +same +\index{constructor!package} +as {\it domain constructor}. +} + +\ourGloss{\glossaryTerm{parameter}}{% +see {\it argument}. +} + +\ourGloss{\glossaryTerm{parameterized datatype}}{% +a domain that is built on another, for example, polynomials with integer +\index{datatype!parameterized} +coefficients. +} + + +\ourGloss{\glossaryTerm{parameterized form}}{% +a expression of the form $f(x,y)$, an {\it application} of a +function. +} + +\ourGloss{\glossaryTerm{parent}}{% +(of a domain or category) a category which is explicitly declared in the source code +definition for the domain either to the left of the {\tt with} or +as an {\it export} of the domain. +See {\it category extension}. +See also a {\bf Cross Reference} page of a constructor in Browse. +} + + +\ourGloss{\glossaryTerm{parse}}{% +1. +(verb) to transform a user input string representing a valid +Axiom expression into an internal representation as a +tree-structure; the +resulting internal representation is then ``interpreted'' by Axiom to +perform some indicated action. +} + +\ourGloss{\glossaryTerm{partially ordered set}}{% +a set with a reflexive, transitive and antisymetric {\it binary} +operation. +} + + +\ourGloss{\glossaryTerm{pattern matching}}{% +1. +(on expressions) Given an expression called the ``subject'' $u$, the +attempt to rewrite $u$ using a set of ``rewrite rules.'' +Each rule has the form $A == B$ where $A$ indicates an +expression called a ``pattern'' and $B$ denotes a ``replacement.'' +The meaning of this rule is ``replace $A$ by $B$.'' +If a given pattern $A$ matches a subexpression of $u$, that +subexpression is replaced by $B$. +Once rewritten, pattern matching continues until no further changes +occur. 2. +(on strings) the attempt to match a string indicating a ``pattern'' to +another string called a ``subject'', for example, for the purpose of +identifying a list of names. +In Browse, users may enter search strings +for the purpose +of identifying constructors, operations, and attributes. +} + +\ourGloss{\glossaryTerm{pile}}{% +alternate syntax for a block, using indentation and column alignment +(see also {\it block}). +} + +\ourGloss{\glossaryTerm{pointer}}{% +a reference implemented by a link directed from one object to another in +the computer memory. +An object is said to {\it refer} to another if it has a pointer to that +other object. +Objects can also refer to themselves (cyclic references are legal). +\index{semantics!pointer} +Also more than one object can refer to the same object. +See also {\it pointer semantics}. +} + + +\ourGloss{\glossaryTerm{pointer semantics}}{% +the programming language semantics used in languages such as LISP that +\index{semantics!copying} +allow objects to be {\it mutable}. +\index{semantics!pointer} +Consider the following sequence of Axiom statements: +\newline +$x : Vector Integer := [1,4,7]$ \newline +$y := x$ \newline +$swap!(x,2,3)$ \newline +The function \spadfunFrom{swap}{Vector} +is used to interchange the second and third value in the list +$x$, producing the value $[1,7,4]$. +What value does $y$ have after evaluation of the third statement? +The answer is different in Axiom than it is in a language with +{\it copying semantics}. +In Axiom, first the vector $[1,2,3]$ is created and the variable +$x$ set to point to this object. +Let's call this object $V$. +Next, the variable $y$ is made to point to $V$ just as +$x$ does. +Now the third statement interchanges the last 2 elements of $V$ +(the {\tt !} at the end of the name \spadfunFrom{swap}{Vector} tells you +that this operation is destructive, that is, it changes the elements {\it +in place}). +\index{operation!destructive} +Both $x$ and $y$ perceive this change to $V$. +\index{semantics!copying} +Thus both $x$ and $y$ then have the value $[1,7,4]$. +In PASCAL, the second statement causes a copy of $V$ to be stored +\index{PASCAL} +under $y$. +Thus the change to $V$ made by the third statement does not affect +$y$. +} + +\newpage + +\ourGloss{\glossaryTerm{polymorphic}}{% +a {\it function} (for example, one implementing an {\it algorithm}) +defined with categorical types so as to be +applicable over a variety of domains +(the domains which are members of the categorical types). +Every Axiom function defined in a domain or package constructor with a +domain-valued parameter is polymorphic. +For example, the same matrix $+$ function is used to add +``matrices over integers'' as ``matrices over matrices over integers.'' +} + +\ourGloss{\glossaryTerm{postfix}}{% +an {\it operator} that follows its single {\it operand}. +Postfix operators are not available in Axiom. +} + +\ourGloss{\glossaryTerm{precedence}}{% +{\it (syntax)} refers to the so-called {\it binding power} of an +operator. For example, $*$ has higher binding power than $+$ so +that the expression $a + b * c$ is equivalent to $a + (b * c)$. +} + +\ourGloss{\glossaryTerm{precision}}{% +the number of digits in the specification of a number. +The operation \spadfunFrom{digits}{Float} sets this for objects +of {\bf Float}. +\index{Float} +} + +\ourGloss{\glossaryTerm{predicate}}{% +1. +a Boolean-valued function, for example, $odd: Integer -> Boolean$. +2. +a Boolean-valued expression. +} + +\ourGloss{\glossaryTerm{prefix}}{% +{\it (syntax)} an {\it operator} such as $-$ +that is written {\it before} its single {\it operand}. +Every function of one argument can be used as a prefix operator. +For example, all of the following have equivalent meaning in +Axiom: $f(x)$, $f x$, and $f.x$. +See also {\it dot notation}. +} + +\ourGloss{\glossaryTerm{quote}}{% +the prefix {\it operator} {\tt '} meaning {\it do not +evaluate}. +} + +\ourGloss{\glossaryTermNoIndex{Record}}{% +(basic domain constructor) a domain constructor used to create an +\index{Record} +inhomogeneous aggregate composed of pairs of +selectors and +values. +A {\bf Record} domain is written in the form +$Record(a1: D1, \ldots, an: Dn)$ ($n > 0$) where +$a1$, \ldots, $an$ are identifiers called the {\it selectors} +of the record, and $D1$, \ldots, $Dn$ are domains indicating +the type of the component stored under selector $an$. +} + +\ourGloss{\glossaryTerm{recurrence relation}}{% +A relation that can be expressed as a function $f$ with some +argument $n$ which depends on the value of $f$ at $k$ +previous values. +In most cases, Axiom will rewrite a recurrence relation on +compilation so as to {\it cache} its previous $k$ values and +therefore make the computation significantly more efficient. +} + + +\ourGloss{\glossaryTerm{recursion}}{% +use of a self-reference within the body of a function. +Indirect recursion is when a function uses a function below it in the +call chain. +} + +\ourGloss{\glossaryTerm{recursive}}{% +1. +A function that calls itself, either directly or indirectly through +another function. +2. +self-referential. +See also {\it recursive}. +} + +\ourGloss{\glossaryTerm{reference}}{% +see {\it pointer} +} + +\ourGloss{\glossaryTerm{relative}}{% +(of a domain) A package that exports operations relating to the domain, +in addition to those exported by the domain. +See a {\bf Cross Reference} page of a constructor in Browse. +} + +%\ourGloss{\glossaryTermNoIndex{Rep}}{% +%a special identifier used as a {\it local variable} of a domain +%\index{Rep} +%constructor body to denote the representation domain for objects of a +%domain. See {\it representation}. +%} + +\ourGloss{\glossaryTerm{representation}}{% +a {\it domain} providing a data structure for elements of a +domain, generally denoted by the special identifier {\it Rep} in +the Axiom programming language. +As domains are abstract datatypes, +this representation is +not available to users of the domain, only to functions defined in the +{\it function body} for a domain constructor. +Any domain can be used as a representation. +} + +\ourGloss{\glossaryTerm{reserved word}}{% +a special sequence of non-blank characters with special meaning in the +Axiom language. +Examples of reserved words are names such as {\tt for}, +\index{for} +$if$, +\index{if} +and $free$, +\index{free} +operator names such as +$+$ and {\bf mod}, special character strings such as +{\tt ==} and {\tt :=}. +} + +\ourGloss{\glossaryTerm{retraction}}{% +to move an object in a parameterized domain back to the underlying +domain, for example to move the object $7$ from a ``fraction of +integers'' (domain {\bf Fraction Integer}) to ``the integers'' (domain +{\bf Integer}). +} + +\ourGloss{\glossaryTerm{return}}{% +when leaving a function, the value of the expression following +{\tt return} +\index{return} +becomes the value of the function. +} + +\ourGloss{\glossaryTerm{ring}}{% +a set with a commutative addition, associative multiplication, a unit +element, where multiplication is distributive over addition and subtraction. +} + +\ourGloss{\glossaryTerm{rule}}{% +{\it (syntax)} 1. +An expression of the form $rule A == B$ indicating a ``rewrite +rule.'' 2. +An expression of the form $rule (R1;...;Rn)$ indicating a set of +``rewrite rules'' $R1$,...,$Rn$. +See {\it pattern matching} for details. +} + + +\ourGloss{\glossaryTerm{run-time}}{% +the time when computation is done. +Contrast with {\it compile-time}, and +{\it dynamic} as opposed to {\it static}. +For example, the decision of the intepreter to build +a structure such as ``matrices with power series entries'' in response to +user input is made at run-time. +} + + +\ourGloss{\glossaryTerm{run-time check}}{% +an error-checking that can be done only when the program receives user +input; for example, confirming that a value is in the proper range for a +computation. +} + +\ourGloss{\glossaryTerm{search string}}{% +a string entered into an {\it input area} on a HyperDoc screen. +} + + +\ourGloss{\glossaryTerm{selector}}{% +an identifier used to address a component value of a +{\bf Record} datatype. +\index{Record} +} + + +\ourGloss{\glossaryTerm{semantics}}{% +the relationships between symbols and their meanings. +The rules for obtaining the {\it meaning} of any syntactically valid +expression. +} + +\ourGloss{\glossaryTerm{semigroup}}{% +{\it (algebra)} a {\it monoid} which need not have an identity; it +is closed and associative. +} + + +\ourGloss{\glossaryTerm{side effect}}{% +action that changes a component or structure of a value. +\index{operation!destructive} +See {\it destructive operation} for details. +} + + +\ourGloss{\glossaryTerm{signature}}{% +{\it (syntax)} an expression describing the type of an {\it operation}. +A signature has the form $name : source -> target$, where +$source$ is the type of the arguments of the operation, and +$target$ is the type of the result. +} + + +\ourGloss{\glossaryTerm{small float}}{% +an object of +the domain {\bf DoubleFloat} +for floating-point arithmetic as provided by the +\index{DoubleFloat} +computer hardware. +} + + +\ourGloss{\glossaryTerm{small integer}}{% +an object of +the domain {\bf SingleInteger} +for integer arithmetic +\index{SingleInteger} +as provided by the computer hardware. +} + + +\ourGloss{\glossaryTerm{source}}{% +the {\it type} of the argument of a {\it function}; the type +expression before the $->$ in a {\it signature}. +For example, the source of $f : (Integer, Integer) -> Integer$ is +$(Integer, Integer)$. +} + + +\ourGloss{\glossaryTerm{sparse}}{% +data structure whose elements are mostly identical (a sparse matrix is +one filled mostly with zeroes). +} + + +\ourGloss{\glossaryTerm{static}}{% +that computation done before run-time, such as compilation. +Contrast {\it dynamic}. +} + + +\ourGloss{\glossaryTerm{step number}}{% +the number that precedes user input lines in an interactive session; +the output of user results is also labeled by this number. +} + + +\ourGloss{\glossaryTerm{stream}}{% +an object of {\bf Stream(R)}, a generalization of a +{\it list} to allow an infinite number of elements. +Elements of a stream are computed ``on demand.'' +Streams are used to implement various forms of power series. +} + + +\ourGloss{\glossaryTerm{string}}{% +an object of domain {\bf String}. +Strings are literals consisting of an arbitrary sequence of +characters surrounded by double-quotes ({\tt "}), +for example, $"Look here!"$. +} + +\ourGloss{\glossaryTerm{subdomain}}{% +{\it (basic concept)} a {\it domain} together with a +{\it predicate} characterizing the members of the domain +that belong to the subdomain. +The exports of a subdomain are usually distinct from the domain +itself. +A fundamental assumption however is that values in the subdomain +are automatically coerceable to values in +the domain. +For example, if $n$ and $m$ are declared to be members +of a subdomain of the integers, then {\it any} {\it binary} +operation from {\bf Integer} is available on $n$ and +$m$. +On the other hand, if the result of that operation is to be +assigned to, say, $k$, also declared to be of that subdomain, +a {\it run-time} check is generally necessary to ensure that +the result belongs to the subdomain. +} + + +\ourGloss{\glossaryTerm{such that clause}}{% +{\it (syntax)} +the use of {\tt |} followed by an expression to filter an +iteration. +} + + +\ourGloss{\glossaryTerm{suffix}}{% +{\it (syntax)} an {\it operator} that is placed after its operand. +Suffix operators are not allowed in the Axiom language. +} + + +\ourGloss{\glossaryTerm{symbol}}{% +objects denoted by {\it identifier} literals; an +element of domain {\bf Symbol}. +The interpreter, by default, converts the symbol $x$ into +{\bf Variable(x)}. +} + + +\ourGloss{\glossaryTerm{syntax}}{% +rules of grammar and punctuation for forming correct expressions. +} + + +\ourGloss{\glossaryTerm{system commands}}{% +top-level Axiom statements that begin with {\tt )}. +System commands allow users to query the database, read files, trace +functions, and so on. +} + + +\ourGloss{\glossaryTerm{tag}}{% +an identifier used to discriminate a branch of a {\bf Union} type. +\index{Union} +} + + +\ourGloss{\glossaryTerm{target}}{% +the {\it type} of the result of a {\it function}; the type +expression following the {\tt ->} in a {\it signature}. +} + +\ourGloss{\glossaryTerm{top-level}}{% +refers to direct user interactions with the Axiom interpreter. +} + + +\ourGloss{\glossaryTerm{totally ordered set}}{% +{\it (algebra)} a partially ordered set where any two elements are +comparable. +} + + +\ourGloss{\glossaryTerm{trace}}{% +use of system function {\tt )trace} to track the arguments passed to +a function and the values returned. +} + + +\ourGloss{\glossaryTerm{tuple}}{% +an expression of two or more other expressions separated by commas, +for example, $4,7,11$. +Tuples are also used for multiple arguments both for +applications (for example, $f(x,y)$) and in +signatures +(for example, $(Integer, Integer) -> +Integer$). A tuple is not a data structure, rather a syntax mechanism for +grouping expressions. +} + + +\ourGloss{\glossaryTerm{type}}{% +The type of any {\it category} is the unique symbol {\it +Category}. +The type of a {\it domain} is any {\it category} to +which the domain belongs. +The type of any other object is either the (unique) domain to +which the object belongs or a {\it subdomain} of that domain. +The type of objects is in general not unique. +} + +\ourGloss{\glossaryTermNoIndex{Type}}{% +a category with no operations or attributes, of which all other categories +\index{Type} +in Axiom are extensions. +} + +\ourGloss{\glossaryTerm{type checking}}{% +a system function that determines whether the datatype of an object is +appropriate for a given operation. +} + + +\ourGloss{\glossaryTerm{type constructor}}{% +a {\it domain constructor} or {\it category constructor}. +} + + +\ourGloss{\glossaryTerm{type inference}}{% +when the interpreter chooses the type for an object based on context. +For example, if the user interactively issues the definition +$f(x) == (x + \%i)**2$ then issues $f(2)$, the +interpreter will infer the type of $f$ to be $Integer -> +Complex Integer$. +} + + +\ourGloss{\glossaryTerm{unary}}{% +operation or function with {\it arity} 1. +} + + +\ourGloss{\glossaryTerm{underlying domain}}{% +for a {\it domain} that has a single domain-valued parameter, the +{\it underlying domain} refers to that parameter. +For example, the domain ``matrices of integers'' ({\bf Matrix +Integer}) has underlying domain {\bf Integer}. +} + + +\ourGloss{\glossaryTermNoIndex{Union}}{% +(basic domain constructor) a domain constructor used to combine any set +\index{Union} +of domains into a single domain. +A {\bf Union} +domain is written in the form $Union(a1: D1, ..., an: Dn)$ +($n > 0$) where $a1$, ..., $an$ are identifiers called the +{\it tags} of the union, and $D1$, ..., $Dn$ are domains called +the {\it branches} of the union. +The tags $$ai$$ are optional, but required when two of the +$$Di$$ are equal, for example, +$Union(inches: Integer, centimeters: Integer)$. In the interpreter, +values of union domains are automatically coerced to values in the +branches and vice-versa as appropriate. +See also {\it case}. +} + + +\ourGloss{\glossaryTerm{unit}}{% +{\it (algebra)} an invertible element. +} + + +\ourGloss{\glossaryTerm{user function}}{% +a function defined by a user during an interactive session. +Contrast {\it built-in function}. +} + + +\ourGloss{\glossaryTerm{user variable}}{% +a variable created by the user at top-level during an interactive +session. +} + + +\ourGloss{\glossaryTerm{value}}{% +1. +the result of evaluating an expression. +2. +a property associated with a {\it variable} in a +{\it binding} in an {\it environment}. +} + +\ourGloss{\glossaryTerm{variable}}{% +a means of referring to an object, but not an object itself. +A variable has a name and an associated {\it binding} created by +{\it evaluation} of Axiom expressions such as +declarations, +assignments, and +definitions. +In the top-level {\it environment} of the +interpreter, variables are +global variables. +Such variables can be freely referenced in user-defined functions +although a $free$ declaration is needed to assign values to +\index{free} +them. See {\it local variable} for details. +} + +\ourGloss{\glossaryTermNoIndex{Void}}{% +the type given when the {\it value} and {\it type} of an +expression are not needed. +\index{Void} +Also used when there is no guarantee at run-time that a value and +predictable mode will result. +} + + +\ourGloss{\glossaryTerm{wild card}}{% +a symbol that matches any substring including the empty string; for +example, the search string ``{\tt *an*}'' matches any word containing the +consecutive letters ``{\tt a}'' and ``{\tt n}''. +} + + +\ourGloss{\glossaryTerm{workspace}}{% +an interactive record of the user input and output held in an +interactive history file. +Each user input and corresponding output expression in the workspace has +a corresponding {\it step number}. +The current output expression in the workspace is referred to as +$\%$. The output expression associated with step number $n$ is +referred to by $\%\%(n)$. +The $k$-th previous output expression relative to the current step +number $n$ is referred to by $\%\%(- k)$. +Each interactive {\it frame} has its own workspace. +} + +}\onecolumn\fussy + + +\vfill +\eject +\setcounter{chapter}{7} % Appendix H +\chapter{License} +%\appendix{License} +\begin{verbatim} +Portions of this document are Copyright by their respective authors. +All rights reserved by the authors. Used by permission. + +Cover art ``Blue Bayou'' Copyright (2004) Jocelyn Guidry + +Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. +Text for this document is released under the license: +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +\end{verbatim} + +\eject +\begin{verbatim} +Testing involves function calls which are intended to fail in +the normal course of testing. In order to distinguish cases +where things fail by design you can issue the command +{\tt )set message test on}. This inhibits the message +"Daly Bug". You can now distinguish between intentional +failures which do not contain the "Daly Bug" message and +unintentional failures which do. + + +RDJ note to RSS: Expressions not statements or lines-- By an +expression I mean any syntactically correct program fragment. +Everything in AXIOM is an expression since every fragment has a value +and a type. In most languages including LISP, a "statement" is +different from an expression: it is executed for side-effect only and +an error is incurred if you assign it a value. This "gimmick" takes +care of incomplete expressions such as "if x > 0 then y" in blocks. +In LISP, "u := (if x > 0 then y)" is illegal but in AXIOM it is legal. +Also, in AXIOM the value of a repeat loop is void even though you +might be be able to prove that it always returns a valid value (you +have an example of this)! This will be considered a bug not a +feature. But it is how things stand. In any case---this point should +be in a box somewhere since it is key to a user's understanding to the +language. I am not sure where. You only gain an appreciation for it +after awhile in chapter 5. + +\end{verbatim} +\eject +\begin{thebibliography}{99} +\bibitem{1} Lamport, Leslie, +{\it LaTeX: A Document Preparation System,} \\ +Reading, Massachusetts, +Addison-Wesley Publishing Company, Inc., +1986. ISBN 0-201-15790-X +\bibitem{2} Knuth, Donald, {\it The \TeX{}book} \\ +Reading, Massachusetts, +Addison-Wesley Publishing Company, Inc., +1984. ISBN 0-201-13448-9 +\bibitem{3} Jenks, Richard D. and Sutor, Robert S.,\\ +{\it Axiom, The Scientific Computation System} \\ +Springer-Verlag, New York, NY 1992 ISBN 0-387-97855-0 +\end{thebibliography} +\end{document} diff --git a/books/bookvol1.pamphlet b/books/bookvol1.pamphlet new file mode 100644 index 0000000..25bb7bc --- /dev/null +++ b/books/bookvol1.pamphlet @@ -0,0 +1,14833 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 1: Axiom Tutorial}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Axiom Features} +\label{ch:axiom features} +\section{Introduction to Axiom} +Welcome to the world of Axiom. +We call Axiom a scientific computation system: +a self-contained toolbox designed to meet +your scientific programming needs, +from symbolics, to numerics, to graphics. + +This introduction is a quick overview of some of the +features Axiom offers. + +\subsection{Symbolic Computation} +Axiom provides a wide range of simple commands for symbolic +mathematical problem solving. Do you need to solve an equation, to +expand a series, or to obtain an integral? If so, just ask Axiom +to do it. + +Given $$\int\left({{1\over{(x^3 \ {(a+b x)}^{1/3})}}}\right)dx$$ +we would enter this into Axiom as: + +\spadcommand{integrate(1/(x**3 * (a+b*x)**(1/3)),x)} +which would give the result: +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +-{2 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{\root{3}\of{a}}\ {{\root{3}\of{{b +\ x}+ a}}^2}}+{{{\root{3}\of{a}}^2}\ {\root{3}\of{{b \ x}+ +a}}}+ a}\right)}}+ +\\ +\\ +\displaystyle +{4 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{{\root{3}\of{a}}^ +2}\ {\root{3}\of{{b \ x}+ a}}}- a}\right)}}+ +\\ +\\ +\displaystyle +{{12}\ {b^2}\ {x^2}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {{\root{3}\of{a}}^ +2}\ {\root{3}\of{{b \ x}+ a}}}+{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ b \ x}-{9 \ a}\right)}\ {\sqrt{3}}\ {\root{3}\of{a}}\ {{\root{3}\of{{b +\ x}+ a}}^2}} +\end{array} +\right)}\over{{18}\ {a^2}\ {x^2}\ {\sqrt{3}}\ {\root{3}\of{a}}} +$$ +\returnType{Type: Union(Expression Integer,...)} +Axiom provides state-of-the-art algebraic machinery to handle your +most advanced symbolic problems. + +\subsection{Numeric Computation} +Axiom has a numerical library that includes operations for linear +algebra, solution of equations, and special functions. For many of +these operations, you can select any number of floating point digits +to be carried out in the computation. + +Solve $x^{49}-49x^4+9$ to 49 digits of accuracy. +First we need to change the default output length of numbers: + +\spadcommand{digits(49)} +and then we execute the command: + +\spadcommand{solve(x**49-49*x**4+9 = 0,1.e-49)} +$$ +\begin{array}{@{}l} +\displaystyle +\left[{x = -{0.6546536706904271136718122105095984761851224331 +556}}, \right. +\\ +\\ +\displaystyle +\left.{x ={1.086921395653859508493939035954893289009213388763}}, + \right. +\\ +\\ +\displaystyle +\left.{x ={0.654653670725527173969468606613676483536148760766 +1}}\right] +\end{array} +$$ + + +\returnType{Type: List Equation Polynomial Float} +The output of a computation can be converted to FORTRAN to be used +in a later numerical computation. +Besides floating point numbers, Axiom provides literally +dozens of kinds of numbers to compute with. +These range from various kinds of integers, to fractions, complex +numbers, quaternions, continued fractions, and to numbers represented +with an arbitrary base. + +What is $10$ to the $90$-th power in base $32$? + +\spadcommand{radix(10**90,32)} +returns: + +%\noindent +{\tt FMM3O955CSEIV0ILKH820CN3I7PICQU0OQMDOFV6TP000000000000000000 } +\returnType{Type: RadixExpansion 32} + +The Axiom numerical library can be enhanced with a +substantial number of functions from the NAG library of numerical and +statistical algorithms. These functions will provide coverage of a wide +range of areas including roots of functions, Fourier transforms, quadrature, +differential equations, data approximation, non-linear optimization, linear +algebra, basic statistics, step-wise regression, analysis of variance, +time series analysis, mathematical programming, and special functions. +Contact the Numerical Algorithms Group Limited, Oxford, England. + +\subsection{Mathematical Structures} +Axiom also has many kinds of mathematical structures. These +range from simple ones (like polynomials and matrices) to more +esoteric ones (like ideals and Clifford algebras). Most structures +allow the construction of arbitrarily complicated ``types.'' + +Even a simple input expression can +result in a type with several levels. + +\spadcommand{matrix [ [x + \%i,0], [1,-2] ]} +$$ +\left[ +\begin{array}{cc} +{x+i} & 0 \\ +1 & -2 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Complex Integer} + +The ``\%i'' is Axiom's notation for $\sqrt{-1}$. + +The Axiom interpreter builds types in response to user input. +Often, the type of the result is changed in order to be applicable to +an operation. + +The inverse operation requires that elements of the above matrices +are fractions. However the original elements are polynomials with +coefficients which are complex numbers ({\bf Complex(Integer)}) in Axiom +terms. Inverse will coerce these to fractions whose numerator and +denominator are polynomials with coefficients which are complex numbers. + +\spadcommand{inverse(\%)} +$$ +\left[ +\begin{array}{cc} +{1 \over {x+i}} & 0 \\ +{1 \over {{2 \ x}+{2 \ i}}} & -{1 \over 2} +\end{array} +\right] +$$ +\returnType{Type: Union(Matrix Fraction Polynomial Complex Integer,...)} + +\subsection{HyperDoc} + +\begin{figure}[htbp] +\includegraphics[bbllx=1, bblly=1, bburx=298, bbury=290]{ps/bookvol1a.ps} +\caption{Hyperdoc opening menu} +\label{fig-intro-br} +\end{figure} + +HyperDoc presents you windows on the world of Axiom, +offering on-line help, examples, tutorials, a browser, and reference +material. HyperDoc gives you on-line access to this document in a +``hypertext'' format. Words that appear in a different font (for +example, {\tt Matrix}, {\bf factor}, and +{\it category}) are generally mouse-active; if you click on one +with your mouse, HyperDoc shows you a new window for that word. + +As another example of a HyperDoc facility, suppose that you want to +compute the roots of $x^{49} - 49x^4 + 9$ to 49 digits (as in our +previous example) and you don't know how to tell Axiom to do this. +The ``basic command'' facility of HyperDoc leads the way. Through the +series of HyperDoc windows +and mouse clicks, you and +HyperDoc generate the correct command to issue to compute the answer. + +\subsection{Interactive Programming } +Axiom's interactive programming language lets you define your +own functions. A simple example of a user-defined function is one +that computes the successive Legendre polynomials. Axiom lets +you define these polynomials in a piece-wise way. +\index{Legendre Polynomial} +The first Legendre polynomial. + +\spadcommand{p(0) == 1} +\returnType{Type: Void} +The second Legendre polynomial. + +\spadcommand{p(1) == x} +\returnType{Type: Void} +The $n$-th Legendre polynomial for $(n > 1)$. + +\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1) * p(n-2))/n} +\returnType{Type: Void} + +In addition to letting you define simple functions like this, the +interactive language can be used to create entire application +packages. + +The above definitions for $p$ do no computation---they simply +tell Axiom how to compute $p(k)$ for some positive integer +$k$. + +To actually get a value of a Legendre polynomial, you ask for it. + +\index{Legendre polynomials} +What is the tenth Legendre polynomial? + +\spadcommand{p(10)} +\begin{verbatim} + Compiling function p with type Integer -> Polynomial Fraction + Integer + Compiling function p as a recurrence relation. +\end{verbatim} +$$ +{{{46189} \over {256}} \ {x \sp {10}}} -{{{109395} \over {256}} \ {x \sp +8}}+{{{45045} \over {128}} \ {x \sp 6}} -{{{15015} \over {128}} \ {x \sp +4}}+{{{3465} \over {256}} \ {x \sp 2}} -{{63} \over {256}} +$$ +\returnType{Type: Polynomial Fraction Integer} +Axiom applies the above pieces for $p$ to obtain the value +of $p(10)$. But it does more: it creates an optimized, compiled +function for $p$. The function is formed by putting the pieces +together into a single piece of code. By {\it compiled}, we mean that +the function is translated into basic machine-code. By {\it +optimized}, we mean that certain transformations are performed on that +code to make it run faster. For $p$, Axiom actually +translates the original definition that is recursive (one that calls +itself) to one that is iterative (one that consists of a simple loop). + +What is the coefficient of $x^{90}$ in $p(90)$? + +\index{coefficient} +\spadcommand{coefficient(p(90),x,90)} +$$ +{5688265542052017822223458237426581853561497449095175} \over +{77371252455336267181195264} +$$ +\returnType{Type: Polynomial Fraction Integer} + +In general, a user function is type-analyzed and compiled on first use. +Later, if you use it with a different kind of object, the function +is recompiled if necessary. + +\subsection{Graphics} + +You may often want to visualize a symbolic formula or draw a graph +from a set of numerical values. To do this, you can call upon the +Axiom graphics capability. + +Axiom is capable of displaying graphs in two or three dimensions +and multiple curves can be drawn on the same graph. The whole +graphics package can be driven from interactive commands. + +Graphs in Axiom are interactive objects you can manipulate with +your mouse. Just click on the graph, and a control panel pops up. +Using this mouse and the control panel, you can translate, rotate, +zoom, change the coloring, lighting, shading, and perspective on the +picture. You can also generate a PostScript copy of your graph to +produce hard-copy output. + +The graphics package runs as a separate process. It interacts with +both the Axiom interpreter and the Hyperdoc facility. In Hyperdoc +you can click on an embedded graph and it will become ``live'' so +you can rotate and translate it. + +For example, there is a differential equation known as {\sl Bessel's +equation} which is +$$ +z^2 \frac{d^2 y}{dz^2} + z \frac{dy}{dz} + (z^2 - v^2)y = 0 +$$ + +We can plot a solution to this equation in Axiom with the command: +\spadcommand{draw(5*besselJ(0,sqrt(x**2+y**2)), x=-20..20, y=-20..20)} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=233, bbury=223]{ps/bookvol1b.ps} +\caption{$J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$} +\end{figure} +Draw $J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$. + +\subsection{Data Structures} + +A variety of data structures are available for interactive use. These +include strings, lists, vectors, sets, multisets, and hash tables. A +particularly useful structure for interactive use is the infinite +stream: + +\index{stream} +Create the infinite stream of derivatives of Legendre polynomials. + +\spadcommand{[D(p(i),x) for i in 1..]} +$$ +\begin{array}{@{}l} +\displaystyle +\left[ 1, {3 \ x}, {{{{15}\over 2}\ {x^2}}-{3 \over 2}}, + {{{{35}\over 2}\ {x^3}}-{{{15}\over 2}\ x}}, {{{{315}\over +8}\ {x^4}}-{{{105}\over 4}\ {x^2}}+{{15}\over 8}}, \right. +\\ +\\ +\displaystyle +\left.{{{{693}\over 8}\ {x^5}}-{{{315}\over 4}\ {x^3}}+{{{105}\over +8}\ x}}, {{{{3003}\over{16}}\ {x^6}}-{{{3465}\over{16}}\ {x^ +4}}+{{{945}\over{16}}\ {x^2}}-{{35}\over{16}}}, \right. +\\ +\\ +\displaystyle +\left.{{{{6435}\over{16}}\ {x^7}}-{{{9009}\over{16}}\ {x^5}}+ +{{{3465}\over{16}}\ {x^3}}-{{{315}\over{16}}\ x}}, \right. +\\ +\\ +\displaystyle +\left.{{{{109395}\over{128}}\ {x^8}}-{{{45045}\over{32}}\ {x^ +6}}+{{{45045}\over{64}}\ {x^4}}-{{{3465}\over{32}}\ {x^2}}+{{3 +15}\over{128}}}, \right. +\\ +\\ +\displaystyle +\left.{{{{230945}\over{128}}\ {x^9}}-{{{109395}\over{32}}\ {x^ +7}}+{{{135135}\over{64}}\ {x^5}}-{{{15015}\over{32}}\ {x^3}}+ +{{{3465}\over{128}}\ x}}, \ldots \right] +\end{array} +$$ +\returnType{Type: Stream Polynomial Fraction Integer} + + +Streams display only a few of their initial elements. Otherwise, they +are ``lazy'': they only compute elements when you ask for them. + +Data structures are an important component for building application +software. Advanced users can represent data for applications in an +optimal fashion. In all, Axiom offers over forty kinds of +aggregate data structures, ranging from mutable structures (such as +cyclic lists and flexible arrays) to storage efficient structures +(such as bit vectors). As an example, streams are used as the +internal data structure for power series. + +\index{series} +What is the series expansion +of $\log(\cot(x))$ +about $x=\pi/2$? +%NOTE: The book has a different answer (see p6) + +\spadcommand{series(log(cot(x)),x = \%pi/2)} +$$ +\begin{array}{@{}l} +\displaystyle +{\log \left({{-{2 \ x}+ \pi}\over 2}\right)}+ +{{1 \over 3}\ {{\left(x -{\pi \over 2}\right)}^2}}+ +{{7 \over{90}}\ {{\left(x -{\pi \over 2}\right)}^4}}+ +{{{62}\over{2835}}\ {{\left(x -{\pi \over 2}\right)}^6}}+ +\\ +\\ +\displaystyle +{{{127}\over{18900}}\ {{\left(x -{\pi \over 2}\right)}^8}}+ +{{{146}\over{66825}}\ {{\left(x -{\pi \over 2}\right)}^{10}}}+ +{O \left({{\left(x -{\pi \over 2}\right)}^{11}}\right)} +\end{array} +$$ +\returnType{Type: GeneralUnivariatePowerSeries(Expression Integer,x,pi/2)} + +Series and streams make no attempt to compute {\it all} their +elements! Rather, they stand ready to deliver elements on demand. + +What is the coefficient of the $50$-th +term of this series? + +\index{coefficient} +\spadcommand{coefficient(\%,50)} +$$ +{44590788901016030052447242300856550965644} \over +{7131469286438669111584090881309360354581359130859375} +$$ +\returnType{Type: Expression Integer} + +\index{\%} +Note the use of ``\%'' here. This means the value of the last +expression we computed. In this case it is the long expression +above. + +\subsection{Pattern Matching} + +A convenient facility for symbolic computation is ``pattern +matching.'' Suppose you have a trigonometric expression and you want +to transform it to some equivalent form. Use a $rule$ command to +describe the transformation rules you \index{rule} need. Then give +the rules a name and apply that name as a function to your +trigonometric expression. + +Here we introduce two rewrite rules. These are given in a ``pile'' +syntax using indentation. We store them in a file in the following +form: + +\spadcommand{sinCosExpandRules := rule\\ +\ \ sin(x+y) == sin(x)*cos(y) + sin(y)*cos(x)\\ +\ \ cos(x+y) == cos(x)*cos(y) - sin(x)*sin(y)\\ +\ \ sin(2*x) == 2*sin(x)*cos(x)\\ +\ \ cos(2*x) == cos(x)**2 - sin(x)**2 +} + +Then we use the {\bf )read} command to read the {\bf input} file. +The {\bf )read} command yields: +\begin{verbatim} + {sin(y + x) == cos(x)sin(y) + cos(y)sin(x), + cos(y + x) == - sin(x)sin(y) + cos(x)cos(y), + sin(2x) == 2cos(x)sin(x), + 2 2 + cos(2x) == - sin(x) + cos(x) } +\end{verbatim} +\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} + +Now we can apply the rules to a simple trigonometric expression. + +\spadcommand{sinCosExpandRules(sin(a+2*b+c))} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left(-{{\cos \left({a}\right)}\ {{\sin \left({b}\right)}^2}}- +{2 \ {\cos \left({b}\right)}\ {\sin \left({a}\right)}\ {\sin +\left({b}\right)}}+{{\cos \left({a}\right)}\ {{\cos \left({b}\right)}^ +2}}\right)}\ {\sin \left({c}\right)}}- +\\ +\\ +\displaystyle +{{\cos \left({c}\right)}\ {\sin \left({a}\right)}\ {{\sin \left({b}\right)}^ +2}}+{2 \ {\cos \left({a}\right)}\ {\cos \left({b}\right)}\ {\cos +\left({c}\right)}\ {\sin \left({b}\right)}}+ +\\ +\\ +\displaystyle +{{{\cos \left({b}\right)}^2}\ {\cos \left({c}\right)}\ {\sin +\left({a}\right)}} +\end{array} +$$ +\returnType{Type: Expression Integer} + + +Using {\bf input} files and the {\bf )read} command, +you can create your own library of transformation +rules relevant to your applications, then selectively apply the rules +you need. + +\subsection{Polymorphic Algorithms} +All components of the Axiom algebra library are written in the +Axiom library language called {\bf Spad}.\footnote{{\bf Spad} is +short for Scratchpad which was the original name of the Axiom system} +This language is similar to the +interactive language except for protocols that authors are obliged to +follow. The library language permits you to write ``polymorphic +algorithms,'' algorithms defined to work in their most natural +settings and over a variety of types. + +Here we define a system of polynomial equations $S$. + +\spadcommand{S := [3*x**3 + y + 1 = 0,y**2 = 4]} +$$ +\left[ +{{y+{3 \ {x \sp 3}}+1}=0}, {{y \sp 2}=4} +\right] +$$ +\returnType{Type: List Equation Polynomial Integer} + +And then we solve the system $S$ using rational number arithmetic and +30 digits of accuracy. + +\index{solve} +\spadcommand{solve(S,1/10**30)} +$$ +\left[ +{\left[ {y=-2}, {x={{1757879671211184245283070414507} \over +{2535301200456458802993406410752}}} +\right]}, + {\left[ {y=2}, {x=-1} +\right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Fraction Integer} + +Or we can solve $S$ with the solutions expressed in radicals. + +\index{radicalSolve} +\spadcommand{radicalSolve(S)} +$$ +\begin{array}{@{}l} +\displaystyle +\left[{\left[{y = 2}, {x = - 1}\right]}, {\left[{y = 2}, +{x ={{-{\sqrt{- 3}}+ 1}\over 2}}\right]}, \right. +\\ +\\ +\displaystyle +\left.{\left[{y = 2}, {x ={{{\sqrt{- 3}}+ 1}\over 2}}\right]}, + {\left[{y = - 2}, {x ={1 \over{\root{3}\of{3}}}}\right]}, + \right. +\\ +\\ +\displaystyle +\left.{\left[{y = - 2}, {x ={{{{\sqrt{- 1}}\ {\sqrt{3}}}- 1}\over{2 +\ {\root{3}\of{3}}}}}\right]}, {\left[{y = - 2}, {x ={{-{{\sqrt{- + 1}}\ {\sqrt{3}}}- 1}\over{2 \ {\root{3}\of{3}}}}}\right]}\right] +\end{array} +$$ +\returnType{Type: List List Equation Expression Integer} + +While these solutions look very different, the results were produced +by the same internal algorithm! The internal algorithm actually works +with equations over any ``field.'' Examples of fields are the +rational numbers, floating point numbers, rational functions, power +series, and general expressions involving radicals. + +\subsection{Extensibility} + +Users and system developers alike can augment the Axiom library, +all using one common language. Library code, like interpreter code, +is compiled into machine binary code for run-time efficiency. + +Using this language, you can create new computational types and new +algorithmic packages. All library code is polymorphic, described in +terms of a database of algebraic properties. By following the +language protocols, there is an automatic, guaranteed interaction +between your code and that of colleagues and system implementers. + +\subsection{Open Source} + +Axiom is completely open source. All of the algebra and all of the +source code for the interpreter, compiler, graphics, browser, and +numerics is shipped with the system. There are several websites +that host Axiom source code. + +Axiom is written using Literate Programming\cite{2} so each file is actually +a document rather than just machine source code. The goal is to make +the whole system completely literate so people can actually read the +system and understand it. This is the first volume in a series of books +that will attempt to reach that goal. + +The primary site is the Axiom wiki (http://wiki.axiom-developer.org). +The wiki is the general support site for Axiom. Wikis allow users to +interactively modify web pages. On the Axiom site it is possible to +type Axiom code directly into the browser to create modified pages +which are recomputed on the fly. + +The wiki also contains links to the Axiom documentation, the Axiom +mailing list (axiom-developer@nongnu.org), and many other facilities. + +Axiom is hosted at the Free Software Foundation site which is\\ +(http://savannah.nongnu.org/projects/axiom). + +Axiom is hosted at the Sourceforge site which is\\ +(http://sourceforge.net/projects/axiom). + +\vfill\eject +\chapter{Ten Fundamental Ideas} +\label{ch:Ten Fundamental Ideas} +Axiom has both an {\it interactive language} for user +interactions and a {\it programming language} for building library +modules. Like Modula 2, \index{Modula 2} PASCAL, \index{PASCAL} +FORTRAN, \index{FORTRAN} and Ada, \index{Ada} the programming language +emphasizes strict type-checking. Unlike these languages, types in +Axiom are dynamic objects: they are created at run-time in +response to user commands. + +Here is the idea of the Axiom programming language in a +nutshell. Axiom types range from algebraic ones (like +polynomials, matrices, and power series) to data structures (like +lists, dictionaries, and input files). Types combine in any +meaningful way. You can build polynomials of matrices, matrices of +polynomials of power series, hash tables with symbolic keys and +rational function entries, and so on. + +{\it Categories} define algebraic properties to ensure mathematical +correctness. They ensure, for example, that matrices of polynomials +are OK, but matrices of input files are not. Through categories, +programs can discover that polynomials of continued fractions have a +commutative multiplication whereas polynomials of matrices do not. + +Categories allow algorithms to be defined in their most natural +setting. For example, an algorithm can be defined to solve polynomial +equations over {\it any} field. Likewise a greatest common divisor +can compute the ``gcd'' of two elements from {\it any} Euclidean +domain. Categories foil attempts to compute meaningless ``gcds'', for +example, of two hashtables. Categories also enable algorithms to be +compiled into machine code that can be run with arbitrary types. + +The Axiom interactive language is oriented towards ease-of-use. +The Axiom interpreter uses type-inferencing to deduce the type +of an object from user input. Type declarations can generally be +omitted for common types in the interactive language. + +So much for the nutshell. +Here are these basic ideas described by ten design principles: + +\subsection{Types are Defined by Abstract Datatype Programs} + +Basic types are called {\it domains of computation}, or, +simply, {\it domains.} +\index{domain} +Domains are defined by Axiom programs of the form: + +\begin{verbatim} +Name(...): Exports == Implementation +\end{verbatim} + +Each domain has a capitalized {\tt Name} that is used to refer to the +class of its members. For example, {\tt Integer} denotes ``the +class of integers,'' {\tt Float}, ``the class of floating point +numbers,'' and {\tt String}, ``the class of strings.'' + +The ``{\tt ...}'' part following {\tt Name} lists zero or more +parameters to the constructor. Some basic ones like {\tt Integer} take +no parameters. Others, like {\tt Matrix}, {\tt Polynomial} and +{\tt List}, take a single parameter that again must be a domain. For +example, {\tt Matrix(Integer)} denotes ``matrices over the integers,'' +{\tt Polynomial (Float)} denotes ``polynomial with floating point +coefficients,'' and {\tt List (Matrix (Polynomial (Integer)))} denotes +``lists of matrices of polynomials over the integers.'' There is no +restriction on the number or type of parameters of a domain +constructor. + +SquareMatrix(2,Integer) is an example of a domain constructor that accepts +both a particular data value as well as an integer. In this case the +number 2 specifies the number of rows and columns the square matrix +will contain. Elements of the matricies are integers. + +The {\tt Exports} part specifies operations for creating and +manipulating objects of the domain. For example, type +{\tt Integer} exports constants $0$ and $1$, and +operations \spadopFrom{+}{Integer}, \spadopFrom{-}{Integer}, and +\spadopFrom{*}{Integer}. While these operations are common, others +such as \spadfunFrom{odd?}{Integer} and \spadfunFrom{bit?}{Integer} +are not. In addition the Exports section can contain symbols that +represent properties that can be tested. For example, the Category +{\tt EntireRing} has the symbol {\tt noZeroDivisors} which asserts +that if a product is zero then one of the factors must be zero. + +The {\tt Implementation} part defines functions that implement the +exported operations of the domain. These functions are frequently +described in terms of another lower-level domain used to represent the +objects of the domain. Thus the operation of adding two vectors of +real numbers can be described and implemented using the addition +operation from {\tt Float}. + +\subsection{The Type of Basic Objects is a Domain or Subdomain} + +Every Axiom object belongs to a {\it unique} domain. The domain +of an object is also called its {\it type.} Thus the integer $7$ +has type {\tt Integer} and the string {\tt "daniel"} has type +{\tt String}. + +The type of an object, however, is not unique. The type of integer +$7$ is not only {\tt Integer} but {\tt NonNegativeInteger}, +{\tt PositiveInteger}, and possibly, in general, any other +``subdomain'' of the domain {\tt Integer}. A {\it subdomain} +\index{subdomain} is a domain with a ``membership predicate''. +{\tt PositiveInteger} is a subdomain of {\tt Integer} with the +predicate ``is the integer $> 0$?''. + +Subdomains with names are defined by abstract datatype programs +similar to those for domains. The {\it Export} part of a subdomain, +however, must list a subset of the exports of the domain. The {\tt +Implementation} part optionally gives special definitions for +subdomain objects. + +\subsection{Domains Have Types Called Categories} + +Domain and subdomains in Axiom are themselves objects that have +types. The type of a domain or subdomain is called a {\it category}. +\index{category} Categories are described by programs of the form: + +\begin{verbatim} +Name(...): Category == Exports +\end{verbatim} +The type of every category is the distinguished symbol {\tt Category.} +The category {\tt Name} is used to designate the class of domains of +that type. For example, category {\tt Ring} designates the class +of all rings. Like domains, categories can take zero or more +parameters as indicated by the ``{\tt ...}'' part following {\tt +Name.} Two examples are {\tt Module(R)} and +{\tt MatrixCategory(R,Row,Col)}. + +The {\tt Exports} part defines a set of operations. For example, +{\tt Ring} exports the operations \spadopFrom{0}{Ring}, +\spadopFrom{1}{Ring}, \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, and +\spadopFrom{*}{Ring}. Many algebraic domains such as +{\tt Integer} and {\tt Polynomial (Float)} are rings. +{\tt String} and {\tt List (R)} (for any domain $R$) +are not. + +Categories serve to ensure the type-correctness. The definition of +matrices states {\tt Matrix(R: Ring)} requiring its single parameter +$R$ to be a ring. Thus a ``matrix of polynomials'' is allowed, +but ``matrix of lists'' is not. + +Categories say nothing about representation. Domains, which are +instances of category types, specify representations. + +\subsection{Operations Can Refer To Abstract Types} + +All operations have prescribed source and target types. Types can be +denoted by symbols that stand for domains, called ``symbolic +domains.'' The following lines of Axiom code use a symbolic +domain $R$: + +\begin{verbatim} +R: Ring +power: (R, NonNegativeInteger): R -> R +power(x, n) == x ** n +\end{verbatim} + +Line 1 declares the symbol $R$ to be a ring. Line 2 declares the +type of $power$ in terms of $R$. From the definition on +line 3, $power(3,2)$ produces 9 for $x = 3$ and $R =$ +{\tt Integer}. Also, $power(3.0,2)$ produces $9.0$ for +$x = 3.0$ and $R =$ {\tt Float}. +$power("oxford",2)$ however fails since $"oxford"$ has type +{\tt String} which is not a ring. + +Using symbolic domains, algorithms can be defined in their most +natural or general setting. + +\subsection{Categories Form Hierarchies} + +Categories form hierarchies (technically, directed-acyclic graphs). A +simplified hierarchical world of algebraic categories is shown below. +At the top of this world is {\tt SetCategory}, the class of +algebraic sets. The notions of parents, ancestors, and descendants is +clear. Thus ordered sets (domains of category {\tt OrderedSet}) +and rings are also algebraic sets. Likewise, fields and integral +domains are rings and algebraic sets. However fields and integral +domains are not ordered sets. + +\begin{verbatim} +SetCategory +---- Ring ---- IntegralDomain ---- Field + | + +---- Finite ---+ + | \ + +---- OrderedSet -----+ OrderedFinite +\end{verbatim} +\begin{center} +Figure 1. A simplified category hierarchy. +\end{center} + +\subsection{Domains Belong to Categories by Assertion} + +A category designates a class of domains. Which domains? You might +think that {\tt Ring} designates the class of all domains that +export $0$, $1$, \spadopFrom{+}{Integer}, +\spadopFrom{-}{Integer}, and \spadopFrom{*}{Integer}. But this is not +so. Each domain must {\it assert} which categories it belongs to. + +The {\tt Export} part of the definition for {\tt Integer} reads, +for example: + +\begin{verbatim} +Join(OrderedSet, IntegralDomain, ...) with ... +\end{verbatim} + +This definition asserts that {\tt Integer} is both an ordered set +and an integral domain. In fact, {\tt Integer} does not +explicitly export constants $0$ and $1$ and operations +\spadopFrom{+}{Ring}, \spadopFrom{-}{Ring} and \spadopFrom{*}{Ring} at +all: it inherits them all from $Ring$! Since +{\tt IntegralDomain} is a descendant of $Ring$, +{\tt Integer} is therefore also a ring. + +Assertions can be conditional. For example, {\tt Complex(R)} +defines its exports by: + +\begin{verbatim} +Ring with ... if R has Field then Field ... +\end{verbatim} +Thus {\tt Complex(Float)} is a field but {\tt Complex(Integer)} +is not since {\tt Integer} is not a field. + +You may wonder: ``Why not simply let the set of operations determine +whether a domain belongs to a given category?''. Axiom allows +operation names (for example, {\bf norm}) to have very different +meanings in different contexts. The meaning of an operation in +Axiom is determined by context. By associating operations with +categories, operation names can be reused whenever appropriate or +convenient to do so. As a simple example, the operation {\tt <} +might be used to denote lexicographic-comparison in an algorithm. +However, it is wrong to use the same {\tt <} with this definition +of absolute-value: $$abs(x) == if\ x < 0\ then -x\ else\ x$$ Such a +definition for {\tt abs} in Axiom is protected by context: +argument $x$ is required to be a member of a domain of category +{\tt OrderedSet}. + +\subsection{Packages Are Clusters of Polymorphic Operations} + +In Axiom, facilities for symbolic integration, solution of +equations, and the like are placed in ``packages''. A {\it package} +\index{package} is a special kind of domain: one whose exported +operations depend solely on the parameters of the constructor and/or +explicit domains. Packages, unlike Domains, do not specify the +representation. + +If you want to use Axiom, for example, to define some algorithms +for solving equations of polynomials over an arbitrary field $F$, +you can do so with a package of the form: + +\begin{verbatim} +MySolve(F: Field): Exports == Implementation +\end{verbatim} +where {\tt Exports} specifies the {\bf solve} operations +you wish to export from the domain and the {\tt Implementation} +defines functions for implementing your algorithms. Once Axiom has +compiled your package, your algorithms can then be used for any {\tt F}: +floating-point numbers, rational numbers, complex rational functions, +and power series, to name a few. + +\subsection{The Interpreter Builds Domains Dynamically} + +The Axiom interpreter reads user input then builds whatever types +it needs to perform the indicated computations. +For example, to create the matrix +$$M = \pmatrix{x^2+1&0\cr0&x / 2\cr}$$ +using the command: + +\spadcommand{M = [ [x**2+1,0],[0,x / 2] ]::Matrix(POLY(FRAC(INT)))} +$$ +M={\left[ +\begin{array}{cc} +x^2+1 & 0 \\ +0 & x/2 +\end{array} +\right]} +$$ +\returnType{Type: Matrix Polynomial Fraction Integer} +the interpreter first loads the modules {\tt Matrix}, +{\tt Polynomial}, {\tt Fraction}, and {\tt Integer} +from the library, then builds the {\it domain tower} ``matrices of +polynomials of rational numbers (i.e. fractions of integers)''. + +You can watch the loading process by first typing + +\spadcommand{)set message autoload on} +In addition to the named +domains above many additional domains and categories are loaded. +Most systems are preloaded with such common types. For efficiency +reasons the most common domains are preloaded but most (there are +more than 1100 domains, categories, and packages) are not. Once these +domains are loaded they are immediately available to the interpreter. + +Once a domain tower is built, it contains all the operations specific +to the type. Computation proceeds by calling operations that exist in +the tower. For example, suppose that the user asks to square the +above matrix. To do this, the function \spadopFrom{*}{Matrix} from +{\tt Matrix} is passed the matrix $M$ to compute $M * M$. +The function is also passed an environment containing $R$ +that, in this case, is {\tt Polynomial (Fraction (Integer))}. +This results in the successive calling of the \spadopFrom{*}{Fraction} +operations from {\tt Polynomial}, then from {\tt Fraction}, +and then finally from {\tt Integer}. + +Categories play a policing role in the building of domains. Because +the argument of {\tt Matrix} is required to be a {\tt Ring}, +Axiom will not build nonsensical types such as ``matrices of +input files''. + +\subsection{Axiom Code is Compiled} + +Axiom programs are statically compiled to machine code, then +placed into library modules. Categories provide an important role in +obtaining efficient object code by enabling: +\begin{itemize} +\item static type-checking at compile time; +\item fast linkage to operations in domain-valued parameters; +\item optimization techniques to be used for partially specified types +(operations for ``vectors of $R$'', for instance, can be open-coded even +though {\tt R} is unknown). +\end{itemize} + +\subsection{Axiom is Extensible} + +Users and system implementers alike use the Axiom language to +add facilities to the Axiom library. The entire Axiom +library is in fact written in the Axiom source code and +available for user modification and/or extension. + +Axiom's use of abstract datatypes clearly separates the exports +of a domain (what operations are defined) from its implementation (how +the objects are represented and operations are defined). Users of a +domain can thus only create and manipulate objects through these +exported operations. This allows implementers to ``remove and +replace'' parts of the library safely by newly upgraded (and, we hope, +correct) implementations without consequence to its users. + +Categories protect names by context, making the same names available +for use in other contexts. Categories also provide for code-economy. +Algorithms can be parameterized categorically to characterize their +correct and most general context. Once compiled, the same machine +code is applicable in all such contexts. + +Finally, Axiom provides an automatic, guaranteed interaction +between new and old code. For example: +\begin{itemize} +\item if you write a new algorithm that requires a parameter to be a +field, then your algorithm will work automatically with every field +defined in the system; past, present, or future. +\item if you introduce a new domain constructor that produces a field, +then the objects of that domain can be used as parameters to any algorithm +using field objects defined in the system; past, present, or future. +\end{itemize} + + +Before embarking on the tour, we need to brief those readers working +interactively with Axiom on some details. + +\chapter{Starting Axiom} +\label{ch:Starting Axiom} + +Welcome to the Axiom environment for interactive computation and +problem solving. Consider this chapter a brief, whirlwind tour of the +Axiom world. We introduce you to Axiom's graphics and the +Axiom language. Then we give a sampling of the large variety of +facilities in the Axiom system, ranging from the various kinds +of numbers, to data types (like lists, arrays, and sets) and +mathematical objects (like matrices, integrals, and differential +equations). We include a discussion of system commands and an +interactive ``undo.'' +\section{Starting Up and Winding Down} +\label{sec:Starting Up and Winding Down} +You need to know how to start the Axiom system and how to stop it. +We assume that Axiom has been correctly installed on your +machine. Information on how to install Axiom is available on +the wiki website\cite{3}. + +To begin using Axiom, issue the command {\bf axiom} to the +operating system shell. +\index{axiom @{\bf axiom}} There is a brief pause, some start-up +messages, and then one or more windows appear. + +If you are not running Axiom under the X Window System, there is +only one window (the console). At the lower left of the screen there +is a prompt that \index{prompt} looks like +\begin{verbatim} +(1) -> +\end{verbatim} + +When you want to enter input to Axiom, you do so on the same +line after the prompt. The ``1'' in ``(1)'', also called the equation +number, is the computation step number and is incremented +\index{step number} after you enter Axiom statements. +Note, however, that a system command such as {\tt )clear all} +may change the step number in other ways. We talk about step numbers +more when we discuss system commands and the workspace history facility. + +If you are running Axiom under the X Window System, there may be +two \index{X Window System} windows: the console window (as just +described) and the HyperDoc main menu. \index{Hyper@{HyperDoc}} +HyperDoc is a multiple-window hypertext system +that lets you \index{window} view Axiom documentation and +examples on-line, execute Axiom expressions, and generate +graphics. If you are in a graphical windowing environment, it is +usually started automatically when Axiom begins. If it is not +running, issue {\tt )hd} to start it. + +To interrupt an Axiom computation, hold down the \index{interrupt} +{\bf Ctrl} (control) key and press {\bf c}. This brings you back to +the Axiom prompt. + +\boxed{4.6in}{ +\vskip 0.1cm +To exit from Axiom, move to the console window, \index{stopping +@{stopping Axiom}} type {\tt )quit} \index{exiting @{exiting +Axiom}} at the input prompt and press the {\bf Enter} key. +You will probably be prompted with the following +message: +\begin{center} +Please enter {\bf y} or {\bf yes} if you really want to leave the \\ +interactive environment and return to the operating system +\end{center} +You should respond {\bf yes}, for example, to exit Axiom.\\ +} + +We are purposely vague in describing exactly what your screen looks +like or what messages Axiom displays. Axiom runs on a number of +different machines, operating systems and window environments, and +these differences all affect the physical look of the system. You can +also change the way that Axiom behaves via {\it system commands} +described later in this chapter and in the Axiom System Commands. +(Chapter \ref{sec:Axiom System Commands} on +page \pageref{sec:Axiom System Commands}) +System commands are special commands, like {\tt )set}, that begin with +a closing parenthesis and are used to change your environment. For +example, you can set a system variable so that you are not prompted +for confirmation when you want to leave Axiom. + +\subsection{Clef} +\label{sec:Clef} +If you are using Axiom under the X Window System, the +\index{Clef} \index{command line editor} Clef command +line editor is probably available and installed. With this editor you +can recall previous lines with the up and down arrow keys. To move +forward and backward on a line, use the right and left arrows. You +can use the {\bf Insert} key to toggle insert mode on or off. When +you are in insert mode, the cursor appears as a large block and if you +type anything, the characters are inserted into the line without +deleting the previous ones. + +If you press the {\bf Home} key, the cursor moves to the beginning of +the line and if you press the {\bf End} key, the cursor moves to the +end of the line. Pressing {\bf Ctrl-End} deletes all the text from +the cursor to the end of the line. + +Clef also provides Axiom operation name completion for +\index{operation name completion} a limited set of operations. If you +enter a few letters and then press the {\bf Tab} key, Clef tries to +use those letters as the prefix of an Axiom operation name. If +a name appears and it is not what you want, press {\bf Tab} again to +see another name. + +\subsection{Typographic Conventions} +\label{sec:Typographic Conventions} +In this document we have followed these typographical conventions: +\begin{itemize} +% +\item Categories, domains and packages are displayed in this font: +{\tt Ring}, {\tt Integer}, {\tt DiophantineSolutionPackage}. +% +\item Prefix operators, infix operators, and punctuation symbols in +the Axiom language are displayed in the text like this: +{\tt +}, {\tt \$}, {\tt +->}. +% +\item Axiom expressions or expression fragments are displayed in this font:\\ +{\tt inc(x) == x + 1}. +% +\item For clarity of presentation, \TeX{} is often used to format expressions\\ +$g(x)=x^2+1$. +% +\item Function names and HyperDoc button names are displayed in the text in +this font: +{\bf factor}, {\bf integrate}, {\bf Lighting}. +% +\item Italics are used for emphasis and for words defined in the glossary: \\ +{\it category}. +\end{itemize} + +This document contains over many examples of Axiom input and output. All +examples were run though Axiom and their output was created in \TeX{} +form. We have deleted system +messages from the example output if those messages are not important +for the discussions in which the examples appear. + +\section{The Axiom Language} +\label{sec:The Axiom Language} +The Axiom language is a rich language for performing interactive +computations and for building components of the Axiom library. +Here we present only some basic aspects of the language that you need +to know for the rest of this chapter. Our discussion here is +intentionally informal, with details unveiled on an ``as needed'' +basis. For more information on a particular construct, we suggest you +consult the index. + +\subsection{Arithmetic Expressions} +\label{sec:Arithmetic Expressions} +For arithmetic expressions, use the ``{\tt +}'' and ``{\tt -}'' operator +as in mathematics. Use ``{\tt *}'' for multiplication, and ``{\tt **}'' +for exponentiation. To create a fraction, use ``{\tt /}''. When an +expression contains several operators, those of highest +{\it precedence} are evaluated first. For arithmetic operators, +``{\tt **}'' has highest precedence, ``{\tt *}'' and ``{\tt /}'' have the +next highest precedence, and ``{\tt +}'' and ``{\tt -}'' have the lowest +precedence. + +Axiom puts implicit parentheses around operations of higher +precedence, and groups those of equal precedence from left to right. +\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1} +$$ +-{{19} \over 4} +$$ +\returnType{Type: Fraction Integer} + +The above expression is equivalent to this. +\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1} +$$ +-{{19} \over 4} +$$ +\returnType{Type: Fraction Integer} + +If an expression contains subexpressions enclosed in parentheses, +the parenthesized subexpressions are evaluated first (from left to +right, from inside out). +\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))} +$$ +{11} \over 4 +$$ +\returnType{Type: Fraction Integer} + +\subsection{Previous Results} +\label{sec:Previous Results} +\index{\%} +\index{\%\%} +Use the percent sign ``{\tt \%}'' to refer to the last result. +\index{result!previous} Also, use ``{\tt \%\%}' to refer to +previous results. \index{percentpercent@{\%\%}} ``{\tt \%\%(-1)}'' is +equivalent to ``{\tt \%}'', ``{\tt \%\%(-2)}'' returns the next to +the last result, and so on. ``{\tt \%\%(1)}'' returns the result from +step number 1, ``{\tt \%\%(2)}'' returns the result from step number 2, +and so on. ``{\tt \%\%(0)}'' is not defined. + +This is ten to the tenth power. +\spadcommand{10 ** 10} +$$ +10000000000 +$$ +\returnType{Type: PositiveInteger} + +This is the last result minus one. +\spadcommand{\% - 1} +$$ +9999999999 +$$ +\returnType{Type: PositiveInteger} + +This is the last result. +\spadcommand{\%\%(-1)} +$$ +9999999999 +$$ +\returnType{Type: PositiveInteger} + +This is the result from step number 1. +\spadcommand{\%\%(1)} +$$ +10000000000 +$$ +\returnType{Type: PositiveInteger} + +\subsection{Some Types} +\label{sec:Some Types} +Everything in Axiom has a type. The type determines what operations +you can perform on an object and how the object can be used. + +Positive integers are given type {\bf PositiveInteger}. +\spadcommand{8} +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +Negative ones are given type {\bf Integer}. This fine +distinction is helpful to the Axiom interpreter. + +\spadcommand{-8} +$$ +-8 +$$ +\returnType{Type: Integer} + +Here a positive integer exponent gives a polynomial result. +\spadcommand{x**8} +$$ +x \sp 8 +$$ +\returnType{Type: Polynomial Integer} + +Here a negative integer exponent produces a fraction. +\spadcommand{x**(-8)} +$$ +1 \over {x \sp 8} +$$ +\returnType{Type: Fraction Polynomial Integer} + +\subsection{Symbols, Variables, Assignments, and Declarations} +\label{sec:Symbols, Variables, Assignments, and Declarations} +A {\it symbol} is a literal used for the input of things like +the ``variables'' in polynomials and power series. + +We use the three symbols $x$, $y$, and $z$ in +entering this polynomial. +\spadcommand{(x - y*z)**2} +$$ +{{y \sp 2} \ {z \sp 2}} -{2 \ x \ y \ z}+{x \sp 2} +$$ +\returnType{Type: Polynomial Integer} + +A symbol has a name beginning with an uppercase or lowercase +alphabetic \index{symbol!naming} character, ``{\tt \%}'', or +``{\tt !}''. Successive characters (if any) can be any of the +above, digits, or ``{\tt ?}''. Case is distinguished: the symbol +{\tt points} is different from the symbol {\tt Points}. + +A symbol can also be used in Axiom as a {\it variable}. A variable +refers to a value. To {\sl assign} a value to a variable, +\index{variable!naming} the operator ``{\tt :=}'' \index{assignment} +is used. Axiom actually has two forms of assignment: +{\it immediate assignment}, +\index{assignment!immediate} +\index{immediate assignment} +as discussed here, and +{\it delayed assignment}. +\index{assignment!delayed} +\index{delayed assignment} +A variable initially has no restrictions on the kinds +of \index{declaration} values to which it can refer. + +This assignment gives the value $4$ (an integer) to +a variable named $x$. +\spadcommand{x := 4} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +This gives the value $z + 3/5$ (a polynomial) to $x$. +\spadcommand{x := z + 3/5} +$$ +z+{3 \over 5} +$$ +\returnType{Type: Polynomial Fraction Integer} + +To restrict the types of objects that can be assigned to a variable, +use a {\it declaration} +\spadcommand{y : Integer} +\returnType{Type: Void} + +After a variable is declared to be of some type, only values +of that type can be assigned to that variable. +\spadcommand{y := 89} +$$ +89 +$$ +\returnType{Type: Integer} + +The declaration for $y$ forces values assigned to $y$ to +be converted to integer values. +\spadcommand{y := sin \%pi} +$$ +0 +$$ +\returnType{Type: Integer} + +If no such conversion is possible, +Axiom refuses to assign a value to $y$. +\spadcommand{y := 2/3} +\begin{verbatim} + Cannot convert right-hand side of assignment + 2 + - + 3 + + to an object of the type Integer of the left-hand side. +\end{verbatim} + +A type declaration can also be given together with an assignment. +The declaration can assist Axiom in choosing the correct +operations to apply. +\spadcommand{f : Float := 2/3} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +Any number of expressions can be given on input line. +Just separate them by semicolons. +Only the result of evaluating the last expression is displayed. + +These two expressions have the same effect as +the previous single expression. + +\spadcommand{f : Float; f := 2/3} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +The type of a symbol is either {\tt Symbol} +or {\tt Variable({\it name})} where {\it name} is the name +of the symbol. + +By default, the interpreter +gives this symbol the type {\tt Variable(q)}. + +\spadcommand{q} +$$ +q +$$ +\returnType{Type: Variable q} + +When multiple symbols are involved, {\tt Symbol} is used. +\spadcommand{[q, r]} +$$ +\left[ +q, r +\right] +$$ +\returnType{Type: List OrderedVariableList [q,r]} + +What happens when you try to use a symbol that is the name of a variable? +\spadcommand{f} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +Use a single quote ``{\tt '}'' before \index{quote} the name to get the symbol. + +\index{quote symbols} +\index{symbol quoting} +\spadcommand{'f} +$$ +f +$$ +\returnType{Type: Variable f} + +Quoting a name creates a symbol by preventing evaluation of the name +as a variable. Experience will teach you when you are most likely +going to need to use a quote. We try to point out the location of +such trouble spots. + +\subsection{Conversion} +\label{sec:Conversion} +\index{conversion} +\index{::} +Objects of one type can usually be ``converted'' to objects of several +other types. To {\sl convert} an object to a new type, use the ``{\tt ::}'' +infix operator. For example, +to display an object, it is necessary to convert the object to type +{\tt OutputForm}. + +This produces a polynomial with rational number coefficients. + +\spadcommand{p := r**2 + 2/3} +$$ +{r \sp 2}+{2 \over 3} +$$ +\returnType{Type: Polynomial Fraction Integer} + +Create a quotient of polynomials with integer coefficients +by using ``{\tt ::}''. + +\spadcommand{p :: Fraction Polynomial Integer } +$$ +{{3 \ {r \sp 2}}+2} \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +Some conversions can be performed automatically when Axiom tries +to evaluate your input. Others conversions must be explicitly +requested. + +\subsection{Calling Functions} +\label{sec:Calling Functions} +As we saw earlier, when you want to add or subtract two values, you +place the arithmetic operator ``{\tt +}'' or ``{\tt -}'' between the two +arguments denoting the values. To use most other Axiom +operations, however, you use another syntax: \index{function!calling} +write the name of the operation first, then an open parenthesis, then +each of the arguments separated by commas, and, finally, a closing +parenthesis. If the operation takes only one argument and the +argument is a number or a symbol, you can omit the parentheses. + +This calls the operation {\bf factor} with the single integer argument $120$. + +\index{factor} +\spadcommand{factor(120)} +$$ +{2 \sp 3} \ 3 \ 5 +$$ +\returnType{Type: Factored Integer} + +This is a call to {\bf divide} with the two integer arguments +$125$ and $7$. + +\index{divide} +\spadcommand{divide(125,7)} +$$ +\left[ +{quotient={17}}, {remainder=6} +\right] +$$ +\returnType{Type: Record(quotient: Integer, remainder: Integer)} + +This calls {\bf quatern} with four floating-point arguments. +\index{quatern} +\spadcommand{quatern(3.4,5.6,2.9,0.1)} +$$ +{3.4}+{{5.6} \ i}+{{2.9} \ j}+{{0.1} \ k} +$$ +\returnType{Type: Quaternion Float} + +This is the same as {\bf factorial}(10). +\index{factorial} +\spadcommand{factorial 10} +$$ +3628800 +$$ +\returnType{Type: PositiveInteger} + +An operations that returns a {\tt Boolean} value (that is, +{\tt true} or {\tt false}) frequently has a name suffixed with +a question mark (``?''). For example, the {\bf even?} +operation returns {\tt true} if its integer argument is an even +number, {\tt false} otherwise. + +An operation that can be destructive on one or more arguments +usually has a name ending in a exclamation point (``!''). +\index{destructive operations} +This actually means that it is {\it allowed} to update its +arguments but it is not {\it required} to do so. For example, +the underlying representation of a collection type may not allow +the very last element to removed and so an empty object may be +returned instead. Therefore, it is important that you use the +object returned by the operation and not rely on a physical +change having occurred within the object. Usually, destructive +operations are provided for efficiency reasons. + +\subsection{Some Predefined Macros} +\label{sec:Some Predefined Macros} +Axiom provides several macros for your convenience. Macros are names +\index{macro!predefined} (or forms) that expand to larger expressions +for commonly used values. + +\begin{center} +\begin{tabular}{ll} +{\it \%i} & The square root of -1. \\ +{\it \%e} & The base of the natural logarithm. \\ +{\it \%pi} & $\pi$. \\ +{\it \%infinity} & $\infty$. \\ +{\it \%plusInfinity} & $+\infty$. \\ +{\it \%minusInfinity} & $-\infty$. +\end{tabular} +\end{center} +\index{\%i} +\index{\%e} +\index{\%pi} +\index{pi@{$\pi$ (= \%pi)}} +\index{\%infinity} +\index{infinity@{$\infty$ (= \%infinity)}} +\index{\%plusInfinity} +\index{\%minusInfinity} + +To display all the macros (along with anything you have +defined in the workspace), issue the system command {\tt )display all}. + +\subsection{Long Lines} +\label{sec:Long Lines} +When you enter Axiom expressions from your keyboard, there will +be times when they are too long to fit on one line. Axiom does +not care how long your lines are, so you can let them continue from +the right margin to the left side of the next line. + +Alternatively, you may want to enter several shorter lines and have +Axiom glue them together. To get this glue, put an underscore +(\_) at the end of each line you wish to continue. +\index{\_ escape} +\begin{verbatim} +2_ ++_ +3 +\end{verbatim} +is the same as if you had entered +\begin{verbatim} +2+3 +\end{verbatim} + +Axiom statements in an {\bf input} file +can use indentation to indicate the program structure . + +\subsection{Comments} +\label{sec:Comments} +Comment statements begin with two consecutive hyphens or two +consecutive plus signs and continue until the end of the line. + +\index{-- comments} +\index{++ comments} +\index{+++ comments} +The comment beginning with ``{\tt --}'' is ignored by Axiom. +\spadcommand{2 + 3 -- this is rather simple, no?} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +There is no way to write long multi-line comments other than starting +each line with ``{\tt --}'' or ``{\tt ++}''. + +\section{Using Axiom as a Pocket Calculator} +At the simplest level Axiom can be used as a pocket calculator +where expressions involving numbers and operators are entered +directly in infix notation. In this sense the more advanced +features of the calculator can be regarded as operators (e.g +{\bf sin}, {\bf cos}, etc). + +\subsection{Basic Arithmetic} +An example of this might be to calculate the cosine of 2.45 (in radians). +To do this one would type: + +\spadcommand{(1)-> cos 2.45} +$$ +-{0.7702312540 473073417} +$$ +\returnType{Type: Float} + +Before proceeding any further it would be best to explain the previous +three lines. Axiom presents a ``(1) {\tt ->} '' prompt (shown here +but omitted elsewhere) +when interacting with the user. The full prompt has other +text preceding this but it is not relevant here. The number in parenthesis +is the step number of the input which may be used to refer to the +{\sl results} of previous calculations. The step number appears at the start +of the second line to tell you which step the result belongs to. Since the +interpreter probably loaded numerous libraries to calculate the result given +above and listed each one in the prcess, there could easily be several pages +of text between your input and the answer. + +The last line contains the type of the result. The type {\tt Float} is used +to represent real numbers of arbitrary size and precision (where the user is +able to define how big arbitrary is -- the default is 20 digits but can be +as large as your computer system can handle). The type of the result can help +track down mistakes in your input if you don't get the answer you expected. + +Other arithmetic operations such as addition, subtraction, and multiplication +behave as expected: + +\spadcommand{6.93 * 4.1328} +$$ +28.640304 +$$ +\returnType{Type: Float} + +\spadcommand{6.93 / 4.1328} +$$ +1.6768292682 926829268 +$$ +\returnType{Type: Float} + +but integer division isn't quite so obvious. For example, if one types: + +\spadcommand{4/6} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Integer} + +a fractional result is obtained. The function used to display fractions +attempts to produce the most readable answer. In the example: + +\spadcommand{4/2} +$$ +2 +$$ +\returnType{Type: Fraction Integer} + +the result is stored as the fraction 2/1 but is displayed as the integer 2. +This fraction could be converted to type {\tt Integer} with no loss of +information but Axiom will not do so automatically. + +\subsection{Type Conversion} +To obtain the floating point value of a fraction one must convert ( +{\bf conversions} are applied by the user and +{\bf coercions} are applied automatically by the interpreter) the result +to type {\tt Float} using the ``::'' operator as follows: + +\index{:: conversion} +\spadcommand{(4.6)::Float} +$$ +4.6 +$$ +\returnType{Type: Float} + +Although Axiom can convert this back to a fraction it might not be the +same fraction you started with due to rounding errors. For example, the +following conversion appears to be without error but others might not: + +\spadcommand{\%::Fraction Integer} +$$ +{23} \over 5 +$$ +\returnType{Type: Fraction Integer} + +\index{\%} +where ``\%'' represents the previous {\it result} (not the calculation). + +Although Axiom has the ability to work with floating-point numbers to +a very high precision it must be remembered that calculations with these +numbers are {\bf not} exact. Since Axiom is a computer algebra package and +not a numerical solutions package this should not create too many problems. +The idea is that the user should use Axiom to do all the necessary symbolic +manipulation and only at the end should actual numerical results be extracted. + +If you bear in mind that Axiom appears to store expressions just as you have +typed them and does not perform any evalutation of them unless forced to then +programming in the system will be much easier. It means that anything you +ask Axiom to do (within reason) will be carried out with complete accuracy. + +\index{:: failure} +In the previous examples the ``::'' operator was used to convert values from +one type to another. This type conversion is not possible for all values. +For instance, it is not possible to convert the number 3.4 to an integer +type since it can't be represented as an integer. The number 4.0 can be +converted to an integer type since it has no fractional part. + +Conversion from floating point values to integers is performed using the +functions {\bf round} and {\bf truncate}. The first of these rounds a +floating point number to the nearest integer while the other truncates +(i.e. removes the fractional part). Both functions return the result as a +{\bf floating point} number. To extract the fractional part of a floating +point number use the function {\bf fractionPart} but note that the sign +of the result depends on the sign of the argument. Axiom obtains the +fractional partof $x$ using $x - truncate(x)$: + +\index{round} +\spadcommand{round(3.77623)} +$$ +4.0 +$$ +\returnType{Type: Float} + +\spadcommand{round(-3.77623)} +$$ +-{4.0} +$$ +\returnType{Type: Float} + +\index{truncate} +\spadcommand{truncate(9.235)} +$$ +9.0 +$$ +\returnType{Type: Float} + +\spadcommand{truncate(-9.654)} +$$ +-{9.0} +$$ +\returnType{Type: Float} + +\index{fractionPart} +\spadcommand{fractionPart(-3.77623)} +$$ +-{0.77623} +$$ +\returnType{Type: Float} + +\subsection{Useful Functions} +To obtain the absolute value of a number the {\bf abs} function can be used. +This takes a single argument which is usually an integer or a floating point +value but doesn't necessarily have to be. The sign of a value can be obtained +via the {\bf sign} function which returns $-1$, $0$, or $1$ depending on the +sign of the argument. + +\index{abs} +\spadcommand{abs(4)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{abs(-3)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{abs(-34254.12314)} +$$ +34254.12314 +$$ +\returnType{Type: Float} + +\index{sign} +\spadcommand{sign(-49543.2345346)} +$$ +-1 +$$ +\returnType{Type: Integer} + +\spadcommand{sign(0)} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\spadcommand{sign(234235.42354)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Tests on values can be done using various functions which are generally more +efficient than using relational operators such as $=$ particularly if the +value is a matrix. Examples of some of these functions are: + +\index{positive?} +\spadcommand{positive?(-234)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{negative?} +\spadcommand{negative?(-234)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{zero?} +\spadcommand{zero?(42)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{one?} +\spadcommand{one?(1)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{odd?} +\spadcommand{odd?(23)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{odd?(9.435)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{even?} +\spadcommand{even?(-42)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{prime?} +\spadcommand{prime?(37)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\spadcommand{prime?(-37)} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +Some other functions that are quite useful for manipulating numerical values +are: + +\begin{verbatim} +sin(x) Sine of x +cos(x) Cosine of x +tan(x) Tangent of x +asin(x) Arcsin of x +acos(x) Arccos of x +atan(x) Arctangent of x +gcd(x,y) Greatest common divisor of x and y +lcm(x,y) Lowest common multiple of x and y +max(x,y) Maximum of x and y +min(x,y) Minimum of x and y +factorial(x) Factorial of x +factor(x) Prime factors of x +divide(x,y) Quotient and remainder of x/y +\end{verbatim} +\index{sin} +\index{cos} +\index{tan} +\index{asin} +\index{acos} +\index{atan} +\index{gcd} +\index{lcm} +\index{max} +\index{min} +\index{factorial} +\index{factor} +\index{divide} + +Some simple infix and prefix operators: +\begin{verbatim} ++ Addition - Subtraction +- Numerical Negation ~ Logical Negation +/\ Conjunction (AND) \/ Disjunction (OR) +and Logical AND (/\) or Logical OR (\/) +not Logical Negation ** Exponentiation +* Multiplication / Division +quo Quotient rem Remainder +< less than > greater than +<= less than or equal >= greater than or equal +\end{verbatim} +\index{$+$ Addition} +\index{$-$ Subtraction} +\index{$-$ Numerical Negation} +\index{$\tilde{}$ Logical Negation} +\index{not Logical Negation} +\index{$**$ Exponentiation} +\index{$*$ Multiplication} +\index{$/$ Division} +\index{quo Quotient} +\index{rem Remainder} +\index{$<$ less than} +\index{$>$ greater than} +\index{$<=$ less than or equal} +\index{$>=$ greater than or equal} + +Some useful Axiom macros: +\begin{verbatim} +%i The square root of -1 +%e The base of the natural logarithm +%pi Pi +%infinity Infinity +%plusInfinity Positive Infinity +%minusInfinity Negative Infinity +\end{verbatim} + +\section{Using Axiom as a Symbolic Calculator} +In the previous section all the examples involved numbers and simple +functions. Also none of the expressions entered were assigned to anything. +In this section we will move on to simple algebra (i.e. expressions involving +symbols and other features available on more sophisticated calculators). + +\subsection{Expressions Involving Symbols} +Expressions involving symbols are entered just as they are written down, +for example: + +\index{immediate assignment} +\index{assignment immediate} +\spadcommand{xSquared := x**2} +$$ +x \sp 2 +$$ +\returnType{Type: Polynomial Integer} + +where the assignment operator ``:='' represents immediate assignment. Later +it will be seen that this form of assignment is not always desirable and +the use of the delayed assignment operator ``=='' will be introduced. The +type of the result is {\tt Polynomial Integer} which is used to represent +polynomials with integer coefficients. Some other examples along similar +lines are: + +\spadcommand{xDummy := 3.21*x**2} +$$ +{3.21} \ {x \sp 2} +$$ +\returnType{Type: Polynomial Float} + +\spadcommand{xDummy := x**2.5} +$$ +{x \sp 2} \ {\sqrt {x}} +$$ +\returnType{Type: Expression Float} + +\spadcommand{xDummy := x**3.3} +$$ +{x \sp 3} \ {{\root {{10}} \of {x}} \sp 3} +$$ +\returnType{Type: Expression Float} + +\spadcommand{xyDummy := x**2 - y**2} +$$ +-{y \sp 2}+{x \sp 2} +$$ +\returnType{Type: Polynomial Integer} + +Given that we can define expressions involving symbols, how do we actually +compute the result when the symbols are assigned values? The answer is to +use the {\bf eval} function which takes an expression as its first argument +followed by a list of assignments. For example, to evaluate the expressions +{\sl XDummy} and {\sl xyDummy} resulting from their respective +assignments above we type: + +\index{eval} +\spadcommand{eval(xDummy,x=3)} +$$ +37.5405075985 29552193 +$$ +\returnType{Type: Expression Float} + +\spadcommand{eval(xyDummy, [x=3, y=2.1])} +$$ +4.59 +$$ +\returnType{Type: Polynomial Float} + +\subsection{Complex Numbers} +For many scientific calculations real numbers aren't sufficient and support +for complex numbers is also required. Complex numbers are handled in an +intuitive manner. Axiom uses the {\bf \%i} macro to represent +the square root of $-1$. Thus expressions involving complex numbers are +entered just like other expressions. + +\index{complex numbers} +\spadcommand{(2/3 + \%i)**3} +$$ +-{{46} \over {27}}+{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +The real and imaginary parts of a complex number can be extracted using +the {\bf real} and {\bf imag} functions and the complex conjugate of a +number can be obtained using {\bf conjugate}: + +\index{real, complex numbers} +\spadcommand{real(3 + 2*\%i)} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\index{imag, complex numbers} +\spadcommand{imag(3+ 2*\%i)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\index{conjugate, complex numbers} +\spadcommand{conjugate(3 + 2*\%i)} +$$ +3 -{2 \ i} +$$ +\returnType{Type: Complex Integer} + +The function {\bf factor} can also be applied to complex numbers but the +results aren't quite so obvious as for factoring integer: + +\spadcommand{144 + 24*\%i} +$$ +{144}+{{24} \ i} +$$ +\returnType{Type: Complex Integer} + +\index{factor, complex numbers} +\spadcommand{factor(\%)} +$$ +i \ {{\left( 1+i +\right)} +\sp 6} \ 3 \ {\left( 6+i +\right)} +$$ +\returnType{Type: Factored Complex Integer} + +\subsection{Number Representations} +By default all numerical results are displayed in decimal with real numbers +shown to 20 significant figures. If the integer part of a number is longer +than 20 digits then nothing after the decimal point is shown and the integer +part is given in full. To alter the number of digits shown the function +{\bf digits} can be called. The result returned by this function is the +previous setting. For example, to find the value of $\pi$ to 40 digits +we type: + +\index{digits function} +\spadcommand{digits(40)} +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{\%pi::Float} +$$ +3.1415926535\ 8979323846\ 2643383279\ 502884197 +$$ +\returnType{Type: Float} + +As can be seen in the example above, there is a gap after every ten digits. +This can be changed using the {\bf outputSpacing} function where the argument +is the number of digits to be displayed before a space is inserted. If no +spaces are desired then use the value $0$. Two other functions controlling +the appearance of real numbers are {\bf outputFloating} and {\bf outputFixed}. +The former causes Axiom to display floating-point values in exponent notation +and the latter causes it to use fixed-point notation. For example: + +\index{outputSpacing} +\index{outputFloating} +\spadcommand{outputFloating(); \%} +$$ +0.3141592653 5897932384 6264338327 9502884197 E 1 +$$ +\returnType{Type: Float} + +\spadcommand{outputFloating(3); 0.00345} +$$ +0.345 E -2 +$$ +\returnType{Type: Float} + +\index{outputFixed} +\spadcommand{outputFixed(); \%} +$$ +0.00345 +$$ +\returnType{Type: Float} + +\spadcommand{outputFixed(3); \%} +$$ +0.003 +$$ +\returnType{Type: Float} + +\index{outputGeneral} +\spadcommand{outputGeneral(); \%} +$$ +0.00345 +$$ +\returnType{Type: Float} + +\index{; output suppression} +Note that the semicolon ``;'' in the examples above allows several +expressions to be entered on one line. The result of the last expression +is displayed. remember also that the percent symbol ``\%'' is used to +represent the result of a previous calculation. + +To display rational numbers in a base other than 10 the function {\bf radix} +is used. The first argument of this function is the expression to be +displayed and the second is the base to be used. + +\index{radix} +\index{RadixExpansion} +\spadcommand{radix(10**10,32)} +$$ +{\rm 9A0NP00 } +$$ +\returnType{Type: RadixExpansion 32} + +\spadcommand{radix(3/21,5)} +$$ +0.{\overline {032412}} +$$ +\returnType{Type: RadixExpansion 5} + +Rational numbers can be represented as a repeated decimal expansion using +the {\bf decimal} function or as a continued fraction using +{\bf continuedFraction}. Any attempt to call these functions with irrational +values will fail. + +\index{decimal} +\index{DecimalExpansion} +\spadcommand{decimal(22/7)} +$$ +3.{\overline {142857}} +$$ +\returnType{Type: DecimalExpansion} + +\index{continuedFraction} +\spadcommand{continuedFraction(6543/210)} +$$ +{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} +$$ +\returnType{Type: ContinuedFraction Integer} + +Finally, partial fractions in compact and expanded form are available via the +functions {\bf partialFraction} and {\bf padicFraction} respectively. The +former takes two arguments, the first being the numerator of the fraction +and the second being the denominator. The latter function takes a fraction +and expands it further while the function {\bf compactFraction} does the +reverse: + +\index{partialFraction} +\spadcommand{partialFraction(234,40)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\index{padicFraction} +\spadcommand{padicFraction(\%)} +$$ +6 -{1 \over 2} -{1 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\index{compactFraction} +\spadcommand{compactFraction(\%)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\spadcommand{padicFraction(234/40)} +$$ +{117} \over {20} +$$ +\returnType{Type: PartialFraction Fraction Integer} + +To extract parts of a partial fraction the function {\bf nthFractionalTerm} +is available and returns a partial fraction of one term. To decompose this +further the numerator can be obtained using {\bf firstNumer} and the +denominator with {\bf firstDenom}. The whole part of a partial fraction can +be retrieved using {\bf wholePart} and the number of fractional parts can +be found using the function {\bf numberOf FractionalTerms}: + +\spadcommand{t := partialFraction(234,40)} +$$ +6 -{3 \over {2 \sp 2}}+{3 \over 5} +$$ +\returnType{Type: PartialFraction Integer} + +\index{wholePart} +\spadcommand{wholePart(t)} +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +\index{numberOfFractionalTerms} +\spadcommand{numberOfFractionalTerms(t)} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +\index{nthFractionalTerm} +\spadcommand{p := nthFractionalTerm(t,1)} +$$ +-{3 \over {2 \sp 2}} +$$ +\returnType{Type: PartialFraction Integer} + +\index{firstNumer} +\spadcommand{firstNumer(p)} +$$ +-3 +$$ +\returnType{Type: Integer} + +\index{firstDenom} +\spadcommand{firstDenom(p)} +$$ +2 \sp 2 +$$ +\returnType{Type: Factored Integer} + +\subsection{Modular Arithmetic} +By using the type constructor {\tt PrimeField} it is possible to do +arithmetic modulo some prime number. For example, arithmetic module $7$ +can be performed as follows: + +\index{PrimeField} +\spadcommand{x : PrimeField 7 := 5} +$$ +5 +$$ +\returnType{Type: PrimeField 7} + +\spadcommand{x**5 + 6} +$$ +2 +$$ +\returnType{Type: PrimeField 7} + +\spadcommand{1/x} +$$ +3 +$$ +\returnType{Type: PrimeField 7} + +The first example should be read as: +\begin{center} +{\tt Let $x$ be of type PrimeField(7) and assign to it the value $5$} +\end{center} + +Note that it is only possible to invert non-zero values if the arithmetic +is performed modulo a prime number. Thus arithmetic modulo a non-prime +integer is possible but the reciprocal operation is undefined and will +generate an error. Attempting to use the {\tt PrimeField} type constructor +with a non-prime argument will generate an error. An example of non-prime +modulo arithmetic is: + +\index{IntegerMod} +\spadcommand{y : IntegerMod 8 := 11} +$$ +3 +$$ +\returnType{Type: IntegerMod 8} + +\spadcommand{y*4 + 27} +$$ +7 +$$ +\returnType{Type: IntegerMod 8} + +Note that polynomials can be constructed in a similar way: + +\spadcommand{(3*a**4 + 27*a - 36)::Polynomial PrimeField 7} +$$ +{3 \ {a \sp 4}}+{6 \ a}+6 +$$ +\returnType{Type: Polynomial PrimeField 7} + +\section{General Points about Axiom} +\subsection{Computation Without Output} +It is sometimes desirable to enter an expression and prevent Axiom from +displaying the result. To do this the expression should be terminated with +a semicolon ``;''. In a previous section it was mentioned that a set of +expressions separated by semicolons would be evaluated and the result +of the last one displayed. Thus if a single expression is followed by a +semicolon no output will be produced (except for its type): + +\index{; output suppression} +\spadcommand{2 + 4*5;} +\returnType{Type: PositiveInteger} + +\subsection{Accessing Earlier Results} +\index{\%} +\index{\%\%} +The ``\%'' macro represents the result of the previous computation. The +``\%\%'' macro is available which takes a single integer argument. If the +argument is positive then it refers to the step number of the calculation +where the numbering begins from one and can be seen at the end of each +prompt (the number in parentheses). If the argument is negative then it +refers to previous results counting backwards from the last result. That is, +``\%\%(-1)'' is the same as ``\%''. The value of ``\%\%(0)'' is not defined and +will generate an error if requested. + +\subsection{Splitting Expressions Over Several Lines} +Although Axiom will quite happily accept expressions that are longer than +the width of the screen (just keep typing without pressing the {\bf Return} +key) it is often preferable to split the expression being entered at a point +where it would result in more readable input. To do this the underscore +``\_'' symbol is placed before the break point and then the {\bf Return} +key is pressed. The rest of the expression is typed on the next line, +can be preceeded by any number of whitespace chars, for example: +\begin{verbatim} +2_ ++_ +3 +\end{verbatim} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +\index{\_ escape} +The underscore symbol is an escape character and its presence alters the +meaning of the characters that follow it. As mentions above whitespace +following an underscore is ignored (the {\bf Return} key generates a +whitespace character). Any other character following an underscore loses +whatever special meaning it may have had. Thus one can create the +identifier ``a+b'' by typing ``a\_+b'' although this might lead to confusions. +Also note the result of the following example: + +\spadcommand{ThisIsAVeryLong\_\\ +VariableName} +$$ +ThisIsAVeryLongVariableName +$$ +\returnType{Type: Variable ThisIsAVeryLongVariableName} + +\subsection{Comments and Descriptions} +\index{-- comments} +\index{++ comments} +\index{+++ comments} +Comments and descriptions are really only of use in files of Axiom code but +can be used when the output of an interactive session is being spooled to +a file (via the system command {\bf )spool}). A comment begins with two +dashes ``- -'' and continues until the end of the line. Multi-line +comments are only possible if each individual line begins with two dashes. + +Descriptions are the same as comments except that the Axiom compiler will +include them in the object files produced and make them available to the +end user for documentation purposes. + +A description is placed {\bf before} a calculation begins with three +``+++'' signs and a description placed after a calculation begins with +two plus signs ``++''. The so-called ``plus plus'' comments are used +within the algebra files and are processed by the compiler to add +to the documentation. The so-called ``minus minus'' comments are ignored +everywhere. + +\subsection{Control of Result Types} +\index{:: conversion} +In earlier sections the type of an expression was converted to another +via the ``::'' operator. However, this is not the only method for +converting between types and two other operators need to be introduced +and explained. + +\index{\$ package calling} +The first operator is ``\$'' and is used to specify the package to be +used to calculate the result. Thus: + +\spadcommand{(2/3)\$Float} +$$ +0.6666666666\ 6666666667 +$$ +\returnType{Type: Float} + +tells Axiom to use the ``/'' operator from the {\tt Float} package to +evaluate the expression $2/3$. This does not necessarily mean that the +result will be of the same type as the domain from which the operator +was taken. In the following example the {\bf sign} operator is taken +from the {\tt Float} package but the result is of type {\tt Integer}. + +\spadcommand{sign(2.3)\$Float} +$$ +1 +$$ +\returnType{Type: Integer} + +\index{type target} +The other operator is ``@'' which is used to tell Axiom what the desired +type of the result of the calculation is. In most situations all three +operators yield the same results but the example below should help +distinguish them. + +\spadcommand{(2 + 3)::String} +$$ +\mbox{\tt "5"} +$$ +\returnType{Type: String} + +\spadcommand{(2 + 3)@String} +\begin{verbatim} +An expression involving @ String actually evaluated to one of + type PositiveInteger . Perhaps you should use :: String . +\end{verbatim} + +\spadcommand{(2 + 3)\$String} +\begin{verbatim} + The function + is not implemented in String . +\end{verbatim} + +If an expression {\sl X} is converted using one of the three operators to +type {\sl T} the interpretations are: + +\index{:: conversion} +{\bf ::} means explicitly convert {\sl X} to type {\sl T} if possible. + +\index{\$ package calling} +{\bf \$} means use the available operators for type {\sl T} to compute {\sl X}. + +\index{type target} +{\bf @} means choose operators to compute {\sl X} so that the result is of +type {\sl T}. + +\subsection{Using system commands} +\label{sec:Using system commands} +We conclude our tour of Axiom with a brief discussion of +{\it system commands}. System commands are special statements +that start with a closing parenthesis ({\tt )}). They are used +to control or display your Axiom environment, start the +HyperDoc system, issue operating system commands and leave +Axiom. For example, {\tt )system}\index{)system} + is used to issue commands +to the operating system from Axiom. Here +is a brief description of some of these commands. + +Perhaps the most important user command is the +{\tt )clear all}\index{)clear} +command that initializes your environment. Every section and +subsection in this document has an invisible {\tt )clear all} that is +read prior to the examples given in the section. {\tt )clear all} +gives you a fresh, empty environment with no user variables defined +and the step number reset to $1$. The {\tt )clear} command +can also be used to selectively clear values and properties of system +variables. + +Another useful system command is {\tt )read}.\index{)read} +A preferred way to +develop an application in Axiom is to put your interactive +commands into a file, say {\bf my.input} file. To get Axiom to +read this file, you use the system command {\tt )read my.input}. +If you need to make changes to your approach or definitions, go into +your favorite editor, change {\bf my.input}, then {\tt )read +my.input} again. + +Other system commands include: {\tt )history},\index{)history} + to display +previous input and/or output lines; {\tt )display},\index{)display} + to display +properties and values of workspace variables; and {\tt )what}.\index{)what} + +Issue {\tt )what} to get a list of Axiom objects that +contain a given substring in their name. +\spadcommand{)what operations integrate} +\begin{verbatim} + +Operations whose names satisfy the above pattern(s): + +HermiteIntegrate algintegrate complexIntegrate +expintegrate extendedIntegrate fintegrate +infieldIntegrate integrate internalIntegrate +internalIntegrate0 lazyGintegrate lazyIntegrate +lfintegrate limitedIntegrate monomialIntegrate +nagPolygonIntegrate palgintegrate pmComplexintegrate +pmintegrate primintegrate tanintegrate + +To get more information about an operation such as +limitedIntegrate , issue the command )display op limitedIntegrate + +\end{verbatim} +\subsection{Using undo} +\label{sec:Undo} +A useful system command is {\tt )undo}. \index{)undo} +Sometimes while computing +interactively with Axiom, you make a mistake and enter an +incorrect definition or assignment. Or perhaps you need to try one of +several alternative approaches, one after another, to find the best +way to approach an application. For this, you will find the +{\it undo} facility of Axiom helpful. + +System command {\tt )undo n} means ``undo back to step +$n$''; it restores the values of user variables to those that +existed immediately after input expression $n$ was evaluated. +Similarly, {\tt )undo -n} undoes changes caused by the last +$n$ input expressions. Once you have done an {\tt )undo}, +you can continue on from there, or make a change and {\bf redo} all +your input expressions from the point of the {\tt )undo} forward. +The {\tt )undo} is completely general: it changes the environment +like any user expression. Thus you can {\tt )undo} any previous +undo. + +Here is a sample dialogue between user and Axiom. + +``Let me define +two mutually dependent functions $f$ and $g$ piece-wise.'' +\spadcommand{f(0) == 1; g(0) == 1} +\returnType{Type: Void} + +``Here is the general term for $f$.'' +\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)} +\returnType{Type: Void} + +``And here is the general term for $g$.'' +\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)} +\returnType{Type: Void} + +``What is value of $f(3)$?'' +\spadcommand{f(3)} +$$ +-{x \sp 3}+{{\left( e+{{1 \over 3} \ d} +\right)} +\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} +-{{1 \over 9} \ {d \sp 2}} +\right)} +\ x}+{{1 \over 8} \ {e \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +``Hmm, I think I want to define $f$ differently. +Undo to the environment right after I defined $f$.'' +\spadcommand{)undo 2} + +``Here is how I think I want $f$ to be defined instead.'' +\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)} +\begin{verbatim} + 1 old definition(s) deleted for function or rule f +\end{verbatim} +\returnType{Type: Void} + +Redo the computation from expression $3$ forward. +\spadcommand{)undo )redo} +\begin{verbatim} +g(n) == -x*f(n-1) + d/3*g(n-1) + + Type: Void +f(3) + + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function f with type Integer -> Polynomial Fraction + Integer + Compiling function f as a recurrence relation. + ++++ |*1;f;1;G82322;AUX| redefined + ++++ |*1;f;1;G82322| redefined +\end{verbatim} +$$ +-{x \sp 3}+{d \ {x \sp 2}} -{{1 \over 3} \ {d \sp 2} \ x}+{{1 \over {27}} +\ {d \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +``I want my old definition of +$f$ after all. Undo the undo and restore +the environment to that immediately after $(4)$.'' +\spadcommand{)undo 4} + +``Check that the value of $f(3)$ is restored.'' +\spadcommand{f(3)} +\begin{verbatim} + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function g with type Integer -> Polynomial Fraction + Integer + Compiling function g as a recurrence relation. + ++++ |*1;g;1;G82322;AUX| redefined + ++++ |*1;g;1;G82322| redefined + Compiling function f with type Integer -> Polynomial Fraction + Integer + Compiling function f as a recurrence relation. + ++++ |*1;f;1;G82322;AUX| redefined + ++++ |*1;f;1;G82322| redefined +\end{verbatim} +$$ +-{x \sp 3}+{{\left( e+{{1 \over 3} \ d} +\right)} +\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} +-{{1 \over 9} \ {d \sp 2}} +\right)} +\ x}+{{1 \over 8} \ {e \sp 3}} +$$ +\returnType{Type: Polynomial Fraction Integer} + +After you have gone off on several tangents, then backtracked to +previous points in your conversation using {\tt )undo}, you might +want to save all the ``correct'' input commands you issued, +disregarding those undone. The system command {\tt )history +)write mynew.input} writes a clean straight-line program onto the file +{\bf mynew.input} on your disk. + +\section{Data Structures in Axiom} +This chapter is an overview of {\sl some} of the data structures provided +by Axiom. +\subsection{Lists} +The Axiom {\tt List} type constructor is used to create homogenous lists of +finite size. The notation for lists and the names of the functions that +operate over them are similar to those found in functional languages such +as ML. + +Lists can be created by placing a comma separated list of values inside +square brackets or if a list with just one element is desired then the +function {\bf list} is available: + +\spadcommand{[4]} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{list} +\spadcommand{list(4)} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{[1,2,3,5,7,11]} +$$ +\left[ +1, 2, 3, 5, 7, {11} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The function {\bf append} takes two lists as arguments and returns the list +consisting of the second argument appended to the first. A single element +can be added to the front of a list using {\bf cons}: + +\index{append} +\spadcommand{append([1,2,3,5],[7,11])} +$$ +\left[ +1, 2, 3, 5, 7, {11} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{cons} +\spadcommand{cons(23,[65,42,19])} +$$ +\left[ +{23}, {65}, {42}, {19} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Lists are accessed sequentially so if Axiom is asked for the value of the +twentieth element in the list it will move from the start of the list over +nineteen elements before it reaches the desired element. Each element of a +list is stored as a node consisting of the value of the element and a pointer +to the rest of the list. As a result the two main operations on a list are +called {\bf first} and {\bf rest}. Both of these functions take a second +optional argument which specifies the length of the first part of the list: + +\index{first} +\spadcommand{first([1,5,6,2,3])} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{first([1,5,6,2,3],2)} +$$ +\left[ +1, 5 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{rest} +\spadcommand{rest([1,5,6,2,3])} +$$ +\left[ +5, 6, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{rest([1,5,6,2,3],2)} +$$ +\left[ +6, 2, 3 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Other functions are {\bf empty?} which tests to see if a list contains no +elements, {\bf member?} which tests to see if the first argument is a member +of the second, {\bf reverse} which reverses the order of the list, {\bf sort} +which sorts a list, and {\bf removeDuplicates} which removes any duplicates. +The length of a list can be obtained using the ``\#'' operator. + +\index{empty?} +\spadcommand{empty?([7,2,-1,2])} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{member?} +\spadcommand{member?(-1,[7,2,-1,2])} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{reverse} +\spadcommand{reverse([7,2,-1,2])} +$$ +\left[ +2, -1, 2, 7 +\right] +$$ +\returnType{Type: List Integer} + +\index{sort} +\spadcommand{sort([7,2,-1,2])} +$$ +\left[ +-1, 2, 2, 7 +\right] +$$ +\returnType{Type: List Integer} + +\index{removeDuplicates} +\spadcommand{removeDuplicates([1,5,3,5,1,1,2])} +$$ +\left[ +1, 5, 3, 2 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{\# list length} +\spadcommand{\#[7,2,-1,2]} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Lists in Axiom are mutable and so their contents (the elements and the links) +can be modified in place. Functions that operate over lists in this way have +names ending in the symbol ``!''. For example, {\bf concat!} takes two lists +as arguments and appends the second argument to the first (except when the +first argument is an empty list) and {\bf setrest!} changes the link +emanating from the first argument to point to the second argument: + +\spadcommand{u := [9,2,4,7]} +$$ +\left[ +9, 2, 4, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{concat\!} +\spadcommand{concat!(u,[1,5,42]); u} +$$ +\left[ +9, 2, 4, 7, 1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{endOfu := rest(u,4)} +$$ +\left[ +1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{partOfu := rest(u,2)} +$$ +\left[ +4, 7, 1, 5, {42} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\index{setrest\!} +\spadcommand{setrest!(endOfu,partOfu); u} +$$ +\left[ +9, 2, {\overline {4, 7, 1}} +\right] +$$ +\returnType{Type: List PositiveInteger} + +From this it can be seen that the lists returned by {\bf first} and {\bf rest} +are pointers to the original list and {\sl not} a copy. Thus great care must +be taken when dealing with lists in Axiom. + +Although the {\sl n}th element of the list {\sl l} can be obtained by +applying the {\bf first} function to $n-1$ applications of {\bf rest} +to {\sl l}, Axiom provides a more useful access method in the form of +the ``.'' operator: + +\spadcommand{u.3} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.5} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.6} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{first rest rest u -- Same as u.3} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u.first} +$$ +9 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{u(3)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +The operation {\sl u.i} is referred to as {\sl indexing into u} or +{\sl elting into u}. The latter term comes from the {\bf elt} function +which is used to extract elements (the first element of the list is at +index $1$). + +\index{elt} +\spadcommand{elt(u,4)} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +If a list has no cycles then any attempt to access an element beyond the +end of the list will generate an error. However, in the example above there +was a cycle starting at the third element so the access to the sixth +element wrapped around to give the third element. Since lists are mutable it +is possible to modify elements directly: + +\spadcommand{u.3 := 42; u} +$$ +\left[ +9, 2, {\overline {{42}, 7, 1}} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Other list operations are: +\spadcommand{L := [9,3,4,7]; \#L} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\index{last} +\spadcommand{last(L)} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{L.last} +$$ +7 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{L.(\#L - 1)} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +Note that using the ``\#'' operator on a list with cycles causes Axiom to +enter an infinite loop. + +Note that any operation on a list {\sl L} that returns a list ${\sl L}L^{'}$ +will, in general, be such that any changes to ${\sl L}L^{'}$ will have the +side-effect of altering {\sl L}. For example: + +\spadcommand{m := rest(L,2)} +$$ +\left[ +4, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{m.1 := 20; L} +$$ +\left[ +9, 3, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n := L} +$$ +\left[ +9, 3, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n.2 := 99; L} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Thus the only safe way of copying lists is to copy each element from one to +another and not use the assignment operator: + +\index{copy} +\spadcommand{p := [i for i in n] -- Same as `p := copy(n)'} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{p.2 := 5; p} +$$ +\left[ +9, 5, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +\spadcommand{n} +$$ +\left[ +9, {99}, {20}, 7 +\right] +$$ +\returnType{Type: List PositiveInteger} + +In the previous example a new way of constructing lists was given. This is +a powerful method which gives the reader more information about the contents +of the list than before and which is extremely flexible. The example + +\spadcommand{[i for i in 1..10]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +should be read as + +\begin{center} +``Using the expression {\sl i}, generate each element of the list by +iterating the symbol {\sl i} over the range of integers [1,10]'' +\end{center} + +To generate the list of the squares of the first ten elements we just use: + +\spadcommand{[i**2 for i in 1..10]} +$$ +\left[ +1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} +\right] +$$ +\returnType{Type: List PositiveInteger} + +For more complex lists we can apply a condition to the elements that are to +be placed into the list to obtain a list of even numbers between 0 and 11: + +\spadcommand{[i for i in 1..10 | even?(i)]} +$$ +\left[ +2, 4, 6, 8, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +This example should be read as: +\begin{center} +``Using the expression {\sl i}, generate each element of the list +by iterating the symbol {\sl i} over the range of integers [1,10] such that +{\sl i} is even'' +\end{center} + +The following achieves the same result: + +\spadcommand{[i for i in 2..10 by 2]} +$$ +\left[ +2, 4, 6, 8, {10} +\right] +$$ +\returnType{Type: List PositiveInteger} + +\subsection{Segmented Lists} +\label{sec:Segmented Lists} +\index{Segmented Lists} +A segmented list is one in which some of the elements are ranges of values. +The {\bf expand} function converts lists of this type into ordinary lists: + +\spadcommand{[1..10]} +$$ +\left[ +{1..{10}} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +\spadcommand{[1..3,5,6,8..10]} +$$ +\left[ +{1..3}, {5..5}, {6..6}, {8..{10}} +\right] +$$ +\returnType{Type: List Segment PositiveInteger} + +\index{expand} +\spadcommand{expand(\%)} +$$ +\left[ +1, 2, 3, 5, 6, 8, 9, {10} +\right] +$$ +\returnType{Type: List Integer} + +If the upper bound of a segment is omitted then a different type of +segmented list is obtained and expanding it will produce a stream (which +will be considered in the next section): + +\index{UniversalSegment} +\spadcommand{[1..]} +$$ +\left[ +{1..} +\right] +$$ +\returnType{Type: List UniversalSegment PositiveInteger} + +\spadcommand{expand(\%)} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\subsection{Streams} +\label{sec:Streams} +\index{Streams} +Streams are infinite lists which have the ability to calculate the next +element should it be required. For example, a stream of positive integers +and a list of prime numbers can be generated by: + +\spadcommand{[i for i in 1..]} +$$ +\left[ +1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +\spadcommand{[i for i in 1.. | prime?(i)]} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream PositiveInteger} + +In each case the first few elements of the stream are calculated for display +purposes but the rest of the stream remains unevaluated. The value of items +in a stream are only calculated when they are needed which gives rise to +their alternative name of ``lazy lists''. + +Another method of creating streams is to use the {\bf generate(f,a)} function. +This applies its first argument repeatedly onto its second to produce the +stream $[a,f(a),f(f(a)),f(f(f(a)))\ldots]$. Given that the function +{\bf nextPrime} returns the lowest prime number greater than its argument we +can generate a stream of primes as follows: + +\index{generate} +\index{nextPrime} +\spadcommand{generate(nextPrime,2)\$Stream Integer} +$$ +\left[ +2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +\index{Fibonacci} +As a longer example a stream of Fibonacci numbers will be computed. The +Fibonacci numbers start at $1$ and each following number is the addition +of the two numbers that precede it so the Fibonacci sequence is: +$$1,1,2,3,5,8,\ldots$$. + +Since the generation of any Fibonacci number only relies on knowing the +previous two numbers we can look at the series through a window of two +elements. To create the series the window is placed at the start over +the values $[1,1]$ and their sum obtained. The window is now shifted to +the right by one position and the sum placed into the empty slot of the +window; the process is then repeated. To implement this we require a +function that takes a list of two elements (the current view of the window), +adds them, and outputs the new window. The result is the function +$[a,b]$~{\tt ->}~$[b,a+b]$: +\spadcommand{win : List Integer -> List Integer} +\returnType{Type: Void} + +\spadcommand{win(x) == [x.2, x.1 + x.2]} +\returnType{Type: Void} + +\spadcommand{win([1,1])} +$$ +\left[ +1, 2 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{win(\%)} +$$ +\left[ +2, 3 +\right] +$$ +\returnType{Type: List Integer} + +Thus it can be seen that by repeatedly applying {\bf win} to the {\sl results} +of the previous invocation each element of the series is obtained. Clearly +{\bf win} is an ideal function to construct streams using the {\bf generate} +function: +\spadcommand{fibs := [generate(win,[1,1])]} +$$ +\left[ +{\left[ 1, 1 +\right]}, + {\left[ 1, 2 +\right]}, + {\left[ 2, 3 +\right]}, + {\left[ 3, 5 +\right]}, + {\left[ 5, 8 +\right]}, + {\left[ 8, {13} +\right]}, + {\left[ {13}, {21} +\right]}, + {\left[ {21}, {34} +\right]}, + {\left[ {34}, {55} +\right]}, + {\left[ {55}, {89} +\right]}, + \ldots +\right] +$$ +\returnType{Type: Stream List Integer} + +This isn't quite what is wanted -- we need to extract the first element of +each list and place that in our series: +\spadcommand{fibs := [i.1 for i in [generate(win,[1,1])] ]} +$$ +\left[ +1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, +\ldots +\right] +$$ +\returnType{Type: Stream Integer} + +Obtaining the 200th Fibonacci number is trivial: +\spadcommand{fibs.200} +$$ +280571172992510140037611932413038677189525 +$$ +\returnType{Type: PositiveInteger} + +\index{complete} +One other function of interest is {\bf complete} which expands a finite +stream derived from an infinite one (and thus was still stored as an +infinite stream) to form a finite stream. + +\subsection{Arrays, Vectors, Strings, and Bits} +The simplest array data structure is the {\sl one-dimensional array} which +can be obtained by applying the {\bf oneDimensionalArray} function to a list: + +\index{OneDimensionalArray} +\spadcommand{oneDimensionalArray([7,2,5,4,1,9])} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +One-dimensional array are homogenous (all elements must have the same type) +and mutable (elements can be changed) like lists but unlike lists they are +constant in size and have uniform access times (it is just as quick to read +the last element of a one-dimensional array as it is to read the first; this +is not true for lists). + +Since these arrays are mutable all the warnings that apply to lists apply to +arrays. That is, it is possible to modify an element in a copy of an array +and change the original: +\spadcommand{x := oneDimensionalArray([7,2,5,4,1,9])} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{y := x} +$$ +\left[ +7, 2, 5, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{y.3 := 20 ; x} +$$ +\left[ +7, 2, {20}, 4, 1, 9 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +Note that because these arrays are of fixed size the {\bf concat!} function +cannot be applied to them without generating an error. If arrays of this +type are required use the {\bf FlexibleArray} constructor. +\index{FlexibleArray} + +One-dimensional arrays can be created using {\bf new} which specifies the size +of the array and the initial value for each of the elements. Other operations +that can be applied to one-dimensional arrays are {\bf map!} which applies +a mapping onto each element, {\bf swap!} which swaps two elements and +{\bf copyInto!(a,b,c)} which copies the array {\sl b} onto {\sl a} starting at +position {\sl c}. +\spadcommand{a : ARRAY1 PositiveInteger := new(10,3)} +$$ +\left[ +3, 3, 3, 3, 3, 3, 3, 3, 3, 3 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +(note that {\tt ARRAY1} is an abbreviation for the type +{\tt OneDimensionalArray}.) Other types based on one-dimensional arrays are +{\tt Vector}\index{Vector}, +{\tt String}\index{String}, +and {\tt Bits}\index{Bits}. + +\index{map\!} +\spadcommand{map!(i +-> i+1,a); a} +$$ +\left[ +4, 4, 4, 4, 4, 4, 4, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{b := oneDimensionalArray([2,3,4,5,6])} +$$ +\left[ +2, 3, 4, 5, 6 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\index{swap\!} +\spadcommand{swap!(b,2,3); b} +$$ +\left[ +2, 4, 3, 5, 6 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\index{copyInto\!} +\spadcommand{copyInto!(a,b,3)} +$$ +\left[ +4, 4, 2, 4, 3, 5, 6, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\spadcommand{a} +$$ +\left[ +4, 4, 2, 4, 3, 5, 6, 4, 4, 4 +\right] +$$ +\returnType{Type: OneDimensionalArray PositiveInteger} + +\index{vector} +\spadcommand{vector([1/2,1/3,1/14])} +$$ +\left[ +{1 \over 2}, {1 \over 3}, {1 \over {14}} +\right] +$$ +\returnType{Type: Vector Fraction Integer} + +\spadcommand{"Hello, World"} +$$ +\mbox{\tt "Hello, World"} +$$ +\returnType{Type: String} + +\index{bits} +\spadcommand{bits(8,true)} +$$ +\mbox{\tt "11111111"} +$$ +\returnType{Type: Bits} + +A vector is similar to a one-dimensional array except that if its +components belong to a ring then arithmetic operations are provided. + +\subsection{Flexible Arrays} +\label{sec:Flexible Arrays} +\index{Flexible Arrays} +Flexible arrays are designed to provide the efficiency of one-dimensional +arrays while retaining the flexibility of lists. They are implemented by +allocating a fixed block of storage for the array. If the array needs to +be expanded then a larger block of storage is allocated and the contents +of the old block are copied into the new one. + +There are several operations that can be applied to this type, most of +which modify the array in place. As a result these functions all have +names ending in ``!''. The {\bf physicalLength} returns the actual length +of the array as stored in memory while the {\bf physicalLength!} allows this +value to be changed by the user. +\spadcommand{f : FARRAY INT := new(6,1)} +$$ +\left[ +1, 1, 1, 1, 1, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{f.1:=4; f.2:=3 ; f.3:=8 ; f.5:=2 ; f} +$$ +\left[ +4, 3, 8, 1, 2, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{insert\!} +\spadcommand{insert!(42,f,3); f} +$$ +\left[ +4, 3, {42}, 8, 1, 2, 1 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{insert!(28,f,8); f} +$$ +\left[ +4, 3, {42}, 8, 1, 2, 1, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{removeDuplicates\!} +\spadcommand{removeDuplicates!(f)} +$$ +\left[ +4, 3, {42}, 8, 1, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{delete\!} +\spadcommand{delete!(f,5)} +$$ +\left[ +4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{g:=f(3..5)} +$$ +\left[ +{42}, 8, 2 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{g.2:=7; f} +$$ +\left[ +4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\spadcommand{insert!(g,f,1)} +$$ +\left[ +{42}, 7, 2, 4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{physicalLength} +\spadcommand{physicalLength(f)} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +\index{physicalLength\!} +\spadcommand{physicalLength!(f,20)} +$$ +\left[ +{42}, 7, 2, 4, 3, {42}, 8, 2, {28} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{merge\!} +\index{sort\!} +\spadcommand{merge!(sort!(f),sort!(g))} +$$ +\left[ +2, 2, 2, 3, 4, 7, 7, 8, {28}, {42}, {42}, +{42} +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{shrinkable} +\spadcommand{shrinkable(false)\$FlexibleArray(Integer)} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +There are several things to point out concerning these +examples. First, although flexible arrays are mutable, making copies +of these arrays creates separate entities. This can be seen by the +fact that the modification of element {\sl g.2} above did not alter +{\sl f}. Second, the {\bf merge!} function can take an extra argument +before the two arrays are merged. The argument is a comparison +function and defaults to ``{\tt <=}'' if omitted. Lastly, +{\bf shrinkable} tells the system whether or not to let flexible arrays +contract when elements are deleted from them. An explicit package +reference must be given as in the example above. + +\section{Functions, Choices, and Loops} +By now the reader should be able to construct simple one-line expressions +involving variables and different data structures. This section builds on +this knowledge and shows how to use iteration, make choices, and build +functions in Axiom. At the moment it is assumed that the reader has a rough +idea of how types are specified and constructed so that they can follow +the examples given. + +From this point on most examples will be taken from input files. + +\subsection{Reading Code from a File} +Input files contain code that will be fed to the command prompt. The +primary different between the command line and an input file is that +indentation matters. In an input file you can specify ``piles'' of code +by using indentation. + +The names of all input files in Axiom should end in ``.input'' otherwise +Axiom will refuse to read them. + +If an input file is named {\bf foo.input} you can feed the contents of +the file to the command prompt (as though you typed them) by writing: +{\bf )read foo.input}\index{)read}. + +It is good practice to start each input file with the {\bf )clear all} +command so that all functions and variables in the current environment +are erased. +\subsection{Blocks} +\label{sec:Blocks} +\index{Blocks} +\index{pile} +The Axiom constructs that provide looping, choices, and user-defined +functions all rely on the notion of blocks. A block is a sequence of +expressions which are evaluated in the order that they appear except +when it is modified by control expressions such as loops. To leave a +block prematurely use an expression of the form: +{\sl BoolExpr}~{\tt =>}~{\sl Expr} +where {\sl BoolExpr} is any Axiom expression that has type {\tt Boolean}. +The value and type of {\sl Expr} determines the value and type returned +by the block. + +If blocks are entered at the keyboard (as opposed to reading them from +a text file) then there is only one way of creating them. The syntax is: +$$( expression1 ; expression2; \ldots ; expressionN )$$ + +In an input file a block can be constructed as above or by placing all the +statements at the same indentation level. When indentation is used to +indicate program structure the block is called a {\sl pile}. As an example +of a simple block a list of three integers can be constructed using +parentheses: +\spadcommand{( a:=4; b:=1; c:=9; L:=[a,b,c])} +$$ +\left[ +4, 1, 9 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Doing the same thing using piles in an input file you could type: +\begin{verbatim} +L := + a:=4 + b:=1 + c:=9 + [a,b,c] +\end{verbatim} +$$ +\left[ +4, 1, 9 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Since blocks have a type and a value they can be used as arguments to +functions or as part of other expressions. It should be pointed out that +the following example is not recommended practice but helps to illustrate +the idea of blocks and their ability to return values: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} + +Note that indentation is {\bf extremely} important. If the example above +had the pile starting at ``a:='' moved left by two spaces so that the +``a'' was under the ``('' of the first line then the interpreter would +signal an error. Furthermore if the closing parenthesis ``)'' is moved +up to give +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c) +\end{verbatim} +\begin{verbatim} + Line 1: sqrt(4.0 + + ....A + Error A: Missing mate. + Line 2: a:=3.0 + Line 3: b:=1.0 + Line 4: c:=a + b + Line 5: c) + .........AB + Error A: (from A up to B) Ignored. + Error B: Improper syntax. + Error B: syntax error at top level + Error B: Possibly missing a ) + 5 error(s) parsing +\end{verbatim} +then the parser will generate errors. If the parenthesis is shifted right +by several spaces so that it is in line with the ``c'' thus: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +\begin{verbatim} + Line 1: sqrt(4.0 + + ....A + Error A: Missing mate. + Line 2: a:=3.0 + Line 3: b:=1.0 + Line 4: c:=a + b + Line 5: c + Line 6: ) + .........A + Error A: (from A up to A) Ignored. + Error A: Improper syntax. + Error A: syntax error at top level + Error A: Possibly missing a ) + 5 error(s) parsing +\end{verbatim} +a similar error will be raised. Finally, the ``)'' must be indented by +at least one space relative to the sqrt thus: +\begin{verbatim} +sqrt(4.0 + + a:=3.0 + b:=1.0 + c:=a + b + c + ) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} +or an error will be generated. + +It can be seen that great care needs to be taken when constructing input +files consisting of piles of expressions. It would seem prudent to add +one pile at a time and check if it is acceptable before adding more, +particularly if piles are nested. However, it should be pointed out that +the use of piles as values for functions is not very readable and so +perhaps the delicate nature of their interpretation should deter programmers +from using them in these situations. Using piles should really be restricted +to constructing functions, etc. and a small amount of rewriting can remove +the need to use them as arguments. For example, the previous block could +easily be implemented as: +\begin{verbatim} +a:=3.0 +b:=1.0 +c:=a + b +sqrt(4.0 + c) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +a:=3.0 +\end{verbatim} +$$ +3.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +b:=1.0 +\end{verbatim} +$$ +1.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +c:=a + b +\end{verbatim} +$$ +4.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +sqrt(4.0 + c) +\end{verbatim} +$$ +2.8284271247\ 461900976 +$$ +\returnType{Type: Float} + +which achieves the same result and is easier to understand. Note that this +is still a pile but it is not as fragile as the previous version. +\subsection{Functions} +Definitions of functions in Axiom are quite simple providing two things +are observed. First, the type of the function must either be completely +specified or completely unspecified. Second, the body of the function is +assigned to the function identifier using the delayed assignment operator +``==''. +\index{delayed assignment} +\index{assignment delayed} + +To specify the type of something the ``:'' operator is used. Thus to define +a variable {\sl x} to be of type {\tt Fraction Integer} we enter: +\spadcommand{x : Fraction Integer} +\returnType{Type: Void} + +For functions the method is the same except that the arguments are +placed in parentheses and the return type is placed after the symbol +``{\tt ->}''. Some examples of function definitions taking zero, one, +two, or three arguments and returning a list of integers are: + +\spadcommand{f : () -> List Integer} +\returnType{Type: Void} + +\spadcommand{g : (Integer) -> List Integer} +\returnType{Type: Void} + +\spadcommand{h : (Integer, Integer) -> List Integer} +\returnType{Type: Void} + +\spadcommand{k : (Integer, Integer, Integer) -> List Integer} +\returnType{Type: Void} + +Now the actual function definitions might be: +\spadcommand{f() == [\ ]} +\returnType{Type: Void} + +\spadcommand{g(a) == [a]} +\returnType{Type: Void} + +\spadcommand{h(a,b) == [a,b]} +\returnType{Type: Void} + +\spadcommand{k(a,b,c) == [a,b,c]} +\returnType{Type: Void} + +with some invocations of these functions: +\spadcommand{f()} +\begin{verbatim} + Compiling function f with type () -> List Integer +\end{verbatim} +$$ +\left[\ +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{g(4)} +\begin{verbatim} + Compiling function g with type Integer -> List Integer +\end{verbatim} +$$ +\left[ +4 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{h(2,9)} +\begin{verbatim} + Compiling function h with type (Integer,Integer) -> List Integer +\end{verbatim} +$$ +\left[ +2, 9 +\right] +$$ +\returnType{Type: List Integer} + +\spadcommand{k(-3,42,100)} +\begin{verbatim} + Compiling function k with type (Integer,Integer,Integer) -> List + Integer +\end{verbatim} +$$ +\left[ +-3, {42}, {100} +\right] +$$ +\returnType{Type: List Integer} + +The value returned by a function is either the value of the last expression +evaluated or the result of a {\bf return} statement. For example, the +following are effectively the same: +\spadcommand{p : Integer -> Integer} +\returnType{Type: Void} + +\spadcommand{p x == (a:=1; b:=2; a+b+x)} +\returnType{Type: Void} + +\spadcommand{p x == (a:=1; b:=2; return(a+b+x))} +\returnType{Type: Void} + +Note that a block (pile) is assigned to the function identifier {\bf p} and +thus all the rules about blocks apply to function definitions. Also there was +only one argument so the parenthese are not needed. + +This is basically all that one needs to know about defining functions in +Axiom -- first specify the complete type and then assign a block to the +function name. The rest of this section is concerned with defining more +complex blocks than those in this section and as a result function definitions +will crop up continually particularly since they are a good way of testing +examples. Since the block structure is more complex we will use the {\bf pile} +notation and thus have to use input files to read the piles. + +\subsection{Choices} +\label{sec:Choices} +\index{Choices} +\index{$=>$ block exit} +\index{if-then-else} +\index{pile} +Apart from the ``{\tt =>}'' operator that allows a block to exit before the end +Axiom provides the standard {\bf if-then-else} construct. The general +syntax is: +{\center{if {\sl BooleanExpr} then {\sl Expr1} else {\sl Expr2}}} + +where ``else {\sl Expr2}'' can be omitted. If the expression {\sl BooleanExpr} +evaluates to {\tt true} then {\sl Expr1} is executed otherwise {\sl Expr2} +(if present) will be executed. An example of piles and {\bf if-then-else} is: +(read from an input file) +\begin{verbatim} +h := 2.0 +if h > 3.1 then + 1.0 + else + z:= cos(h) + max(x,0.5) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +h := 2.0 +\end{verbatim} +$$ +2.0 +$$ +\returnType{Type: Float} + +\begin{verbatim} +if h > 3.1 then + 1.0 + else + z:= cos(h) + max(x,0.5) +\end{verbatim} +$$ +x +$$ +\returnType{Type: Polynomial Float} + +Note the indentation -- the ``else'' must be indented relative to the ``if'' +otherwise it will generate an error (Axiom will think there are two piles, +the second one beginning with ``else''). + +Any expression that has type {\tt Boolean} can be used as {\tt BooleanExpr} +and the most common will be those involving the relational operators ``$>$'', +``$<$'', and ``=''. Usually the type of an expression involving the equality +operator ``='' will be {\bf Boolean} but in those situations when it isn't +you may need to use the ``@'' operator to ensure that it is. + +\subsection{Loops} +\label{sec:Loops} +\index{Loops} +Loops in Axiom are regarded as expressions containing another expression +called the {\sl loop body}. The loop body is executed zero or more times +depending on the kind of loop. Loops can be nested to any depth. + +\index{Loops repeat} +\index{repeat Loops} +\subsubsection{The {\tt repeat} loop} +The simplest kind of loop provided by Axiom is the {\bf repeat} loop. The +general syntax of this is: +{\center{{\bf repeat} {\sl loopBody}}} + +\index{break} +\index{return} +This will cause Axiom to execute {\sl loopBody} repeatedly until either a +{\bf break} or {\bf return} statement is encountered. If {\sl loopBody} +contains neither of these statements then it will loop forever. The +following piece of code will display the numbers from $1$ to $4$: +\begin{verbatim} +i:=1 +repeat + if i > 4 then break + output(i) + i:=i+1 +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +i:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +repeat + if i > 4 then break + output(i) + i:=i+1 + + 1 + 2 + 3 + 4 +\end{verbatim} +\returnType{Type: Void} + +\index{break} +\index{return} +\index{$=>$ block exit} +It was mentioned that loops will only be left when either a {\bf break} or +{\bf return} statement is encountered so why can't one use the ``{\tt =>}'' +operator? The reason is that the ``{\tt =>}'' operator tells Axiom to leave the +current block whereas {\bf break} leaves the current loop. The {\bf return} +statement leaves the current function. + +\index{iterate} +To skip the rest of a loop body and continue the next iteration of the loop +use the {\bf iterate} statement (the -- starts a comment in Axiom) +\begin{verbatim} +i := 0 +repeat + i := i + 1 + if i > 6 then break + -- Return to start if i is odd + if odd?(i) then iterate + output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +i := 0 +\end{verbatim} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +\begin{verbatim} +repeat + i := i + 1 + if i > 6 then break + -- Return to start if i is odd + if odd?(i) then iterate + output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +\subsubsection{The {\tt while} loop} +\label{sec:The while loop} +\index{while} +\index{repeat} +\index{break} +\index{$=>$ block exit} +The while statement extends the basic {\bf repeat} loop to place the control +of leaving the loop at the start rather than have it buried in the middle. +Since the body of the loop is still part of a {\bf repeat} loop, {\bf break} +and ``{\tt =>}'' work in the same way as in the previous section. The general +syntax of a {\bf while} loop is: +{\center{while {\sl BoolExpr} repeat {\sl loopBody}}} + +As before, {\sl BoolExpr} must be an expression of type {\bf Boolean}. Before +the body of the loop is executed {\sl BoolExpr} is tested. If it evaluates to +{\tt true} then the loop body is entered otherwise the loop is terminated. +Multiple conditions can be applied using the logical operators such as +{\bf and} or by using several {\bf while} statements before the {\bf repeat}. + +By using {\tt and} in the test we get +\begin{verbatim} +x:=1 +y:=1 +while x < 4 and y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +x:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 and y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] +\end{verbatim} +\returnType{Type: Void} + +We could use two parallel whiles +\begin{verbatim} +x:=1 +y:=1 +while x < 4 while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +x:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] +\end{verbatim} +\returnType{Type: Void} + +Note that the last example using two {\bf while} statements is {\sl not} a +nested loop but the following one is: +\begin{verbatim} +x:=1 +y:=1 +while x < 4 repeat + while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 +\end{verbatim} +the {\bf )read} yields: +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +y:=1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while x < 4 repeat + while y < 10 repeat + output [x,y] + x := x + 1 + y := y + 2 + + [1,1] + [2,3] + [3,5] + [4,7] + [5,9] +\end{verbatim} +\returnType{Type: Void} + +Suppose that, given a matrix of arbitrary size, we find the position and +value of the first negative element by examining the matrix in row-major +order: +\begin{verbatim} +m := matrix [ [ 21, 37, 53, 14 ],_ + [ 8, 22,-24, 16 ],_ + [ 2, 10, 15, 14 ],_ + [ 26, 33, 55,-13 ] ] + +lastrow := nrows(m) +lastcol := ncols(m) +r := 1 +while r <= lastrow repeat + c := 1 -- Index of first column + while c <= lastcol repeat + if elt(m,r,c) < 0 then + output [r,c,elt(m,r,c)] + r := lastrow + break -- Don't look any further + c := c + 1 + r := r + 1 +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +m := matrix [ [ 21, 37, 53, 14 ],_ + [ 8, 22,-24, 16 ],_ + [ 2, 10, 15, 14 ],_ + [ 26, 33, 55,-13 ] ] +\end{verbatim} +$$ +\left[ +\begin{array}{cccc} +{21} & {37} & {53} & {14} \\ +8 & {22} & -{24} & {16} \\ +2 & {10} & {15} & {14} \\ +{26} & {33} & {55} & -{13} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +\begin{verbatim} +lastrow := nrows(m) +\end{verbatim} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +lastcol := ncols(m) +\end{verbatim} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +r := 1 +\end{verbatim} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\begin{verbatim} +while r <= lastrow repeat + c := 1 -- Index of first column + while c <= lastcol repeat + if elt(m,r,c) < 0 then + output [r,c,elt(m,r,c)] + r := lastrow + break -- Don't look any further + c := c + 1 + r := r + 1 + + [2,3,- 24] +\end{verbatim} +\returnType{Type: Void} + +\subsubsection{The {\tt for} loop} +\label{sec:The for loop} +\index{for} +\index{for list} +\index{for segment} +\index{iterate} +The last loop statement of interest is the {\bf for} loop. There are two +ways of creating a {\bf for} loop. The first way uses either a list or +a segment: +\begin{center} +for {\sl var} in {\sl seg} repeat {\sl loopBody}\\ +for {\sl var} in {\sl list} repeat {\sl loopBody} +\end{center} +where {\sl var} is an index variable which is iterated over the values in +{\sl seg} or {\sl list}. The value {\sl seg} is a segment such as $1\ldots10$ +or $1\ldots$ and {\sl list} is a list of some type. For example: + +We can {\sl iterate} the block thus: +\begin{verbatim} +for i in 1..10 repeat + ~prime?(i) => iterate + output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for i in 1..10 repeat + ~prime?(i) => iterate + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +We can iterate over a list +\begin{verbatim} +for w in ["This", "is", "your", "life!"] repeat + output(w) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for w in ["This", "is", "your", "life!"] repeat + output(w) + + This + is + your + life! +\end{verbatim} +\returnType{Type: Void} + +\index{such that} +The second form of the {\bf for} loop syntax includes a ``{\bf such that}'' +clause which must be of type {\bf Boolean}: +\begin{center} +for {\sl var} | {\sl BoolExpr} in {\sl seg} repeat {\sl loopBody}\\ +for {\sl var} | {\sl BoolExpr} in {\sl list} repeat {\sl loopBody} +\end{center} +We can iterate over a segment +\begin{verbatim} +for i in 1..10 | prime?(i) repeat + output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for i in 1..10 | prime?(i) repeat + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +or over a list +\begin{verbatim} +for i in [1,2,3,4,5,6,7,8,9,10] | prime?(i) repeat + output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for i in [1,2,3,4,5,6,7,8,9,10] | prime?(i) repeat + output(i) + + 2 + 3 + 5 + 7 +\end{verbatim} +\returnType{Type: Void} + +You can also use a {\bf while} clause: +\begin{verbatim} +for i in 1.. while i < 7 repeat + if even?(i) then output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for i in 1.. while i < 7 repeat + if even?(i) then output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +Using the ``{\bf such that}'' clause makes this appear simpler: +\begin{verbatim} +for i in 1.. | even?(i) while i < 7 repeat + output(i) +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for i in 1.. | even?(i) while i < 7 repeat + output(i) + + 2 + 4 + 6 +\end{verbatim} +\returnType{Type: Void} + +You can use multiple {\bf for} clauses to iterate over several sequences +in parallel: +\begin{verbatim} +for a in 1..4 for b in 5..8 repeat + output [a,b] +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for a in 1..4 for b in 5..8 repeat + output [a,b] + + [1,5] + [2,6] + [3,7] + [4,8] +\end{verbatim} +\returnType{Type: Void} + +As a general point it should be noted that any symbols referred to in the +``{\bf such that}'' and {\bf while} clauses must be pre-defined. This +either means that the symbols must have been defined in an outer level +(e.g. in an enclosing loop) or in a {\bf for} clause appearing before the +``{\bf such that}'' or {\bf while}. For example: +\begin{verbatim} +for a in 1..4 repeat + for b in 7..9 | prime?(a+b) repeat + output [a,b,a+b] +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for a in 1..4 repeat + for b in 7..9 | prime?(a+b) repeat + output [a,b,a+b] + + [2,9,11] + [3,8,11] + [4,7,11] + [4,9,13] +\end{verbatim} +\returnType{Type: Void} + +\index{for by} +\index{by for} +Finally, the {\bf for} statement has a {\bf by} clause to specify the +step size. This makes it possible to iterate over the segment in +reverse order: +\begin{verbatim} +for a in 1..4 for b in 8..5 by -1 repeat + output [a,b] +\end{verbatim} +the {\bf )read} yields: +\begin{verbatim} +for a in 1..4 for b in 8..5 by -1 repeat + output [a,b] + + [1,8] + [2,7] + [3,6] + [4,5] +\end{verbatim} +\returnType{Type: Void} + +Note that without the ``by -1'' the segment 8..5 is empty so there is +nothing to iterate over and the loop exits immediately. + +%\setcounter{chapter}{0} % Chapter 1 + +\hyphenation{ +multi-set +Uni-var-iate-Poly-nomial +Mul-ti-var-iate-Poly-nomial +Distributed-Mul-ti-var-iate-Poly-nomial +Homo-gen-eous-Distributed-Mul-ti-var-iate-Poly-nomial +New-Distributed-Mul-ti-var-iate-Poly-nomial +General-Distributed-Mul-ti-var-iate-Poly-nomial +} + + +\section{Numbers} +\label{sec:Numbers} +Axiom distinguishes very carefully between different kinds of +numbers, how they are represented and what their properties are. Here +are a sampling of some of these kinds of numbers and some things you +can do with them. + +Integer arithmetic is always exact. +\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3} +$$ +25387751112538918594666224484237298 +$$ +\returnType{Type: PositiveInteger} + +Integers can be represented in factored form. +\index{factor} +\spadcommand{factor 643238070748569023720594412551704344145570763243} +$$ +{{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} \sp 5} \ {{23} +\sp 3} \ {{29} \sp 2} +$$ +\returnType{Type: Factored Integer} + +Results stay factored when you do arithmetic. +Note that the $12$ is automatically factored for you. + +\spadcommand{\% * 12} +$$ +{2 \sp 2} \ 3 \ {{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} +\sp 5} \ {{23} \sp 3} \ {{29} \sp 2} +$$ +\returnType{Type: Factored Integer} + +Integers can also be displayed to bases other than 10. +This is an integer in base 11. + +\index{radix} +\spadcommand{radix(25937424601,11)} +$$ +10000000000 +$$ +\returnType{Type: RadixExpansion 11} + +Roman numerals are also available for those special occasions. +\index{Roman numerals} +\index{roman} +\spadcommand{roman(1992)} +$$ +{\rm MCMXCII } +$$ +\returnType{Type: RomanNumeral} + +Rational number arithmetic is also exact. + +\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9} +$$ +{55739} \over {2520} +$$ +\returnType{Type: Fraction Integer} + +To factor fractions, you have to pmap {\bf factor} onto the numerator +and denominator. + +\index{map} +\index{factor} +\spadcommand{map(factor,r)} +$$ +{{139} \ {401}} \over {{2 \sp 3} \ {3 \sp 2} \ 5 \ 7} +$$ +\returnType{Type: Fraction Factored Integer} + +{\tt SingleInteger} refers to machine word-length integers. +\index{SingleInteger} +In English, this expression means ``$11$ as a small integer''. +\spadcommand{11@SingleInteger} +$$ +11 +$$ +\returnType{Type: SingleInteger} + +Machine double-precision floating-point numbers are also available for +numeric and graphical applications. +\index{DoubleFloat} +\spadcommand{123.21@DoubleFloat} +$$ +123.21000000000001 +$$ +\returnType{Type: DoubleFloat} + +The normal floating-point type in Axiom, {\tt Float}, is a +software implementation of floating-point numbers in which the +exponent and the mantissa may have any number of digits. +The types {\tt Complex(Float)} and +{\tt Complex(DoubleFloat)} are the corresponding software +implementations of complex floating-point numbers. + +This is a floating-point approximation to about twenty digits. +\index{floating point} The ``{\tt ::}'' is used here to change from +one kind of object (here, a rational number) to another (a +floating-point number). + +\spadcommand{r :: Float} +$$ +22.1186507936 50793651 +$$ +\returnType{Type: Float} + +Use \spadfunFrom{digits}{Float} to change the number of digits in +the representation. +This operation returns the previous value so you can reset it +later. +\index{digits} +\spadcommand{digits(22)} +$$ +20 +$$ +\returnType{Type: PositiveInteger} + +To $22$ digits of precision, the number +$e^{\pi {\sqrt {163.0}}}$ appears to be an integer. +\index{exp} +\index{\%pi} +\index{sqrt} +\spadcommand{exp(\%pi * sqrt 163.0)} +$$ +26253741 2640768744.0 +$$ +\returnType{Type: Float} + +Increase the precision to forty digits and try again. +\spadcommand{digits(40); exp(\%pi * sqrt 163.0)} +$$ +26253741\ 2640768743.9999999999\ 9925007259\ 76 +$$ +\returnType{Type: Float} + +Here are complex numbers with rational numbers as real and +\index{complex numbers} imaginary parts. +\spadcommand{(2/3 + \%i)**3} +$$ +-{{46} \over {27}}+{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +The standard operations on complex numbers are available. +\index{conjugate} +\spadcommand{conjugate \% } +$$ +-{{46} \over {27}} -{{1 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +You can factor complex integers. +\index{factor} +\spadcommand{factor(89 - 23 * \%i)} +$$ +-{{\left( 1+i +\right)} +\ {{\left( 2+i +\right)} +\sp 2} \ {{\left( 3+{2 \ i} +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +Complex numbers with floating point parts are also available. +\index{exp} +\spadcommand{exp(\%pi/4.0 * \%i)} +$$ +{0.7071067811\ 8654752440\ 0844362104\ 8490392849} + +$$ +$$ +{{0.7071067811\ 8654752440\ 0844362104\ 8490392848} \ i} +$$ +\returnType{Type: Complex Float} + +The real and imaginary parts can be symbolic. +\index{complex} +\spadcommand{complex(u,v)} +$$ +u+{v \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Of course, you can do complex arithmetic with these also. +\spadcommand{\% ** 2} +$$ +-{v \sp 2}+{u \sp 2}+{2 \ u \ v \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Every rational number has an exact representation as a +repeating decimal expansion +\index{decimal} +\spadcommand{decimal(1/352)} +$$ +0.{00284}{\overline {09}} +$$ +\returnType{Type: DecimalExpansion} + +A rational number can also be expressed as a continued fraction. + +\index{continuedFraction} +\spadcommand{continuedFraction(6543/210)} +$$ +{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} +$$ +\returnType{Type: ContinuedFraction Integer} + +Also, partial fractions can be used and can be displayed in a +compact format +\index{partialFraction} +\index{fraction!partial} +\spadcommand{partialFraction(1,factorial(10))} +$$ +{{159} \over {2 \sp 8}} -{{23} \over {3 \sp 4}} -{{12} \over {5 \sp 2}}+{1 +\over 7} +$$ +\returnType{Type: PartialFraction Integer} + +or expanded format. +\index{padicFraction} +\spadcommand{padicFraction(\%)} +$$ +{1 \over 2}+{1 \over {2 \sp 4}}+{1 \over {2 \sp 5}}+{1 \over {2 \sp 6}}+{1 +\over {2 \sp 7}}+{1 \over {2 \sp 8}} -{2 \over {3 \sp 2}} -{1 \over {3 \sp +3}} -{2 \over {3 \sp 4}} -{2 \over 5} -{2 \over {5 \sp 2}}+{1 \over 7} +$$ +\returnType{Type: PartialFraction Integer} + +Like integers, bases (radices) other than ten can be used for rational +numbers. +Here we use base eight. +\index{radix} +\spadcommand{radix(4/7, 8)} +$$ +0.{\overline 4} +$$ +\returnType{Type: RadixExpansion 8} + +Of course, there are complex versions of these as well. +Axiom decides to make the result a complex rational number. +\spadcommand{\% + 2/3*\%i} +$$ +{4 \over 7}+{{2 \over 3} \ i} +$$ +\returnType{Type: Complex Fraction Integer} + +You can also use Axiom to manipulate fractional powers. +\index{radical} +\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)} +$$ +\root {3} \of {{{{14} \ {\sqrt {7}}}+5}} +$$ +\returnType{Type: AlgebraicNumber} + +You can also compute with integers modulo a prime. +\index{PrimeField} +\spadcommand{x : PrimeField 7 := 5} +$$ +5 +$$ +\returnType{Type: PrimeField 7} + +Arithmetic is then done modulo $7$. +\spadcommand{x**3} +$$ +6 +$$ +\returnType{Type: PrimeField 7} + +Since $7$ is prime, you can invert nonzero values. +\spadcommand{1/x} +$$ +3 +$$ +\returnType{Type: PrimeField 7} + +You can also compute modulo an integer that is not a prime. +\index{IntegerMod} +\spadcommand{y : IntegerMod 6 := 5} +$$ +5 +$$ +\returnType{Type: IntegerMod 6} + +All of the usual arithmetic operations are available. +\spadcommand{y**3} +$$ +5 +$$ +\returnType{Type: IntegerMod 6} + +Inversion is not available if the modulus is not a prime number. + +\spadcommand{1/y} +\begin{verbatim} + There are 12 exposed and 13 unexposed library operations named / + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op / + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named / + with argument type(s) + PositiveInteger + IntegerMod 6 + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +This defines $a$ to be an algebraic number, that is, +a root of a polynomial equation. +\index{rootOf} +\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)} +$$ +a +$$ +\returnType{Type: Expression Integer} + +Computations with $a$ are reduced according to the polynomial equation. +\spadcommand{(a + 1)**10} +$$ +-{{85} \ {a \sp 4}} -{{264} \ {a \sp 3}} -{{378} \ {a \sp 2}} -{{458} \ +a} -{287} +$$ +\returnType{Type: Expression Integer} + +Define $b$ to be an algebraic number involving $a$. +\spadcommand{b := rootOf(b**4 + a,b)} +$$ +b +$$ +\returnType{Type: Expression Integer} + +Do some arithmetic. +\spadcommand{2/(b - 1)} +$$ +2 \over {b -1} +$$ +\returnType{Type: Expression Integer} + +To expand and simplify this, call {\it ratDenom} +to rationalize the denominator. +\index{ratDenom} +\spadcommand{ratDenom(\%)} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 1 +\end{array} +$$ + +\returnType{Type: Expression Integer} + +If we do this, we should get $b$. +\spadcommand{2/\%+1} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 3 +\end{array} +\right)}\over{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ +4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ +\\ +\\ +\displaystyle +{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ +3}+{2 \ {a^2}}- a + 1 +\end{array} +\right)} +$$ +\returnType{Type: Expression Integer} + +But we need to rationalize the denominator again. + +\spadcommand{ratDenom(\%)} +$$ +b +$$ +\returnType{Type: Expression Integer} + +Types {\tt Quaternion} and {\tt Octonion} are also available. +Multiplication of quaternions is non-commutative, as expected. +\index{Quaternion} +\index{Octonion} +\index{quatern} +\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} +$$ +-{8 \ i}+{{16} \ j} -{8 \ k} +$$ +\returnType{Type: Quaternion Integer} + +\section{Data Structures} +\label{sec:Data Structures} +Axiom has a large variety of data structures available. Many +data structures are particularly useful for interactive computation +and others are useful for building applications. The data structures +of Axiom are organized into {\sl category hierarchies}. + +A {\it list} is the most commonly used data structure in +Axiom for holding objects all of the same type. The name {\it list} is +short for ``linked-list of nodes.'' Each node consists of a value +(\spadfunFrom{first}{List}) and a link (\spadfunFrom{rest}{List}) that +points to the next node, or to a distinguished value denoting the +empty list. To get to, say, the third element, Axiom starts at the +front of the list, then traverses across two links to the third node. + +Write a list of elements using square brackets with commas separating +the elements. +\spadcommand{u := [1,-7,11]} +$$ +\left[ +1, -7, {11} +\right] +$$ +\returnType{Type: List Integer} + +This is the value at the third node. Alternatively, you can say $u.3$. +\index{first} +\index{rest} +\spadcommand{first rest rest u} +$$ +11 +$$ +\returnType{Type: PositiveInteger} + +Many operations are defined on lists, such as: {\bf empty?}, to test +that a list has no elements; {\bf cons}$(x,l)$, to create a new list +with {\bf first} element $x$ and {\bf rest} $l$; {\bf reverse}, to +create a new list with elements in reverse order; and {\bf sort}, to +arrange elements in order. + +An important point about lists is that they are ``mutable'': their +constituent elements and links can be changed ``in place.'' +To do this, use any of the operations whose names end with the +character ``{\tt !}''. + +The operation \spadfunFrom{concat!}{List}$(u,v)$ replaces the +last link of the list $u$ to point to some other list $v$. +Since $u$ refers to the original list, this change is seen by $u$. +\index{concat\!} +\spadcommand{concat!(u,[9,1,3,-4]); u} +$$ +\left[ +1, -7, {11}, 9, 1, 3, -4 +\right] +$$ +\returnType{Type: List Integer} + +A {\it cyclic list} is a list with a ``cycle'': \index{list!cyclic} a +link pointing back to an earlier node of the list. \index{cyclic +list} To create a cycle, first get a node somewhere down the list. +\spadcommand{lastnode := rest(u,3)} +$$ +\left[ +9, 1, 3, -4 +\right] +$$ +\returnType{Type: List Integer} + +Use \spadfunFrom{setrest!}{List} to change the link emanating from +that node to point back to an earlier part of the list. + +\index{setrest\!} +\spadcommand{setrest!(lastnode,rest(u,2)); u} +$$ +\left[ +1, -7, {\overline {{11}, 9}} +\right] +$$ +\returnType{Type: List Integer} + +\index{stream} +A {\it stream} is a structure that (potentially) has an infinite +number of distinct elements. Think of a stream as an +``infinite list'' where elements are computed successively. + +Create an infinite stream of factored integers. Only a certain number +of initial elements are computed and displayed. + +\index{factor} +\spadcommand{[factor(i) for i in 2.. by 2]} +$$ +\left[ +2, {2 \sp 2}, {2 \ 3}, {2 \sp 3}, {2 \ 5}, {{2 \sp 2} \ 3}, + {2 \ 7}, {2 \sp 4}, {2 \ {3 \sp 2}}, {{2 \sp 2} \ 5}, +\ldots +\right] +$$ +\returnType{Type: Stream Factored Integer} + +Axiom represents streams by a collection of already-computed +elements together with a function to compute the next element ``on +demand.'' Asking for the $n$-th element causes elements +$1$ through $n$ to be evaluated. +\spadcommand{\%.36} +$$ +{2 \sp 3} \ {3 \sp 2} +$$ +\returnType{Type: Factored Integer} + +Streams can also be finite or cyclic. +They are implemented by a linked list structure similar to lists +and have many of the same operations. +For example, {\bf first} and {\bf rest} are used to access +elements and successive nodes of a stream. + +A {\it one-dimensional array} is another data structure used to hold +objects of the same type. +Unlike lists, one-dimensional arrays are inflexible---they are +\index{array!one-dimensional} implemented using a fixed block of +storage. Their advantage is that they give quick and equal access +time to any element. + +A simple way to create a one-dimensional array is to apply the +operation {\bf oneDimensionalArray} to a list of elements. +\index{oneDimensionalArray} +\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]} +$$ +\left[ +1, -7, 3, {3 \over 2} +\right] +$$ +\returnType{Type: OneDimensionalArray Fraction Integer} + +One-dimensional arrays are also mutable: you can change their +constituent elements ``in place.'' +\spadcommand{a.3 := 11; a} +$$ +\left[ +1, -7, {11}, {3 \over 2} +\right] +$$ +\returnType{Type: OneDimensionalArray Fraction Integer} + +However, one-dimensional arrays are not flexible structures. +You cannot destructively {\bf concat!} them together. +\spadcommand{concat!(a,oneDimensionalArray [1,-2])} +\begin{verbatim} + There are 5 exposed and 0 unexposed library operations named concat! + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op concat! + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the arguments + will allow you to apply the operation. + + Cannot find a definition or applicable library operation named + concat! with argument type(s) + OneDimensionalArray Fraction Integer + OneDimensionalArray Integer + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +Examples of datatypes similar to {\tt OneDimensionalArray} +are: {\tt Vector}\index{Vector} +(vectors are mathematical structures +implemented by one-dimensional arrays), +{\tt String}\index{String} +(arrays of ``characters,'' represented by byte vectors), and +{\tt Bits}\index{Bits} +(represented by ``bit vectors''). + +A vector of 32 bits, each representing the {\bf Boolean} value +${\tt true}$. +\index{bits} +\spadcommand{bits(32,true)} +$$ +\mbox{\tt "11111111111111111111111111111111"} +$$ +\returnType{Type: Bits} + +A {\it flexible array} is +a cross between a list \index{array!flexible} and a one-dimensional +array. Like a one-dimensional array, a flexible array occupies a fixed +block of storage. Its block of storage, however, has room to expand. +When it gets full, it grows (a new, larger block of storage is +allocated); when it has too much room, it contracts. + +Create a flexible array of three elements. +\index{flexibleArray} +\spadcommand{f := flexibleArray [2, 7, -5]} +$$ +\left[ +2, 7, -5 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +Insert some elements between the second and third elements. +\index{insert\!} +\spadcommand{insert!(flexibleArray [11, -3],f,2)} +$$ +\left[ +2, {11}, -3, 7, -5 +\right] +$$ +\returnType{Type: FlexibleArray Integer} + +\index{heap} +\index{priority queue} +Flexible arrays are used to implement ``heaps.'' A {\it heap} is an +example of a data structure called a {\it priority queue}, where +elements are ordered with respect to one another. A heap +is organized so as to optimize insertion +and extraction of maximum elements. The {\bf extract!} operation +returns the maximum element of the heap, after destructively removing +that element and reorganizing the heap so that the next maximum +element is ready to be delivered. + +An easy way to create a heap is to apply the operation {\it heap} +to a list of values. +\index{heap} +\spadcommand{h := heap [-4,7,11,3,4,-7]} +$$ +\left[ +{11}, 4, 7, -4, 3, -7 +\right] +$$ +\returnType{Type: Heap Integer} + +This loop extracts elements one-at-a-time from $h$ until the heap +is exhausted, returning the elements as a list in the order they were +extracted. +\index{extract\!} +\index{empty?} +\spadcommand{[extract!(h) while not empty?(h)]} +$$ +\left[ +{11}, 7, 4, 3, -4, -7 +\right] +$$ +\returnType{Type: List Integer} + +A {\it binary tree} is a ``tree'' with at most two branches +\index{tree} per node: it is either empty, or else is a node +consisting of a value, and a left and right subtree (again, binary +trees). Examples of binary tree types are +{\tt BinarySearchTree},\index{BinarySearchTree} +{\tt PendantTree}, \index{PendantTree} +{\tt TournamentTree}, \index{TournamentTree} and +{\tt BalancedBinaryTree}.\index{BalancedBinaryTree} + +\index{binary search tree} +A {\it binary search tree} is a binary tree such that, +\index{tree!binary search} for each node, the value of the node is +\index{binary search tree} greater than all values (if any) in the +left subtree, and less than or equal all values (if any) in the right +subtree. +\index{binarySearchTree} +\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} +$$ +\left[ +{\left[ 2, 3, 4 +\right]}, + 5, {\left[ 7, 9, {11} +\right]} +\right] +$$ +\returnType{Type: BinarySearchTree PositiveInteger} + +\index{balanced binary tree} +A {\it balanced binary tree} is useful for doing modular computations. +\index{balanced binary tree} Given a list $lm$ of moduli, +\index{tree!balanced binary} {\bf modTree}$(a,lm)$ produces +a balanced binary tree with the values $a \bmod m$ at its leaves. +\index{modTree} +\spadcommand{modTree(8,[2,3,5,7])} +$$ +\left[ +0, 2, 3, 1 +\right] +$$ +\returnType{Type: List Integer} + +\index{set} +A {\it set} is a collection of elements where duplication and order is +irrelevant. Sets are always finite and have no +corresponding structure like streams for infinite collections. + +Create sets using braces ``\{`` and ``\}'' rather than brackets. + +\index{set} +\spadcommand{fs := set[1/3,4/5,-1/3,4/5]} +$$ +\left\{ +-{1 \over 3}, {1 \over 3}, {4 \over 5} +\right\} +$$ +\returnType{Type: Set Fraction Integer} + +\index{multiset} +A {\it multiset} is a set that keeps track of the number of duplicate +values. + +For all the primes $p$ between 2 and 1000, find the +distribution of $p \bmod 5$. +\index{multiset} +\index{primes} +\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} +$$ +\left\{ +0, {{42} \mbox{\rm : } 3}, {{40} \mbox{\rm : } 1}, {{38} \mbox{\rm : +} 4}, {{47} \mbox{\rm : } 2} +\right\} +$$ +\returnType{Type: Multiset Integer} + +\index{table} +A {\it table} is conceptually a set of ``key--value'' pairs and is a +generalization of a multiset. For examples of tables, see +{\tt AssociationList}, \index{AssociationList} +{\tt HashTable}, \index{HashTable} +{\tt KeyedAccessFile}, \index{KeyedAccessFile} +{\tt Library}, \index{Library} +{\tt SparseTable}, \index{SparseTable} +{\tt StringTable}, and \index{StringTable} +{\tt Table}. The \index{Table} +domain +{\tt Table(Key, Entry)} provides a general-purpose type for +tables with {\it values} of type $Entry$ indexed by {\it keys} of type +$Key$. + +Compute the above distribution of primes using tables. First, let +$t$ denote an empty table of keys and values, each of type {\tt Integer}. +\index{Table} +\spadcommand{t : Table(Integer,Integer) := empty()} +$$ +{\rm table}() +$$ +\returnType{Type: Table(Integer,Integer)} + +We define a function {\bf howMany} to return the number of values +of a given modulus $k$ seen so far. It calls +{\bf search}$(k,t)$ which returns the number of values +stored under the key $k$ in table $t$, or {\tt ``failed''} +if no such value is yet stored in $t$ under $k$. + +\index{howMany} +\index{search} +In English, this says ``Define $howMany(k)$ as follows. +First, let $n$ be the value of {\it search}$(k,t)$. +Then, if $n$ has the value $"failed"$, return the value +$1$; otherwise return $n + 1$.'' +\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)} +\returnType{Type: Void} + +Run through the primes to create the table, then print the table. +The expression {\tt t.m := howMany(m)} updates the value in table $t$ +stored under key $m$. +\index{primes} +\index{rem} +\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t} +\begin{verbatim} + Compiling function howMany with type Integer -> Integer +\end{verbatim} +$$ +{\rm table } +\left( +{{2={47}}, {4={38}}, {1={40}}, {3={42}}, {0=1}} +\right) +$$ +\returnType{Type: Table(Integer,Integer)} + +\index{record} +A {\it record} is an example of an inhomogeneous collection of +objects.A record consists of a +set of named {\it selectors} that can be used to access its +components. \index{Record@{\sf Record}} + +Declare that $daniel$ can only be +assigned a record with two prescribed fields. +\index{Record} +\spadcommand{daniel : Record(age : Integer, salary : Float)} +\returnType{Type: Void} + +Give $daniel$ a value, using square brackets to enclose the values of +the fields. +\spadcommand{daniel := [28, 32005.12]} +$$ +\left[ +{age={28}}, {salary={32005.12}} +\right] +$$ +\returnType{Type: Record(age: Integer,salary: Float)} + +Give $daniel$ a raise. +\spadcommand{daniel.salary := 35000; daniel} +$$ +\left[ +{age={28}}, {salary={35000.0}} +\right] +$$ +\returnType{Type: Record(age: Integer,salary: Float)} + +\index{union} +A {\it union} is a data structure used when objects have multiple +types.\index{Union@{\sf Union}} + +Let $dog$ be either an integer or a string value. +\index{Union} +\spadcommand{dog: Union(licenseNumber: Integer, name: String)} +\returnType{Type: Void} + +Give $dog$ a name. +\spadcommand{dog := "Whisper"} +$$ +\mbox{\tt "Whisper"} +$$ +\returnType{Type: Union(name: String,...)} + +All told, there are over forty different data structures in Axiom. +Using the domain constructors you can add your own data structure or +extend an existing one. Choosing the right data structure for your +application may be the key to obtaining good performance. + +\section{Expanding to Higher Dimensions} +\label{sec:Expanding to Higher Dimensions} +To get higher dimensional aggregates, you can create one-dimensional +aggregates with elements that are themselves aggregates, for example, +lists of lists, one-dimensional arrays of lists of multisets, and so +on. For applications requiring two-dimensional homogeneous +aggregates, you will likely find {\it two-dimensional arrays} +\index{matrix} and {\it matrices} most useful. +\index{array!two-dimensional} + +The entries in +{\tt TwoDimensionalArray} \index{TwoDimensionalArray} +and {\tt Matrix} \index{Matrix} objects are +all the same type, except that those for {\tt Matrix} must belong to a +{\tt Ring}.\index{Ring} +You create and access elements in roughly the same way. +Since matrices have an understood algebraic structure, certain +algebraic operations are available for matrices but not for arrays. +Because of this, we limit our discussion here to {\tt Matrix}, that +can be regarded as an extension of {\tt TwoDimensionalArray}. See {\tt +TwoDimensionalArray} for more information about arrays. There are also +Axiom's linear algebra facilities like, see {\tt Matrix}, +{\tt Permanent}, \index{Permanent} +{\tt SquareMatrix}, \index{SquareMatrix} +{\tt Vector}, \index{Vector} + +You can create a matrix from a list of lists, \index{matrix!creating} +where each of the inner lists represents a row of the matrix. +\index{matrix} +\spadcommand{m := matrix([ [1,2], [3,4] ])} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +The ``collections'' construct is useful for creating matrices whose +entries are given by formulas. \index{matrix!Hilbert} +\spadcommand{matrix([ [1/(i + j - x) for i in 1..4] for j in 1..4])} +$$ +\left[ +\begin{array}{cccc} +-{1 \over {x -2}} & -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} \\ +-{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} \\ +-{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} \\ +-{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} & -{1 \over {x -8}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Polynomial Integer} + +Let $vm$ denote the three by three Vandermonde matrix. +\spadcommand{vm := matrix [ [1,1,1], [x,y,z], [x*x,y*y,z*z] ]} +$$ +\left[ +\begin{array}{ccc} +1 & 1 & 1 \\ +x & y & z \\ +{x \sp 2} & {y \sp 2} & {z \sp 2} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +Use this syntax to extract an entry in the matrix. + +\spadcommand{vm(3,3)} +$$ +z \sp 2 +$$ +\returnType{Type: Polynomial Integer} + +You can also pull out a {\bf row} or a {\bf column}. + +\spadcommand{column(vm,2)} +$$ +\left[ +1, y, {y \sp 2} +\right] +$$ +\returnType{Type: Vector Polynomial Integer} + +You can do arithmetic. + +\spadcommand{vm * vm} +$$ +\left[ +\begin{array}{ccc} +{{x \sp 2}+x+1} & {{y \sp 2}+y+1} & {{z \sp 2}+z+1} \\ +{{{x \sp 2} \ z}+{x \ y}+x} & {{{y \sp 2} \ z}+{y \sp 2}+x} & {{z \sp +3}+{y \ z}+x} \\ +{{{x \sp 2} \ {z \sp 2}}+{x \ {y \sp 2}}+{x \sp 2}} & {{{y \sp 2} \ {z \sp +2}}+{y \sp 3}+{x \sp 2}} & {{z \sp 4}+{{y \sp 2} \ z}+{x \sp 2}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +You can perform operations such as +{\bf transpose}, \index{transpose} +{\bf trace}, and \index{trace} +{\bf determinant}. \index{determinant} +\index{factor} +\spadcommand{factor determinant vm} +$$ +{\left( y -x +\right)} +\ {\left( z -y +\right)} +\ {\left( z -x +\right)} +$$ +\returnType{Type: Factored Polynomial Integer} + +\section{Writing Your Own Functions} +\label{sec:Writing Your Own Functions} +Axiom provides you with a very large library of predefined +operations and objects to compute with. You can use the Axiom +library of constructors to create new objects dynamically of quite +arbitrary complexity. For example, you can make lists of matrices of +fractions of polynomials with complex floating point numbers as +coefficients. Moreover, the library provides a wealth of operations +that allow you to create and manipulate these objects. + +For many applications, you need to interact with the interpreter and +write some Axiom programs to tackle your application. +Axiom allows you to write functions interactively, +\index{function} thereby effectively extending the system library. +Here we give a few simple examples. + +\index{factorial} +We begin by looking at several ways that you can define the +``factorial'' function in Axiom. The first way is to give a +\index{function!piece-wise definition} piece-wise definition of the +function. \index{piece-wise function definition} This method is best +for a general recurrence relation since the pieces are gathered +together and compiled into an efficient iterative function. +Furthermore, enough previously computed values are automatically saved +so that a subsequent call to the function can pick up from where it +left off. + +Define the value of {\bf fact} at $0$. +\spadcommand{fact(0) == 1} +\returnType{Type: Void} + +Define the value of {\bf fact}(n) for general $n$. +\spadcommand{fact(n) == n*fact(n-1)} +\returnType{Type: Void} + +Ask for the value at $50$. The resulting function created by +Axiom computes the value by iteration. + +\spadcommand{fact(50)} +\begin{verbatim} + Compiling function fact with type Integer -> Integer + Compiling function fact as a recurrence relation. +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A second definition uses an {\tt if-then-else} and recursion. +\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)} +\returnType{Type: Void} + +This function is less efficient than the previous version since +each iteration involves a recursive function call. +\spadcommand{fac(50)} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A third version directly uses iteration. +\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)} +\returnType{Type: Void} + +This is the least space-consumptive version. +\spadcommand{fa(50)} +\begin{verbatim} + Compiling function fac with type Integer -> Integer +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +A final version appears to construct a large list and then reduces over +it with multiplication. +\index{reduce} +\spadcommand{f(n) == reduce(*,[i for i in 2..n])} +\returnType{Type: Void} + +In fact, the resulting computation is optimized into an efficient +iteration loop equivalent to that of the third version. +\spadcommand{f(50)} +\begin{verbatim} +Compiling function f with type + PositiveInteger -> PositiveInteger +\end{verbatim} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +The library version uses an algorithm that is different from the four +above because it highly optimizes the recurrence relation definition of +{\bf factorial}. + +\index{factorial} +\spadcommand{factorial(50)} +$$ +30414093201713378043612608166064768844377641568960512000000000000 +$$ +\returnType{Type: PositiveInteger} + +Remember you are not limited to one-line functions in Axiom. If you +place your function definitions in {\bf .input} files +\index{file!input}, you can have multi-line functions that use +indentation for grouping. + +\index{diagonalMatrix} +Given $n$ elements, {\bf diagonalMatrix} creates an +$n$ by $n$ matrix with those elements down the diagonal. +This function uses a permutation matrix +\index{permutation matrix} +that interchanges the $i$th and $j$th rows of a matrix +by which it is right-multiplied. + +This function definition shows a style of definition that can be used +in {\bf .input} files. Indentation is used to create {\sl blocks}: +sequences of expressions that are evaluated in sequence except as +modified by control statements such as {\tt if-then-else} and {\tt return}. + +\begin{verbatim} +permMat(n, i, j) == + m := diagonalMatrix + [(if i = k or j = k then 0 else 1) + for k in 1..n] + m(i,j) := 1 + m(j,i) := 1 + m +\end{verbatim} + +This creates a four by four matrix that interchanges the second and third +rows. +\spadcommand{p := permMat(4,2,3)} +\begin{verbatim} + Compiling function permMat with type (PositiveInteger, + PositiveInteger,PositiveInteger) -> Matrix Integer +\end{verbatim} +$$ +\left[ +\begin{array}{cccc} +1 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 \\ +0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 1 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Create an example matrix to permute. +\spadcommand{m := matrix [ [4*i + j for j in 1..4] for i in 0..3]} +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +5 & 6 & 7 & 8 \\ +9 & {10} & {11} & {12} \\ +{13} & {14} & {15} & {16} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +Interchange the second and third rows of m. +\spadcommand{permMat(4,2,3) * m} +$$ +\left[ +\begin{array}{cccc} +1 & 2 & 3 & 4 \\ +9 & {10} & {11} & {12} \\ +5 & 6 & 7 & 8 \\ +{13} & {14} & {15} & {16} +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +A function can also be passed as an argument to another function, +which then applies the function or passes it off to some other +function that does. You often have to declare the type of a function +that has functional arguments. + +This declares {\bf t} to be a two-argument function that returns a +{\tt Float}. The first argument is a function that takes one +{\tt Float} argument and returns a {\tt Float}. + +\spadcommand{t : (Float -> Float, Float) -> Float} +\returnType{Type: Void} + +This is the definition of {\bf t}. + +\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2} +\returnType{Type: Void} + +We have not defined a {\bf cos} in the workspace. The one from the +Axiom library will do. + +\spadcommand{t(cos, 5.2058)} +$$ +1.0 +$$ +\returnType{Type: Float} + +Here we define our own (user-defined) function. +\spadcommand{cosinv(y) == cos(1/y)} +\returnType{Type: Void} + +Pass this function as an argument to {\bf t}. +\spadcommand{t(cosinv, 5.2058)} +$$ +1.7392237241\ 8005164925\ 4147684772\ 932520785 +$$ +\returnType{Type: Float} + +Axiom also has pattern matching capabilities for +\index{simplification} +simplification +\index{pattern matching} +of expressions and for defining new functions by rules. +For example, suppose that you want to apply regularly a transformation +that groups together products of radicals: +$$\sqrt{a}\sqrt{b} \mapsto \sqrt{ab}, \quad +(\forall a)(\forall b)$$ +Note that such a transformation is not generally correct. +Axiom never uses it automatically. + +Give this rule the name {\bf groupSqrt}. +\index{groupSqrt} +\index{rule} +\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))} +$$ +{ \%C \ {\sqrt {a}} \ {\sqrt {b}}} \mbox{\rm == } { \%C \ {\sqrt {{a \ +b}}}} +$$ +\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} + +Here is a test expression. +\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y}}}+ +{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x}}}\right)}\ {\sqrt{z}}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x}}\ {\sqrt{y}}}+ +{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ +y}+{x^2} +\end{array} +$$ +\returnType{Type: Expression Integer} + +The rule +{\bf groupSqrt} successfully simplifies the expression. +\spadcommand{groupSqrt a} +$$ +\begin{array}{@{}l} +\displaystyle +{{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y \ z}}}+ +{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x \ z}}}+ + +\\ +\\ +\displaystyle +{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x \ y}}}+ +{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ +y}+{x^2} +\end{array} +$$ +\returnType{Type: Expression Integer} + +\section{Polynomials} +\label{sec:Polynomials} +Polynomials are the commonly used algebraic types in symbolic +computation. \index{polynomial} Interactive users of Axiom +generally only see one type of polynomial they can use: {\tt Polynomial(R)}. +\index{Polynomial(R)} +This type represents polynomials in any number of unspecified +variables over a particular coefficient domain $R$. This type +represents its coefficients {\sl sparsely}: only terms with non-zero +coefficients are represented. + +In building applications, many other kinds of polynomial +representations are useful. Polynomials may have one variable or +multiple variables, the variables can be named or unnamed, the +coefficients can be stored sparsely or densely. So-called +``distributed multivariate polynomials'' store polynomials as +coefficients paired with vectors of exponents. This type is +particularly efficient for use in algorithms for solving systems of +non-linear polynomial equations. + +The polynomial constructor most familiar to the interactive user +is {\tt Polynomial}. +\spadcommand{(x**2 - x*y**3 +3*y)**2} +$$ +{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp +3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} +$$ +\returnType{Type: Polynomial Integer} + +If you wish to restrict the variables used, +{\tt UnivariatePolynomial} provides polynomials in one variable. +\index{UnivariatePolynomial} +\spadcommand{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)} +$$ +{{18} \ {x \sp 3}}+{{60} \ {x \sp 2}} -{{46} \ x}+8 +$$ +\returnType{Type: UnivariatePolynomial(x,Integer)} + +The constructor {\tt MultivariatePolynomial}, which can be abbreviated as +MPOLY, provides polynomials in one or more specified variables. +\index{MultivariatePolynomial} +\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2} +$$ +{x \sp 4} -{2 \ {y \sp 3} \ {x \sp 3}}+{{\left( {y \sp 6}+{6 \ y} +\right)} +\ {x \sp 2}} -{6 \ {y \sp 4} \ x}+{9 \ {y \sp 2}} +$$ +\returnType{Type: MultivariatePolynomial([x,y],Integer)} + +You can change the way the polynomial appears by modifying the variable +ordering in the explicit list. +\spadcommand{m :: MPOLY([y,x],INT)} +$$ +{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp +3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} +$$ +\returnType{Type: MultivariatePolynomial([y,x],Integer)} + +The constructor {\tt DistributedMultivariatePolynomial}, which +can be abbreviated as DMP, provides +polynomials in one or more specified variables with the monomials +ordered lexicographically. +\index{DistributedMultivariatePolynomial} +\spadcommand{m :: DMP([y,x],INT)} +$$ +{{y \sp 6} \ {x \sp 2}} -{6 \ {y \sp 4} \ x} -{2 \ {y \sp 3} \ {x \sp +3}}+{9 \ {y \sp 2}}+{6 \ y \ {x \sp 2}}+{x \sp 4} +$$ +\returnType{Type: DistributedMultivariatePolynomial([y,x],Integer)} + +The constructor \index{HomogeneousDistributedMultivariatePolynomial} +{\tt HomogeneousDistributedMultivariatePolynomial}, which can be +abbreviated as HDMP, is similar +except that the monomials are ordered by total order refined by +reverse lexicographic order. +\spadcommand{m :: HDMP([y,x],INT)} +$$ +{{y \sp 6} \ {x \sp 2}} -{2 \ {y \sp 3} \ {x \sp 3}} -{6 \ {y \sp 4} \ +x}+{x \sp 4}+{6 \ y \ {x \sp 2}}+{9 \ {y \sp 2}} +$$ +\returnType{Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)} + +More generally, the domain constructor +{\tt GeneralDistributedMultivariatePolynomial} allows the user to +provide an arbitrary predicate to define his own term ordering. These +last three constructors are typically used in Gr\"{o}bner basis +\index{Gr\"{o}bner} applications and +when a flat (that is, non-recursive) display is wanted and the term +ordering is critical for controlling the computation. + +\section{Limits} +\label{sec:Limits} +\index{limit} +Axiom's {\bf limit} function is usually used to evaluate +limits of quotients where the numerator and denominator \index{limit} +both tend to zero or both tend to infinity. To find the limit of an +expression $f$ as a real variable $x$ tends to a limit +value $a$, enter {\tt limit(f, x=a)}. Use +{\bf complexLimit} if the variable is complex. +\index{complexLimit} + +You can take limits of functions with parameters. +\index{limit!of function with parameters} +\spadcommand{g := csc(a*x) / csch(b*x)} +$$ +{\csc +\left( +{{a \ x}} +\right)} +\over {\csch +\left( +{{b \ x}} +\right)} +$$ +\returnType{Type: Expression Integer} + +As you can see, the limit is expressed in terms of the parameters. +\spadcommand{limit(g,x=0)} +$$ +b \over a +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} +\index{OrderedCompletion} + +A variable may also approach plus or minus infinity: +\spadcommand{h := (1 + k/x)**x} +$$ +{{x+k} \over x} \sp x +$$ +\returnType{Type: Expression Integer} + +\index{\%plusInfinity} +\index{\%minusInfinity} +Use {\tt \%plusInfinity} and {\tt \%minusInfinity} to +denote $\infty$ and $-\infty$. +\spadcommand{limit(h,x=\%plusInfinity)} +$$ +e \sp k +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +A function can be defined on both sides of a particular value, but +may tend to different limits as its variable approaches that value from the +left and from the right. + +\index{limit} +\index{sqrt} +\spadcommand{limit(sqrt(y**2)/y,y = 0)} +$$ +\left[ +{leftHandLimit=-1}, {rightHandLimit=1} +\right] +$$ +\returnType{Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} + +As $x$ approaches $0$ along the real axis, {\tt exp(-1/x**2)} +tends to $0$. + +\index{limit} +\index{exp} +\spadcommand{limit(exp(-1/x**2),x = 0)} +$$ +0 +$$ +\returnType{Type: Union(OrderedCompletion Expression Integer,...)} + +However, if $x$ is allowed to approach $0$ along any path in the +complex plane, the limiting value of {\tt exp(-1/x**2)} depends on the +path taken because the function has an essential singularity at $x=0$. +This is reflected in the error message returned by the function. +\index{complexLimit} +\spadcommand{complexLimit(exp(-1/x**2),x = 0)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\section{Series} +\label{sec:Series} + +Axiom also provides power series. \index{series!power} By default, +Axiom tries to compute and display the first ten elements of a series. +Use {\tt )set streams calculate} to change the default value to +something else. For the purposes of this document, we have used this +system command to display fewer than ten terms. +\index{)set streams calculate} + +You can convert a functional expression to a power series by using the +operation {\bf series}. In this example, {\tt sin(a*x)} is +expanded in powers of $(x - 0)$, that is, in powers of $x$. + +\index{series} +\index{sin} +\index{UnivariatePuiseuxSeries} +\spadcommand{series(sin(a*x),x = 0)} +$$ +{a \ x} -{{{a \sp 3} \over 6} \ {x \sp 3}}+{{{a \sp 5} \over {120}} \ {x +\sp 5}} -{{{a \sp 7} \over {5040}} \ {x \sp 7}}+{{{a \sp 9} \over {362880}} +\ {x \sp 9}} -{{{a \sp {11}} \over {39916800}} \ {x \sp {11}}}+{O +\left( +{{x \sp {12}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +This expression expands {\tt sin(a*x)} in powers of {\tt (x - \%pi/4)}. +\spadcommand{series(sin(a*x),x = \%pi/4)} +$$ +{\sin +\left({{{a \ \pi} \over 4}}\right)}+ +{a \ {\cos \left({{{a \ \pi} \over 4}} \right)} +\ {\left( x -{\pi \over 4} \right)}}- +\hbox{\hskip 2.0cm} +$$ +$$ +{{{{a \sp 2} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over 2} +\ {{\left( x -{\pi \over 4} \right)}\sp 2}} - +{{{{a \sp 3} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over 6} +\ {{\left( x -{\pi \over 4} \right)}\sp 3}} + +$$ +$$ +{{{{a \sp 4} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {24}} +\ {{\left( x -{\pi \over 4} \right)}\sp 4}} + +{{{{a \sp 5} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {120}} +\ {{\left( x -{\pi \over 4} \right)}\sp 5}} - +$$ +$$ +{{{{a \sp 6} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {720}} +\ {{\left( x -{\pi \over 4} \right)}\sp 6}} - +{{{{a \sp 7} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {5040}} +\ {{\left( x -{\pi \over 4} \right)}\sp 7}} + +$$ +$$ +{{{{a \sp 8} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {40320}} +\ {{\left( x -{\pi \over 4} \right)}\sp 8}} + +{{{{a \sp 9} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {362880}} +\ {{\left( x -{\pi \over 4} \right)}\sp 9}} - +$$ +$$ +{{{{a \sp {10}} \ {\sin \left({{{a \ \pi} \over 4}} \right)}} +\over {3628800}} +\ {{\left( x -{\pi \over 4} \right)}\sp {10}}} + +{O \left({{{\left( x -{\pi \over 4} \right)}\sp {11}}} \right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,pi/4)} + +Axiom provides \index{series!Puiseux} {\it Puiseux series:} +\index{Puiseux series} series with rational number exponents. The +first argument to {\bf series} is an in-place function that +computes the $n$-th coefficient. (Recall that the +``{\tt +->}'' is an infix operator meaning ``maps to.'') + +\index{factorial} +\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x=0,4/3..,2)} +%%NOTE: the paper book shows O(x^4) but Axiom computes O(x^5) +$$ +{x \sp {4 \over 3}} -{{1 \over 6} \ {x \sp {{10} \over 3}}}+{O +\left( +{{x \sp 5}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +Once you have created a power series, you can perform arithmetic +operations on that series. We compute the Taylor expansion of $1/(1-x)$. +\index{series!Taylor} +\spadcommand{f := series(1/(1-x),x = 0)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +Compute the square of the series. +\spadcommand{f ** 2} +$$ +1+{2 \ x}+{3 \ {x \sp 2}}+{4 \ {x \sp 3}}+{5 \ {x \sp 4}}+{6 \ {x \sp +5}}+{7 \ {x \sp 6}}+{8 \ {x \sp 7}}+{9 \ {x \sp 8}}+{{10} \ {x \sp +9}}+{{11} \ {x \sp {10}}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +The usual elementary functions +({\bf log}, {\bf exp}, trigonometric functions, and so on) +are defined for power series. +\spadcommand{f := series(1/(1-x),x = 0)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\index{log} +\spadcommand{g := log(f)} +$$ +\begin{array}{@{}l} +x+ +{{1 \over 2} \ {x \sp 2}}+ +{{1 \over 3} \ {x \sp 3}}+ +{{1 \over 4} \ {x \sp 4}}+ +{{1 \over 5} \ {x \sp 5}}+ +{{1 \over 6} \ {x \sp 6}}+ +{{1 \over 7} \ {x \sp 7}}+ +\\ +\\ +\displaystyle +{{1 \over 8} \ {x \sp 8}}+ +{{1 \over 9} \ {x \sp 9}}+ +{{1 \over {10}} \ {x \sp {10}}}+ +{{1 \over {11}} \ {x \sp {11}}}+ +{O \left({{x \sp {12}}} \right)} +\end{array} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +\index{exp} +\spadcommand{exp(g)} +$$ +1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x +\sp 9}+{x \sp {10}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} + +% Warning: currently there are (interpreter) problems with converting +% rational functions and polynomials to power series. + +Here is a way to obtain numerical approximations of +$e$ from the Taylor series expansion of {\bf exp}(x). +First create the desired Taylor expansion. +\index{taylor} +\spadcommand{f := taylor(exp(x))} +$$ +1+x+{{1 \over 2} \ {x \sp 2}}+{{1 \over 6} \ {x \sp 3}}+{{1 \over {24}} \ +{x \sp 4}}+{{1 \over {120}} \ {x \sp 5}}+{{1 \over {720}} \ {x \sp 6}} + +\hbox{\hskip 1.0cm} +$$ +$$ +{{1 +\over {5040}} \ {x \sp 7}} + +{{1 \over {40320}} \ {x \sp 8}}+{{1 \over +{362880}} \ {x \sp 9}}+{{1 \over {3628800}} \ {x \sp {10}}}+{O +\left( +{{x \sp {11}}} +\right)} +$$ +\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} + +Evaluate the series at the value $1.0$. +% Warning: syntax for evaluating power series may change. +As you see, you get a sequence of partial sums. +\index{eval} +\spadcommand{eval(f,1.0)} +$$ +\left[ +{1.0}, {2.0}, {2.5}, {2.6666666666 666666667}, \hbox{\hskip 3.0cm} +\right. +$$ +$${2.7083333333 333333333}, {2.7166666666 666666667}, \hbox{\hskip 1.0cm} +$$ +$${2.7180555555 555555556}, {2.7182539682 53968254}, \hbox{\hskip 1.1cm} +$$ +$$\left. +{2.7182787698 412698413}, {2.7182815255 731922399}, \ldots +\hbox{\hskip 0.4cm} +\right] +$$ +\returnType{Type: Stream Expression Float} + +\section{Derivatives} +\label{sec:Derivatives} +\index{Derivatives} +Use the Axiom function {\bf D} to differentiate an +\index{derivative} expression. \index{differentiation} + +To find the derivative of an expression $f$ with respect to a +variable $x$, enter {\bf D}(f, x). + +\index{exp} +\spadcommand{f := exp exp x} +$$ +e \sp {e \sp x} +$$ +\returnType{Type: Expression Integer} + +\index{D Derivatives} +\spadcommand{D(f, x)} +$$ +{e \sp x} \ {e \sp {e \sp x}} +$$ +\returnType{Type: Expression Integer} + +An optional third argument $n$ in {\bf D} asks Axiom for the $n$-th +derivative of $f$. This finds the fourth derivative of $f$ with +respect to $x$. + +\spadcommand{D(f, x, 4)} +$$ +{\left( {{e \sp x} \sp 4}+{6 \ {{e \sp x} \sp 3}}+{7 \ {{e \sp x} \sp +2}}+{e \sp x} +\right)} +\ {e \sp {e \sp x}} +$$ +\returnType{Type: Expression Integer} + +You can also compute partial derivatives by specifying the order of +\index{differentiation!partial} +differentiation. +\index{sin} +\spadcommand{g := sin(x**2 + y)} +$$ +\sin +\left( +{{y+{x \sp 2}}} +\right) +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(g, y)} +$$ +\cos +\left( +{{y+{x \sp 2}}} +\right) +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(g, [y, y, x, x])} +$$ +{4 \ {x \sp 2} \ {\sin +\left( +{{y+{x \sp 2}}} +\right)}} +-{2 \ {\cos +\left( +{{y+{x \sp 2}}} +\right)}} +$$ +\returnType{Type: Expression Integer} + +Axiom can manipulate the derivatives (partial and iterated) of +\index{differentiation!formal} expressions involving formal operators. +All the dependencies must be explicit. + +This returns $0$ since F (so far) does not explicitly depend on $x$. + +\spadcommand{D(F,x)} +$$ +0 +$$ +\returnType{Type: Polynomial Integer} + +Suppose that we have F a function of $x$, $y$, and $z$, +where $x$ and $y$ are themselves functions of $z$. + +Start by declaring that $F$, $x$, and $y$ are operators. +\index{operator} + +\index{operator function} +\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y} +$$ +y +$$ +\returnType{Type: BasicOperator} + +You can use F, $x$, and $y$ in expressions. + +\spadcommand{a := F(x z, y z, z**2) + x y(z+1)} +$$ +{x +\left( +{{y +\left( +{{z+1}} +\right)}} +\right)}+{F +\left( +{{x +\left( +{z} +\right)}, + {y +\left( +{z} +\right)}, + {z \sp 2}} +\right)} +$$ +\returnType{Type: Expression Integer} + +Differentiate formally with respect to $z$. +The formal derivatives appearing in $dadz$ are not just formal symbols, +but do represent the derivatives of $x$, $y$, and F. + +\spadcommand{dadz := D(a, z)} +$$ +\begin{array}{@{}l} +\displaystyle +{2 \ z \ {{F_{, 3}}\left({{x \left({z}\right)}, {y \left({z}\right)}, + {z^2}}\right)}}+{{{y_{\ }^{,}}\left({z}\right)}\ {{F_{, 2}}\left({{x +\left({z}\right)}, {y \left({z}\right)}, {z^2}}\right)}}+ + +\\ +\\ +\displaystyle +{{{x_{\ }^{,}}\left({z}\right)}\ {{F_{, 1}}\left({{x \left({z}\right)}, + {y \left({z}\right)}, {z^2}}\right)}}+{{{x_{\ }^{,}}\left({y +\left({z + 1}\right)}\right)}\ {{y_{\ }^{,}}\left({z + 1}\right)}} +\end{array} +$$ +\returnType{Type: Expression Integer} + +You can evaluate the above for particular functional values of +F, $x$, and $y$. If $x(z)$ is {\bf exp}(z) and $y(z)$ is {\bf log}(z+1), +then evaluates {\tt dadz}. + +\index{eval} +\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^z}, + {\log \left({z + 1}\right)}, {z^2}}\right)}}+ +\\ +\\ +\displaystyle +{{F_{, 2}}\left({{e^ +z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ +\\ +\\ +\displaystyle +{{\left(z + 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log +\left({z + 1}\right)}, {z^2}}\right)}}+ z + 1 +\end{array} +\right)}\over{z + 1} +$$ +\returnType{Type: Expression Integer} + +You obtain the same result by first evaluating $a$ and +then differentiating. + +\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))} +$$ +{F +\left( +{{e \sp z}, {\log +\left( +{{z+1}} +\right)}, + {z \sp 2}} +\right)}+z+2 +$$ +\returnType{Type: Expression Integer} + +\spadcommand{D(\%, z)} +$$ +{\left( +\begin{array}{@{}l} +\displaystyle +{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^ +z}, {\log \left({z + 1}\right)}, {z^2}}\right)}}+ +\\ +\\ +\displaystyle +{{F_{, 2}}\left({{e^z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ +\\ +\\ +\displaystyle +{{\left(z ++ 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log \left({z ++ 1}\right)}, {z^2}}\right)}}+ z + 1 +\end{array} +\right)} +\over{z + 1} +$$ +\returnType{Type: Expression Integer} + +\section{Integration} +\label{sec:Integration} + +Axiom has extensive library facilities for integration. +\index{integration} + +The first example is the integration of a fraction with denominator +that factors into a quadratic and a quartic irreducible polynomial. +The usual partial fraction approach used by most other computer +algebra systems either fails or introduces expensive unneeded +algebraic numbers. + +We use a factorization-free algorithm. +\index{integrate} +\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} +$$ +{\arctan +\left( +{{{x \sp 3}+{3 \ {x \sp 2}}+{3 \ x}+1}} +\right)} +\over 3 +$$ +\returnType{Type: Union(Expression Integer,...)} + +When real parameters are present, the form of the integral can depend on +the signs of some expressions. + +Rather than query the user or make sign assumptions, Axiom returns +all possible answers. +\spadcommand{integrate(1/(x**2 + a),x)} +$$ +\left[ +{{\log +\left( +{{{{{\left( {x \sp 2} -a +\right)} +\ {\sqrt {-a}}}+{2 \ a \ x}} \over {{x \sp 2}+a}}} +\right)} +\over {2 \ {\sqrt {-a}}}}, {{\arctan +\left( +{{{x \ {\sqrt {a}}} \over a}} +\right)} +\over {\sqrt {a}}} +\right] +$$ +\returnType{Type: Union(List Expression Integer,...)} + +The {\bf integrate} operation generally assumes that all +parameters are real. The only exception is when the integrand has +complex valued quantities. + +If the parameter is complex instead of real, then the notion of sign +is undefined and there is a unique answer. You can request this +answer by ``prepending'' the word ``complex'' to the command name: + +\index{complexIntegrate} +\spadcommand{complexIntegrate(1/(x**2 + a),x)} +%%NOTE: the expression in the book is different but they differentiate +%%to exactly the same answer. +$$ +{{\log +\left( +{{{{x \ {\sqrt {-a}}}+a} \over {\sqrt {-a}}}} +\right)} +-{\log +\left( +{{{{x \ {\sqrt {-a}}} -a} \over {\sqrt {-a}}}} +\right)}} +\over {2 \ {\sqrt {-a}}} +$$ +\returnType{Type: Expression Integer} + +The following two examples illustrate the limitations of table-based +approaches. The two integrands are very similar, but the answer to +one of them requires the addition of two new algebraic numbers. + +This one is the easy one. +The next one looks very similar +but the answer is much more complicated. +\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} +$$ +{{\left( {{120} \ {b \sp 3} \ {x \sp 3}} -{{135} \ a \ {b \sp 2} \ {x +\sp 2}}+{{162} \ {a \sp 2} \ b \ x} -{{243} \ {a \sp 3}} +\right)} +\ {{\root {3} \of {{{b \ x}+a}}} \sp 2}} \over {{440} \ {b \sp 4}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Only an algorithmic approach is guaranteed to find what new constants +must be added in order to find a solution. + +\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} +$$ +\left( +\begin{array}{@{}l} +-{2 \ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log +\left( +{{{{\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp 2}}+{{{\root +{3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}}+a}} +\right)}}+ +\\ +\\ +\displaystyle +{4\ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log +\left( +{{{{{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}} -a}} +\right)}}+ +\\ +\\ +\displaystyle +{{12}\ {b \sp 2} \ {x \sp 2} \ {\arctan +\left( +{{{{2 \ {\sqrt {3}} \ {{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ +x}+a}}}}+{a \ {\sqrt {3}}}} \over {3 \ a}}} +\right)}}+ +\\ +\\ +\displaystyle +{{\left( +{{12} \ b \ x} -{9 \ a} +\right)} +\ {\sqrt {3}} \ {\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp +2}} +\end{array} +\right) +\over {{18} \ {a \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\root {3} \of +{a}}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Some computer algebra systems use heuristics or table-driven +approaches to integration. When these systems cannot determine the +answer to an integration problem, they reply ``I don't know.'' Axiom +uses an algorithm which is a {\sl decision procedure} for integration. +If Axiom returns the original integral that conclusively proves that +an integral cannot be expressed in terms of elementary functions. + +When Axiom returns an integral sign, it has proved that no answer +exists as an elementary function. + +\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} +$$ +\int \sp{\displaystyle x} {{{\log +\left( +{{{\sqrt {{b+{ \%Q \ a}}}}+1}} +\right)} +\over \%Q} \ {d \%Q}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Axiom can handle complicated mixed functions much beyond what you +can find in tables. + +Whenever possible, Axiom tries to express the answer using the +functions present in the integrand. + +\index{sinh} +\index{sqrt} +\index{cosh} +\index{integrate} +\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} +%%NOTE: the book has the same answer with a trailing ``+4'' term. +%%This term is not generated by Axiom +$$ +{2 \ {\log +\left( +{{{-{2 \ {\cosh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)}} +-{2 \ x}} \over {{\sinh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)} +-{\cosh +\left( +{{{\sqrt {{x+b}}}+1}} +\right)}}}} +\right)}} +-{2 \ {\sqrt {{x+b}}}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +A strong structure-checking algorithm in Axiom finds hidden algebraic +relationships between functions. + +\index{tan} +\index{atan} +\spadcommand{integrate(tan(atan(x)/3),x)} +%%NOTE: the book has a trailing ``+16'' term in the numerator +%%This is not generated by Axiom +$$ +\left( +\begin{array}{@{}l} +{8 \ {\log +\left( +{{{3 \ {{\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)} +\sp 2}} -1}} +\right)}} +-{3 \ {{\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)} +\sp 2}}+ +\\ +\\ +\displaystyle +{{18} \ x \ {\tan +\left( +{{{\arctan +\left( +{x} +\right)} +\over 3}} +\right)}} +\end{array} +\right) +\over {18} +$$ +\returnType{Type: Union(Expression Integer,...)} + +The discovery of this algebraic relationship is necessary for correct +integration of this function. +Here are the details: +\begin{enumerate} +\item +If $x=\tan t$ and $g=\tan (t/3)$ then the following +algebraic relation is true: $${g^3-3xg^2-3g+x=0}$$ +\item +Integrate $g$ using this algebraic relation; this produces: +$${{(24g^2 - 8)\log(3g^2 - 1) + (81x^2 + 24)g^2 + 72xg - 27x^2 - 16} +\over{54g^2 - 18}}$$ +\item +Rationalize the denominator, producing: +$${8\log(3g^2-1) - 3g^2 + 18xg + 16} \over {18}$$ +Replace $g$ by the initial definition +$g = \tan(\arctan(x)/3)$ +to produce the final result. +\end{enumerate} +\index{arctan} + +This is an example of a mixed function where +the algebraic layer is over the transcendental one. +\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)} +$$ +-{{2 \ {\sqrt {{{\log +\left( +{x} +\right)}+x}}}} +\over {{\log +\left( +{x} +\right)}+x}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +While incomplete for non-elementary functions, Axiom can +handle some of them. +\index{exp} +\index{erf} +\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} +$$ +{{{\left( {\erf +\left( +{x} +\right)} +-1 +\right)} +\ {\sqrt {\pi}} \ {\log +\left( +{{{{\erf +\left( +{x} +\right)} +-1} \over {{\erf +\left( +{x} +\right)}+1}}} +\right)}} +-{2 \ {\sqrt {\pi}}}} \over {{8 \ {\erf +\left( +{x} +\right)}} +-8} +$$ +\returnType{Type: Union(Expression Integer,...)} + +\section{Differential Equations} +\label{sec:Differential Equations} +The general approach used in integration also carries over to the +solution of linear differential equations. + +Let's solve some differential equations. +Let $y$ be the unknown function in terms of $x$. +\index{operator function} +\spadcommand{y := operator 'y} +$$ +y +$$ +\returnType{Type: BasicOperator} +\index{BasicOperator} + +Here we solve a third order equation with polynomial coefficients. +\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4} +$$ +{{{x \sp 3} \ {{y \sb {{\ }} \sp {,,,}} +\left( +{x} +\right)}}+{{x +\sp 2} \ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}} +-{2 \ x \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}}+{2 +\ {y +\left( +{x} +\right)}}}={2 +\ {x \sp 4}} +$$ +\returnType{Type: Equation Expression Integer} +\index{Equation} + +\index{solve} +\spadcommand{solve(deq, y, x)} +%%NOTE: the book has a different solution and it appears to be +%%less complicated than this one. +$$ +\begin{array}{@{}l} +\left[ +{particular={{{x \sp 5} -{{10} \ {x \sp 3}}+{{20} \ {x \sp 2}}+4} \over +{{15} \ x}}}, +\right. +\\ +\\ +\displaystyle +\left. +{basis={\left[ {{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+1} +\over x}, {{{x \sp 3} -1} \over x}, {{{x \sp 3} -{3 \ {x \sp 2}} -1} +\over x} +\right]}} +\right] +\end{array} +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} + + +Here we find all the algebraic function solutions of the equation. +\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0} +$$ +{{{\left( {x \sp 2}+1 +\right)} +\ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}}+{3 +\ x \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}}+{y +\left( +{x} +\right)}}=0 +$$ +\returnType{Type: Equation Expression Integer} + +\index{solve} +\spadcommand{solve(deq, y, x)} +$$ +\left[ +{particular=0}, +{basis={\left[ {1 \over {\sqrt {{{x \sp 2}+1}}}}, +{{\log +\left( +{{{\sqrt {{{x \sp 2}+1}}} -x}} +\right)} +\over {\sqrt {{{x \sp 2}+1}}}} +\right]}} +\right] +$$ +\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} + +Coefficients of differential equations can come from arbitrary +constant fields. For example, coefficients can contain algebraic +numbers. + +This example has solutions whose logarithmic derivative is an +algebraic function of degree two. + +\spadcommand{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x} +$$ +{2 \ {x \sp 3} \ {{y \sb {{\ }} \sp {,,}} +\left( +{x} +\right)}}+{3 +\ {x \sp 2} \ {{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}} +-{2 \ {y +\left( +{x} +\right)}} +$$ +\returnType{Type: Expression Integer} + +\index{solve} +\spadcommand{solve(eq,y,x).basis} +$$ +\left[ +{e \sp {\left( -{2 \over {\sqrt {x}}} +\right)}}, + {e \sp {2 \over {\sqrt {x}}}} +\right] +$$ +\returnType{Type: List Expression Integer} + +Here's another differential equation to solve. +\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)} +$$ +{{y \sb {{\ }} \sp {,}} +\left( +{x} +\right)}={{y +\left( +{x} +\right)} +\over {{{y +\left( +{x} +\right)} +\ {\log +\left( +{{y +\left( +{x} +\right)}} +\right)}}+x}} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{solve(deq, y, x)} +$$ +{{{y +\left( +{x} +\right)} +\ {{\log +\left( +{{y +\left( +{x} +\right)}} +\right)} +\sp 2}} -{2 \ x}} \over {2 \ {y +\left( +{x} +\right)}} +$$ +\returnType{Type: Union(Expression Integer,...)} + +Rather than attempting to get a closed form solution of +a differential equation, you instead might want to find an +approximate solution in the form of a series. + +Let's solve a system of nonlinear first order equations and get a +solution in power series. Tell Axiom that $x$ is also an +operator. + +\index{operator function} +\spadcommand{x := operator 'x} +$$ +x +$$ +\returnType{Type: BasicOperator} + +Here are the two equations forming our system. +\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2} +$$ +{{x \sb {{\ }} \sp {,}} +\left( +{t} +\right)}={{{x +\left( +{t} +\right)} +\sp 2}+1} +$$ +\returnType{Type: Equation Expression Integer} + +\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)} +$$ +{{y \sb {{\ }} \sp {,}} +\left( +{t} +\right)}={{x +\left( +{t} +\right)} +\ {y +\left( +{t} +\right)}} +$$ +\returnType{Type: Equation Expression Integer} + +We can solve the system around $t = 0$ with the initial +conditions $x(0) = 0$ and $y(0) = 1$. Notice that since +we give the unknowns in the order $[x, y]$, the answer is a list +of two series in the order +$[{\rm series\ for\ }x(t), {\rm series\ for\ }y(t)]$. + +\index{seriesSolve} +\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])} +$$ +\left[ +{\ t+ +{{1 \over 3} \ {t \sp 3}}+ +{{2 \over {15}} \ {t \sp 5}}+ +{{{17} \over {315}} \ {t \sp 7}}+ +{{{62} \over {2835}} \ {t \sp 9}}+ +{O \left({{t \sp {11}}} \right)}}, +\right. +\hbox{\hskip 2.0cm} +$$ +$$ +\hbox{\hskip 0.4cm} +\left. +{1+ +{{1 \over 2} \ {t \sp 2}}+ +{{5 \over {24}} \ {t \sp 4}}+ +{{{61} \over {720}} \ {t \sp 6}}+ +{{{277} \over {8064}} \ {t \sp 8}}+ +{{{50521} \over {3628800}} \ {t \sp {10}}}+ +{O \left({{t \sp {11}}}\right)}} +\right] +$$ +\returnType{Type: List UnivariateTaylorSeries(Expression Integer,t,0)} +\index{UnivariateTaylorSeries} + +\section{Solution of Equations} +\label{sec:Solution of Equations} +Axiom also has state-of-the-art algorithms for the solution of +systems of polynomial equations. When the number of equations and +unknowns is the same, and you have no symbolic coefficients, you can +use {\bf solve} for real roots and {\bf complexSolve}\index{complexSolve} +for complex roots. In each case, you tell Axiom how accurate you +want your result to be. All operations in the {\it solve} family +return answers in the form of a list of solution sets, where each +solution set is a list of equations. + +A system of two equations involving a symbolic parameter $t$. +\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]} +\returnType{Type: Void} + +Find the real roots of $S(19)$ with +rational arithmetic, correct to within $1/10^{20}$. +\spadcommand{solve(S(19),1/10**20)} +$$ +\left[ +{\left[ {y=5}, {x=-{{2451682632253093442511} \over +{295147905179352825856}}} +\right]}, +\right. +$$ +$$ +\left. +{\left[ {y=5}, {x={{2451682632253093442511} \over +{295147905179352825856}}} +\right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Fraction Integer} + +Find the complex roots of $S(19)$ with floating +point coefficients to $20$ digits accuracy in the mantissa. + +\index{complexSolve} +\spadcommand{complexSolve(S(19),10.e-20)} +$$ +\left[ +{\left[ {y={5.0}}, {x={8.3066238629 180748526}} \right]}, +\right. +$$ +$$ +{\left[ {y={5.0}}, {x=-{8.3066238629 180748526}} \right]}, +$$ +$$ +\left. +{\left[ {y=-{{3.0} \ i}}, {x={1.0}} \right]}, +{\left[ {y={{3.0} \ i}}, {x={1.0}} \right]} +\right] +$$ +\returnType{Type: List List Equation Polynomial Complex Float} + +If a system of equations has symbolic coefficients and you want +a solution in radicals, try {\bf radicalSolve}. +\index{radicalSolve} +\spadcommand{radicalSolve(S(a),[x,y])} +$$ +\left[ +{\left[ {x=-{\sqrt {{a+{50}}}}}, {y=5} \right]}, +{\left[ {x={\sqrt {{a+{50}}}}}, {y=5} \right]}, +\right. +$$ +$$ +\hbox{\hskip 0.7cm} +\left. +{\left[ {x=1}, {y={\sqrt {{{-a+1} \over 2}}}} \right]}, +{\left[ {x=1}, {y=-{\sqrt {{{-a+1} \over 2}}}} \right]} +\right] +$$ +\returnType{Type: List List Equation Expression Integer} + +For systems of equations with symbolic coefficients, you can apply +{\bf solve}, listing the variables that you want Axiom to +solve for. For polynomial equations, a solution cannot usually be +expressed solely in terms of the other variables. Instead, the +solution is presented as a ``triangular'' system of equations, where +each polynomial has coefficients involving only the succeeding +variables. This is analogous to converting a linear system of +equations to ``triangular form''. + +A system of three equations in five variables. +\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]} +$$ +\left[ +{z -y+{x \sp 2}}, {{{x \sp 2} \ z} -{b \ y}+{x \sp 4}}, {{{y \sp 2} \ +z} -{b \ x} -a} +\right] +$$ +\returnType{Type: List Polynomial Integer} + +Solve the system for unknowns $[x,y,z]$, +reducing the solution to triangular form. +\spadcommand{solve(eqns,[x,y,z])} +$$ +\left[ +{\left[ {x=-{a \over b}}, {y=0}, {z=-{{a \sp 2} \over {b \sp 2}}} +\right]}, +\right. +\hbox{\hskip 10.0cm} +$$ +$$ +\left. +\begin{array}{@{}l} +\left[ +{x={{{z \sp 3}+{2 \ b \ {z \sp 2}}+{{b \sp 2} \ z} -a} \over b}}, +{y={z+b}}, +\right. +\hbox{\hskip 10.0cm} +\\ +\\ +\displaystyle +{z \sp 6}+{4 \ b \ {z \sp 5}}+ +{6 \ {b \sp 2} \ {z \sp 4}}+ +{{\left( {4 \ {b \sp 3}} -{2 \ a} \right)}\ {z \sp 3}}+ +{{\left( {b \sp 4} -{4 \ a \ b} \right)}\ {z \sp 2}}- +\hbox{\hskip 4.0cm} +\\ +\\ +\displaystyle +\left. +{2 \ a \ {b \sp 2} \ z} -{b \sp 3}+{a \sp 2}=0 +\right] +\end{array} +\right] +\hbox{\hskip 7.0cm} +$$ +\returnType{Type: List List Equation Fraction Polynomial Integer} + +\chapter{Graphics} +\label{sec:Graphics} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1c.ps} +\caption{An Axiom Graphic} +\end{figure} +Axiom has a two- and three-dimensional drawing and rendering +\index{graphics} package that allows you to draw, shade, color, +rotate, translate, map, clip, scale and combine graphic output of +Axiom computations. The graphics interface is capable of +plotting functions of one or more variables and plotting parametric +surfaces. Once the graphics figure appears in a window, move your +mouse to the window and click. A control panel appears immediately +and allows you to interactively transform the object. Refer to the +original Axiom book\cite{1} and the input files included with Axiom +for additional examples. + +This is an example of Axiom's graphics. From the Control Panel you can +rescale the plot, turn axes and units on and off and save the image, +among other things. Axiom is capable of many different kinds of graphs +in both 2D and 3D settings. Points, lines, planes, wireframe, solids, +shaded solids, multiple graphs, parametric graphs, tubes, and many +other kinds of objects can be created and manipulated by the algebra +and on the control panels. + +This is an example of Axiom's three-dimensional plotting. +It is a graph of the complex arctangent function. +The image displayed was rotated and had the ``shade'' and ``outline'' +display options set from the 3D Control Panel. +The PostScript output was produced by clicking on the +{\bf save} 3D Control Panel button and then +clicking on the {\bf PS} button. + +\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=175, bbury=185]{ps/bookvol1d.ps} +\caption{$(x,y) \rightarrow real atan complex(x,y), -\pi\ldots\pi, -\pi\ldots\pi$} +\end{figure} + +\subsection{Plotting 2D graphs} +There are three kinds of 2D graphs of curves defined by +\begin{enumerate} +\item a function $y=f(x)$ over a finite interval of $x$ +(page~\pageref{sec:2Dcase1}) +\item parametric equations $x=f(t)$ $y=g(t)$ +(page~\pageref{sec:2Dcase2}) +\item nonsingular solutions in a rectangular region +(page~\pageref{sec:2Dcase3}) +\end{enumerate} + +PostScript +\index{PostScript} output is available so that Axiom images can be +printed.\footnote{PostScript is a trademark of Adobe Systems +Incorporated, registered in the United States.} + +\subsubsection{Plotting 2D graphs of 1 variable} +\label{sec:2Dcase1} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a function defined by a formula $f(x)$ is: +\begin{center} +{\tt draw(f(x), x = a..b, {\it options})} +\end{center} + +where $a..b$ defines the range of $x$, and where {\it options} +prescribes zero or more options as described in +\ref{sec:Two-Dimensional Options} on +page~\pageref{sec:Two-Dimensional Options}. An +example of an option is $curveColor == bright red().$ An alternative +format involving functions $f$ and $g$ is also available.\\ +} +Give the names of the functions and +drop the variable name specification in the second argument. +Axiom supplies a default title if one is not given. + +\index{sin} +\index{tan} +\spadgraph{draw(sin(tan(x)) - tan(sin(x)), x=0..6) } +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=216, bbury=226]{ps/bookvol1e.ps} +\caption{$sin(tan(x)) - tan(sin(x)), x=0\ldots6$} +\end{figure} + +\subsubsection{Plotting 2D parametric plane curves} +\label{sec:2Dcase2} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a two-di\-men\-sion\-al plane curve defined by +parametric formulas $x = f(t)$ and $y = g(t)$ is: +\begin{center} +{\tt draw(curve(f(t), g(t)), t = a..b, {\it options})} +\end{center} + +where $a..b$ defines the range of the independent variable $t$, and +where {\it options} prescribes zero or more options as described in +\ref{sec:Three-Dimensional Options} on +page~\pageref{sec:Three-Dimensional Options}. An +example of an option is $curveColor == bright red().$\\ } + +The second kind of two-di\-men\-sion\-al graph are +\index{parametric plane curve} curves produced by parametric +equations. \index{curve!parametric plane} Let $x = f(t)$ and +$y = g(t)$ be formulas of two functions $f$ and $g$ as the parameter $t$ +ranges over an interval $[a,b]$. The function {\bf curve} takes the +two functions $f$ and $g$ as its parameters. + +\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)} +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=216, bbury=226]{ps/bookvol1f.ps} +\caption{$curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t=0..2*\pi$} +\end{figure} + +\subsubsection{Plotting 2D algebraic curves} +\label{sec:2Dcase3} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a non-singular solution curve given by +a polynomial of the form $p(x,y) = 0$ is: +\begin{center} +{\tt draw(p(x,y) = 0, x, y, range == [a..b, c..d], {\it options})} +\end{center} + +where the second and third arguments name the first and second +independent variables of $p$. A {\tt range} option is always given to +designate a bounding rectangular region of the plane +$a \leq x \leq b, c \leq y \leq d$. +Zero or more additional options as described in +\ref{sec:Two-Dimensional Options} on +page~\pageref{sec:Two-Dimensional Options} may be given.\\ +} + +A third kind of two-di\-men\-sion\-al graph is a non-singular +``solution curve'' \index{curve!plane algebraic} in a rectangular +region of the plane. For example: + +\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1)-4*x-1) } +$$ +{y \sp 4}+{{\left( {2 \ {x \sp 2}} -{{16} \ x} -6 +\right)} +\ {y \sp 2}}+{x \sp 4} -{{16} \ {x \sp 3}}+{{58} \ {x \sp 2}} -{{12} \ x} +-6 +$$ +\returnType{Type: Polynomial Integer} +\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7]) } +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1g.ps} +\caption{$p = 0, x, y, range == [-1..11, -7..7]$} +\end{figure} + +A solution curve is a curve defined by a +polynomial equation $p(x,y) = 0$. \index{plane algebraic curve} +Non-singular means that the curve is ``smooth'' in that it does not +cross itself or come to a point (cusp). Algebraically, this means +that for any point $(x,y)$ on the curve, that is, a point such that +$p(x,y) = 0$, the partial derivatives +${{\partial p}\over{\partial x}}(x,y)$ and +${{\partial p}\over{\partial y}}(x,y)$ are not both zero. +\index{curve!smooth} \index{curve!non-singular} \index{smooth curve} +\index{non-singular curve} +We require that the polynomial has rational or integral coefficients. + +The first argument is always expressed as an equation of the form $p = 0$ +where $p$ is a polynomial. + +\subsubsection{Colors} +\label{sec:colors} +\index{Colors} +\index{Palettes} +The domain {\tt Color} \index{Color} provides operations for +manipulating \index{graphics!color} colors in two-di\-men\-sion\-al +graphs. \index{color} Colors are objects of {\tt Color}. Each color +has a {\it hue} and a {\it weight}. \index{hue} Hues are represented +by integers that range from $1$ to the +\spadfunFrom{numberOfHues()}{Color}, normally +\index{graphics!color!number of hues} $27$. \index{weight} Weights +are floats and have the value $1.0$ by default. + +\begin{description} + +\item[{\bf color}]\funArgs{integer} +creates a color of hue {\it integer} and weight $1.0$. +\index{graphics!color!color function} +\index{color} + +\item[{\bf hue}]\funArgs{color} +returns the hue of {\it color} as an integer. +\index{graphics!color!hue function} +\index{hue} + +\item[{\bf red}]\funArgs{} +\funSyntax{blue}{}, +\funSyntax{green}{}, and \funSyntax{yellow}{} +\index{graphics!color!primary color functions} +\index{red} +\index{green} +\index{blue} +create colors of that hue with weight $1.0$. + +\item[$\hbox{\it color}_{1}$ {\tt +} $\hbox{\it color}_{2}$] returns the +color that results from additively combining the indicated +$\hbox{\it color}_{1}$ and $\hbox{\it color}_{2}$. +Color addition is not commutative: changing the order of the arguments +produces different results. + +\item[{\it integer} {\tt *} {\it color}] +changes the weight of {\it color} by {\it integer} +without affecting its hue. +\index{graphics!color!multiply function} +For example, +$red() + 3*yellow()$ produces a color closer to yellow than to red. +Color multiplication is not associative: changing the order of grouping +\index{color!multiplication} +produces different results. +\end{description} + +These functions can be used to change the point and curve colors +for two- and three-di\-men\-sion\-al graphs. +Use the {\tt pointColor} option for points. + +\subsubsection{Two-Dimensional Options} +\label{sec:Two-Dimensional Options} + +The {\bf draw} commands take an optional list of options, such as {\tt +title} shown above. Each option is given by the syntax: +{\it name} {\tt ==} {\it value}. +Here is a list of the available options in the +order that they are described below. + +\begin{tabular}{ll} +adaptive & The {\tt adaptive} option turns adaptive plotting on or off.\\ +\index{adaptive} + & Adaptive plotting uses an algorithm that traverses a graph\\ + & and computes more points for those parts of the graph \\ + & with high curvature. The higher the curvature of a region is,\\ + & the more points the algorithm computes.\\ + & {\tt adaptive == true} or {\tt adaptive == false}\\ +clip & The {\tt clip} option turns clipping on or off. If on,\\ +\index{clip} + & large values are cut off according to {\bf clipPointsDefault}\\ + & {\tt clip == true} or {\tt clip == false} or a range\\ + & {\tt clip == [-2*\%pi..2*\%pi,\%pi..\%pi]}\\ +unit & The {\tt unit} option sets the intervals to which the axis\\ +\index{unit} + & units are plotted according to the indicated steps\\ + & {\tt unit == [2.0, 1.0]}\\ +curveColor & The {\tt curveColor} option sets the color of the graph curves\\ +\index{curveColor} + & or lines to be the indicated palette and color\\ + & {\tt curveColor == bright red()} (see pp \pageref{sec:colors} + and \pageref{sec:Palette})\\ +range & The {\tt range} option sets the range of variables in a graph\\ +\index{range} + & to be within the ranges for solving plane algebraic curve plots\\ + & {\tt range=[-2..2,-2..1]}\\ +toScale & The {\tt toScale} option does plotting to scale if {\tt true}\\ +\index{toScale} + & or uses the entire viewport if {\tt false}. The default can\\ + & be determined using {\bf drawToScale}\\ + & {\tt toScale == true} or {\tt toScale == false}\\ +pointColor & The {\tt pointColor} option sets the color of the graph curves\\ +\index{pointColor} + & or lines to be the indicated palette and color\\ + & {\tt pointColor == bright red()} (see page \pageref{sec:colors})\\ +coordinates & The {\tt coordinates} option indicates the coordinate system\\ +\index{coordinates} + & in which the graph is plotted. This can be one of:\\ + & {\bf bipolar}, {\bf bipolarCylindrical}, {\bf cartesian},\\ + & {\bf conical}, {\bf cylindrical}, {\bf elliptic},\\ + & {\bf ellipticCylindrical}, {\bf oblateSpheroidal}, \\ + & {\bf parabolic}, {\bf parabolicCylindrical}, {\bf paraboloidal}\\ + & {\bf polar}, {\bf prolateSpheroidal}, {\bf spherical}, and\\ + & {\bf toroidal}\\ + & {\tt coordinates == polar} +\end{tabular} + +\subsection{Palette} +\label{sec:Palette} +\index{graphics!palette} + +Domain {\tt Palette} is the domain of shades of colors: +{\bf dark}, {\bf dim}, {\bf bright}, {\bf pastel}, and {\bf light}, +designated by the integers $1$ through $5$, respectively. +\index{Palette} + +Colors are normally ``bright.'' + +\spadcommand{shade red()} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +To change the shade of a color, apply the name of a shade to it. +\index{color!shade} +\index{shade} + +\spadcommand{myFavoriteColor := dark blue() } +$$ +[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Dark \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +The expression $shade(color)$ +returns the value of a shade of $color$. + +\spadcommand{shade myFavoriteColor } +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +The expression $hue(color)$ returns its hue. + +\spadcommand{hue myFavoriteColor } +$$ +\mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0} +$$ +\returnType{Type: Color} + +Palettes can be used in specifying colors in two-di\-men\-sion\-al graphs. + +\spadgraph{draw(x**2,x=-1..1,curveColor == dark blue())} + +% window was 300 x 300 +%\epsffile[0 0 295 295]{ps/23dpal.ps} + + +\subsection{Two-Dimensional Control-Panel} +\label{sec:Two-Dimensional Control-Panel} + +\index{graphics!2D control-panel} +Once you have created a viewport, move your mouse to the viewport and click +with your left mouse button to display a control-panel. +The panel is displayed on the side of the viewport closest to +where you clicked. Each of the buttons which toggle on and off show the +current state of the graph. + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=251, bbury=415]{ps/bookvol1h.ps} +\caption{Two-dimensional control-panel.} +\end{figure} + +\subsubsection{Transformations} +\index{graphics!2D control-panel!transformations} + +Object transformations are executed from the control-panel by mouse-activated +potentiometer windows. +% +\begin{description} +% +\item[Scale:] To scale a graph, click on a mouse button +\index{graphics!2D control-panel!scale} +within the {\bf Scale} window in the upper left corner of the control-panel. +The axes along which the scaling is to occur are indicated by setting the +toggles above the arrow. +With {\tt X On} and {\tt Y On} appearing, both axes are selected and scaling +is uniform. +If either is not selected, for example, if {\tt X Off} appears, scaling is +non-uniform. +% +\item[Translate:] To translate a graph, click the mouse in the +\index{graphics!2D control-panel!translate} +{\bf Translate} window in the direction you wish the graph to move. +This window is located in the upper right corner of the control-panel. +Along the top of the {\bf Translate} window are two buttons for selecting +the direction of translation. +Translation along both coordinate axes results when {\tt X On} and {\tt Y +On} appear or along one axis when one is on, for example, {\tt X On} and +{\tt Y Off} appear. +\end{description} + +\subsubsection{Messages} +\index{graphics!2D control-panel!messages} + +The window directly below the transformation potentiometer windows is +used to display system messages relating to the viewport and the control-panel. +The following format is displayed: \newline +% +\begin{center} +[scaleX, scaleY] $>$graph$<$ [translateX, translateY] \newline +\end{center} +The two values to the left show the scale factor along the {\tt X} and +{\tt Y} coordinate axes. The two values to the right show the distance of +translation from the center in the {\tt X} and {\tt Y} directions. The number +in the center shows which graph in the viewport this data pertains to. +When multiple graphs exist in the same viewport, +the graph must be selected (see ``Multiple Graphs,'' below) in +order for its transformation data to be shown, otherwise the number +is 1. + +\subsubsection{Multiple Graphs} + +\index{graphics!2D control-panel!multiple graphs} +The {\bf Graphs} window contains buttons that allow the placement +of two-di\-men\-sion\-al graphs into one of nine available slots in any other +two-di\-men\-sion\-al viewport. +In the center of the window are numeral buttons from one to nine +that show whether a graph is displayed in the viewport. +Below each number button is a button showing whether a graph +that is present is selected for application of some +transformation. +When the caret symbol is displayed, then the graph in that slot +will be manipulated. +Initially, the graph for which the viewport is created occupies +the first slot, is displayed, and is selected. +% +% +\begin{description} +% +\item[Clear:] The {\bf Clear} button deselects every viewport graph slot. +\index{graphics!2D control-panel!clear} +A graph slot is reselected by selecting the button below its number. +% +\item[Query:] The {\bf Query} button is used to display the scale and +\index{graphics!2D control-panel!query} +translate data for the indicated graph. When this button is selected the +message ``Click on the graph to query'' appears. Select a slot +number button from the {\bf Graphs} window. The scaling factor and translation +offset of the graph are then displayed in the message window. +% +\item[Pick:] The {\bf Pick} button is used to select a graph +\index{graphics!2D control-panel!pick} +to be placed or dropped into the indicated viewport. When this button is +selected, the message ``Click on the graph to pick'' appears. +Click on the slot with the graph number of the desired +graph. The graph information is held waiting for +you to execute a {\bf Drop} in some other graph. +% +\item[Drop:] Once a graph has been picked up using the {\bf Pick} button, +\index{graphics!2D control-panel!drop} +the {\bf Drop} button places it into a new viewport slot. +The message ``Click on the graph to drop'' appears in the message +window when the {\bf Drop} button is selected. +By selecting one of the slot number buttons in the {\bf Graphs} +window, the graph currently being held is dropped into this slot +and displayed. +\end{description} + +\subsubsection{Buttons} +\index{graphics!2D control-panel!buttons} + +% +\begin{description} +% +\item[Axes] turns the coordinate axes on or off. +\index{graphics!2D control-panel!axes} +% +\item[Units] turns the units along the {\tt x} +and {\tt y} axis on or off. +\index{graphics!2D control-panel!units} +% +\item[Box] encloses the area of the viewport graph +in a bounding box, or removes the box if already enclosed. +\index{graphics!2D control-panel!box} +% +\item[Pts] turns on or off the display of points. +\index{graphics!2D control-panel!points} +% +\item[Lines] turns on or off the display +of lines connecting points. +\index{graphics!2D control-panel!lines} +% +\item[PS] writes the current viewport contents to +\index{graphics!2D control-panel!ps} +a file {\bf axiom2d.ps} or to a name specified in the user's {\bf +\index{graphics!.Xdefaults!PostScript file name} +.Xdefaults} file. +\index{file!.Xdefaults @{\bf .Xdefaults}} +The file is placed in the directory from which Axiom or the {\bf +viewalone} program was invoked. +\index{PostScript} +% +\item[Reset] resets the object transformation +characteristics and attributes back to their initial states. +\index{graphics!2D control-panel!reset} +% +\item[Hide] makes the control-panel disappear. +\index{graphics!2D control-panel!hide} +% +\item[Quit] queries whether the current viewport +\index{graphics!2D control-panel!quit} +session should be terminated. +\end{description} + +\subsection{Operations for Two-Dimensional Graphics} +\label{sec:Operations for Two-Dimensional Graphics} + +Here is a summary of useful Axiom operations for two-di\-men\-sion\-al +graphics. +Each operation name is followed by a list of arguments. +Each argument is written as a variable informally named according +to the type of the argument (for example, {\it integer}). +If appropriate, a default value for an argument is given in +parentheses immediately following the name. + +% +\begin{description} +% +\item[{\bf adaptive}]\funArgs{\optArg{boolean\argDef{true}}} +\index{adaptive plotting} +sets or indicates whether graphs are plotted +\index{graphics!set 2D defaults!adaptive} +according to the adaptive refinement algorithm. +% +\item[{\bf axesColorDefault}]\funArgs{\optArg{color\argDef{dark blue()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!axes color} +axes in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf clipPointsDefault}]\funArgs{\optArg{boolean\argDef{false}}} +sets or +indicates whether point clipping is +\index{graphics!set 2D defaults!clip points} +to be applied as the default for graph plots. +% +\item[{\bf drawToScale}]\funArgs{\optArg{boolean\argDef{false}}} +sets or +indicates whether the plot of a graph +\index{graphics!set 2D defaults!to scale} +is ``to scale'' or uses the entire viewport space as the default. +% +\item[{\bf lineColorDefault}]\funArgs{\optArg{color\argDef{pastel yellow()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!line color} +lines or curves in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf maxPoints}]\funArgs{\optArg{integer\argDef{500}}} +sets or indicates +the default maximum number of +\index{graphics!set 2D defaults!max points} +possible points to be used when constructing a two-di\-men\-sion\-al graph. +% +\item[{\bf minPoints}]\funArgs{\optArg{integer\argDef{21}}} +sets or indicates the default minimum number of +\index{graphics!set 2D defaults!min points} +possible points to be used when constructing a two-di\-men\-sion\-al graph. +% +\item[{\bf pointColorDefault}]\funArgs{\optArg{color\argDef{bright red()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!point color} +points in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf pointSizeDefault}]\funArgs{\optArg{integer\argDef{5}}} +sets or indicates the default size of the +\index{graphics!set 2D defaults!point size} +dot used to plot points in a two-di\-men\-sion\-al graph. +% +\item[{\bf screenResolution}]\funArgs{\optArg{integer\argDef{600}}} +sets or indicates the default screen +\index{graphics!set 2D defaults!screen resolution} +resolution constant used in setting the computation limit of adaptively +\index{adaptive plotting} +generated curve plots. +% +\item[{\bf unitsColorDefault}]\funArgs{\optArg{color\argDef{dim green()}}} +sets or indicates the default color of the +\index{graphics!set 2D defaults!units color} +unit labels in a two-di\-men\-sion\-al graph viewport. +% +\item[{\bf viewDefaults}]\funArgs{} +resets the default settings for the following +\index{graphics!set 2D defaults!reset viewport} +attributes: point color, line color, axes color, units color, point size, +viewport upper left-hand corner position, and the viewport size. +% +\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[100,100]}}} +sets or indicates the default position of the +\index{graphics!set 2D defaults!viewport position} +upper left-hand corner of a two-di\-men\-sion\-al viewport, relative to the +display root window. +The upper left-hand corner of the display is considered to be at the +(0, 0) position. +% +\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[200,200]}}} +sets or +indicates the default size in which two +\index{graphics!set 2D defaults!viewport size} +dimensional viewport windows are shown. +It is defined by a width and then a height. +% +\item[{\bf viewWriteAvailable}] +\funArgs{\optArg{list\argDef{["pixmap","bitmap", "postscript", "image"]}}} +indicates the possible file types +\index{graphics!2D defaults!available viewport writes} +that can be created with the \spadfunFrom{write}{TwoDimensionalViewport} function. +% +\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} +sets or indicates the default types of files, in +\index{graphics!set 2D defaults!write viewport} +addition to the {\bf data} file, that are created when a +{\bf write} function is executed on a viewport. +% +\item[{\bf units}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} +turns the units on or off for the graph with index {\it integer}. +% +\item[{\bf axes}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +turns the axes on +\index{graphics!2D commands!axes} +or off for the graph with index {\it integer}. +% +\item[{\bf close}]\funArgs{viewport} +closes {\it viewport}. +\index{graphics!2D commands!close} +% +\item[{\bf connect}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +declares whether lines +\index{graphics!2D commands!connect} +connecting the points are displayed or not. +% +\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} +declares +whether the two-di\-men\-sion\-al control-panel is automatically displayed +or not. +% +\item[{\bf graphs}]\funArgs{viewport} +returns a list +\index{graphics!2D commands!graphs} +describing the state of each graph. +If the graph state is not being used this is shown by {\tt "undefined"}, +otherwise a description of the graph's contents is shown. +% +\item[{\bf graphStates}]\funArgs{viewport} +displays +\index{graphics!2D commands!state of graphs} +a list of all the graph states available for {\it viewport}, giving the +values for every property. +% +\item[{\bf key}]\funArgs{viewport} +returns the process +\index{graphics!2D commands!key} +ID number for {\it viewport}. +% +\item[{\bf move}]\funArgs{viewport, +$integer_{x}$(viewPosDefault), +$integer_{y}$(viewPosDefault)} +moves {\it viewport} on the screen so that the +\index{graphics!2D commands!move} +upper left-hand corner of {\it viewport} is at the position {\it (x,y)}. +% +\item[{\bf options}]\funArgs{\it viewport} +returns a list +\index{graphics!2D commands!options} +of all the {\tt DrawOption}s used by {\it viewport}. +% +\item[{\bf points}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} +specifies whether the graph points for graph {\it integer} are +\index{graphics!2D commands!points} +to be displayed or not. +% +\item[{\bf region}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} +declares whether graph {\it integer} is or is not to be displayed +with a bounding rectangle. +% +\item[{\bf reset}]\funArgs{viewport} +resets all the properties of {\it viewport}. +% +\item[{\bf resize}]\funArgs{viewport, +$integer_{width}$,$integer_{height}$} +\index{graphics!2D commands!resize} +resizes {\it viewport} with a new {\it width} and {\it height}. +% +\item[{\bf scale}]\funArgs{viewport, $integer_{n}$\argDef{1}, +$integer_{x}$\argDef{0.9}, $integer_{y}$\argDef{0.9}} +scales values for the +\index{graphics!2D commands!scale} +{\it x} and {\it y} coordinates of graph {\it n}. +% +\item[{\bf show}]\funArgs{viewport, $integer_{n}$\argDef{1}, +string\argDef{"on"}} +indicates if graph {\it n} is shown or not. +% +\item[{\bf title}]\funArgs{viewport, string\argDef{"Axiom 2D"}} +designates the title for {\it viewport}. +% +\item[{\bf translate}]\funArgs{viewport, +$integer_{n}$\argDef{1}, +$float_{x}$\argDef{0.0}, $float_{y}$\argDef{0.0}} +\index{graphics!2D commands!translate} +causes graph {\it n} to be moved {\it x} and {\it y} units in the respective directions. +% +\item[{\bf write}]\funArgs{viewport, $string_{directory}$, +\optArg{strings}} +if no third argument is given, writes the {\bf data} file onto the directory +with extension {\bf data}. +The third argument can be a single string or a list of strings with some or +all the entries {\tt "pixmap"}, {\tt "bitmap"}, {\tt "postscript"}, and +{\tt "image"}. +\end{description} + +\subsection{Building Two-Dimensional Graphs Manually} +\label{sec:Building Two-Dimensional Graphs Manually} + +In this section we demonstrate how to create two-di\-men\-sion\-al graphs from +lists of points and give an example showing how to read the lists +of points from a file. + +\subsubsection{Creating a Two-Dimensional Viewport from a List of Points} + +Axiom creates lists of points in a two-di\-men\-sion\-al viewport by utilizing +the {\tt GraphImage} and {\tt TwoDimensionalViewport} domains. +The \spadfunFrom{makeGraphImage}{GraphImage} +function takes a list of lists of points parameter, a list of colors for +each point in the graph, a list of colors for each line in the graph, and +a list of sizes for each point in the graph. +% + +The following expressions create a list of lists of points which will be read +by Axiom and made into a two-di\-men\-sion\-al viewport. + +\spadcommand{p1 := point [1,1]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p2 := point [0,1]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p3 := point [0,0]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p4 := point [1,0]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p5 := point [1,.5]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p6 := point [.5,0]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p7 := point [0,0.5]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p8 := point [.5,1]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p9 := point [.25,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p10 := point [.25,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p11 := point [.75,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p12 := point [.75,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +Finally, here is the list. + +\spadcommand{llp := [ [p1,p2], [p2,p3], [p3,p4], [p4,p1], [p5,p6], [p6,p7], [p7,p8], [p8,p5], [p9,p10], [p10,p11], [p11,p12], [p12,p9] ] } +$$ +\left[ +{\left[ {\left[ {1.0}, {1.0} +\right]}, + {\left[ {0.0}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {0.0}, {1.0} +\right]}, + {\left[ {0.0}, {0.0} +\right]} +\right]}, + {\left[ {\left[ {0.0}, {0.0} +\right]}, + {\left[ {1.0}, {0.0} +\right]} +\right]}, +\right. +$$ +$$ +\left. + {\left[ {\left[ {1.0}, {0.0} +\right]}, + {\left[ {1.0}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {1.0}, {0.5} +\right]}, + {\left[ {0.5}, {0.0} +\right]} +\right]}, + {\left[ {\left[ {0.5}, {0.0} +\right]}, + {\left[ {0.0}, {0.5} +\right]} +\right]}, +\right. +$$ +$$ +\left. + {\left[ {\left[ {0.0}, {0.5} +\right]}, + {\left[ {0.5}, {1.0} +\right]} +\right]}, + {\left[ {\left[ {0.5}, {1.0} +\right]}, + {\left[ {1.0}, {0.5} +\right]} +\right]}, + {\left[ {\left[ {0.25}, {0.25} +\right]}, + {\left[ {0.25}, {0.75} +\right]} +\right]}, +\right. +$$ +$$ +\left. + {\left[ {\left[ {0.25}, {0.75} +\right]}, + {\left[ {0.75}, {0.75} +\right]} +\right]}, + {\left[ {\left[ {0.75}, {0.75} +\right]}, + {\left[ {0.75}, {0.25} +\right]} +\right]}, + {\left[ {\left[ {0.75}, {0.25} +\right]}, + {\left[ {0.25}, {0.25} +\right]} +\right]} +\right] +$$ +\returnType{Type: List List Point DoubleFloat} + +Now we set the point sizes for all components of the graph. + +\spadcommand{size1 := 6::PositiveInteger } +$$ +6 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{size2 := 8::PositiveInteger } +$$ +8 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{size3 := 10::PositiveInteger } + +\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] } +$$ +\left[ +6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10 +\right] +$$ +\returnType{Type: List PositiveInteger} + +Here are the colors for the points. + +\spadcommand{pc1 := pastel red() } +$$ +[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{pc2 := dim green() } +$$ +[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Dim \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{pc3 := pastel yellow() } +$$ +[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } +Pastel \mbox{\rm palette} +$$ +\returnType{Type: Palette} + +\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] } +$$ +\left[ +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} } +\right] +$$ +\returnType{Type: List Palette} + +Here are the colors for the lines. + +\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] } +$$ +\left[ +{[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Light \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Bright \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Light \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Bright \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dark \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Pastel \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Light \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Dim \mbox{\rm palette} }, +\right. +$$ +$$ +\left. +{[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} +\mbox{\rm ] from the } Light \mbox{\rm palette} } +\right] +$$ +\returnType{Type: List Palette} + +Now the {\tt GraphImage} is created according to the component +specifications indicated above. + +\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE } + +The \spadfunFrom{makeViewport2D}{TwoDimensionalViewport} function now +creates a {\tt TwoDimensionalViewport} for this graph according to the +list of options specified within the brackets. + +\spadgraph{makeViewport2D(g,[title("Lines")])\$VIEW2D } + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1i.ps} +\caption{The Hand Constructed Line Graph} +\end{figure} + +This example demonstrates the use of the {\tt GraphImage} functions +\spadfunFrom{component}{GraphImage} and \spadfunFrom{appendPoint}{GraphImage} +in adding points to an empty {\tt GraphImage}. + +\spadcommand{g := graphImage()\$GRIMAGE } +$$ +\mbox{\rm Graph with } 0 \mbox{\rm point lists} +$$ +\returnType{Type: GraphImage} + +\spadcommand{p1 := point [0,0]\$(Point DFLOAT) } +$$ +\left[ +{0.0}, {0.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p2 := point [.25,.25]\$(Point DFLOAT) } +$$ +\left[ +{0.25}, {0.25} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p3 := point [.5,.5]\$(Point DFLOAT) } +$$ +\left[ +{0.5}, {0.5} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p4 := point [.75,.75]\$(Point DFLOAT) } +$$ +\left[ +{0.75}, {0.75} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{p5 := point [1,1]\$(Point DFLOAT) } +$$ +\left[ +{1.0}, {1.0} +\right] +$$ +\returnType{Type: Point DoubleFloat} + +\spadcommand{component(g,p1)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{component(g,p2)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p3)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p4)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{appendPoint(g,p5)\$GRIMAGE} +\returnType{Type: Void} + +\spadcommand{g1 := makeGraphImage(g)\$GRIMAGE } +\spadgraph{makeViewport2D(g1,[title("Graph Points")])\$VIEW2D } +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=128, bbury=128]{ps/bookvol1j.ps} +\caption{Graph Points} +\end{figure} + +A list of points can also be made into a {\tt GraphImage} by using +the operation \spadfunFrom{coerce}{GraphImage}. It is equivalent to adding +each point to $g2$ using \spadfunFrom{component}{GraphImage}. + +\spadcommand{g2 := coerce([ [p1],[p2],[p3],[p4],[p5] ])\$GRIMAGE } + +Now, create an empty {\tt TwoDimensionalViewport}. + +\spadcommand{v := viewport2D()\$VIEW2D } + +\spadcommand{options(v,[title("Just Points")])\$VIEW2D } + +Place the graph into the viewport. + +\spadcommand{putGraph(v,g2,1)\$VIEW2D } + +Take a look. + +\spadgraph{makeViewport2D(v)\$VIEW2D } + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=135, bbury=135]{ps/bookvol1k.ps} +\caption{Just Points} +\end{figure} + +\subsubsection{Creating a Two-Dimensional Viewport of a List of Points from a File} + +The following three functions read a list of points from a +file and then draw the points and the connecting lines. The +points are stored in the file in readable form as floating point numbers +(specifically, {\tt DoubleFloat} values) as an alternating +stream of $x$- and $y$-values. For example, +\begin{verbatim} +0.0 0.0 1.0 1.0 2.0 4.0 +3.0 9.0 4.0 16.0 5.0 25.0 +\end{verbatim} + +\begin{verbatim} +drawPoints(lp:List Point DoubleFloat):VIEW2D == + g := graphImage()$GRIMAGE + for p in lp repeat + component(g,p,pointColorDefault(),lineColorDefault(), + pointSizeDefault()) + gi := makeGraphImage(g)$GRIMAGE + makeViewport2D(gi,[title("Points")])$VIEW2D + +drawLines(lp:List Point DoubleFloat):VIEW2D == + g := graphImage()$GRIMAGE + component(g, lp, pointColorDefault(), lineColorDefault(), + pointSizeDefault())$GRIMAGE + gi := makeGraphImage(g)$GRIMAGE + makeViewport2D(gi,[title("Points")])$VIEW2D + +plotData2D(name, title) == + f:File(DFLOAT) := open(name,"input") + lp:LIST(Point DFLOAT) := empty() + while ((x := readIfCan!(f)) case DFLOAT) repeat + y : DFLOAT := read!(f) + lp := cons(point [x,y]$(Point DFLOAT), lp) + lp + close!(f) + drawPoints(lp) + drawLines(lp) +\end{verbatim} +% +This command will actually create the viewport and the graph if +the point data is in the file $"file.data"$. +\begin{verbatim} +plotData2D("file.data", "2D Data Plot") +\end{verbatim} + +\subsection{Appending a Graph to a Viewport Window Containing a Graph} +\label{sec:Appending a Graph to a Viewport Window Containing a Graph} + +This section demonstrates how to append a two-di\-men\-sion\-al graph to a viewport +already containing other graphs. +The default {\bf draw} command places a graph into the first +{\tt GraphImage} slot position of the {\tt TwoDimensionalViewport}. + +We createa a graph in the first slot of a viewport. + +\spadcommand{v1 := draw(sin(x),x=0..2*\%pi) } + +Then we create a second graph. + +\spadcommand{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red()) } + +The operation \spadfunFrom{getGraph}{TwoDimensionalViewport} +retrieves the {\tt GraphImage} $g1$ from the first slot position +in the viewport $v1$. + +\spadcommand{g1 := getGraph(v1,1) } + +Now \spadfunFrom{putGraph}{TwoDimensionalViewport} +places $g1$ into the the second slot position of $v2$. + +\spadcommand{putGraph(v2,g1,2) } + +Display the new {\tt TwoDimensionalViewport} containing both graphs. + +\spadgraph{makeViewport2D(v2) } +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=135, bbury=135]{ps/bookvol1l.ps} +\caption{Two graphs on one viewport} +\end{figure} + +In general you can plot up to 9 graphs on the 2D viewport. Each +graph can be manipulated separately using the 2D control panel. + +The {\bf Pick} and {\bf Drop} buttons on the 2D control panel work +like cut and paste mechanisms in a windowing environment (except that +they don't use the clipboard). So it is possible to pick one graph +and drop it on a different graph. + +\subsection{Plotting 3D Graphs} +There are 3 kinds of three dimensional graphs you can generate: +\begin{enumerate} +\item surfaces defined by a function of two real variables +(page~\pageref{sec:3Dcase1}) +\item space curves and tubes defined by parametric equations +(page~\pageref{sec:3Dcase2}) +\item surfaces defined by parametric equations +(page~\pageref{sec:3Dcase3}) +\end{enumerate} +\subsubsection{Plotting 3D functions of 2 variables} +\label{sec:3Dcase1} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a surface defined by a formula $f(x,y)$ +of two variables $x$ and $y$ is: +% +\begin{center} +{\tt draw(f(x,y), x = a..b, y = c..d, {\it options})} +\end{center} +where $a..b$ and $c..d$ define the range of $x$ +and $y$, and where {\it options} prescribes zero or more +options as described in \ref{sec:Three-Dimensional Options} +on page~\pageref{sec:Three-Dimensional Options}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving a function $f$ is also +available.\\ +} +The simplest way to plot a function of two variables is to use a formula. +With formulas you always precede the range specifications with +the variable name and an {\tt =} sign. + +\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)} + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1m.ps} +\caption{$cos(x*y),x=-3..3,y=-3..3$} +\end{figure} + + +\subsubsection{Plotting 3D parametric space curves} +\label{sec:3Dcase2} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a three-di\-men\-sion\-al space curve defined by +parametric formulas $x = f(t)$, $y = g(t)$, and +$z = h(t)$ is: +% +\begin{center} +{\tt draw(curve(f(t),g(t),h(t)), t = a..b, {\it options})} +\end{center} +where $a..b$ defines the range of the independent variable +$t$, and where {\it options} prescribes zero or more options +as described in \ref{sec:Three-Dimensional Options} +on page~\pageref{sec:Three-Dimensional Options}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving functions $f$, $g$ and +$h$ is also available.\\ +} + +If you use explicit formulas to draw a space curve, always precede +the range specification with the variable name and an +{\tt =} sign. + +\spadgraph{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)} + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1n.ps} +\caption{$curve(5*cos(t), 5*sin(t),t), t=-12..12$} +\end{figure} + + +\subsubsection{Plotting 3D parametric surfaces} +\label{sec:3Dcase3} +\boxed{4.6in}{ +\vskip 0.1cm +The general format for drawing a three-di\-men\-sion\-al graph defined by +parametric formulas $x = f(u,v)$, $y = g(u,v)$, +and $z = h(u,v)$ is: +% +\begin{center} +{\tt draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d, {\it options})} +\end{center} +where $a..b$ and $c..d$ define the range of the +independent variables $u$ and $v$, and where +{\it options} prescribes zero or more options as described in +\ref{sec:Three-Dimensional Options} on +page~\pageref{sec:Three-Dimensional Options}. +An example of an option is $title == "Title of Graph".$ +An alternative format involving functions $f$, $g$ and +$h$ is also available.\\ +} +This example draws a graph of a surface plotted using the +parabolic cylindrical coordinate system option. +\index{coordinate system!parabolic cylindrical} +The values of the functions supplied to {\bf surface} are +\index{parabolic cylindrical coordinate system} +interpreted in coordinates as given by a {\tt coordinates} option, +here as parabolic cylindrical coordinates. + +\spadgraph{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)} + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=215, bbury=215]{ps/bookvol1o.ps} +\caption{$surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\pi$} +\end{figure} + +\subsection{Three-Dimensional Options} +\label{sec:Three-Dimensional Options} +\index{graphics!3D options} +The {\bf draw} commands optionally take an optional list of options such +as {\tt coordinates} as shown in the last example. +Each option is given by the syntax: $name$ {\tt ==} $value$. +Here is a list of the available options in the order that they are +described below: + +\begin{tabular}{ll} +title & The {\tt title} option gives a title to the graph\\ + & {\tt title == "Title of Graph"}\\ +coordinates & The {\tt coordinates} option indicates the coordinate system\\ + & in which the graph is plotted. This can be one of:\\ + & {\bf bipolar}, {\bf bipolarCylindrical}, {\bf cartesian},\\ + & {\bf conical}, {\bf cylindrical}, {\bf elliptic},\\ + & {\bf ellipticCylindrical}, {\bf oblateSpheroidal}, \\ + & {\bf parabolic}, {\bf parabolicCylindrical}, {\bf paraboloidal}\\ + & {\bf polar}, {\bf prolateSpheroidal}, {\bf spherical}, and\\ + & {\bf toroidal}\\ + & {\tt coordinates == polar}\\ +var1Steps & The {\tt var1Steps} option specifies the number of intervals\\ + & to divide a surface plot for the first parameter\\ + & {\tt var1Steps == 30}\\ +var2Steps & The {\tt var1Steps} option specifies the number of intervals\\ + & to divide a surface plot for the second parameter\\ + & {\tt var2Steps == 30}\\ +style & The {\tt style} determines which of four rendering algorithms\\ + & is used for the graph. The choices are {\bf wireMesh},\\ + & {\bf solid}, {\bf shade}, {\bf smooth}\\ + & {\tt style == "smooth"}\\ +colorFunction & The {\tt colorFunction} names a function that will be called\\ + & to determine the color of each point. If we have the function\\ + & {\tt color2(u,v) == u**2 - v**2} we can call it with\\ + & {\tt colorFunction == color2}\\ +tubeRadius & The {\tt tubeRadius} option specifies the radius of the tube\\ + & that encircles the specified space curve.\\ + & {\tt tubeRadius == .3}\\ +tubePoints & The {\tt tubePoints} option specifies the number of vertices\\ + & defining the polygon that is used to create a tube around the\\ + & specified space curve. The larger this number is the more\\ + & cylindrical the tube becomes.\\ + & {\tt tubePoints == 3}\\ +space & The {\tt space} option lets you build multiple graphs in\\ + & three space. To use this option, first create an empty \\ + & three-space object calling {\bf create3Space} as in:\\ + & {\tt s:=create3Space()\$(ThreeSpace SF)}\\ + & and then use the space option thereafter.\\ + & {\tt space == s} +\end{tabular} + +\subsection{Three-Dimensional Control-Panel} +\label{sec:Three-Dimensional Control-Panel} + +\index{graphics!3D control-panel} +Once you have created a viewport, move your mouse to the viewport +and click with your left mouse button. +This displays a control-panel on the side of the viewport +that is closest to where you clicked. + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=165, bbury=215]{ps/bookvol1p.ps} +\caption{Three-dimensional control-panel.} +\end{figure} + +\subsubsection{Transformations} + +We recommend you first select the {\bf Bounds} button while +\index{graphics!3D control-panel!transformations} +executing transformations since the bounding box displayed +indicates the object's position as it changes. +% +\begin{description} +% +\item[Rotate:] A rotation transformation occurs by clicking the mouse +\index{graphics!3D control-panel!rotate} +within the {\bf Rotate} window in the upper left corner of the +control-panel. +The rotation is computed in spherical coordinates, using the +horizontal mouse position to increment or decrement the value of +the longitudinal angle $\theta$ within the +range of 0 to 2$\pi$ and the vertical mouse position +to increment or decrement the value of the latitudinal angle +$\phi$ within the range of -$\pi$ +to $\pi$. +The active mode of rotation is displayed in green on a color +monitor or in clear text on a black and white monitor, while the +inactive mode is displayed in red for color display or a mottled +pattern for black and white. +% +\begin{description} +% +\item[origin:] The {\bf origin} button indicates that the +rotation is to occur with respect to the origin of the viewing space, that is +indicated by the axes. +% +\item[object:] The {\bf object} button indicates that the +rotation is to occur with respect to the center of volume of the object, +independent of the axes' origin position. +\end{description} +% +\item[Scale:] A scaling transformation occurs by clicking the mouse +\index{graphics!3D control-panel!scale} +within the {\bf Scale} window in the upper center of the +control-panel, containing a zoom arrow. +The axes along which the scaling is to occur are indicated by +selecting the appropriate button above the zoom arrow window. +The selected axes are displayed in green on a color monitor or in +clear text on a black and white monitor, while the unselected axes +are displayed in red for a color display or a mottled pattern for +black and white. +% +\begin{description} +% +\item[uniform:] Uniform scaling along the {\tt x}, {\tt y} +and {\tt z} axes occurs when all the axes buttons are selected. +% +\item[non-uniform:] If any of the axes buttons are +not selected, non-uniform scaling occurs, that is, scaling occurs only in the +direction of the axes that are selected. +\end{description} +% +\item[Translate:] Translation occurs by indicating with the mouse in the +\index{graphics!3D control-panel!translate} +{\bf Translate} window the direction you want the graph to move. +This window is located in the upper right corner of the +control-panel and contains a potentiometer with crossed arrows +pointing up, down, left and right. +Along the top of the {\bf Translate} window are three buttons +({\bf XY}, +{\bf XZ}, and {\bf YZ}) indicating the three orthographic projection planes. +Each orientates the group as a view into that plane. +Any translation of the graph occurs only along this plane. +\end{description} + +\subsubsection{Messages} + +\index{graphics!3D control-panel!messages} + +The window directly below the potentiometer windows for transformations is +used to display system messages relating to the viewport, the control-panel +and the current graph displaying status. + +\subsubsection{Colormap} + +\index{graphics!3D control-panel!color map} + +Directly below the message window is the colormap range indicator +window. +\index{colormap} +The Axiom Colormap shows a sampling of the spectrum from +which hues can be drawn to represent the colors of a surface. +The Colormap is composed of five shades for each of the hues along +this spectrum. +By moving the markers above and below the Colormap, the range of +hues that are used to color the existing surface are set. +The bottom marker shows the hue for the low end of the color range +and the top marker shows the hue for the upper end of the range. +Setting the bottom and top markers at the same hue results in +monochromatic smooth shading of the graph when {\bf Smooth} mode is selected. +At each end of the Colormap are {\bf +} and {\bf -} buttons. +When clicked on, these increment or decrement the top or bottom +marker. + +\subsubsection{Buttons} +\index{graphics!3D control-panel!buttons} + +Below the Colormap window and to the left are located various +buttons that determine the characteristics of a graph. +The buttons along the bottom and right hand side all have special +meanings; the remaining buttons in the first row indicate the mode +or style used to display the graph. +The second row are toggles that turn on or off a property of the +graph. +On a color monitor, the property is on if green (clear text, on a +monochrome monitor) and off if red (mottled pattern, on a +monochrome monitor). +Here is a list of their functions. +% +\begin{description} +% +\item[Wire] displays surface and tube plots as a +\index{graphics!3D control-panel!wire} +wireframe image in a single color (blue) with no hidden surfaces removed, +or displays space curve plots in colors based upon their parametric variables. +This is the fastest mode for displaying a graph. +This is very useful when you +want to find a good orientation of your graph. +% +\item[Solid] displays the graph with hidden +\index{graphics!3D control-panel!solid} +surfaces removed, drawing each polygon beginning with the furthest +from the viewer. +The edges of the polygons are displayed in the hues specified by +the range in the Colormap window. +% +\item[Shade] displays the graph with hidden +\index{graphics!3D control-panel!shade} +surfaces removed and with the polygons shaded, drawing each +polygon beginning with the furthest from the viewer. +Polygons are shaded in the hues specified by the range in the +Colormap window using the Phong illumination model. +\index{Phong!illumination model} +% +\item[Smooth] displays the graph using a +\index{graphics!3D control-panel!smooth} +renderer that computes the graph one line at a time. +The location and color of the graph at each visible point on the +screen are determined and displayed using the Phong illumination +\index{Phong!illumination model} +model. +Smooth shading is done in one of two ways, depending on the range +selected in the colormap window and the number of colors available +from the hardware and/or window manager. +When the top and bottom markers of the colormap range are set to +different hues, the graph is rendered by dithering between the +\index{dithering} +transitions in color hue. +When the top and bottom markers of the colormap range are set to +the same hue, the graph is rendered using the Phong smooth shading +model. +\index{Phong!smooth shading model} +However, if enough colors cannot be allocated for this purpose, +the renderer reverts to the color dithering method until a +sufficient color supply is available. +For this reason, it may not be possible to render multiple Phong +smooth shaded graphs at the same time on some systems. +% +\item[Bounds] encloses the entire volume of the +viewgraph within a bounding box, or removes the box if previously selected. +\index{graphics!3D control-panel!bounds} +The region that encloses the entire volume of the viewport graph is displayed. +% +\item[Axes] displays Cartesian +\index{graphics!3D control-panel!axes} +coordinate axes of the space, or turns them off if previously selected. +% +\item[Outline] causes +\index{graphics!3D control-panel!outline} +quadrilateral polygons forming the graph surface to be outlined in black when +the graph is displayed in {\bf Shade} mode. +% +\item[BW] converts a color viewport to black and white, or vice-versa. +\index{graphics!3D control-panel!bw} +When this button is selected the +control-panel and viewport switch to an immutable colormap composed of a range +of grey scale patterns or tiles that are used wherever shading is necessary. +% +\item[Light] takes you to a control-panel described below. +% +\item[ViewVolume] takes you to another control-panel as described below. +\index{graphics!3D control-panel!save} +% +\item[Save] creates a menu of the possible file types that can +be written using the control-panel. +The {\bf Exit} button leaves the save menu. +The {\bf Pixmap} button writes an Axiom pixmap of +\index{graphics!3D control-panel!pixmap} +the current viewport contents. The file is called {\bf axiom3d.pixmap} and is +located in the directory from which Axiom or {\bf viewalone} was +started. +The {\bf PS} button writes the current viewport contents to +\index{graphics!3D control-panel!ps} +PostScript output rather than to the viewport window. +By default the file is called {\bf axiom3d.ps}; however, if a file +\index{file!.Xdefaults @{\bf .Xdefaults}} +name is specified in the user's {\bf .Xdefaults} file it is +\index{graphics!.Xdefaults!PostScript file name} +used. +The file is placed in the directory from which the Axiom or +{\bf viewalone} session was begun. +See also the \spadfunFrom{write}{ThreeDimensionalViewport} +function. +\index{PostScript} +% +\item[Reset] returns the object transformation +\index{graphics!3D control-panel!reset} +characteristics back to their initial states. +% +\item[Hide] causes the control-panel for the +\index{graphics!3D control-panel!hide} +corresponding viewport to disappear from the screen. +% +\item[Quit] queries whether the current viewport +\index{graphics!3D control-panel!quit} +session should be terminated. +\end{description} + +\subsubsection{Light} + +\index{graphics!3D control-panel!light} + +%>>>\begin{figure}[htbp] +%>>>\begin{picture}(183,252)(-125,0) +%>>>\special{psfile=ps/3dlight.ps} +%>>>\end{picture} +%>>>\caption{Three-Dimensional Lighting Panel.} +%>>>\end{figure} + +The {\bf Light} button changes the control-panel into the +{\bf Lighting Control-Panel}. At the top of this panel, the three axes +are shown with the same orientation as the object. A light vector from +the origin of the axes shows the current position of the light source +relative to the object. At the bottom of the panel is an {\bf Abort} +button that cancels any changes to the lighting that were made, and a +{\bf Return} button that carries out the current set of lighting changes +on the graph. +% +\begin{description} +% +\item[XY:] The {\bf XY} lighting axes window is below the +\index{graphics!3D control-panel!move xy} +{\bf Lighting Control-Panel} title and to the left. +This changes the light vector within the {\bf XY} view plane. +% +\item[Z:] The {\bf Z} lighting axis window is below the +\index{graphics!3D control-panel!move z} +{\bf Lighting Control-Panel} title and in the center. This +changes the {\bf Z} +location of the light vector. +% +\item[Intensity:] +Below the {\bf Lighting Control-Panel} title +\index{graphics!3D control-panel!intensity} +and to the right is the light intensity meter. +Moving the intensity indicator down decreases the amount of +light emitted from the light source. +When the indicator is at the top of the meter the light source is +emitting at 100\% intensity. +At the bottom of the meter the light source is emitting at a level +slightly above ambient lighting. +\end{description} + +\subsubsection{View Volume} + +\index{graphics!3D control-panel!view volume} + +The {\bf View Volume} button changes the control-panel into +the {\bf Viewing Volume Panel}. +At the bottom of the viewing panel is an {\bf Abort} button that +cancels any changes to the viewing volume that were made and a +{\it Return} button that carries out the current set of +viewing changes to the graph. +% +%>>>\begin{figure}[htbp] +%>>>\begin{picture}(183,252)(-125,0) +%>>>\special{psfile=ps/3dvolume.ps} +%>>>\end{picture} +%>>>\caption{Three-Dimensional Volume Panel.} +%>>>\end{figure} + +\begin{description} + +\item[Eye Reference:] At the top of this panel is the +\index{graphics!3D control-panel!eye reference} +{\bf Eye Reference} window. +It shows a planar projection of the viewing pyramid from the eye +of the viewer relative to the location of the object. +This has a bounding region represented by the rectangle on the +left. +Below the object rectangle is the {\bf Hither} window. +By moving the slider in this window the hither clipping plane sets +\index{hither clipping plane} +the front of the view volume. +As a result of this depth clipping all points of the object closer +to the eye than this hither plane are not shown. +The {\bf Eye Distance} slider to the right of the {\bf Hither} +slider is used to change the degree of perspective in the image. +% +\item[Clip Volume:] The {\bf Clip Volume} window is at the +\index{graphics!3D control-panel!clip volume} +bottom of the {\bf Viewing Volume Panel}. +On the right is a {\bf Settings} menu. +In this menu are buttons to select viewing attributes. +Selecting the {\bf Perspective} button computes the image using +perspective projection. +\index{graphics!3D control-panel!perspective} +The {\bf Show Region} button indicates whether the clipping region +of the +\index{graphics!3D control-panel!show clip region} +volume is to be drawn in the viewport and the {\bf Clipping On} +button shows whether the view volume clipping is to be in effect +when the image +\index{graphics!3D control-panel!clipping on} +is drawn. +The left side of the {\bf Clip Volume} window shows the clipping +\index{graphics!3D control-panel!clip volume} +boundary of the graph. +Moving the knobs along the {\bf X}, {\bf Y}, and {\bf Z} sliders +adjusts the volume of the clipping region accordingly. +\end{description} + +\subsection{Operations for Three-Dimensional Graphics} +\label{sec:Operations for Three-Dimensional Graphics} + + +Here is a summary of useful Axiom operations for three-di\-men\-sion\-al +graphics. +Each operation name is followed by a list of arguments. +Each argument is written as a variable informally named according +to the type of the argument (for example, {\it integer}). +If appropriate, a default value for an argument is given in +parentheses immediately following the name. + +% +\bgroup\hbadness = 10001\sloppy +\begin{description} +% +\item[{\bf adaptive3D?}]\funArgs{} +tests whether space curves are to be plotted +\index{graphics!plot3d defaults!adaptive} +according to the +\index{adaptive plotting} +adaptive refinement algorithm. + +% +\item[{\bf axes}]\funArgs{viewport, string\argDef{"on"}} +turns the axes on and off. +\index{graphics!3D commands!axes} + +% +\item[{\bf close}]\funArgs{viewport} +closes the viewport. +\index{graphics!3D commands!close} + +% +\item[{\bf colorDef}]\funArgs{viewport, +$\hbox{\it color}_{1}$\argDef{1}, $\hbox{\it color}_{2}$\argDef{27}} +sets the colormap +\index{graphics!3D commands!define color} +range to be from +$\hbox{\it color}_{1}$ to $\hbox{\it color}_{2}$. + +% +\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} +declares whether the +\index{graphics!3D commands!control-panel} +control-panel for the viewport is to be displayed or not. + +% +\item[{\bf diagonals}]\funArgs{viewport, string\argDef{"off"}} +declares whether the +\index{graphics!3D commands!diagonals} +polygon outline includes the diagonals or not. + +% +\item[{\bf drawStyle}]\funArgs{viewport, style} +selects which of four drawing styles +\index{graphics!3D commands!drawing style} +are used: {\tt "wireMesh", "solid", "shade",} or {\tt "smooth".} + +% +\item[{\bf eyeDistance}]\funArgs{viewport,float\argDef{500}} +sets the distance of the eye from the origin of the object +\index{graphics!3D commands!eye distance} +for use in the \spadfunFrom{perspective}{ThreeDimensionalViewport}. + +% +\item[{\bf key}]\funArgs{viewport} +returns the operating +\index{graphics!3D commands!key} +system process ID number for the viewport. + +% +\item[{\bf lighting}]\funArgs{viewport, +$float_{x}$\argDef{-0.5}, +$float_{y}$\argDef{0.5}, $float_{z}$\argDef{0.5}} +sets the Cartesian +\index{graphics!3D commands!lighting} +coordinates of the light source. + +% +\item[{\bf modifyPointData}]\funArgs{viewport,integer,point} +replaces the coordinates of the point with +\index{graphics!3D commands!modify point data} +the index {\it integer} with {\it point}. + +% +\item[{\bf move}]\funArgs{viewport, +$integer_{x}$\argDef{viewPosDefault}, +$integer_{y}$\argDef{viewPosDefault}} +moves the upper +\index{graphics!3D commands!move} +left-hand corner of the viewport to screen position +\allowbreak +({\small $integer_{x}$, $integer_{y}$}). + +% +\item[{\bf options}]\funArgs{viewport} +returns a list of all current draw options. + +% +\item[{\bf outlineRender}]\funArgs{viewport, string\argDef{"off"}} +turns polygon outlining +\index{graphics!3D commands!outline} +off or on when drawing in {\tt "shade"} mode. + +% +\item[{\bf perspective}]\funArgs{viewport, string\argDef{"on"}} +turns perspective +\index{graphics!3D commands!perspective} +viewing on and off. + +% +\item[{\bf reset}]\funArgs{viewport} +resets the attributes of a viewport to their +\index{graphics!3D commands!reset} +initial settings. + +% +\item[{\bf resize}]\funArgs{viewport, +$integer_{width}$ \argDef{viewSizeDefault}, +$integer_{height}$ \argDef{viewSizeDefault}} +resets the width and height +\index{graphics!3D commands!resize} +values for a viewport. + +% +\item[{\bf rotate}]\funArgs{viewport, +$number_{\theta}$\argDef{viewThetaDefapult}, +$number_{\phi}$\argDef{viewPhiDefault}} +rotates the viewport by rotation angles for longitude +({\it $\theta$}) and +latitude ({\it $\phi$}). +Angles designate radians if given as floats, or degrees if given +\index{graphics!3D commands!rotate} +as integers. + +% +\item[{\bf setAdaptive3D}]\funArgs{boolean\argDef{true}} +sets whether space curves are to be plotted +\index{graphics!plot3d defaults!set adaptive} +according to the adaptive +\index{adaptive plotting} +refinement algorithm. + +% +\item[{\bf setMaxPoints3D}]\funArgs{integer\argDef{1000}} + sets the default maximum number of possible +\index{graphics!plot3d defaults!set max points} +points to be used when constructing a three-di\-men\-sion\-al space curve. + +% +\item[{\bf setMinPoints3D}]\funArgs{integer\argDef{49}} +sets the default minimum number of possible +\index{graphics!plot3d defaults!set min points} +points to be used when constructing a three-di\-men\-sion\-al space curve. + +% +\item[{\bf setScreenResolution3D}]\funArgs{integer\argDef{49}} +sets the default screen resolution constant +\index{graphics!plot3d defaults!set screen resolution} +used in setting the computation limit of adaptively +\index{adaptive plotting} +generated three-di\-men\-sion\-al space curve plots. + +% +\item[{\bf showRegion}]\funArgs{viewport, string\argDef{"off"}} +declares whether the bounding +\index{graphics!3D commands!showRegion} +box of a graph is shown or not. +% +\item[{\bf subspace}]\funArgs{viewport} +returns the space component. +% +\item[{\bf subspace}]\funArgs{viewport, subspace} +resets the space component +\index{graphics!3D commands!subspace} +to {\it subspace}. + +% +\item[{\bf title}]\funArgs{viewport, string} +gives the viewport the +\index{graphics!3D commands!title} +title {\it string}. + +% +\item[{\bf translate}]\funArgs{viewport, +$float_{x}$\argDef{viewDeltaXDefault}, +$float_{y}$\argDef{viewDeltaYDefault}} +translates +\index{graphics!3D commands!translate} +the object horizontally and vertically relative to the center of the viewport. + +% +\item[{\bf intensity}]\funArgs{viewport,float\argDef{1.0}} +resets the intensity {\it I} of the light source, +\index{graphics!3D commands!intensity} +$0 \le I \le 1.$ + +% +\item[{\bf tubePointsDefault}]\funArgs{\optArg{integer\argDef{6}}} +sets or indicates the default number of +\index{graphics!3D defaults!tube points} +vertices defining the polygon that is used to create a tube around +a space curve. + +% +\item[{\bf tubeRadiusDefault}]\funArgs{\optArg{float\argDef{0.5}}} +sets or indicates the default radius of +\index{graphics!3D defaults!tube radius} +the tube that encircles a space curve. + +% +\item[{\bf var1StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} +sets or indicates the default number of +\index{graphics!3D defaults!var1 steps} +increments into which the grid defining a surface plot is subdivided with +respect to the first parameter declared in the surface function. + +% +\item[{\bf var2StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} +sets or indicates the default number of +\index{graphics!3D defaults!var2 steps} +increments into which the grid defining a surface plot is subdivided with +respect to the second parameter declared in the surface function. + +% +\item[{\bf viewDefaults}]\funArgs{{\tt [}$integer_{point}$, +$integer_{line}$, $integer_{axes}$, +$integer_{units}$, $float_{point}$, +\allowbreak$list_{position}$, +$list_{size}${\tt ]}} +resets the default settings for the +\index{graphics!3D defaults!reset viewport defaults} +point color, line color, axes color, units color, point size, +viewport upper left-hand corner position, and the viewport size. + +% +\item[{\bf viewDeltaXDefault}]\funArgs{\optArg{float\argDef{0}}} +resets the default horizontal offset +\index{graphics!3D commands!deltaX default} +from the center of the viewport, +or returns the current default offset if no argument is given. + +% +\item[{\bf viewDeltaYDefault}]\funArgs{\optArg{float\argDef{0}}} +resets the default vertical offset +\index{graphics!3D commands!deltaY default} +from the center of the viewport, +or returns the current default offset if no argument is given. + +% +\item[{\bf viewPhiDefault}]\funArgs{\optArg{float\argDef{-$\pi$/4}}} +resets the default latitudinal view angle, +or returns the current default angle if no argument is given. +\index{graphics!3D commands!phi default} +$\phi$ is set to this value. + +% +\item[{\bf viewpoint}]\funArgs{viewport, $float_{x}$, +$float_{y}$, $float_{z}$} +sets the viewing position in Cartesian coordinates. + +% +\item[{\bf viewpoint}]\funArgs{viewport, +$float_{\theta}$, +$Float_{\phi}$} +sets the viewing position in spherical coordinates. + +% +\item[{\bf viewpoint}]\funArgs{viewport, +$Float_{\theta}$, +$Float_{\phi}$, +$Float_{scaleFactor}$, +$Float_{xOffset}$, $Float_{yOffset}$} +sets the viewing position in spherical coordinates, +the scale factor, and offsets. +\index{graphics!3D commands!viewpoint} +$\theta$ (longitude) and +$\phi$ (latitude) are in radians. + +% +\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[0,0]}}} +sets or indicates the position of the upper +\index{graphics!3D defaults!viewport position} +left-hand corner of a two-di\-men\-sion\-al viewport, +relative to the display root +window (the upper left-hand corner of the display is $[0, 0]$). + +% +\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[400,400]}}} +sets or indicates the width and height dimensions +\index{graphics!3D defaults!viewport size} +of a viewport. + +% +\item[{\bf viewThetaDefault}]\funArgs{\optArg{float\argDef{$\pi$/4}}} +resets the default longitudinal view angle, +or returns the current default angle if no argument is given. +\index{graphics!3D commands!theta default} +When a parameter is specified, the default longitudinal view angle +$\theta$ is set to this value. + +% +\item[{\bf viewWriteAvailable}]\funArgs{\optArg{list\argDef{["pixmap", +"bitmap", "postscript", "image"]}}} +indicates the possible file types +\index{graphics!3D defaults!available viewport writes} +that can be created with the +\spadfunFrom{write}{ThreeDimensionalViewport} function. + +% +\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} +sets or indicates the default types of files +that are created in addition to the {\bf data} file when a +\spadfunFrom{write}{ThreeDimensionalViewport} command +\index{graphics!3D defaults!viewport writes} +is executed on a viewport. + +% +\item[{\bf viewScaleDefault}]\funArgs{\optArg{float}} +sets the default scaling factor, or returns +\index{graphics!3D commands!scale default} +the current factor if no argument is given. + +% +\item[{\bf write}]\funArgs{viewport, directory, \optArg{option}} +writes the file {\bf data} for {\it viewport} +in the directory {\it directory}. +An optional third argument specifies a file type (one of {\tt +pixmap}, {\tt bitmap}, {\tt postscript}, or {\tt image}), or a +list of file types. +An additional file is written for each file type listed. + +% +\item[{\bf scale}]\funArgs{viewport, float\argDef{2.5}} +specifies the scaling factor. +\index{graphics!3D commands!scale} +\index{scaling graphs} +\end{description} +\egroup + +\subsection{Customization using .Xdefaults} +\label{sec:Customization using .Xdefaults} + +\index{graphics!.Xdefaults} + +Both the two-di\-men\-sion\-al and +three-di\-men\-sion\-al drawing facilities consult +the {\bf .Xdefaults} file for various defaults. +\index{file!.Xdefaults @{\bf .Xdefaults}} +The list of defaults that are recognized by the graphing routines +is discussed in this section. +These defaults are preceded by {\tt Axiom.3D.} +for three-di\-men\-sion\-al viewport defaults, {\tt Axiom.2D.} +for two-di\-men\-sion\-al viewport defaults, or {\tt Axiom*} (no dot) for +those defaults that are acceptable to either viewport type. + +% +\begin{description} +% +\item[{\tt Axiom*buttonFont:\ \it font}] \ \newline +This indicates which +\index{graphics!.Xdefaults!button font} +font type is used for the button text on the control-panel. +{\bf Rom11} +% +\item[{\tt Axiom.2D.graphFont:\ \it font}] \quad (2D only) \newline +This indicates +\index{graphics!.Xdefaults!graph number font} +which font type is used for displaying the graph numbers and +slots in the {\bf Graphs} section of the two-di\-men\-sion\-al control-panel. +{\bf Rom22} +% +\item[{\tt Axiom.3D.headerFont:\ \it font}] \ \newline +This indicates which +\index{graphics!.Xdefaults!graph label font} +font type is used for the axes labels and potentiometer +header names on three-di\-men\-sion\-al viewport windows. +This is also used for two-di\-men\-sion\-al control-panels for indicating +which font type is used for potentionmeter header names and +multiple graph title headers. +%for example, {\tt Axiom.2D.headerFont: 8x13}. +{\bf Itl14} +% +\item[{\tt Axiom*inverse:\ \it switch}] \ \newline +This indicates whether the +\index{graphics!.Xdefaults!inverting background} +background color is to be inverted from white to black. +If {\tt on}, the graph viewports use black as the background +color. +If {\tt off} or no declaration is made, the graph viewports use a +white background. +{\bf off} +% +\item[{\tt Axiom.3D.lightingFont:\ \it font}] \quad (3D only) \newline +This indicates which font type is used for the {\bf x}, +\index{graphics!.Xdefaults!lighting font} +{\bf y}, and {\bf z} labels of the two lighting axes potentiometers, and for +the {\bf Intensity} title on the lighting control-panel. +{\bf Rom10} +% +\item[{\tt Axiom.2D.messageFont, Axiom.3D.messageFont:\ \it font}] \ \newline +These indicate the font type +\index{graphics!.Xdefaults!message font} +to be used for the text in the control-panel message window. +{\bf Rom14} +% +\item[{\tt Axiom*monochrome:\ \it switch}] \ \newline +This indicates whether the +\index{graphics!.Xdefaults!monochrome} +graph viewports are to be displayed as if the monitor is black and +white, that is, a 1 bit plane. +If {\tt on} is specified, the viewport display is black and white. +If {\tt off} is specified, or no declaration for this default is +given, the viewports are displayed in the normal fashion for the +monitor in use. +{\bf off} +% +\item[{\tt Axiom.2D.postScript:\ \it filename}] \ \newline +This specifies +\index{graphics!.Xdefaults!PostScript file name} +the name of the file that is generated when a 2D PostScript graph +\index{PostScript} +is saved. +{\bf axiom2d.ps} +% +\item[{\tt Axiom.3D.postScript:\ \it filename}] \ \newline +This specifies +\index{graphics!.Xdefaults!PostScript file name} +the name of the file that is generated when a 3D PostScript graph +\index{PostScript} +is saved. +{\bf axiom3d.ps} +% +\item[{\tt Axiom*titleFont \it font}] \ \newline +This +\index{graphics!.Xdefaults!title font} +indicates which font type is used +for the title text and, for three-di\-men\-sion\-al graphs, +in the lighting and viewing-volume control-panel windows. +\index{graphics!Xdefaults!2d} +{\bf Rom14} +% +\item[{\tt Axiom.2D.unitFont:\ \it font}] \quad (2D only) \newline +This indicates +\index{graphics!.Xdefaults!unit label font} +which font type is used for displaying the unit labels on +two-di\-men\-sion\-al viewport graphs. +{\bf 6x10} +% +\item[{\tt Axiom.3D.volumeFont:\ \it font}] \quad (3D only) \newline +This indicates which font type is used for the {\bf x}, +\index{graphics!.Xdefaults!volume label font} +{\bf y}, and {\bf z} labels of the clipping region sliders; for the +{\bf Perspective}, {\bf Show Region}, and {\bf Clipping On} buttons under +{\bf Settings}, and above the windows for the {\bf Hither} and +{\bf Eye Distance} sliders in the {\bf Viewing Volume Panel} of the +three-di\-men\-sion\-al control-panel. +{\bf Rom8} +\end{description} + + +\chapter{Using Types and Modes} +\label{sec:Using Types and Modes} + +\index{type} +\index{mode} +In this chapter we look at the key notion of {\it type} and its +generalization {\it mode}. We show that every Axiom object has a type +that determines what you can do with the object. In particular, we +explain how to use types to call specific functions from particular +parts of the library and how types and modes can be used to create new +objects from old. We also look at {\tt Record} \index{Record} and +{\tt Union} \index{Union} types and the special type {\tt Any} +\index{Any}. Finally, we give you an idea of how Axiom manipulates +types and modes internally to resolve ambiguities. + + +\section{The Basic Idea} +\label{sec:The Basic Idea} + +The Axiom world deals with many kinds of objects. There are +mathematical objects such as numbers and polynomials, data structure +objects such as lists and arrays, and graphics objects such as points +and graphic images. Functions are objects too. + +\index{domain} +Axiom organizes objects using the notion of domain of computation, or +simply {\it domain}. Each domain denotes a class of objects. The +class of objects it denotes is usually given by the name of the +domain: {\tt Integer} \index{Integer} +for the integers, {\tt Float} \index{Float} +for floating-point +numbers, and so on. The convention is that the first letter of a +domain name is capitalized. Similarly, the domain +{\tt Polynomial(Integer)} \index{Polynomial(Integer)} +denotes ``polynomials with integer +coefficients.'' Also, {\tt Matrix(Float)} \index{Matrix(Float)} +denotes ``matrices with +floating-point entries.'' + +Every basic Axiom object belongs to a unique domain. The integer $3$ +belongs to the domain {\tt Integer} and the polynomial $x + 3$ belongs +to the domain {\tt Polynomial(Integer)}. The domain of an object is +also called its {\it type}. Thus we speak of ``the type +{\tt Integer}'' and ``the type {\tt Polynomial(Integer)}.'' + +After an Axiom computation, the type is displayed toward the +right-hand side of the page (or screen). +\spadcommand{-3} +$$ +-3 +$$ +\returnType{Type: Integer} + +Here we create a rational number but it looks like the last result. +The type however tells you it is different. You cannot identify the +type of an object by how Axiom displays the object. +\spadcommand{-3/1} +$$ +-3 +$$ +\returnType{Type: Fraction Integer} + +When a computation produces a result of a simpler type, Axiom leaves +the type unsimplified. Thus no information is lost. +\spadcommand{x + 3 - x} +$$ +3 +$$ +\returnType{Type: Polynomial Integer} + +\index{retract} +This seldom matters since Axiom retracts the answer to the +simpler type if it is necessary. +\spadcommand{factorial(\%)} +$$ +6 +$$ +\returnType{Type: Expression Integer} + +\index{PositiveInteger} +When you issue a positive number, the type {\tt PositiveInteger} is +printed. Surely, $3$ also has type {\tt Integer}! The curious reader +may now have two questions. First, is the type of an object not +unique? Second, how is {\tt PositiveInteger} related to {\tt +Integer}? +\spadcommand{3} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +\index{subdomain} +Any domain can be refined to a {\it subdomain} by a membership +{\tt predicate}. A {\tt predicate} is a function that, when applied to an +object of the domain, returns either {\tt true} or {\tt false}. For +example, the domain {\tt Integer} can be refined to the subdomain +{\tt PositiveInteger}, the set of integers $x$ such that $x > 0$, by giving +the Axiom predicate $x +-> x > 0$. Similarly, Axiom can define +subdomains such as ``the subdomain of diagonal matrices,'' ``the +subdomain of lists of length two,'' ``the subdomain of monic +irreducible polynomials in $x$,'' and so on. Trivially, any domain is +a subdomain of itself. + +While an object belongs to a unique domain, it can belong to any +number of subdomains. Any subdomain of the domain of an object can be +used as the {\it type} of that object. The type of $3$ is indeed both +{\tt Integer} and {\tt PositiveInteger} as well as any other subdomain +of integer whose predicate is satisfied, such as ``the prime +integers,'' ``the odd positive integers between 3 and 17,'' and so on. + +\subsection{Domain Constructors} +\label{sec:Domain Constructors} + +In Axiom, domains are objects. You can create them, pass them to +functions, and, as we'll see later, test them for certain properties. + +You ask for a value of a function by applying its name +to a set of arguments. + +To ask for ``the factorial of $7$'' you enter this expression to +Axiom. This applies the function {\tt factorial} to the value $7$ to +compute the result. +\index{factorial} +\spadcommand{factorial(7)} +$$ +5040 +$$ +\returnType{Type: PositiveInteger} + +Enter the type {\tt Polynomial (Integer)} as an expression to Axiom. +This looks much like a function call as well. It is! The result is +appropriately stated to be of type {\tt Domain}, which according to +our usual convention, denotes the class of all domains. +\spadcommand{Polynomial(Integer)} +$$ +\mbox{\rm Polynomial Integer} +$$ +\returnType{Type: Domain} +\index{Domain} + +\index{domain constructor} +The most basic operation involving domains is that of building a new +domain from a given one. To create the domain of ``polynomials over +the integers,'' Axiom applies the function {\tt Polynomial} to the +domain {\tt Integer}. A function like {\tt Polynomial} is called a +{\it domain constructor} or, \index{constructor!domain} more simply, a +{\it constructor}. A domain constructor is a function that creates a +domain. An argument to a domain constructor can be another domain or, +in general, an arbitrary kind of object. {\tt Polynomial} takes a +single domain argument while {\tt SquareMatrix} takes a positive +integer as a first argument to give the matrix +dimension and a domain as a second argument to give +the type of its components. + +What kinds of domains can you use as the argument to {\tt List} +or {\tt Polynomial} or {\tt SquareMatrix}? Well, the last two are +mathematical in nature. You want to be able to perform algebraic +operations like ``{\tt +}'' and ``{\tt *}'' on polynomials and square +matrices, and operations such as {\bf determinant} on square +matrices. So you want to allow polynomials of integers {\it and} +polynomials of square matrices with complex number coefficients and, +in general, anything that ``makes sense.'' At the same time, you don't +want Axiom to be able to build nonsense domains such as ``polynomials +of strings!'' + +In contrast to algebraic structures, data structures can hold any kind +of object. Operations on lists such as \spadfunFrom{insert}{List}, +\spadfunFrom{delete}{List}, and \spadfunFrom{concat}{List} just +manipulate the list itself without changing or operating on its +elements. Thus you can build {\tt List} over almost any datatype, +including itself. + +Create a complicated algebraic domain. +\spadcommand{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))} +$$ +\mbox{\rm List List Matrix Polynomial Complex Fraction Integer} +$$ +\returnType{Type: Domain} + +Try to create a meaningless domain. +\spadcommand{Polynomial(String)} +\begin{verbatim} + Polynomial String is not a valid type. +\end{verbatim} + +\index{category} +Evidently from our last example, Axiom has some mechanism that tells +what a constructor can use as an argument. This brings us to the +notion of {\it category}. As domains are objects, they too have a +domain. The domain of a domain is a category. A category is simply a +type whose members are domains. + +\index{Ring} +A common algebraic category is {\tt Ring}, the class of all domains +that are ``rings.'' A ring is an algebraic structure with constants +$0$ and $1$ and operations \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, +and \spadopFrom{*}{Ring}. These operations are assumed ``closed'' +with respect to the domain, meaning that they take two objects of the +domain and produce a result object also in the domain. The operations +are understood to satisfy certain ``axioms,'' certain mathematical +principles providing the algebraic foundation for rings. For example, +the {\it additive inverse axiom} for rings states: \begin{center} +Every element $x$ has an additive inverse $y$ such that $x + y = 0$. +\end{center} The prototypical example of a domain that is a ring is +the integers. Keep them in mind whenever we mention {\tt Ring}. + +Many algebraic domain constructors such as {\tt Complex}, +{\tt Polynomial}, {\tt Fraction}, take rings as arguments and return rings +as values. You can use the infix operator ``$has$'' to ask a domain +if it belongs to a particular category. + +All numerical types are rings. Domain constructor {\tt Polynomial} +builds ``the ring of polynomials over any other ring.'' +\spadcommand{Polynomial(Integer) has Ring} +$$ +{\rm true} +$$ +\returnType{Type: Boolean} + +Constructor {\tt List} never produces a ring. +\spadcommand{List(Integer) has Ring} +$$ +{\rm false} +$$ +\returnType{Type: Boolean} + +The constructor {\tt Matrix(R)} builds ``the domain of all matrices +over the ring $R$.'' This domain is never a ring since the operations +``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' on matrices of arbitrary +shapes are undefined. +\spadcommand{Matrix(Integer) has Ring} +$$ +{\rm false} +$$ +\returnType{Type: Boolean} + +Thus you can never build polynomials over matrices. +\spadcommand{Polynomial(Matrix(Integer))} +\begin{verbatim} + Polynomial Matrix Integer is not a valid type. +\end{verbatim} + +\index{SquareMatrix} +Use {\tt SquareMatrix(n,R)} instead. For any positive integer $n$, it +builds ``the ring of $n$ by $n$ matrices over $R$.'' +\spadcommand{Polynomial(SquareMatrix(7,Complex(Integer)))} +$$ +\mbox{\rm Polynomial SquareMatrix(7,Complex Integer)} +$$ +\returnType{Type: Domain} + +\index{Field} +Another common category is {\tt Field}, the class of all fields. +\index{field} A field is a ring with additional operations. For +example, a field has commutative multiplication and a closed operation +\spadopFrom{/}{Field} for the division of two elements. {\tt Integer} +is not a field since, for example, $3/2$ does not have an integer +result. The prototypical example of a field is the rational numbers, +that is, the domain {\tt Fraction(Integer)}. \index{Fraction(Integer)} +In general, the +constructor {\tt Fraction} \index{Fraction} +takes an IntegralDomain, \index{IntegralDomain} +which is a ring +with additional properties, as an argument and returns a field. +Other domain constructors, such as {\tt Complex}, build fields only if their +argument domain is a field. + +\index{Gaussian Integers} +The complex integers (often called the ``Gaussian integers'') do not form +a field. +\spadcommand{Complex(Integer) has Field} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{Fraction(Complex(Integer))} +But fractions of complex integers do. +\spadcommand{Fraction(Complex(Integer)) has Field} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{Complex(Fraction(Integer))} +The algebraically equivalent domain of complex rational numbers is a field +since domain constructor {\tt Complex} produces a field whenever its +argument is a field. +\spadcommand{Complex(Fraction(Integer)) has Field} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +\index{Type} +\index{Category} +The most basic category is {\tt Type}. \index{Type} It denotes the +class of all domains and subdomains. Note carefully that {\tt Type} +does not denote the class of all types. The type of all categories is +{\tt Category}. The type of {\tt Type} itself is undefined. Domain +constructor {\tt List} is able to build ``lists of elements from +domain $D$'' for arbitrary $D$ simply by requiring that $D$ belong to +category {\tt Type}. + +Now, you may ask, what exactly is a category? \index{category} Like +domains, categories can be defined in the Axiom language. A category +is defined by three components: +% +\begin{enumerate} +\item a name (for example, {\tt Ring}), +used to refer to the class of domains that the category represents; +\item a set of operations, used to refer to the operations that +the domains of this class support +(for example, ``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' for rings); and +\item an optional list of other categories that this category extends. +\end{enumerate} +% +This last component is a new idea. And it is key to the design of +Axiom. Because categories can extend one another, they form +hierarchies. All categories are extensions of {\tt Type} and that {\tt Field} +is an extension of {\tt Ring}. + +\index{exports!category} +\index{category exports} +The operations supported by the domains of a category are called the +{\sl exports} of that category because these are the operations made +available for system-wide use. The exports of a domain of a given +category are not only the ones explicitly mentioned by the category. +Since a category extends other categories, the operations of these +other categories---and all categories these other categories +extend---are also exported by the domains. + +For example, polynomial domains belong to {\tt PolynomialCategory}. +This category explicitly mentions some twenty-nine operations on +polynomials, but it extends eleven other categories (including +{\tt Ring}). As a result, the current system has over one hundred +operations on polynomials. + +If a domain belongs to a category that extends, say, {\tt Ring}, it is +convenient to say that the domain exports {\tt Ring}. The name of the +category thus provides a convenient shorthand for the list of +operations exported by the category. Rather than listing operations +such as \spadopFrom{+}{Ring} and \spadopFrom{*}{Ring} of {\tt Ring} +each time they are needed, the definition of a type simply asserts +that it exports category {\tt Ring}. + +The category name, however, is more than a shorthand. The name +{\tt Ring}, in fact, implies that the operations exported by rings are +required to satisfy a set of ``axioms'' associated with the name +{\tt Ring}. This subtle but important feature distinguishes Axiom from +other abstract datatype designs. + +Why is it not correct to assume that some type is a ring if it exports +all of the operations of {\tt Ring}? Here is why. Some languages +such as {\bf APL} \index{APL} denote the {\tt Boolean} constants +{\tt true} and {\tt false} by the integers $1$ and $0$ respectively, then use +``{\tt +}'' and ``{\tt *}'' to denote the logical operators {\bf or} and +{\bf and}. But with these definitions {\tt Boolean} is not a +ring since the additive inverse axiom is violated. That is, there is +no inverse element $a$ such that $1 + a = 0$, or, in the usual terms: +{\tt true or a = false}. This alternative definition of {\tt Boolean} +can be easily and correctly implemented in Axiom, since {\tt Boolean} +simply does not assert that it is of category {\tt Ring}. This +prevents the system from building meaningless domains such as +{\tt Polynomial(Boolean)} and then wrongfully applying algorithms that +presume that the ring axioms hold. + +Enough on categories. We now return to our discussion of domains. + +\index{exports!Domain} +Domains {\it export} a set of operations to make them available for +system-wide use. {\tt Integer}, for example, exports the operations +\spadopFrom{+}{Integer} and \spadopFrom{=}{Integer} given by the +signatures \spadopFrom{+}{Integer}: +\spadsig{(Integer,Integer)}{Integer} and \spadopFrom{=}{Integer}: +\spadsig{(Integer,Integer)}{Boolean}, respectively. Each of these +operations takes two {\tt Integer} arguments. The +\spadopFrom{+}{Integer} operation also returns an {\tt Integer} but +\spadopFrom{=}{Integer} returns a {\tt Boolean}: {\tt true} or {\tt false}. +The operations exported by a domain usually manipulate objects of the +domain---but not always. + +The operations of a domain may actually take as arguments, and return +as values, objects from any domain. For example, {\tt Fraction +(Integer)} exports the operations \spadopFrom{/}{Fraction}: +\spadsig{(Integer,Integer)}{Fraction(Integer)} and +\spadfunFrom{characteristic}{Fraction}: +\spadsig{}{NonNegativeInteger}. + +Suppose all operations of a domain take as arguments and return as +values, only objects from {\it other} domains. \index{package} This +kind of domain \index{constructor!package} is what Axiom calls a {\it +package}. + +A package does not designate a class of objects at all. Rather, a +package is just a collection of operations. Actually the bulk of the +Axiom library of algorithms consists of packages. The facilities for +factorization; integration; solution of linear, polynomial, and +differential equations; computation of limits; and so on, are all +defined in packages. Domains needed by algorithms can be passed to a +package as arguments or used by name if they are not ``variable.'' +Packages are useful for defining operations that convert objects of +one type to another, particularly when these types have different +parameterizations. As an example, the package +{\tt PolynomialFunction2(R,S)} \index{PolynomialFunction2(R,S)} +defines operations that convert polynomials +over a domain $R$ to polynomials over $S$. To convert an object from +{\tt Polynomial(Integer)} to {\tt Polynomial(Float)}, Axiom builds the +package {\tt PolynomialFunctions2(Integer,Float)} in order to create +the required conversion function. (This happens ``behind the scenes'' +for you.) + +Axiom categories, domains and packages and all their contained +functions are written in the Axiom programming language, called the +{\bf Spad} language, and have been +compiled into machine code. This is what comprises the Axiom +{\it library}. We will show you how to use these +domains and their functions and how to write your own functions. + +\index{Aldor} +\index{Aldor!Spad} +\index{Spad} +\index{Spad!Aldor} +There is a second language, called {\bf Aldor}\cite{4} that is +compatible with the {\bf Spad} language. They both can create +programs than can execute under Axiom. Aldor is a standalone +version of the {\bf Spad} language and contains some additional +syntax to support standalone programs. In addition, {\bf Aldor} +includes some new ideas such as post-facto domain extensions. + +\section{Writing Types and Modes} +\label{sec:Writing Types and Modes} + +We have already seen in the last section several examples of types. +Most of these +examples had either no arguments (for example, {\tt Integer}) or one +argument (for example, {\tt Polynomial (Integer)}). In this section +we give details about writing arbitrary types. We then define modes +and discuss how to write them. We conclude the section with a +discussion on constructor abbreviations. + +When might you need to write a type or mode? You need to do so when +you declare variables. +\index{PositiveInteger} +\spadcommand{a : PositiveInteger} +\returnType{Type: Void} + +You need to do so when you declare functions + +\spadcommand{f : Integer -> String} +\returnType{Type: Void} + +You need to do so when you convert an object from one type to another. +\index{:: conversion} +\index{Complex(Integer)} +\spadcommand{factor(2 :: Complex(Integer))} +$$ +-{i \ {{\left( 1+i +\right)} +\sp 2}} +$$ +\returnType{Type: Factored Complex Integer} + +\index{\$ package call} +\spadcommand{(2 = 3)\$Integer} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\index{target type} +You need to do so when you give computation target type information. +\spadcommand{(2 = 3)@Boolean} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\subsection{Types with No Arguments} +\label{sec:Types with No Arguments} + +A constructor with no arguments can be written either +\index{type!using parentheses} with or without +\index{parentheses!using with types} trailing opening and closing +parentheses ``{\tt ()}''. + +\begin{center} +{\tt Boolean()} \index{Boolean} is the same as {\tt Boolean} \\ +{\tt Integer()} \index{Integer} is the same as {\tt Integer} \\ +{\tt String()} \index{String} is the same as {\tt String} \\ +{\tt Void()} \index{Void} is the same as {\tt Void} +\end{center} + +It is customary to omit the parentheses. + +\subsection{Types with One Argument} +\label{sec:Types with One Argument} + +A constructor with one argument can frequently be +\index{type!using parentheses} written with no +\index{parentheses!using with types} parentheses. Types nest from +right to left so that {\tt Complex Fraction Polynomial Integer} +is the same as {\tt Complex (Fraction (Polynomial (Integer)))}. +You need to use parentheses to force the application of a constructor +to the correct argument, but you need not use any more than is necessary +to remove ambiguities. + +Here are some guidelines for using parentheses (they are possibly slightly +more restrictive than they need to be). + +If the argument is an expression like $2 + 3$ +then you must enclose the argument in parentheses. +\spadcommand{e : PrimeField(2 + 3)} +\returnType{Type: Void} + +If the type is to be used with package calling +then you must enclose the argument in parentheses. +\spadcommand{content(2)\$Polynomial(Integer)} +$$ +2 +$$ +\returnType{Type: Integer} + +Alternatively, you can write the type without parentheses +then enclose the whole type expression with parentheses. +\spadcommand{content(2)\$(Polynomial Complex Fraction Integer)} +$$ +2 +$$ +\returnType{Type: Complex Fraction Integer} + +If you supply computation target type information +then you should enclose the argument in parentheses. +\spadcommand{(2/3)@Fraction(Polynomial(Integer))} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +If the type itself has parentheses around it and we are not in the +case of the first example above, then the parentheses can usually be +omitted. +\spadcommand{(2/3)@Fraction(Polynomial Integer)} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Polynomial Integer} + +If the type is used in a declaration and the argument is a single-word +type, integer or symbol, then the parentheses can usually be omitted. +\spadcommand{(d,f,g) : Complex Polynomial Integer} +\returnType{Type: Void} + +\subsection{Types with More Than One Argument} +\label{sec:Types with More Than One Argument} + +If a constructor \index{type!using parentheses} has more than +\index{parentheses!using with types} one argument, you must use +parentheses. Some examples are \\ + +{\tt UnivariatePolynomial(x, Float)} \index{UnivariatePolynomial} \\ +{\tt MultivariatePolynomial([z,w,r], Complex Float)} \\ +\index{MultivariatePolynomial} +{\tt SquareMatrix(3, Integer)} \index{SquareMatrix} \\ +{\tt FactoredFunctions2(Integer,Fraction Integer)} +\index{FactoredFunctions2} + +\subsection{Modes} +\label{sec:Modes} +\index{mode} +A {\it mode} is a type that possibly is a question mark ({\tt ?}) or +contains one in an argument position. For example, the following are +all modes.\\ + +{\tt ?} \\ +{\tt Polynomial ?} \\ +{\tt Matrix Polynomial ?} \\ +{\tt SquareMatrix(3,?)} \\ +{\tt Integer} \\ +{\tt OneDimensionalArray(Float)} +\index{OneDimensionalArray} + +As is evident from these examples, a mode is a type with a part that +is not specified (indicated by a question mark). Only one ``{\tt ?}'' is +allowed per mode and it must appear in the most deeply nested argument +that is a type. Thus {\tt ?(Integer)}, {\tt Matrix(? (Polynomial))}, +{\tt SquareMatrix(?, Integer)} (it requires a numeric argument) +and {\tt SquareMatrix(?, ?)} are all +invalid. The question mark must take the place of a domain, not data. +This rules out, for example, the two {\tt SquareMatrix} expressions. + +Modes can be used for declarations and conversions. However, you +cannot use a mode for package calling or giving target type information. + +\subsection{Abbreviations} +\label{sec:Abbreviations} + +Every constructor has an abbreviation that +\index{abbreviation!constructor} you can freely +\index{constructor!abbreviation} substitute for the constructor name. +In some cases, the abbreviation is nothing more than the capitalized +version of the constructor name. + +\boxed{4.6in}{ + +\vskip 0.1cm +Aside from allowing types to be written more concisely, abbreviations +are used by Axiom to name various system files for constructors (such +as library filenames, test input files and example files). Here are +some common abbreviations. + +\begin{center} +\begin{tabular}{ll} +\small{\tt COMPLEX} abbreviates {\tt Complex} & +\small{\tt DFLOAT} abbreviates {\tt DoubleFloat} \\ +\small{\tt EXPR} abbreviates {\tt Expression} & +\small{\tt FLOAT} abbreviates {\tt Float} \\ +\small{\tt FRAC} abbreviates {\tt Fraction} & +\small{\tt INT} abbreviates {\tt Integer} \\ +\small{\tt MATRIX} abbreviates {\tt Matrix} & +\small{\tt NNI} abbreviates {\tt NonNegativeInteger} \\ +\small{\tt PI} abbreviates {\tt PositiveInteger} & +\small{\tt POLY} abbreviates {\tt Polynomial} \\ +\small{\tt STRING} abbreviates {\tt String} & +\small{\tt UP} abbreviates {\tt UnivariatePolynomial}\\ +\end{tabular} +\end{center} +\vskip 0.1cm +} + +You can combine both full constructor names and abbreviations in a +type expression. Here are some types using abbreviations. + +\begin{center} +\begin{tabular}{rcl} +{\tt POLY INT} & is the same as & {\tt Polynomial(INT)} \\ +{\tt POLY(Integer)} & is the same as & {\tt Polynomial(Integer)} \\ +{\tt POLY(Integer)} & is the same as & {\tt Polynomial(INT)} \\ +{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt Fraction Complex Integer} \\ +{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt FRAC(Complex Integer)} +\end{tabular} +\end{center} + +There are several ways of finding the names of constructors and their +abbreviations. For a specific constructor, use {\tt )abbreviation +query}. \index{abbreviation} You can also use the {\tt )what} system +command to see the names and abbreviations of constructors. +\index{what} + +\index{)abbreviation} +\index{)abb} +{\tt )abbreviation query} can be abbreviated (no pun intended) to +{\tt )abb q}. +\spadcommand{)abb q Integer} +\begin{verbatim} + INT abbreviates domain Integer +\end{verbatim} + +The {\tt )abbreviation query} command lists the constructor name if +you give the abbreviation. Issue {\tt )abb q} if you want to see the +names and abbreviations of all Axiom constructors. +\spadcommand{)abb q DMP} +\begin{verbatim} + DMP abbreviates domain DistributedMultivariatePolynomial +\end{verbatim} + +Issue this to see all packages whose +names contain the string ``ode''. \index{what packages} +\spadcommand{)what packages ode} +\begin{verbatim} +---------------------- Packages ----------------------- + +Packages with names matching patterns: + ode + + EXPRODE ExpressionSpaceODESolver + FCPAK1 FortranCodePackage1 + GRAY GrayCode + LODEEF ElementaryFunctionLODESolver + NODE1 NonLinearFirstOrderODESolver + ODECONST ConstantLODE + ODEEF ElementaryFunctionODESolver + ODEINT ODEIntegration + ODEPAL PureAlgebraicLODE + ODERAT RationalLODE + ODERED ReduceLODE + ODESYS SystemODESolver + ODETOOLS ODETools + UTSODE UnivariateTaylorSeriesODESolver + UTSODETL UTSodetools +\end{verbatim} + +\section{Declarations} +\label{sec:Declarations} +\index{declarations} +\index{: declaration} +A {\it declaration} is an expression used to restrict the type of +values that can be assigned to variables. A colon ``{\tt :}'' is always +used after a variable or list of variables to be declared. + +\boxed{4.6in}{ +\vskip 0.1cm +For a single variable, the syntax for declaration is +\begin{center} +{\it variableName $:$ typeOrMode} +\end{center} + +For multiple variables, the syntax is +\begin{center} +{\tt ($\hbox{\it variableName}_{1}$, $\hbox{\it variableName}_{2}$, +\ldots $\hbox{\it variableName}_{N}$): {\it typeOrMode}} +\end{center} +\vskip 0.1cm +} + +You can always combine a declaration with an assignment. When you do, +it is equivalent to first giving a declaration statement, then giving +an assignment. + +This declares one variable to have a type. +\spadcommand{a : Integer} +\returnType{Type: Void} + +This declares several variables to have a type. +\spadcommand{(b,c) : Integer} +\returnType{Type: Void} + +$a$, $b$ and $c$ can only hold integer values. +\spadcommand{a := 45} +$$ +45 +$$ +\returnType{Type: Integer} + +If a value cannot be converted to a declared type, +an error message is displayed. +\spadcommand{b := 4/5} +\begin{verbatim} + + Cannot convert right-hand side of assignment + 4 + - + 5 + + to an object of the type Integer of the left-hand side. +\end{verbatim} + +This declares a variable with a mode. +\spadcommand{n : Complex ?} +\returnType{Type: Void} + +This declares several variables with a mode. +\spadcommand{(p,q,r) : Matrix Polynomial ?} +\returnType{Type: Void} + +This complex object has integer real and imaginary parts. +\spadcommand{n := -36 + 9 * \%i} +$$ +-{36}+{9 \ i} +$$ +\returnType{Type: Complex Integer} + +This complex object has fractional symbolic real and imaginary parts. +\spadcommand{n := complex(4/(x + y),y/x)} +$$ +{4 \over {y+x}}+{{y \over x} \ i} +$$ +\returnType{Type: Complex Fraction Polynomial Integer} + +This matrix has entries that are polynomials with integer +coefficients. +\spadcommand{p := [ [1,2],[3,4],[5,6] ]} +$$ +\left[ +\begin{array}{cc} +1 & 2 \\ +3 & 4 \\ +5 & 6 +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Integer} + +This matrix has a single entry that is a polynomial with +rational number coefficients. +\spadcommand{q := [ [x - 2/3] ]} +$$ +\left[ +\begin{array}{c} +{x -{2 \over 3}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Fraction Integer} + +This matrix has entries that are polynomials with complex integer +coefficients. + +\spadcommand{r := [ [1-\%i*x,7*y+4*\%i] ]} +$$ +\left[ +\begin{array}{cc} +{-{i \ x}+1} & {{7 \ y}+{4 \ i}} +\end{array} +\right] +$$ +\returnType{Type: Matrix Polynomial Complex Integer} + +Note the difference between this and the next example. +This is a complex object with polynomial real and imaginary parts. + +\spadcommand{f : COMPLEX POLY ? := (x + y*\%i)**2} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +This is a polynomial with complex integer coefficients. The objects +are convertible from one to the other. + +\spadcommand{g : POLY COMPLEX ? := (x + y*\%i)**2} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +\section{Records} +\label{sec:Records} + +A {\tt Record} is an object composed of one or more other objects, +\index{Record} each of which is referenced \index{selector!record} +with \index{record!selector} a {\it selector}. Components can all +belong to the same type or each can have a different type. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Record} type is \begin{center} {\tt +Record($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, +$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, +$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} \end{center} You must be +careful if a selector has the same name as a variable in the +workspace. If this occurs, precede the selector name by a single +\index{quote} quote.\\ +} + +Record components are implicitly ordered. All the components of a +record can be set at once by assigning the record a bracketed {\it +tuple} of values of the proper length. For example: +\spadcommand{r : Record(a:Integer, b: String) := [1, "two"]} +$$ +\left[ +{a=1}, {b= \mbox{\tt "two"} } +\right] +$$ +\returnType{Type: Record(a: Integer,b: String)} +To access a component of a record $r$, write the name $r$, followed by +a period, followed by a selector. +\index{. Record selector} + +The object returned by this computation is a record with two components: a +$quotient$ part and a $remainder$ part. +\spadcommand{u := divide(5,2)} +$$ +\left[ +{quotient=2}, {remainder=1} +\right] +$$ +\returnType{Type: Record(quotient: Integer,remainder: Integer)} + +This is the quotient part. +\spadcommand{u.quotient} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +This is the remainder part. +\spadcommand{u.remainder} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +You can use selector expressions on the left-hand side of an assignment +to change destructively the components of a record. +\spadcommand{u.quotient := 8978} +$$ +8978 +$$ +\returnType{Type: PositiveInteger} + +The selected component $quotient$ has the value $8978$, which is what +is returned by the assignment. Check that the value of $u$ was +modified. +\spadcommand{u} +$$ +\left[ +{quotient={8978}}, {remainder=1} +\right] +$$ +\returnType{Type: Record(quotient: Integer,remainder: Integer)} + +Selectors are evaluated. Thus you can use variables that evaluate to +selectors instead of the selectors themselves. +\spadcommand{s := 'quotient} +$$ +quotient +$$ +\returnType{Type: Variable quotient} + +Be careful! A selector could have the same name as a variable in the +workspace. If this occurs, precede the selector name by a single +quote, as in $u.'quotient$. \index{selector!quoting} +\spadcommand{divide(5,2).s} +$$ +2 +$$ +\returnType{Type: PositiveInteger} + +Here we declare that the value of $bd$ has two components: a string, +to be accessed via {\tt name}, and an integer, to be accessed via +{\tt birthdayMonth}. +\spadcommand{bd : Record(name : String, birthdayMonth : Integer)} +\returnType{Type: Void} + +You must initially set the value of the entire {\tt Record} at once. +\spadcommand{bd := ["Judith", 3]} +$$ +\left[ +{name= \mbox{\tt "Judith"} }, {birthdayMonth=3} +\right] +$$ +\returnType{Type: Record(name: String,birthdayMonth: Integer)} + +Once set, you can change any of the individual components. +\spadcommand{bd.name := "Katie"} +$$ +\mbox{\tt "Katie"} +$$ +\returnType{Type: String} + +Records may be nested and the selector names can be shared at +different levels. +\spadcommand{r : Record(a : Record(b: Integer, c: Integer), b: Integer)} +\returnType{Type: Void} + +The record $r$ has a $b$ selector at two different levels. +Here is an initial value for $r$. +\spadcommand{r := [ [1,2], 3 ]} +$$ +\left[ +{a={\left[ {b=1}, {c=2} +\right]}}, + {b=3} +\right] +$$ +\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} + +This extracts the $b$ component from the $a$ component of $r$. +\spadcommand{r.a.b} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +This extracts the $b$ component from $r$. +\spadcommand{r.b} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +You can also use spaces or parentheses to refer to {\tt Record} +components. This is the same as $r.a$. +\spadcommand{r(a)} +$$ +\left[ +{b=1}, {c=2} +\right] +$$ +\returnType{Type: Record(b: Integer,c: Integer)} +This is the same as $r.b$. +\spadcommand{r b} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +This is the same as $r.b := 10$. +\spadcommand{r(b) := 10} +$$ +10 +$$ +\returnType{Type: PositiveInteger} + +Look at $r$ to make sure it was modified. +\spadcommand{r} +$$ +\left[ +{a={\left[ {b=1}, {c=2} +\right]}}, + {b={10}} +\right] +$$ +\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} + +\section{Unions} +\label{sec:Unions} + +Type {\tt Union} is used for objects that can be of any of a specific +finite set of types. \index{Union} Two versions of unions are +available, one with selectors (like records) and one without. +\index{union} + +\subsection{Unions Without Selectors} +\label{sec:Unions Without Selectors} + +The declaration $x : Union(Integer, String, Float)$ states that $x$ +can have values that are integers, strings or ``big'' floats. If, for +example, the {\tt Union} object is an integer, the object is said to +belong to the {\tt Integer} {\it branch} of the {\tt Union}. Note +that we are being a bit careless with the language here. Technically, +the type of $x$ is always {\tt Union(Integer, String, Float)}. If it +belongs to the {\tt Integer} branch, $x$ may be converted to an object +of type {\tt Integer}. + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Union} type without selectors is +\begin{center} +{\tt Union($\hbox{\it type}_{1}$, $\hbox{\it type}_{2}$, +\ldots, $\hbox{\it type}+{N}$)} +\end{center} +The types in a union without selectors must be distinct.\\ +} + +It is possible to create unions like {\tt Union(Integer, PositiveInteger)} +but they are difficult to work with because of the overlap in the branch +types. See below for the rules Axiom uses for converting something into +a union object. + +The {\tt case} infix \index{case} operator returns a {\tt Boolean} and can +be used to determine the branch in which an object lies. + +This function displays a message stating in which branch of the +{\tt Union} the object (defined as $x$ above) lies. + +\begin{verbatim} +sayBranch(x : Union(Integer,String,Float)) : Void == + output + x case Integer => "Integer branch" + x case String => "String branch" + "Float branch" +\end{verbatim} + +This tries {\bf sayBranch} with an integer. +\spadcommand{sayBranch 1} +\begin{verbatim} +Compiling function sayBranch with type Union(Integer,String,Float) + -> Void + Integer branch +\end{verbatim} +\returnType{Type: Void} + +This tries {\bf sayBranch} with a string. +\spadcommand{sayBranch "hello"} +\begin{verbatim} + String branch +\end{verbatim} +\returnType{Type: Void} + +This tries {\bf sayBranch} with a floating-point number. +\spadcommand{sayBranch 2.718281828} +\begin{verbatim} + Float branch +\end{verbatim} +\returnType{Type: Void} + +There are two things of interest about this particular +example to which we would like to draw your attention. +\begin{enumerate} +\item Axiom normally converts a result to the target value +before passing it to the function. +If we left the declaration information out of this function definition +then the {\bf sayBranch} call would have been attempted with an +{\tt Integer} rather than a {\tt Union}, and an error would have +resulted. +\item The types in a {\tt Union} are searched in the order given. +So if the type were given as + +%\noindent +{\tt sayBranch(x: Union(String,Integer,Float,Any)): Void} + +then the result would have been ``String branch'' because there +is a conversion from {\tt Integer} to {\tt String}. +\end{enumerate} + +Sometimes {\tt Union} types can have extremely long names. Axiom +therefore abbreviates the names of unions by printing the type of the +branch first within the {\tt Union} and then eliding the remaining +types with an ellipsis ({\tt ...}). + +Here the {\tt Integer} branch is displayed first. Use ``{\tt ::}'' to +create a {\tt Union} object from an object. +\spadcommand{78 :: Union(Integer,String)} +$$ +78 +$$ +\returnType{Type: Union(Integer,...)} + +Here the {\tt String} branch is displayed first. +\spadcommand{s := "string" :: Union(Integer,String)} +$$ +\mbox{\tt "string"} +$$ +\returnType{Type: Union(String,...)} + +Use {\tt typeOf} to see the full and actual {\tt Union} type. \index{typeOf} +\spadcommand{typeOf s} +$$ +Union(Integer,String) +$$ +\returnType{Type: Domain} + +A common operation that returns a union is \spadfunFrom{exquo}{Integer} +which returns the ``exact quotient'' if the quotient is exact, +\spadcommand{three := exquo(6,2)} +$$ +3 +$$ +\returnType{Type: Union(Integer,...)} + +and {\tt "failed"} if the quotient is not exact. +\spadcommand{exquo(5,2)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +A union with a {\tt "failed"} is frequently used to indicate the failure +or lack of applicability of an object. As another example, assign an +integer a variable $r$ declared to be a rational number. +\spadcommand{r: FRAC INT := 3} +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +The operation \spadfunFrom{retractIfCan}{Fraction} tries to retract +the fraction to the underlying domain {\tt Integer}. It produces a +union object. Here it succeeds. +\spadcommand{retractIfCan(r)} +$$ +3 +$$ +\returnType{Type: Union(Integer,...)} + +Assign it a rational number. +\spadcommand{r := 3/2} +$$ +3 \over 2 +$$ +\returnType{Type: Fraction Integer} + +Here the retraction fails. +\spadcommand{retractIfCan(r)} +$$ +\mbox{\tt "failed"} +$$ +\returnType{Type: Union("failed",...)} + +\subsection{Unions With Selectors} +\label{sec:Unions With Selectors} + +Like records, +you can write {\tt Union} types \index{selector!union} with selectors. +\index{union!selector} + +\boxed{4.6in}{ +\vskip 0.1cm +The syntax for writing a {\tt Union} type with selectors is +\begin{center} +{\tt Union($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, +$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, +$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} +\end{center} +You must be careful if a selector has the same name as a variable in +the workspace. If this occurs, precede the selector name by a single +\index{quote} quote. \index{selector!quoting} It is an error to use a +selector that does not correspond to the branch of the {\tt Union} in +which the element actually lies. \\ +} + +Be sure to understand the difference between records and unions with +selectors. \index{union!difference from record} Records can have more +than one component and the selectors are used to refer to the +components. \index{record!difference from union} Unions always have +one component but the type of that one component can vary. An object +of type {\tt Record(a: Integer, b: Float, c: String)} contains an +integer {\it and} a float {\it and} a string. An object of type +{\tt Union(a: Integer, b: Float, c: String)} contains an integer +{\it or} a float {\it or} a string. + +Here is a version of the {\bf sayBranch} function that +works with a union with selectors. It displays a message stating in +which branch of the {\tt Union} the object lies. + +\begin{verbatim} +sayBranch(x:Union(i:Integer,s:String,f:Float)):Void== + output + x case i => "Integer branch" + x case s => "String branch" + "Float branch" +\end{verbatim} + +Note that {\tt case} uses the selector name as its right-hand argument. +\index{case} If you accidentally use the branch type on the right-hand +side of {\tt case}, {\tt false} will be returned. + +Declare variable $u$ to have a union type with selectors. +\spadcommand{u : Union(i : Integer, s : String)} +\returnType{Type: Void} + +Give an initial value to $u$. +\spadcommand{u := "good morning"} +$$ +\mbox{\tt "good morning"} +$$ +\returnType{Type: Union(s: String,...)} + +Use $case$ to determine in which branch of a {\tt Union} an object lies. +\spadcommand{u case i} +$$ +{\tt false} +$$ +\returnType{Type: Boolean} + +\spadcommand{u case s} +$$ +{\tt true} +$$ +\returnType{Type: Boolean} + +To access the element in a particular branch, use the selector. +\spadcommand{u.s} +$$ +\mbox{\tt "good morning"} +$$ +\returnType{Type: String} + +\section{The ``Any'' Domain} +\label{sec:The ``Any'' Domain} + +With the exception of objects of type {\tt Record}, all Axiom data +structures are homogenous, that is, they hold objects all of the same +type. \index{Any} If you need to get around this, you can use type +{\tt Any}. Using {\tt Any}, for example, you can create lists whose +elements are integers, rational numbers, strings, and even other +lists. + +Declare $u$ to have type {\tt Any}. +\spadcommand{u: Any} +\returnType{Type: Void} + +Assign a list of mixed type values to $u$ +\spadcommand{u := [1, 7.2, 3/2, x**2, "wally"]} +$$ +\left[ +1, {7.2}, {3 \over 2}, {x \sp 2}, \mbox{\tt "wally"} +\right] +$$ +\returnType{Type: List Any} + +When we ask for the elements, Axiom displays these types. +\spadcommand{u.1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +Actually, these objects belong to {\tt Any} but Axiom +automatically converts them to their natural types for you. +\spadcommand{u.3} +$$ +3 \over 2 +$$ +\returnType{Type: Fraction Integer} + +Since type {\tt Any} can be anything, it can only belong to type +{\tt Type}. Therefore it cannot be used in algebraic domains. +\spadcommand{v : Matrix(Any)} +\begin{verbatim} + Matrix Any is not a valid type. +\end{verbatim} + +Perhaps you are wondering how Axiom internally represents objects of +type {\tt Any}. An object of type {\tt Any} consists not only a data +part representing its normal value, but also a type part (a +{\it badge}) giving \index{badge} its type. For example, the value $1$ of +type {\tt PositiveInteger} as an object of type {\tt Any} internally +looks like $[1,{\tt PositiveInteger()}]$. + +When should you use {\tt Any} instead of a {\tt Union} type? For a +{\tt Union}, you must know in advance exactly which types you are +going to +allow. For {\tt Any}, anything that comes along can be accommodated. + +\section{Conversion} +\label{sec:Conversion} +\index{:: conversion} +\boxed{4.6in}{ +\vskip 0.1cm +Conversion is the process of changing an object of one type into an +object of another type. The syntax for conversion is: +$$ +{\it object} {\tt ::} {\it newType} +$$ +} + +By default, $3$ has the type {\tt PositiveInteger}. +\spadcommand{3} +$$ +3 +$$ +\returnType{Type: PositiveInteger} + +We can change this into an object of type {\tt Fraction Integer} +by using ``{\tt ::}''. +\spadcommand{3 :: Fraction Integer} +$$ +3 +$$ +\returnType{Type: Fraction Integer} + +A {\it coercion} is a special kind of conversion that Axiom is allowed +to do automatically when you enter an expression. Coercions are +usually somewhat safer than more general conversions. The Axiom +library contains operations called {\bf coerce} and {\bf convert}. +Only the {\bf coerce} operations can be used by the interpreter to +change an object into an object of another type unless you explicitly +use a {\tt ::}. + +By now you will be quite familiar with what types and modes look like. +It is useful to think of a type or mode as a pattern for what you want +the result to be. + +Let's start with a square matrix of polynomials with complex rational +number coefficients. \index{SquareMatrix} +\spadcommand{m : SquareMatrix(2,POLY COMPLEX FRAC INT)} +\returnType{Type: Void} + +\spadcommand{m := matrix [ [x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5] ]} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} + +We first want to interchange the {\tt Complex} and {\tt Fraction} +layers. We do the conversion by doing the interchange in the type +expression. +\spadcommand{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \ i} \over 4}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{{3 \ i} \over 7} \ {y \sp 4}} -x} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Fraction Complex Integer)} + +Interchange the {\tt Polynomial} and the {\tt Fraction} levels. +\spadcommand{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{{3 \ i \ {y \sp 4}} -{7 \ x}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Polynomial Complex Integer)} + +Interchange the {\tt Polynomial} and the {\tt Complex} levels. +\spadcommand{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} + +All the entries have changed types, although in comparing the +last two results only the entry in the lower left corner looks different. +We did all the intermediate steps to show you what Axiom can do. + +In fact, we could have combined all these into one conversion. +\spadcommand{m :: SquareMatrix(2,FRAC COMPLEX POLY INT)} +$$ +\left[ +\begin{array}{cc} +{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ +{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} + +There are times when Axiom is not be able to do the conversion in one +step. You may need to break up the transformation into several +conversions in order to get an object of the desired type. + +We cannot move either {\tt Fraction} or {\tt Complex} above (or to the +left of, depending on how you look at it) {\tt SquareMatrix} because +each of these levels requires that its argument type have commutative +multiplication, whereas {\tt SquareMatrix} does not. That is because +{\tt Fraction} requires that its argument belong to the category +{\tt IntegralDomain} \index{IntegralDomain} +and \index{category} {\tt Complex} requires that its +argument belong to {\tt CommutativeRing}. \index{CommutativeRing} +The {\tt Integer} level did not move +anywhere because it does not allow any arguments. We also did not +move the {\tt SquareMatrix} part anywhere, but we could have. + +Recall that $m$ looks like this. + +\spadcommand{m} +$$ +\left[ +\begin{array}{cc} +{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ +{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} +\end{array} +\right] +$$ +\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} + +If we want a polynomial with matrix coefficients rather than a matrix +with polynomial entries, we can just do the conversion. + +\spadcommand{m :: POLY SquareMatrix(2,COMPLEX FRAC INT)} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \over 7} \ i} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \over 4} \ i} & {1 \over 2} \\ +0 & {{12} -{{9 \over 5} \ i}} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} + +We have not yet used modes for any conversions. Modes are a great +shorthand for indicating the type of the object you want. Instead of +using the long type expression in the last example, we could have +simply said this. + +\spadcommand{m :: POLY ?} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \over 7} \ i} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \over 4} \ i} & {1 \over 2} \\ +0 & {{12} -{{9 \over 5} \ i}} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} + +We can also indicate more structure if we want the entries of the +matrices to be fractions. + +\spadcommand{m :: POLY SquareMatrix(2,FRAC ?)} +$$ +{{\left[ +\begin{array}{cc} +0 & 1 \\ +0 & 0 +\end{array} +\right]} +\ {y \sp 2} \ z}+{{\left[ +\begin{array}{cc} +0 & 0 \\ +{{3 \ i} \over 7} & 0 +\end{array} +\right]} +\ {y \sp 4}}+{{\left[ +\begin{array}{cc} +1 & 0 \\ +-1 & 0 +\end{array} +\right]} +\ x}+{\left[ +\begin{array}{cc} +-{{3 \ i} \over 4} & {1 \over 2} \\ +0 & {{{60} -{9 \ i}} \over 5} +\end{array} +\right]} +$$ +\returnType{Type: Polynomial SquareMatrix(2,Fraction Complex Integer)} + +\section{Subdomains Again} +\label{sec:Subdomains Again} +\index{subdomains} +A {\it subdomain} {\rm S} of a domain {\rm D} is a domain consisting of +\begin{enumerate} +\item those elements of {\rm D} that satisfy some +{\it predicate} (that is, a test that returns {\tt true} or {\tt false}) and +\item a subset of the operations of {\rm D}. +\end{enumerate} +Every domain is a subdomain of itself, trivially satisfying the +membership test: {\tt true}. + +Currently, there are only two system-defined subdomains in Axiom that +receive substantial use. {\tt PositiveInteger} and +{\tt NonNegativeInteger} are subdomains of {\tt Integer}. An element $x$ +of {\tt NonNegativeInteger} is an integer that is greater than or +equal to zero, that is, satisfies $x >= 0$. An element $x$ of +{\tt PositiveInteger} is a nonnegative integer that is, in fact, greater +than zero, that is, satisfies $x > 0$. Not all operations from +{\tt Integer} are available for these subdomains. For example, negation +and subtraction are not provided since the subdomains are not closed +under those operations. When you use an integer in an expression, +Axiom assigns to it the type that is the most specific subdomain whose +predicate is satisfied. + +This is a positive integer. +\spadcommand{5} +$$ +5 +$$ +\returnType{Type: PositiveInteger} + +This is a nonnegative integer. +\spadcommand{0} +$$ +0 +$$ +\returnType{Type: NonNegativeInteger} + +This is neither of the above. +\spadcommand{-5} +$$ +-5 +$$ +\returnType{Type: Integer} + +Furthermore, unless you are assigning an integer to a declared variable +or using a conversion, any integer result has as type the most +specific subdomain. +\spadcommand{(-2) - (-3)} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +\spadcommand{0 :: Integer} +$$ +0 +$$ +\returnType{Type: Integer} + +\spadcommand{x : NonNegativeInteger := 5} +$$ +5 +$$ +\returnType{Type: NonNegativeInteger} + +When necessary, Axiom converts an integer object into one belonging to +a less specific subdomain. For example, in $3-2$, the arguments to +\spadopFrom{-}{Integer} are both elements of {\tt PositiveInteger}, +but this type does not provide a subtraction operation. Neither does +{\tt NonNegativeInteger}, so $3$ and $2$ are viewed as elements of +{\tt Integer}, where their difference can be calculated. The result +is $1$, which Axiom then automatically assigns the type +{\tt PositiveInteger}. + +Certain operations are very sensitive to the subdomains to which their +arguments belong. This is an element of {\tt PositiveInteger}. +\spadcommand{2 ** 2} +$$ +4 +$$ +\returnType{Type: PositiveInteger} + +This is an element of {\tt Fraction Integer}. +\spadcommand{2 ** (-2)} +$$ +1 \over 4 +$$ +\returnType{Type: Fraction Integer} + +It makes sense then that this is a list of elements of {\tt +PositiveInteger}. +\spadcommand{[10**i for i in 2..5]} +$$ +\left[ +{100}, {1000}, {10000}, {100000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +What should the type of {\tt [10**(i-1) for i in 2..5]} be? On one hand, +$i-1$ is always an integer greater than zero as $i$ ranges from $2$ to +$5$ and so $10**i$ is also always a positive integer. On the other, +$i-1$ is a very simple function of $i$. Axiom does not try to analyze +every such function over the index's range of values to determine +whether it is always positive or nowhere negative. For an arbitrary +Axiom function, this analysis is not possible. + +So, to be consistent no such analysis is done and we get this. +\spadcommand{[10**(i-1) for i in 2..5]} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List Fraction Integer} + +To get a list of elements of {\tt PositiveInteger} instead, you have +two choices. You can use a conversion. + +\spadcommand{[10**((i-1) :: PI) for i in 2..5]} +\begin{verbatim} +Compiling function G82696 with type Integer -> Boolean +Compiling function G82708 with type NonNegativeInteger -> Boolean +\end{verbatim} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +Or you can use {\tt pretend}. \index{pretend} +\spadcommand{[10**((i-1) pretend PI) for i in 2..5]} +$$ +\left[ +{10}, {100}, {1000}, {10000} +\right] +$$ +\returnType{Type: List PositiveInteger} + +The operation {\tt pretend} is used to defeat the Axiom type system. +The expression {\tt object pretend D} means ``make a new object +(without copying) of type {\tt D} from {\tt object}.'' If +{\tt object} were an integer and you told Axiom to pretend it was a list, +you would probably see a message about a fatal error being caught and +memory possibly being damaged. Lists do not have the same internal +representation as integers! + +You use {\tt pretend} at your peril. \index{peril} + +Use $pretend$ with great care! Axiom trusts you that the value is of +the specified type. + +\spadcommand{(2/3) pretend Complex Integer} +$$ +2+{3 \ i} +$$ +\returnType{Type: Complex Integer} + +\section{Package Calling and Target Types} +\label{sec:Package Calling and Target Types} +\index{\$ package call} +Axiom works hard to figure out what you mean by an expression without +your having to qualify it with type information. Nevertheless, there +are times when you need to help it along by providing hints (or even +orders!) to get Axiom to do what you want. + +Declarations using types and modes control the type of the results +produced. For example, we can either produce a complex object with +polynomial real and imaginary parts or a polynomial with complex +integer coefficients, depending on the declaration. + +Package calling is used to tell Axiom to use a particular function +from a particular part of the library. + +Use the \spadopFrom{/}{Fraction} from {\tt Fraction Integer} to create +a fraction of two integers. +\spadcommand{2/3} +$$ +2 \over 3 +$$ +\returnType{Type: Fraction Integer} + +If we wanted a floating point number, we can say ``use the +\spadopFrom{/}{Float} in {\tt Float}.'' +\spadcommand{(2/3)\$Float} +$$ +0.6666666666 6666666667 +$$ +\returnType{Type: Float} + +Perhaps we actually wanted a fraction of complex integers. +\spadcommand{(2/3)\$Fraction(Complex Integer)} +$$ +2 \over 3 +$$ +\returnType{Type: Float} + +In each case, Axiom used the indicated operations, sometimes first +needing to convert the two integers into objects of the appropriate type. +In these examples, ``/'' is written as an infix operator. + +\boxed{4.6in}{ +\vskip 0.1cm +To use package calling with an infix operator, use the following syntax: +$$(\ arg_1{\rm \ op\ }arg_2\ )\$type$$ +} + +We used, for example, $(2/3)\${\rm Float}$. The expression $2+3+4$ +is equivalent to $(2+3)+4$. Therefore in the expression +$(2+3+4)\${\rm Float}$ the second ``+'' comes from the {\rm Float} +domain. The first ``+'' comes from {\rm Float} because the package +call causes Axiom to convert $(2+3)$ and $4$ to type +{\rm Float}. Before the sum is converted, it is given a target type +of {\rm Float} by Axiom and then evaluated. The target type causes the +``+'' from {\tt Float} to be used. + +\boxed{4.6in}{ +\vskip 0.1cm +For an operator written before its arguments, you must use parentheses +around the arguments (even if there is only one), and follow the closing +parenthesis by a ``\$'' and then the type. +$$ fun\ (\ arg_1, arg_2, \ldots, arg_N\ )\$type$$ +} + +For example, to call the ``minimum'' function from {\rm SmallFloat} on two +integers, you could write {\bf min}(4,89)\${\tt SmallFloat}. Another use of +package calling is to tell Axiom to use a library function rather than a +function you defined. + +\index{target type} +Sometimes rather than specifying where an operation comes from, you +just want to say what type the result should be. We say that you provide a +{\sl target type} for the expression. Instead of using a ``\$'', use a ``@'' +to specify the requested target type. Otherwise, the syntax is the same. +Note that giving a target type is not the same as explicitly doing a +conversion. The first says ``try to pick operations so that the result has +such-and-such a type.'' The second says ``compute the result and then convert +to an object of such-and-such a type.'' + +Sometimes it makes sense, as in this expression, to say ``choose the +operations in this expression so that the final result is {\rm Float}. +\spadcommand{(2/3)@Float} +$$ +0.6666666666 6666666667 +$$ +\returnType{Type: Float} + +Here we used ``{\tt @}'' to say that the target type of the left-hand side +was {\tt Float}. In this simple case, there was no real difference +between using ``{\tt \$}'' and ``{\tt @}''. +You can see the difference if you try the following. + +This says to try to choose ``{\tt +}'' so that the result is a string. +Axiom cannot do this. +\spadcommand{(2 + 3)@String} +\begin{verbatim} +An expression involving @ String actually evaluated to one of + type PositiveInteger . Perhaps you should use :: String . +\end{verbatim} + +This says to get the {\tt +} from {\tt String} and apply it to the two +integers. Axiom also cannot do this because there is no {\tt +} +exported by {\tt String}. +\spadcommand{(2 + 3)\$String} +\begin{verbatim} + The function + is not implemented in String . +\end{verbatim} + +(By the way, the operation \spadfunFrom{concat}{String} or juxtaposition +is used to concatenate two strings.) +\index{String} + +When we have more than one operation in an expression, the difference +is even more evident. The following two expressions show that Axiom +uses the target type to create different objects. +The ``{\tt +}'', ``{\tt *}'' and ``{\tt **}'' operations are all +chosen so that an object of the correct final type is created. + +This says that the operations should be chosen so that the result is a +{\tt Complex} object. +\spadcommand{((x + y * \%i)**2)@(Complex Polynomial Integer)} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +This says that the operations should be chosen so that the result is a +{\tt Polynomial} object. +\spadcommand{((x + y * \%i)**2)@(Polynomial Complex Integer)} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +What do you think might happen if we left off all target type and +package call information in this last example? +\spadcommand{(x + y * \%i)**2} +$$ +-{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} +$$ +\returnType{Type: Polynomial Complex Integer} + +We can convert it to {\tt Complex} as an afterthought. But this is +more work than just saying making what we want in the first place. +\spadcommand{\% :: Complex ?} +$$ +-{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} +$$ +\returnType{Type: Complex Polynomial Integer} + +Finally, another use of package calling is to qualify fully an +operation that is passed as an argument to a function. + +Start with a small matrix of integers. +\spadcommand{h := matrix [ [8,6],[-4,9] ]} +$$ +\left[ +\begin{array}{cc} +8 & 6 \\ +-4 & 9 +\end{array} +\right] +$$ +\returnType{Type: Matrix Integer} + +We want to produce a new matrix that has for entries the +multiplicative inverses of the entries of $h$. One way to do this is +by calling \spadfunFrom{map}{MatrixCategoryFunctions2} with the +\spadfunFrom{inv}{Fraction} function from {\tt Fraction (Integer)}. + +\spadcommand{map(inv\$Fraction(Integer),h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +We could have been a bit less verbose and used abbreviations. +\spadcommand{map(inv\$FRAC(INT),h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +As it turns out, Axiom is smart enough to know what we mean anyway. +We can just say this. +\spadcommand{map(inv,h)} +$$ +\left[ +\begin{array}{cc} +{1 \over 8} & {1 \over 6} \\ +-{1 \over 4} & {1 \over 9} +\end{array} +\right] +$$ +\returnType{Type: Matrix Fraction Integer} + +\section{Resolving Types} +\label{sec:Resolving Types} + +In this section we briefly describe an internal process by which +\index{resolve} Axiom determines a type to which two objects of +possibly different types can be converted. We do this to give you +further insight into how Axiom takes your input, analyzes it, and +produces a result. + +What happens when you enter $x + 1$ to Axiom? Let's look at what you +get from the two terms of this expression. + +This is a symbolic object whose type indicates the name. +\spadcommand{x} +$$ +x +$$ +\returnType{Type: Variable x} + +This is a positive integer. +\spadcommand{1} +$$ +1 +$$ +\returnType{Type: PositiveInteger} + +There are no operations in {\tt PositiveInteger} that add positive +integers to objects of type {\tt Variable(x)} nor are there any in +{\tt Variable(x)}. Before it can add the two parts, Axiom must come +up with a common type to which both $x$ and $1$ can be converted. We +say that Axiom must {\it resolve} the two types into a common type. +In this example, the common type is {\tt Polynomial(Integer)}. + +Once this is determined, both parts are converted into polynomials, +and the addition operation from {\tt Polynomial(Integer)} is used to +get the answer. +\spadcommand{x + 1} +$$ +x+1 +$$ +\returnType{Type: Polynomial Integer} + +Axiom can always resolve two types: if nothing resembling the original +types can be found, then {\tt Any} is be used. \index{Any} This is +fine and useful in some cases. + +\spadcommand{["string",3.14159]} +$$ +\left[ +\mbox{\tt "string"} , {3.14159} +\right] +$$ +\returnType{Type: List Any} + +In other cases objects of type {\tt Any} can't be used by the +operations you specified. +\spadcommand{"string" + 3.14159} +\begin{verbatim} +There are 11 exposed and 5 unexposed library operations named + + having 2 argument(s) but none was determined to be applicable. + Use HyperDoc Browse, or issue + )display op + + to learn more about the available operations. Perhaps + package-calling the operation or using coercions on the + arguments will allow you to apply the operation. + +Cannot find a definition or applicable library operation named + + with argument type(s) + String + Float + + Perhaps you should use "@" to indicate the required return type, + or "$" to specify which version of the function you need. +\end{verbatim} + +Although this example was contrived, your expressions may need to be +qualified slightly to help Axiom resolve the types involved. You may +need to declare a few variables, do some package calling, provide some +target type information or do some explicit conversions. + +We suggest that you just enter the expression you want evaluated and +see what Axiom does. We think you will be impressed with its ability +to ``do what I mean.'' If Axiom is still being obtuse, give it some +hints. As you work with Axiom, you will learn where it needs a little +help to analyze quickly and perform your computations. + +\section{Exposing Domains and Packages} +\label{sec:Exposing Domains and Packages} + +In this section we discuss how Axiom makes some operations available +to you while hiding others that are meant to be used by developers or +only in rare cases. If you are a new user of Axiom, it is likely that +everything you need is available by default and you may want to skip +over this section on first reading. + +Every \index{constructor!exposed} domain and package in the Axiom +library \index{constructor!hidden} is \index{exposed!constructor} +either exposed (meaning that you can use its operations without doing +anything special) or it is {\it hidden} (meaning you have to either +package call the operations it contains or +explicitly expose it to use the operations). The initial exposure +status for a constructor is set in the file {\bf exposed.lsp} (see the +{\it Installer's Note} \index{exposed.lsp @{\bf exposed.lsp}} for +Axiom \index{file!exposed.lsp @{\bf exposed.lsp}} if you need to know +the location of this file). Constructors are collected together in +\index{group!exposure} {\it exposure groups}. \index{exposure!group} +Categories are all in the exposure group ``categories'' and the bulk +of the basic set of packages and domains that are exposed are in the +exposure group ``basic.'' Here is an abbreviated sample of the file +(without the Lisp parentheses): + +\begin{verbatim} +basic + AlgebraicNumber AN + AlgebraGivenByStructuralConstants ALGSC + Any ANY + AnyFunctions1 ANY1 + BinaryExpansion BINARY + Boolean BOOLEAN + CardinalNumber CARD + CartesianTensor CARTEN + Character CHAR + CharacterClass CCLASS + CliffordAlgebra CLIF + Color COLOR + Complex COMPLEX + ContinuedFraction CONTFRAC + DecimalExpansion DECIMAL + ... +\end{verbatim} +\begin{verbatim} +categories + AbelianGroup ABELGRP + AbelianMonoid ABELMON + AbelianMonoidRing AMR + AbelianSemiGroup ABELSG + Aggregate AGG + Algebra ALGEBRA + AlgebraicallyClosedField ACF + AlgebraicallyClosedFunctionSpace ACFS + ArcHyperbolicFunctionCategory AHYP + ... +\end{verbatim} + +For each constructor in a group, the full name and the abbreviation is +given. There are other groups in {\bf exposed.lsp} but initially only +the constructors in exposure groups ``basic'' ``categories'' +``naglink'' and ``anna'' are exposed. + +As an interactive user of Axiom, you do not need to modify this file. +Instead, use {\tt )set expose} to expose, hide or query the exposure +status of an individual constructor or exposure group. \index{set expose} +The reason for having exposure groups is to be able to expose +or hide multiple constructors with a single command. For example, you +might group together into exposure group ``quantum'' a number of +domains and packages useful for quantum mechanical computations. +These probably should not be available to every user, but you want an +easy way to make the whole collection visible to Axiom when it is +looking for operations to apply. + +If you wanted to hide all the basic constructors available by default, +you would issue {\tt )set expose drop group basic}. +\index{set expose drop group} We do not recommend that you do this. +If, however, you discover that you have hidden all the basic constructors, +you should issue {\tt )set expose add group basic} to restore your default +environment. \index{set expose add group} + +It is more likely that you would want to expose or hide individual +constructors. We use several operations from +{\tt OutputForm}, a domain usually hidden. To avoid package calling every +operation from {\tt OutputForm}, we expose the domain and let Axiom +conclude that those operations should be used. Use {\tt )set expose +add constructor} and {\tt )set expose drop constructor} to expose and +hide a constructor, respectively. \index{set expose drop constructor} +You should use the constructor name, not the abbreviation. The +{\tt )set expose} command guides you through these options. +\index{set expose add constructor} + +If you expose a previously hidden constructor, Axiom exhibits new +behavior (that was your intention) though you might not expect the +results that you get. {\tt OutputForm} is, in fact, one of the worst +offenders in this regard. \index{OutputForm} This domain is meant to +be used by other domains for creating a structure that Axiom knows how +to display. It has functions like \spadopFrom{+}{OutputForm} that +form output representations rather than do mathematical calculations. +Because of the order in which Axiom looks at constructors when it is +deciding what operation to apply, {\tt OutputForm} might be used +instead of what you expect. + +This is a polynomial. +\spadcommand{x + x} +$$ +2 \ x +$$ +\returnType{Type: Polynomial Integer} + +Expose {\tt OutputForm}. +\spadcommand{)set expose add constructor OutputForm } +\begin{verbatim} + OutputForm is now explicitly exposed in frame G82322 +\end{verbatim} + +This is what we get when {\tt OutputForm} is automatically available. +\spadcommand{x + x} +$$ +x+x +$$ +\returnType{Type: OutputForm} + +Hide {\tt OutputForm} so we don't run into problems with any later examples! +\spadcommand{)set expose drop constructor OutputForm } +\begin{verbatim} + OutputForm is now explicitly hidden in frame G82322 +\end{verbatim} + +Finally, exposure is done on a frame-by-frame basis. A {\it frame} +\index{frame!exposure and} is one of possibly several logical Axiom +workspaces within a physical one, each having its own environment (for +example, variables and function definitions). If you have several +Axiom workspace windows on your screen, they are all different frames, +automatically created for you by HyperDoc. Frames can be manually +created, made active and destroyed by the {\tt )frame} system command. +\index{frame} They do not share exposure information, so you need to +use {\tt )set expose} in each one to add or drop constructors from +view. + +\section{Commands for Snooping} +\label{sec:Commands for Snooping} + +To conclude this chapter, we introduce you to some system commands +that you can use for getting more information about domains, packages, +categories, and operations. The most powerful Axiom facility for +getting information about constructors and operations is the Browse +component of HyperDoc. + +Use the {\tt )what} system command to see lists of system objects +whose name contain a particular substring (uppercase or lowercase is +not significant). \index{what} + +Issue this to see a list of all operations with ``{\tt complex}'' in +their names. \index{what operation} +\spadcommand{)what operation complex} +\begin{verbatim} + +Operations whose names satisfy the above pattern(s): + +complex complex? +complexEigenvalues complexEigenvectors +complexElementary complexExpand +complexForm complexIntegrate +complexLimit complexNormalize +complexNumeric complexNumericIfCan +complexRoots complexSolve +complexZeros createLowComplexityNormalBasis +createLowComplexityTable doubleComplex? +drawComplex drawComplexVectorField +fortranComplex fortranDoubleComplex +pmComplexintegrate + +To get more information about an operation such as +complexZeros, issue the command )display op complexZeros +\end{verbatim} + +If you want to see all domains with ``{\tt matrix}'' in their names, +issue this. \index{what domain} +\spadcommand{)what domain matrix} +\begin{verbatim} +----------------------- Domains ----------------------- + +Domains with names matching patterns: + matrix + + DHMATRIX DenavitHartenbergMatrix + DPMM DirectProductMatrixModule + IMATRIX IndexedMatrix + LSQM LieSquareMatrix + M3D ThreeDimensionalMatrix + MATCAT- MatrixCategory& + MATRIX Matrix + RMATCAT- RectangularMatrixCategory& + RMATRIX RectangularMatrix + SMATCAT- SquareMatrixCategory& + SQMATRIX SquareMatrix +\end{verbatim} + +Similarly, if you wish to see all packages whose names contain ``{\tt +gauss}'', enter this. \index{what packages} +\spadcommand{)what package gauss} +\begin{verbatim} +---------------------- Packages ----------------------- + +Packages with names matching patterns: + gauss + + GAUSSFAC GaussianFactorizationPackage +\end{verbatim} + +This command shows all the operations that {\tt Any} provides. +Wherever {\tt \$} appears, it means ``{\tt Any}''. \index{show} +\spadcommand{)show Any} +\begin{verbatim} + Any is a domain constructor + Abbreviation for Any is ANY + This constructor is exposed in this frame. + Issue )edit /usr/local/axiom/mnt/algebra/any.spad + to see algebra source code for ANY + +--------------------- Operations ---------------------- + ?=? : (%,%) -> Boolean + any : (SExpression,None) -> % + coerce : % -> OutputForm + dom : % -> SExpression + domainOf : % -> OutputForm + hash : % -> SingleInteger + latex : % -> String + obj : % -> None + objectOf : % -> OutputForm + ?~=? : (%,%) -> Boolean + showTypeInOutput : Boolean -> String + +\end{verbatim} + +This displays all operations with the name {\tt complex}. +\index{display operation} +\spadcommand{)display operation complex} +\begin{verbatim} +There is one exposed function called complex : + [1] (D1,D1) -> D from D if D has COMPCAT D1 and D1 has COMRING +\end{verbatim} + +Let's analyze this output. + +First we find out what some of the abbreviations mean. +\spadcommand{)abbreviation query COMPCAT} +\begin{verbatim} + COMPCAT abbreviates category ComplexCategory +\end{verbatim} + +\spadcommand{)abbreviation query COMRING} +\begin{verbatim} + COMRING abbreviates category CommutativeRing +\end{verbatim} + +So if {\tt D1} is a commutative ring (such as the integers or floats) and +{\tt D} belongs to {\tt ComplexCategory D1}, then there is an operation +called {\bf complex} that takes two elements of {\tt D1} and creates an +element of {\tt D}. The primary example of a constructor implementing +domains belonging to {\tt ComplexCategory} is {\tt Complex}. + +%\setcounter{chapter}{2} + +\chapter{Using HyperDoc} +\label{sec:Using HyperDoc} + +\begin{figure}[htbp] +\includegraphics[bbllx=14, bblly=14, bburx=365, bbury=240]{ps/bookvol1q.ps} +\caption{The HyperDoc root window page} +\end{figure} + +HyperDoc is the gateway to Axiom. \index{HyperDoc} It's both an +on-line tutorial and an on-line reference manual. It also enables you +to use Axiom simply by using the mouse and filling in templates. +HyperDoc is available to you if you are running Axiom under the X +Window System. + +Pages usually have active areas, marked in {\bf this font} (bold +face). As you move the mouse pointer to an active area, the pointer +changes from a filled dot to an open circle. The active areas are +usually linked to other pages. When you click on an active area, you +move to the linked page. + +\section{Headings} +\label{sec:Headings} +Most pages have a standard set of buttons at the top of the page. +This is what they mean: + +\begin{description} + +\item[\HelpBitmap] Click on this to get help. The button only appears +if there is specific help for the page you are viewing. You can get +{\it general} help for HyperDoc by clicking the help button on the +home page. + +\item[\UpBitmap] Click here to go back one page. +By clicking on this button repeatedly, you can go back several pages and +then take off in a new direction. + +\item[\ReturnBitmap] Go back to the home page, that is, the page on +which you started. Use HyperDoc to explore, to make forays into new +topics. Don't worry about how to get back. HyperDoc remembers where +you came from. Just click on this button to return. + +\item[\ExitBitmap] From the root window (the one that is displayed +when you start the system) this button leaves the HyperDoc program, +and it must be restarted if you want to use it again. From any other +HyperDoc window, it just makes that one window go away. You {\it must} +use this button to get rid of a window. If you use the window +manager ``Close'' button, then all of HyperDoc goes away. + +\end{description} + +The buttons are not displayed if they are not applicable to the page +you are viewing. For example, there is no \ReturnBitmap button on the +top-level menu. + +\section{Key Definitions} +\label{sec:Key Definitions} + +The following keyboard definitions are in effect throughout HyperDoc. + +\begin{description} +\item[F1] Display the main help page. +\item[F3] Same as \ExitBitmap{}, makes the window go away if you are not at the top-level window or quits the HyperDoc facility if you are at the top-level. +\item[F5] Rereads the HyperDoc database, if necessary (for system developers). +\item[F9] Displays this information about key definitions. +\item[F12] Same as {\bf F3}. +\item[Up Arrow] Scroll up one line. +\item[Down Arrow] Scroll down one line. +\item[Page Up] Scroll up one page. +\item[Page Down] Scroll down one page. +\end{description} + +\section{Scroll Bars} +\label{sec:Scroll Bars} + +Whenever there is too much text to fit on a page, a +{\it scroll \index{scroll bar} bar} +automatically appears along the right side. + +With a scroll bar, your page becomes an aperture, that is, a window +into a larger amount of text than can be displayed at one time. The +scroll bar lets you move up and down in the text to see different +parts. It also shows where the aperture is relative to the whole +text. The aperture is indicated by a strip on the scroll bar. + +Move the cursor with the mouse to the ``down-arrow'' at the bottom of +the scroll bar and click. See that the aperture moves down one line. +Do it several times. Each time you click, the aperture moves down one +line. Move the mouse to the ``up-arrow'' at the top of the scroll bar +and click. The aperture moves up one line each time you click. + +Next move the mouse to any position along the middle of the scroll bar +and click. HyperDoc attempts to move the top of the aperture to this +point in the text. + +You cannot make the aperture go off the bottom edge. When the +aperture is about half the size of text, the lowest you can move the +aperture is halfway down. + +To move up or down one screen at a time, use the \fbox{\bf PageUp} and +\fbox{\bf PageDown} keys on your keyboard. They move the visible part of the +region up and down one page each time you press them. + +If the HyperDoc page does not contain an input area, you can also use +the \fbox{\bf Home} and \fbox{$\uparrow$} and \fbox{$\downarrow$} +arrow keys to navigate. When you press the \fbox{\bf Home} key, the +screen is positioned at the very top of the page. Use the +\fbox{$\uparrow$} and \fbox{$\downarrow$} arrow keys to move the +screen up and down one line at a time, respectively. + +\section{Input Areas} +\label{sec:Input Areas} + +Input areas are boxes where you can put data. + +To enter characters, first move your mouse cursor to somewhere within +the HyperDoc page. Characters that you type are inserted in front of +the underscore. This means that when you type characters at your +keyboard, they go into this first input area. + +The input area grows to accommodate as many characters as you type. +Use the \fbox{\bf Backspace} key to erase characters to the left. To +modify what you type, use the right-arrow \fbox{$\rightarrow$} and +left-arrow keys \fbox{$\leftarrow$} and the keys \fbox{\bf Insert}, +\fbox{\bf Delete}, \fbox{\bf Home} and \fbox{\bf End}. These keys are +found immediately on the right of the standard IBM keyboard. + +If you press the \fbox{\bf Home} key, the cursor moves to the +beginning of the line and if you press the \fbox{\bf End} key, the +cursor moves to the end of the line. Pressing +\fbox{\bf Ctrl}--\fbox{\bf End} deletes all the text from the +cursor to the end of the line. + +A page may have more than one input area. Only one input area has an +underscore cursor. When you first see apage, the top-most input area +contains the cursor. To type information into another input area, use +the \fbox{\bf Enter} or \fbox{\bf Tab} key to move from one input area to +xanother. To move in the reverse order, use \fbox{\bf Shift}--\fbox{\bf Tab}. + +You can also move from one input area to another using your mouse. +Notice that each input area is active. Click on one of the areas. +As you can see, the underscore cursor moves to that window. + +\section{Radio Buttons and Toggles} +\label{sec:Radio Buttons and Toggles} + +Some pages have {\it radio buttons} and {\it toggles}. +Radio buttons are a group of buttons like those on car radios: you can +select only one at a time. + +Once you have selected a button, it appears to be inverted and +contains a checkmark. To change the selection, move the cursor with +the mouse to a different radio button and click. + +A toggle is an independent button that displays some on/off state. +When ``on'', the button appears to be inverted and contains a +checkmark. When ``off'', the button is raised. + +Unlike radio buttons, you can set a group of them any way you like. +To change toggle the selection, move the cursor with the mouse to the +button and click. + +\section{Search Strings} +\label{sec:Search Strings} + +A {\it search string} is used for searching some database. To learn +about search strings, we suggest that you bring up the HyperDoc +glossary. To do this from the top-level page of HyperDoc: +\begin{enumerate} +\item Click on Reference, bringing up the Axiom Reference page. +\item Click on Glossary, bringing up the glossary. +\end{enumerate} + +The glossary has an input area at its bottom. We review the various +kinds of search strings you can enter to search the glossary. + +The simplest search string is a word, for example, {\tt operation}. A +word only matches an entry having exactly that spelling. Enter the +word {\tt operation} into the input area above then click on +{\bf Search}. As you can see, {\tt operation} matches only one entry, +namely with {\tt operation} itself. + +Normally matching is insensitive to whether the alphabetic characters +of your search string are in uppercase or lowercase. Thus +{\tt operation} and {\tt OperAtion} both have the same effect. +%If you prefer that matching be case-sensitive, issue the command +%{\tt set HHyperName mixedCase} command to the interpreter. + +You will very often want to use the wildcard ``{\tt *}'' in your search +string so as to match multiple entries in the list. The search key +``{\tt *}'' matches every entry in the list. You can also use ``{\tt *}'' +anywhere within a search string to match an arbitrary substring. Try +``{\tt cat*}'' for example: enter ``{\tt cat*}'' into the input area and click +on {\bf Search}. This matches several entries. + +You use any number of wildcards in a search string as long as they are +not adjacent. Try search strings such as ``{\tt *dom*}''. As you see, +this search string matches ``{\tt domain}'', ``{\tt domain constructor}'', +``{\tt subdomain}'', and so on. + +\subsection{Logical Searches} +\label{sec:Logical Searches} + +For more complicated searches, you can use ``{\tt and}'', ``{\tt or}'', and +``{\tt not}'' with basic search strings; write logical expressions using +these three operators just as in the Axiom language. For example, +{\tt domain or package} matches the two entries {\tt domain} and +{\tt package}. Similarly, ``{\tt dom* and *con*}'' matches +``{\tt domain constructor}'' and others. Also ``{\tt not *a*}'' matches +every entry that does not contain the letter ``{\tt a}'' somewhere. + +Use parentheses for grouping. For example, ``{\tt dom* and (not *con*)}'' +matches ``{\tt domain}'' but not ``{\tt domain constructor}''. + +There is no limit to how complex your logical expression can be. +For example, +\begin{center} +{\tt a* or b* or c* or d* or e* and (not *a*)} +\end{center} +is a valid expression. + +\section{Example Pages} +\label{sec:Example Pages} + +Many pages have Axiom example commands. + +Each command has an active ``button'' along the left margin. When you +click on this button, the output for the command is ``pasted-in.'' +Click again on the button and you see that the pasted-in output +disappears. + +Maybe you would like to run an example? To do so, just click on any +part of its text! When you do, the example line is copied into a new +interactive Axiom buffer for this HyperDoc page. + +Sometimes one example line cannot be run before you run an earlier one. +Don't worry---HyperDoc automatically runs all the necessary +lines in the right order! + +The new interactive Axiom buffer disappears when you leave HyperDoc. +If you want to get rid of it beforehand, use the {\bf Cancel} button +of the X Window manager or issue the Axiom system command +{\tt )close.} \index{close} + +\section{X Window Resources for HyperDoc} +\label{sec:X Window Resources for HyperDoc} + +You can control the appearance of HyperDoc while running under Version +11 \index{HyperDoc X Window System defaults} of the X Window System by +placing the following resources \index{X Window System} in the file +{\bf .Xdefaults} in your home directory. \index{file!.Xdefaults} +In what follows, {\it font} is any valid X11 font name +\index{font} (for example, {\tt Rom14}) and {\it color} is any valid +X11 color \index{color} specification (for example, {\tt NavyBlue}). +For more information about fonts and colors, refer to the X Window +documentation for your system. + +\begin{description} +\item[{\tt Axiom.hyperdoc.RmFont:} {\it font}] \ \newline +This is the standard text font. +The default value is {\tt Rom14} +\item[{\tt Axiom.hyperdoc.RmColor:} {\it color}] \ \newline +This is the standard text color. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.ActiveFont:} {\it font}] \ \newline +This is the font used for HyperDoc link buttons. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.ActiveColor:} {\it color}] \ \newline +This is the color used for HyperDoc link buttons. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.AxiomFont:} {\it font}] \ \newline +This is the font used for active Axiom commands. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.AxiomColor:} {\it color}] \ \newline +This is the color used for active Axiom commands. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.BoldFont:} {\it font}] \ \newline +This is the font used for bold face. +The default value is {\tt Bld14} +\item[{\tt Axiom.hyperdoc.BoldColor:} {\it color}] \ \newline +This is the color used for bold face. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.TtFont:} {\it font}] \ \newline +This is the font used for Axiom output in HyperDoc. +This font must be fixed-width. +The default value is {\tt Rom14} +\item[{\tt Axiom.hyperdoc.TtColor:} {\it color}] \ \newline +This is the color used for Axiom output in HyperDoc. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.EmphasizeFont:} {\it font}] \ \newline +This is the font used for italics. +The default value is {\tt Itl14} +\item[{\tt Axiom.hyperdoc.EmphasizeColor:} {\it color}] \ \newline +This is the color used for italics. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.InputBackground:} {\it color}] \ \newline +This is the color used as the background for input areas. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.InputForeground:} {\it color}] \ \newline +This is the color used as the foreground for input areas. +The default value is {\tt white} +\item[{\tt Axiom.hyperdoc.BorderColor:} {\it color}] \ \newline +This is the color used for drawing border lines. +The default value is {\tt black} +\item[{\tt Axiom.hyperdoc.Background:} {\it color}] \ \newline +This is the color used for the background of all windows. +The default value is {\tt white} +\end{description} +\vfill +\eject + +%\setcounter{chapter}{3} + +\chapter{Input Files and Output Styles} +\label{sec:Input Files and Output Styles} + +In this chapter we discuss how to collect Axiom statements +and commands into files and then read the contents into the +workspace. +We also show how to display the results of your computations in +several different styles including \TeX{}, FORTRAN and +monospace two-dimensional format.\footnote{\TeX{} is a +trademark of the American Mathematical Society.} + +The printed version of this book uses the Axiom \TeX{} output formatter. +When we demonstrate a particular output style, we will need to turn +\TeX{} formatting off and the output style on so that the correct output +is shown in the text. + +\section{Input Files} +\label{sec:Input Files} + +In this section we explain what an {\it input file} is and +\index{file!input} why you would want to know about it. We discuss +where Axiom looks for input files and how you can direct it to look +elsewhere. We also show how to read the contents of an input file +into the {\it workspace} and how to use the {\it history} facility to +generate an input file from the statements you have entered directly +into the workspace. + +An {\it input} file contains Axiom expressions and system commands. +Anything that you can enter directly to Axiom can be put into an input +file. This is how you save input functions and expressions that you +wish to read into Axiom more than one time. + +To read an input file into Axiom, use the {\tt )read} system command. +\index{read} For example, you can read a file in a particular +directory by issuing +\begin{verbatim} +)read /spad/src/input/matrix.input +\end{verbatim} + +The ``{\bf .input}'' is optional; this also works: +\begin{verbatim} +)read /spad/src/input/matrix +\end{verbatim} + +What happens if you just enter {\tt )read matrix.input} or even {\tt +)read matrix}? Axiom looks in your current working directory for +input files that are not qualified by a directory name. Typically, +this directory is the directory from which you invoked Axiom. + +To change the current working directory, use the {\tt )cd} system +command. The command {\tt {)cd}} by itself shows the current working +\index{directory!default for searching} directory. \index{cd} To +change it to \index{file!input!where found} the {\tt {src/input}} +subdirectory for user ``babar'', issue +\begin{verbatim} +)cd /u/babar/src/input +\end{verbatim} +Axiom looks first in this directory for an input file. If it is not +found, it looks in the system's directories, assuming you meant some +input file that was provided with Axiom. + +\boxed{4.6in}{ +\vskip 0.1cm +If you have the Axiom history facility turned on (which it is +by default), you can save all the lines you have entered into the +workspace by entering + +)history )write + +\index{history )write} + +Axiom tells you what input file to edit to see your statements. The +file is in your home directory or in the directory you specified with +\index{cd} {\tt {)cd}}.\\ +} + +\section{The .axiom.input File} +\label{sec:The .axiom.input File} + +When Axiom starts up, it tries to read the input file {\bf +.axiom.input}\footnote{{\bf.axiom.input} used to be called +{\bf axiom.input} in the NAG version} +from your home \index{start-up profile file} +directory. \index{file!start-up profile} It +there is no {\bf .axiom.input} in your home directory, it reads the +copy located in its own {\bf src/input} directory. +\index{file!.axiom.input @{\bf .axiom.input}} The file usually +contains system commands to personalize your Axiom environment. In +the remainder of this section we mention a few things that users +frequently place in their {\bf .axiom.input} files. + +In order to have FORTRAN output always produced from your +computations, place the system command {\tt )set output fortran on} in +{\bf .axiom.input}. \index{quit} If you do not want to be prompted +for confirmation when you issue the {\tt )quit} system command, place +{\tt )set quit unprotected} in {\bf .axiom.input}. +\index{set quit unprotected} +If you then decide that you do want to be prompted, issue +{\tt )set quit protected}. \index{set quit protected} This is the +default setting so that new users do not leave Axiom +inadvertently. The system command {\tt )pquit} always +prompts you for confirmation. +\section{Common Features of Using Output Formats} +\label{sec:Common Features of Using Output Formats} + +In this section we discuss how to start and stop the display +\index{output formats!common features} of the different output formats +and how to send the output to the screen or to a file. +\index{file!sending output to} To fix ideas, we use FORTRAN output +format for most of the examples. + +You can use the {\tt )set output} system \index{output +formats!starting} command to \index{output formats!stopping} toggle or +redirect the different kinds of output. \index{set output} The name +of the kind of output follows ``output'' in the command. The names are + +\begin{tabular}{@{}ll} +{\bf fortran} & for FORTRAN output. \\ +{\bf algebra} & for monospace two-dimensional mathematical output. \\ +{\bf tex} & for \TeX{} output. \\ +{\bf script} & for IBM Script Formula Format output. +\end{tabular} + +For example, issue {\tt {)set output fortran on}} to turn on FORTRAN +format and issue {\tt {)set output fortran off}} to turn it off. By +default, {\tt algebra} is {\tt on} and all others are {\tt off}. +\index{set output fortran} When output is started, it is sent to the +screen. To send the output to a file, give the file name without +\index{output formats!sending to file} directory or extension. Axiom +appends a file extension depending on the kind of output being +produced. + +Issue this to redirect FORTRAN output to, for example, the file +{\bf linalg.sfort}. +\spadcommand{)set output fortran linalg} +\begin{verbatim} + FORTRAN output will be written to file linalg.sfort . +\end{verbatim} + +You must {\it also} turn on the creation of FORTRAN output. +The above just says where it goes if it is created. +\spadcommand{)set output fortran on} + +In what directory is this output placed? It goes into the directory +from which you started Axiom, or if you have used the {\tt {)cd}} +system command, the one that you specified with {\tt {)cd}}. +\index{cd} You should use {\tt )cd} before you send the output to the file. + +You can always direct output back to the screen by issuing this. +\index{output formats!sending to screen} +\spadcommand{)set output fortran console} + +Let's make sure FORTRAN formatting is off so that nothing we +do from now on produces FORTRAN output. +\spadcommand{)set output fortran off} + +We also delete the demonstrated output file we created. +\spadcommand{)system rm linalg.sfort} + +You can abbreviate the words ``{\tt on},'' ``{\tt off},'' and +``{\tt console}'' to the minimal number of characters needed to distinguish +them. Because of this, you cannot send output to files called +{\bf on.sfort, off.sfort, of.sfort, console.sfort, consol.sfort} and so on. + +The width of the output on the page is set by \index{output +formats!line length} {\tt )set output length} for all formats except +FORTRAN. \index{set output length} Use {\tt )set fortran fortlength} +to change the FORTRAN line length from its default value of $72$. + +\section{Monospace Two-Dimensional Mathematical Format} +\label{sec:Monospace Two-Dimensional Mathematical Format} + +This is the default output format for Axiom. +It is usually on when you start the system. +\index{set output algebra} +\index{output formats!monospace 2D} +\index{monospace 2D output format} + +If it is not, issue this. +\spadcommand{)set output algebra on} + +Since the printed version of this book (as opposed to the HyperDoc +version) shows output produced by the \TeX{} output formatter, let us +temporarily turn off \TeX{} output. +\spadcommand{)set output tex off} + +Here is an example of what it looks like. +\spadcommand{matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]} +\begin{verbatim} + + + 3 3 2+ + |3%i y + x 3%i y + 2x | + (1) | | + | 4 4 2| + +4%i y + x 4%i y + 2x + +\end{verbatim} +\returnType{Type: Matrix Polynomial Complex Integer} + +Issue this to turn off this kind of formatting. +\spadcommand{)set output algebra off} + +Turn \TeX{} output on again. +\spadcommand{)set output tex on} + +The characters used for the matrix brackets above are rather ugly. +You get this character set when you issue \index{character set} +{\tt )set output characters plain}. \index{set output characters} This +character set should be used when you are running on a machine that +does not support the IBM extended ASCII character set. If you are +running on an IBM workstation, for example, issue +{\tt )set output characters default} to get better looking output. + +\section{TeX Format} +\label{sec:TeX Format} + +Axiom can produce \TeX{} output for your \index{output formats!TeX +@{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format} +The output is produced using macros from the \LaTeX{} document +preparation system by Leslie Lamport\cite{5}. The printed version +of this book was produced using this formatter. + +To turn on \TeX{} output formatting, issue this. +\index{set output tex} +\spadcommand{)set output tex on} + +Here is an example of its output. +\begin{verbatim} +matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4] + +$$ +\left[ +\begin{array}{cc} +{{3 \ i \ {y \sp 3}}+x} & +{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ +{{4 \ i \ {y \sp 4}}+x} & +{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} +\end{array} +\right] +$$ + +\end{verbatim} +This formats as +$$ +\left[ +\begin{array}{cc} +{{3 \ i \ {y \sp 3}}+x} & +{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ +{{4 \ i \ {y \sp 4}}+x} & +{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} +\end{array} +\right] +$$ + +To turn \TeX{} output formatting off, issue +{\tt {)set output tex off}}. +The \LaTeX macros in the output generated by Axiom +are all standard except for the following definitions: +\begin{verbatim} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} +\end{verbatim} + +\section{IBM Script Formula Format} +\label{sec:IBM Script Formula Format} + +Axiom can \index{output formats!IBM Script Formula Format} produce IBM +Script Formula Format output for your +\index{IBM Script Formula Format} expressions. + +To turn IBM Script Formula Format on, issue this. +\index{set output script} +\spadcommand{)set output script on} + +Here is an example of its output. +\begin{verbatim} +matrix [ [i*x**i + j*%i*y**j for i in 1..2] for j in 3..4] + +.eq set blank @ +:df. + >+x> here < <3 @@ %i @@ + >+<2 @@ > > > habove < < <4 @@ %i @@ + >+x> here < <4 @@ %i @@ >+<2 @@ + > > > > right rb> +:edf. +\end{verbatim} + +To turn IBM Script Formula Format output formatting off, issue this. +\spadcommand{)set output script off} + +\section{FORTRAN Format} +\label{sec:FORTRAN Format} + +In addition to turning FORTRAN output on and off and stating where the +\index{output formats!FORTRAN} output should be placed, there are many +options that control the \index{FORTRAN output format} appearance of +the generated code. In this section we describe some of the basic +options. Issue {\tt )set fortran} to see a full list with their +current settings. + +The output FORTRAN expression usually begins in column 7. If the +expression needs more than one line, the ampersand character {\tt \&} +is used in column 6. Since some versions of FORTRAN have restrictions +on the number of lines per statement, Axiom breaks long expressions +into segments with a maximum of 1320 characters (20 lines of 66 +characters) per segment. \index{set fortran} If you want to change +this, say, to 660 characters, issue the system command +\index{set fortran explength} {\tt )set fortran explength 660}. +\index{FORTRAN output format!breaking into multiple statements} +You can turn off the line breaking by issuing {\tt )set fortran segment off}. +\index{set fortran segment} Various code optimization levels are available. + +FORTRAN output is produced after you issue this. +\index{set output fortran} +\spadcommand{)set output fortran on} + +For the initial examples, we set the optimization level to 0, which is the +lowest level. +\index{set fortran optlevel} +\spadcommand{)set fortran optlevel 0} + +The output is usually in columns 7 through 72, although fewer columns +are used in the following examples so that the output +\index{FORTRAN output format!line length} +fits nicely on the page. +\spadcommand{)set fortran fortlength 60} + +By default, the output goes to the screen and is displayed before the +standard Axiom two-dimensional output. In this example, an assignment +to the variable $R1$ was generated because this is the result of step 1. +\spadcommand{(x+y)**3} +\begin{verbatim} + R1=y**3+3*x*y*y+3*x*x*y+x**3 +\end{verbatim} +$$ +{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +Here is an example that illustrates the line breaking. +\spadcommand{(x+y+z)**3} +\begin{verbatim} + R2=z**3+(3*y+3*x)*z*z+(3*y*y+6*x*y+3*x*x)*z+y**3+3*x*y + &*y+3*x*x*y+x**3 +\end{verbatim} +$$ +{z \sp 3}+{{\left( {3 \ y}+{3 \ x} +\right)} +\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} +\right)} +\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +Note in the above examples that integers are generally converted to +\index{FORTRAN output format!integers vs. floats} floating point +numbers, except in exponents. This is the default behavior but can be +turned off by issuing {\tt )set fortran ints2floats off}. +\index{set fortran ints2floats} The rules governing when the conversion +is done are: +\begin{enumerate} +\item If an integer is an exponent, convert it to a floating point +number if it is greater than 32767 in absolute value, otherwise leave it +as an integer. +\item Convert all other integers in an expression to floating point numbers. +\end{enumerate} +These rules only govern integers in expressions. Numbers generated by +Axiom for $DIMENSION$ statements are also integers. + +To set the type of generated FORTRAN data, +\index{FORTRAN output format!data types} +use one of the following: +\begin{verbatim} +)set fortran defaulttype REAL +)set fortran defaulttype INTEGER +)set fortran defaulttype COMPLEX +)set fortran defaulttype LOGICAL +)set fortran defaulttype CHARACTER +\end{verbatim} + +When temporaries are created, they are given a default type of {\tt REAL.} +Also, the {\tt REAL} versions of functions are used by default. +\spadcommand{sin(x)} +\begin{verbatim} + R3=DSIN(x) +\end{verbatim} +$$ +\sin +\left( +{x} +\right) +$$ +\returnType{Type: Expression Integer} + +At optimization level 1, Axiom removes common subexpressions. +\index{FORTRAN output format!optimization level} +\index{set fortran optlevel} +\spadcommand{)set fortran optlevel 1} + +\spadcommand{(x+y+z)**3} +\begin{verbatim} + T2=y*y + T3=x*x + R4=z**3+(3*y+3*x)*z*z+(3*T2+6*x*y+3*T3)*z+y**3+3*x*T2+ + &3*T3*y+x**3 +\end{verbatim} +$$ +{z \sp 3}+{{\left( {3 \ y}+{3 \ x} +\right)} +\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} +\right)} +\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} +$$ +\returnType{Type: Polynomial Integer} + +This changes the precision to {\tt DOUBLE}. \index{set fortran +precision double} Substitute {\tt single} for {\tt double} +\index{FORTRAN output format!precision} to return to single precision. +\index{set fortran precision single} + +\spadcommand{)set fortran precision double} + +Complex constants display the precision. +\spadcommand{2.3 + 5.6*\%i } +\begin{verbatim} + R5=(2.3D0,5.6D0) +\end{verbatim} +$$ +{2.3}+{{5.6} \ i} +$$ +\returnType{Type: Complex Float} + +The function names that Axiom generates depend on the chosen precision. +\spadcommand{sin \%e} +%%NOTE: the book shows DSIN(DEXP(1.0D0)) +\begin{verbatim} + R6=DSIN(DEXP(1)) +\end{verbatim} +$$ +\sin +\left( +{e} +\right) +$$ +\returnType{Type: Expression Integer} + +Reset the precision to {\tt single} and look at these two examples again. +\spadcommand{)set fortran precision single} + +\spadcommand{2.3 + 5.6*\%i} +\begin{verbatim} + R7=(2.3,5.6) +\end{verbatim} +$$ +{2.3}+{{5.6} \ i} +$$ +\returnType{Type: Complex Float} + +\spadcommand{sin \%e} +%%NOTE: the book shows SIN(EXP(1.)) +\begin{verbatim} + R8=SIN(EXP(1)) +\end{verbatim} +$$ +\sin +\left( +{e} +\right) +$$ +\returnType{Type: Expression Integer} +Expressions that look like lists, streams, sets or matrices cause +array code to be generated. +\spadcommand{[x+1,y+1,z+1]} +\begin{verbatim} + T1(1)=x+1 + T1(2)=y+1 + T1(3)=z+1 + R9=T1 +\end{verbatim} +$$ +\left[ +{x+1}, {y+1}, {z+1} +\right] +$$ +\returnType{Type: List Polynomial Integer} + + +A temporary variable is generated to be the name of the array. +\index{FORTRAN output format!arrays} This may have to be changed in +your particular application. +\spadcommand{set[2,3,4,3,5]} +\begin{verbatim} + T1(1)=2 + T1(2)=3 + T1(3)=4 + T1(4)=5 + R10=T1 +\end{verbatim} +$$ +\left\{ +2, 3, 4, 5 +\right\} +$$ +\returnType{Type: Set PositiveInteger} + +By default, the starting index for generated FORTRAN arrays is $0$. +\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} +\begin{verbatim} + T1(0,0)=2.3 + T1(0,1)=9.7 + T1(1,0)=0.0 + T1(1,1)=18.778 + T1 +\end{verbatim} +$$ +\left[ +\begin{array}{cc} +{2.3} & {9.7} \\ +{0.0} & {18.778} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + +To change the starting index for generated FORTRAN arrays to be $1$, +\index{set fortran startindex} issue this. This value can only be $0$ +or $1$. +\spadcommand{)set fortran startindex 1} + +Look at the code generated for the matrix again. +\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} +\begin{verbatim} + T1(1,1)=2.3 + T1(1,2)=9.7 + T1(2,1)=0.0 + T1(2,2)=18.778 + T1 +\end{verbatim} +$$ +\left[ +\begin{array}{cc} +{2.3} & {9.7} \\ +{0.0} & {18.778} +\end{array} +\right] +$$ +\returnType{Type: Matrix Float} + + +%\setcounter{chapter}{0} % Appendix A + +\providecommand{\lanb}{{\tt [}} +\providecommand{\ranb}{{\tt ]}} +\providecommand{\vertline}{$|$} + +\chapter{Axiom System Commands} +\label{sec:Axiom System Commands} + +This chapter describes system commands, the command-line +facilities used to control the Axiom environment. +The first section is an introduction and discusses the common +syntax of the commands available. + +\section{Introduction} +\label{sec:Introduction} + +System commands are used to perform Axiom environment +management. +Among the commands are those that display what has been defined or +computed, set up multiple logical Axiom environments +(frames), clear definitions, read files of expressions and +commands, show what functions are available, and terminate +Axiom. + +Some commands are restricted: the commands +\index{set userlevel interpreter} +\index{set userlevel compiler} +\index{set userlevel development} +\begin{verbatim} +)set userlevel interpreter +)set userlevel compiler +)set userlevel development +\end{verbatim} +set the user-access level to the three possible choices. +All commands are available at {\tt development} level and the fewest +are available at {\tt interpreter} level. +The default user-level is {\tt interpreter}. +\index{user-level} +In addition to the {\tt )set} command +you can use the HyperDoc settings facility to change the {\it user-level.} + + +Each command listing begins with one or more syntax pattern descriptions +plus examples of related commands. +The syntax descriptions are intended to be easy to read and do not +necessarily represent the most compact way of specifying all +possible arguments and options; the descriptions may occasionally +be redundant. + +All system commands begin with a right parenthesis which should be in +the first available column of the input line (that is, immediately +after the input prompt, if any). +System commands may be issued directly to Axiom or be +included in {\bf .input} files. +\index{file!input} + +A system command {\it argument} is a word that directly +follows the command name and is not followed or preceded by a +right parenthesis. +A system command {\it option} follows the system command and +is directly preceded by a right parenthesis. +Options may have arguments: they directly follow the option. +This example may make it easier to remember what is an option and +what is an argument: + +\begin{center} +{\tt )syscmd {\it arg1 arg2} )opt1 {\it opt1arg1 opt1arg2} )opt2 {\it opt2arg1} ...} +\end{center} + +In the system command descriptions, optional arguments and options are +enclosed in brackets (``\lanb'' and ``\ranb''). +If an argument or option name is in italics, it is +meant to be a variable and must have some actual value substituted +for it when the system command call is made. +For example, the syntax pattern description + +\noindent +{\tt )read} {\it fileName} {\tt \lanb{})quietly\ranb{}} + +\noindent +would imply that you must provide an actual file name for +{\it fileName} but need not use the {\tt )quietly} option. +Thus +\begin{verbatim} +)read matrix.input +\end{verbatim} +is a valid instance of the above pattern. + +System command names and options may be abbreviated and may be in +upper or lower case. +The case of actual arguments may be significant, depending on the +particular situation (such as in file names). +System command names and options may be abbreviated to the minimum +number of starting letters so that the name or option is unique. +Thus +\begin{verbatim} +)s Integer +\end{verbatim} +is not a valid abbreviation for the {\tt )set} command, +because both {\tt )set} and {\tt )show} +begin with the letter ``s''. +Typically, two or three letters are sufficient for disambiguating names. +In our descriptions of the commands, we have used no abbreviations for +either command names or options. + +In some syntax descriptions we use a vertical line ``\vertline'' +to indicate that you must specify one of the listed choices. +For example, in +\begin{verbatim} +)set output fortran on | off +\end{verbatim} +only {\tt on} and {\tt off} are acceptable words for following +{\tt boot}. +We also sometimes use ``...'' to indicate that additional arguments +or options of the listed form are allowed. +Finally, in the syntax descriptions we may also list the syntax of +related commands. + +\section{)abbreviation} + +\index{abbreviation} + + +\par\noindent{\bf User Level Required:} compiler + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )abbreviation query \lanb{}{\it nameOrAbbrev}\ranb{}} + +\item {\tt )abbreviation category {\it abbrev fullname} \lanb{})quiet\ranb{}} + +\item {\tt )abbreviation domain {\it abbrev fullname} \lanb{})quiet\ranb{}} + +\item {\tt )abbreviation package {\it abbrev fullname} \lanb{})quiet\ranb{}} + +\item {\tt )abbreviation remove {\it nameOrAbbrev}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to query, set and remove abbreviations for category, +domain and package constructors. +Every constructor must have a unique abbreviation. +This abbreviation is part of the name of the subdirectory +under which the components of the compiled constructor are +stored. +%% BEGIN OBSOLETE +% It is this abbreviation that is used to bring compiled code into +% Axiom with the {\tt )load} command. +%% END OBSOLETE +Furthermore, by issuing this command you +let the system know what file to load automatically if you use a new +constructor. +Abbreviations must start with a letter and then be followed by +up to seven letters or digits. +Any letters appearing in the abbreviation must be in uppercase. + +When used with the {\tt query} argument, +\index{abbreviation query} +this command may be used to list the name +associated with a particular abbreviation or the abbreviation for a +constructor. +If no abbreviation or name is given, the names and corresponding +abbreviations for {\it all} constructors are listed. + +The following shows the abbreviation for the constructor {\tt List}: +\begin{verbatim} +)abbreviation query List +\end{verbatim} +The following shows the constructor name corresponding to the +abbreviation {\tt NNI}: +\begin{verbatim} +)abbreviation query NNI +\end{verbatim} +The following lists all constructor names and their abbreviations. +\begin{verbatim} +)abbreviation query +\end{verbatim} + +To add an abbreviation for a constructor, use this command with +{\tt category}, {\tt domain} or {\tt package}. +\index{abbreviation package} +\index{abbreviation domain} +\index{abbreviation category} +The following add abbreviations to the system for a +category, domain and package, respectively: +\begin{verbatim} +)abbreviation domain SET Set +)abbreviation category COMPCAT ComplexCategory +)abbreviation package LIST2MAP ListToMap +\end{verbatim} +If the {\tt )quiet} option is used, +no output is displayed from this command. +You would normally only define an abbreviation in a library source file. +If this command is issued for a constructor that has already been loaded, the +constructor will be reloaded next time it is referenced. In particular, you +can use this command to force the automatic reloading of constructors. + +To remove an abbreviation, the {\tt remove} argument is used. +\index{abbreviation remove} +This is usually +only used to correct a previous command that set an abbreviation for a +constructor name. +If, in fact, the abbreviation does exist, you are prompted +for confirmation of the removal request. +Either of the following commands +will remove the abbreviation {\tt VECTOR2} and the +constructor name {\tt VectorFunctions2} from the system: +\begin{verbatim} +)abbreviation remove VECTOR2 +)abbreviation remove VectorFunctions2 +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )compile} + +\section{)boot} + +\index{)boot} +\index{boot} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {{\tt )boot} {\it bootExpression}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to execute +expressions written in the BOOT language. +For example, +\begin{verbatim} +)boot times3(x) == 3*x +\end{verbatim} +creates and compiles the Common Lisp function ``times3'' +obtained by translating the BOOT code. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{)fin} +{\tt )lisp} \index{)lisp}, +{\tt )set} \index{)set}, and +{\tt )system} \index{)system}. + + +\section{)cd} +\index{)cd} + +\index{cd} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {{\tt )cd} {\it directory}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command sets the Axiom working current directory. +The current directory is used for looking for +input files (for {\tt )read}), +Axiom library source files (for {\tt )compile}), +saved history environment files (for {\tt )history )restore}), +compiled Axiom library files (for {\tt )library}), and +files to edit (for {\tt )edit}). +It is also used for writing +spool files (via {\tt )spool}), +writing history input files (via {\tt )history )write}) and +history environment files (via {\tt )history )save}),and +compiled Axiom library files (via {\tt )compile}). +\index{read} +\index{compile} +\index{history )restore} +\index{edit} +\index{spool} +\index{history )write} +\index{history )save} + +If issued with no argument, this command sets the Axiom +current directory to your home directory. +If an argument is used, it must be a valid directory name. +Except for the ``{\tt )}'' at the beginning of the command, +this has the same syntax as the operating system {\tt cd} command. + +\par\noindent{\bf Also See:} +{\tt )compile} \index{)compile}, +{\tt )edit} \index{)edit}, +{\tt )history} \index{)history}, +{\tt )library} \index{)library}, +{\tt )read} \index{)read}, and +{\tt )spool} \index{)spool}. + +\section{)close} +\index{)close} +\index{close} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )close} +\item{\tt )close )quietly} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to close down interpreter client processes. +Such processes are started by HyperDoc to run Axiom examples +when you click on their text. When you have finished examining or modifying the +example and you do not want the extra window around anymore, issue +\begin{verbatim} +)close +\end{verbatim} +to the Axiom prompt in the window. + +If you try to close down the last remaining interpreter client +process, Axiom will offer to close down the entire Axiom +session and return you to the operating system by displaying something +like +\begin{verbatim} + This is the last Axiom session. Do you want to kill Axiom? +\end{verbatim} +Type ``{\tt y}'' (followed by the Return key) if this is what you had in mind. +Type ``{\tt n}'' (followed by the Return key) to cancel the command. + +You can use the {\tt )quietly} option to force Axiom to +close down the interpreter client process without closing down +the entire Axiom session. + +\par\noindent{\bf Also See:} +{\tt )quit} \index{)quit} and +{\tt )pquit} \index{)pquit} + + + +\section{)clear} +\index{)clear} +\index{clear} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )clear all} +\item{\tt )clear completely} +\item{\tt )clear properties all} +\item{\tt )clear properties} {\it obj1 \lanb{}obj2 ...\ranb{}} +\item{\tt )clear value all} +\item{\tt )clear value} {\it obj1 \lanb{}obj2 ...\ranb{}} +\item{\tt )clear mode all} +\item{\tt )clear mode} {\it obj1 \lanb{}obj2 ...\ranb{}} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to remove function and variable declarations, definitions +and values from the workspace. +To empty the entire workspace and reset the +step counter to 1, issue +\begin{verbatim} +)clear all +\end{verbatim} +To remove everything in the workspace but not reset the step counter, issue +\begin{verbatim} +)clear properties all +\end{verbatim} +To remove everything about the object {\tt x}, issue +\begin{verbatim} +)clear properties x +\end{verbatim} +To remove everything about the objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear properties x y f +\end{verbatim} + +The word {\tt properties} may be abbreviated to the single letter +``{\tt p}''. +\begin{verbatim} +)clear p all +)clear p x +)clear p x y f +\end{verbatim} +All definitions of functions and values of variables may be removed by either +\begin{verbatim} +)clear value all +)clear v all +\end{verbatim} +This retains whatever declarations the objects had. To remove definitions and +values for the specific objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear value x y f +)clear v x y f +\end{verbatim} +To remove the declarations of everything while leaving the definitions and +values, issue +\begin{verbatim} +)clear mode all +)clear m all +\end{verbatim} +To remove declarations for the specific objects {\tt x, y} and {\tt f}, issue +\begin{verbatim} +)clear mode x y f +)clear m x y f +\end{verbatim} +The {\tt )display names} and {\tt )display properties} commands may be used +to see what is currently in the workspace. + +The command +\begin{verbatim} +)clear completely +\end{verbatim} +does everything that {\tt )clear all} does, and also clears the internal +system function and constructor caches. + +\par\noindent{\bf Also See:} +{\tt )display} \index{)display}, +{\tt )history} \index{)history}, and +{\tt )undo} \index{)undo}. + + +\section{)compile} +\index{)compile} +\index{compile} + + +\par\noindent{\bf User Level Required:} compiler + +\par\noindent{\bf Command Syntax:} + +\begin{list}{} +\item {\tt )compile} +\item {\tt )compile {\it fileName}} +\item {\tt )compile {\it fileName}.as} +\item {\tt )compile {\it directory/fileName}.as} +\item {\tt )compile {\it fileName}.ao} +\item {\tt )compile {\it directory/fileName}.ao} +\item {\tt )compile {\it fileName}.al} +\item {\tt )compile {\it directory/fileName}.al} +\item {\tt )compile {\it fileName}.lsp} +\item {\tt )compile {\it directory/fileName}.lsp} +\item {\tt )compile {\it fileName}.spad} +\item {\tt )compile {\it directory/fileName}.spad} +\item {\tt )compile {\it fileName} )new} +\item {\tt )compile {\it fileName} )old} +\item {\tt )compile {\it fileName} )translate} +\item {\tt )compile {\it fileName} )quiet} +\item {\tt )compile {\it fileName} )noquiet} +\item {\tt )compile {\it fileName} )moreargs} +\item {\tt )compile {\it fileName} )onlyargs} +\item {\tt )compile {\it fileName} )break} +\item {\tt )compile {\it fileName} )nobreak} +\item {\tt )compile {\it fileName} )library} +\item {\tt )compile {\it fileName} )nolibrary} +\item {\tt )compile {\it fileName} )vartrace} +\item {\tt )compile {\it fileName} )constructor} {\it nameOrAbbrev} +\end{list} + +\par\noindent{\bf Command Description:} + +\index{spad} +\index{aldor} +You use this command to invoke the {\bf Spad} compiler or the +{\bf Aldor} compiler. +The {\tt )compile} system command is actually a combination of +Axiom processing and a call to the Aldor compiler. +It is performing double-duty, acting as a front-end to +both the Aldor compiler and the Spad compiler. +(The Spad compiler is written in Lisp and is +an integral part of the Axiom environment. +The Aldor compiler is written in C and executed by the operating system +when called from within Axiom.) + +The command compiles files with file extensions {\it .as, .ao} +and {\it .al} with the +Aldor compiler and files with file extension {\it .spad} with the +Spad compiler. +It also can compile files with file extension {\it .lsp}. These +are assumed to be Lisp files genererated by the Aldor +compiler. +If you omit the file extension, the command looks to see if you +have specified the {\tt )new} or {\tt )old} option. +If you have given one of these options, the corresponding compiler +is used. +Otherwise, the command first looks in the standard system +directories for files with extension {\it .as, .ao} and {\it +.al} and then files with extension {\it .spad}. +The first file found has the appropriate compiler invoked on it. +If the command cannot find a matching file, an error message is +displayed and the command terminates. + +The {\tt )translate} option is used to invoke a special version +of the Spad compiler that will translate a {\it .spad} file +to a {\it .as} file. That is, the {\it .spad} file will be parsed and +analyzed and a file using the new syntax will be created. By default, +the {\it .as} file is created in the same directory as the +{\it .spad} file. If that directory is not writable, the current +directory is used. If the current directory is not writable, an +error message is given and the command terminates. +Note that {\tt )translate} implies the {\tt )old} option so the +file extension can safely be omitted. If {\tt )translate} is +given, all other options are ignored. +Please be aware that the translation is not necessarily one +hundred percent complete or correct. +You should attempt to compile the output with the Aldor compiler +and make any necessary corrections. + +We now describe the options for the new Aldor compiler. + +The first thing {\tt )compile} does is look for a source code +filename among its arguments. +Thus +\begin{verbatim} +)compile mycode.as +)compile /u/jones/as/mycode.as +)compile mycode +\end{verbatim} +all invoke {\tt )compiler} on the file {\tt +/u/jones/as/mycode.as} if the current Axiom working +directory is {\tt /u/jones/as.} (Recall that you can set the +working directory via the {\tt )cd} command. If you don't set it +explicitly, it is the directory from which you started +Axiom.) + +This is frequently all you need to compile your file. +This simple command: +\begin{enumerate} +\item Invokes the Aldor compiler and produces Lisp output. +\item Calls the Lisp compiler if the Aldor compilation was +successful. +\item Uses the {\tt )library} command to tell Axiom about +the contents of your compiled file and arrange to have those +contents loaded on demand. +\end{enumerate} + +Should you not want the {\tt )library} command automatically +invoked, call {\tt )compile} with the {\tt )nolibrary} option. +For example, +\begin{verbatim} +)compile mycode.as )nolibrary +\end{verbatim} + +The general description of Aldor command line arguments is in +the Aldor documentation. +The default options used by the {\tt )compile} command can be +viewed and set using the {\tt )set compiler args} Axiom +system command. +The current defaults are +\begin{verbatim} +-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom +\end{verbatim} +These options mean: +\begin{itemize} +\item {\tt -O}: perform all optimizations, +\item {\tt -Fasy}: generate a {\tt .asy} file, +\item {\tt -Fao}: generate a {\tt .ao} file, +\item {\tt -Flsp}: generate a {\tt .lsp} (Lisp) +file, +\index{Lisp!code generation} +\item {\tt -laxiom}: use the {\tt axiom} library {\tt libaxiom.al}, +\item {\tt -Mno-AXL\_W\_WillObsolete}: do not display messages +about older generated files becoming obsolete, and +\item {\tt -DAxiom}: define the global assertion {\tt Axiom} so that the +Aldor libraries for generating stand-alone code +are not accidentally used with Axiom. +\end{itemize} + +To supplement these default arguments, use the {\tt )moreargs} option on +{\tt )compile.} +For example, +\begin{verbatim} +)compile mycode.as )moreargs "-v" +\end{verbatim} +uses the default arguments and appends the {\tt -v} (verbose) +argument flag. +The additional argument specification {\bf must be enclosed in +double quotes.} + +To completely replace these default arguments for a particular +use of {\tt )compile}, use the {\tt )onlyargs} option. +For example, +\begin{verbatim} +)compile mycode.as )onlyargs "-v -O" +\end{verbatim} +only uses the {\tt -v} (verbose) and {\tt -O} (optimize) +arguments. +The argument specification {\bf must be enclosed in double quotes.} +In this example, Lisp code is not produced and so the compilation +output will not be available to Axiom. + +To completely replace the default arguments for all calls to {\tt +)compile} within your Axiom session, use {\tt )set compiler args.} +For example, to use the above arguments for all compilations, issue +\begin{verbatim} +)set compiler args "-v -O" +\end{verbatim} +Make sure you include the necessary {\tt -l} and {\tt -Y} +arguments along with those needed for Lisp file creation. +As above, {\bf the argument specification must be enclosed in double +quotes.} + +By default, the {\tt )library} system command {\it exposes} all +domains and categories it processes. +This means that the Axiom intepreter will consider those +domains and categories when it is trying to resolve a reference +to a function. +Sometimes domains and categories should not be exposed. +For example, a domain may just be used privately by another +domain and may not be meant for top-level use. +The {\tt )library} command should still be used, though, so that +the code will be loaded on demand. +In this case, you should use the {\tt )nolibrary} option on {\tt +)compile} and the {\tt )noexpose} option in the {\tt )library} +command. For example, +\begin{verbatim} +)compile mycode.as )nolibrary +)library mycode )noexpose +\end{verbatim} + +Once you have established your own collection of compiled code, +you may find it handy to use the {\tt )dir} option on the +{\tt )library} command. +This causes {\tt )library} to process all compiled code in the +specified directory. For example, +\begin{verbatim} +)library )dir /u/jones/as/quantum +\end{verbatim} +You must give an explicit directory after {\tt )dir}, even if you +want all compiled code in the current working directory +processed, e.g. +\begin{verbatim} +)library )dir . +\end{verbatim} + +The {\tt )compile} command works with several file extensions. We saw +above what happens when it is invoked on a file with extension {\tt +.as.} A {\tt .ao} file is a portable binary compiled version of a +{\tt .as} file, and {\tt )compile} simply passes the {\tt .ao} file +onto Aldor. The generated Lisp file is compiled and {\tt )library} +is automatically called, just as if you had specified a {\tt .as} file. + +A {\tt .al} file is an archive file containing {\tt .ao} files. The +archive is created (on Unix systems) with the {\tt ar} program. When +{\tt )compile} is given a {\tt .al} file, it creates a directory whose +name is based on that of the archive. For example, if you issue +\begin{verbatim} +)compile mylib.al +\end{verbatim} +the directory {\tt mylib.axldir} is created. All +members of the archive are unarchived into the +directory and {\tt )compile} is called on each {\tt .ao} file found. It +is your responsibility to remove the directory and its contents, if you +choose to do so. + +A {\tt .lsp} file is a Lisp source file, presumably, in our context, +generated by Aldor when called with the {\tt -Flsp} option. When +{\tt )compile} is used with a {\tt .lsp} file, the Lisp file is +compiled and {\tt )library} is called. You must also have present a +{\tt .asy} generated from the same source file. + +The following are descriptions of options for the Spad compiler. + +You can compile category, domain, and package constructors +contained in files with file extension {\it .spad}. +You can compile individual constructors or every constructor +in a file. + +The full filename is remembered between invocations of this command and +{\tt )edit} commands. +The sequence of commands +\begin{verbatim} +)compile matrix.spad +)edit +)compile +\end{verbatim} +will call the compiler, edit, and then call the compiler again +on the file {\bf matrix.spad.} +If you do not specify a {\it directory,} the working current +directory is searched for the file. +If the file is not found, the standard system directories are searched. + +If you do not give any options, all constructors within a file are +compiled. +Each constructor should have an {\tt )abbreviation} command in +the file in which it is defined. +We suggest that you place the {\tt )abbreviation} commands at the +top of the file in the order in which the constructors are +defined. +The list of commands serves as a table of contents for the file. +\index{abbreviation} + +The {\tt )library} option causes directories containing the +compiled code for each constructor +to be created in the working current directory. +The name of such a directory consists of the constructor +abbreviation and the {\bf .nrlib} file extension. +For example, the directory containing the compiled code for +the {\tt MATRIX} constructor is called {\bf MATRIX.nrlib.} +The {\tt )nolibrary} option says that such files should not +be created. +The default is {\tt )library.} +Note that the semantics of {\tt )library} and {\tt )nolibrary} +for the new Aldor compiler and for the Spad compiler are +completely different. + +The {\tt )vartrace} option causes the compiler to generate +extra code for the constructor to support conditional tracing of +variable assignments. Without +this option, this code is suppressed and one cannot use +the {\tt )vars} option for the trace command. + +The {\tt )constructor} option is used to +specify a particular constructor to compile. +All other constructors in the file are ignored. +The constructor name or abbreviation follows {\tt )constructor.} +Thus either +\begin{verbatim} +)compile matrix.spad )constructor RectangularMatrix +\end{verbatim} +or +\begin{verbatim} +)compile matrix.spad )constructor RMATRIX +\end{verbatim} +compiles the {\tt RectangularMatrix} constructor +defined in {\bf matrix.spad.} + +The {\tt )break} and {\tt )nobreak} options determine what +the Spad compiler does when it encounters an error. +{\tt )break} is the default and it indicates that processing +should stop at the first error. +The value of the {\tt )set break} variable then controls what happens. + + +%% BEGIN OBSOLTE +% It is important for you to realize that it does not suffice to compile a +% constructor to use the new code in the interpreter. +% After compilation, the {\tt )load} command with the +% {\tt )update} option should be used to bring in the new code +% and update internal system tables with information about the +% constructor. +%% END OBSOLTE + +\par\noindent{\bf Also See:} +{\tt )abbreviation} \index{)abbreviation}, +{\tt )edit} \index{)edit}, and +{\tt )library} \index{)library}. + + +\section{)display} +\index{)display} +\index{display} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )display all} +\item {\tt )display properties} +\item {\tt )display properties all} +\item {\tt )display properties} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display value all} +\item {\tt )display value} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display mode all} +\item {\tt )display mode} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}} +\item {\tt )display names} +\item {\tt )display operations} {\it opName} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to display the contents of the workspace and +signatures of functions with a given name. + +The command +\begin{verbatim} +)display names +\end{verbatim} +lists the names of all user-defined objects in the workspace. This is useful +if you do not wish to see everything about the objects and need only be +reminded of their names. + +The commands +\begin{verbatim} +)display all +)display properties +)display properties all +\end{verbatim} +all do the same thing: show the values and types and declared modes of all +variables in the workspace. If you have defined functions, their signatures +and definitions will also be displayed. + +To show all information about a particular variable or user functions, +for example, something named {\tt d}, issue +\begin{verbatim} +)display properties d +\end{verbatim} +To just show the value (and the type) of {\tt d}, issue +\begin{verbatim} +)display value d +\end{verbatim} +To just show the declared mode of {\tt d}, issue +\begin{verbatim} +)display mode d +\end{verbatim} + +All modemaps for a given operation may be +displayed by using {\tt )display operations}. +A {\it modemap} is a collection of information about a particular +reference +to an operation. This includes the types of the arguments and the return +value, the location of the implementation and any conditions on the types. +The modemap may contain patterns. The following displays the modemaps for the +operation \spadfunFrom{complex}{ComplexCategory}: +\begin{verbatim} +)d op complex +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )clear} \index{)clear}, +{\tt )history} \index{)history}, +{\tt )set} \index{)set}, +{\tt )show} \index{)show}, and +{\tt )what} \index{)what}. + + +\section{)edit} +\index{)edit} +\index{edit} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{{\tt )edit} [{\it filename}]} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to edit files. +It works in conjunction with the {\tt )read} +and {\tt )compile} commands to remember the name +of the file on which you are working. +By specifying the name fully, you can edit any file you wish. +Thus +\begin{verbatim} +)edit /u/julius/matrix.input +\end{verbatim} +will place you in an editor looking at the file +{\tt /u/julius/matrix.input}. +\index{editing files} +By default, the editor is {\tt vi}, +\index{vi} +but if you have an EDITOR shell environment variable defined, that editor +will be used. +When Axiom is running under the X Window System, +it will try to open a separate {\tt xterm} running your editor if +it thinks one is necessary. +\index{Korn shell} +For example, under the Korn shell, if you issue +\begin{verbatim} +export EDITOR=emacs +\end{verbatim} +then the emacs +\index{emacs} +editor will be used by {\tt )edit}. + +If you do not specify a file name, the last file you edited, +read or compiled will be used. +If there is no ``last file'' you will be placed in the editor editing +an empty unnamed file. + +It is possible to use the {\tt )system} command to edit a file directly. +For example, +\begin{verbatim} +)system emacs /etc/rc.tcpip +\end{verbatim} +calls {\tt emacs} to edit the file. +\index{emacs} + +\par\noindent{\bf Also See:} +{\tt )system} \index{)system}, +{\tt )compile} \index{)compile}, and +{\tt )read} \index{)read}. + +\section{)fin} +\index{)fin} +\index{fin} + + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )fin} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used by Axiom +developers to leave the Axiom system and return +to the underlying Common Lisp system. +To return to Axiom, issue the +``{\tt (|spad|)}'' +function call to Common Lisp. + +\par\noindent{\bf Also See:} +{\tt )pquit} \index{)pquit} and +{\tt )quit} \index{)quit}. + +\section{)frame} +\label{sec:frame} +\index{)frame} +\index{frame} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )frame new {\it frameName}} + +\item{\tt )frame drop [{\it frameName}]} + +\item{\tt )frame next} + +\item{\tt )frame last} + +\item{\tt )frame names} + +\item{\tt )frame import {\it frameName} + {\it {\tt [}objectName1 {\tt [}objectName2 ...{\tt]]}}} + +\item{\tt )set message frame on | off} + +\item{\tt )set message prompt frame} +\end{list} + +\par\noindent{\bf Command Description:} + +A {\it frame} can be thought of as a logical session within the +physical session that you get when you start the system. You can +have as many frames as you want, within the limits of your computer's +storage, paging space, and so on. +Each frame has its own {\it step number}, {\it environment} and {\it history.} +You can have a variable named {\tt a} in one frame and it will +have nothing to do with anything that might be called {\tt a} in +any other frame. + +Some frames are created by the HyperDoc program and these can +have pretty strange names, since they are generated automatically. +\index{frame names} +To find out the names +of all frames, issue +\begin{verbatim} +)frame names +\end{verbatim} +It will indicate the name of the current frame. + +You create a new frame +\index{frame new} +``{\bf quark}'' by issuing +\begin{verbatim} +)frame new quark +\end{verbatim} +The history facility can be turned on by issuing either +{\tt )set history on} or {\tt )history )on}. +If the history facility is on and you are saving history information +in a file rather than in the Axiom environment +then a history file with filename {\bf quark.axh} will +be created as you enter commands. +If you wish to go back to what +you were doing in the +\index{frame next} +``{\bf initial}'' frame, use +\index{frame last} +\begin{verbatim} +)frame next +\end{verbatim} +or +\begin{verbatim} +)frame last +\end{verbatim} +to cycle through the ring of available frames to get back to +``{\bf initial}''. + +If you want to throw +away a frame (say ``{\bf quark}''), issue +\begin{verbatim} +)frame drop quark +\end{verbatim} +If you omit the name, the current frame is dropped. +\index{frame drop} + +If you do use frames with the history facility on and writing to a file, +you may want to delete some of the older history files. +\index{file!history} +These are directories, so you may want to issue a command like +{\tt rm -r quark.axh} to the operating system. + +You can bring things from another frame by using +\index{frame import} +{\tt )frame import}. +For example, to bring the {\tt f} and {\tt g} from the frame ``{\bf quark}'' +to the current frame, issue +\begin{verbatim} +)frame import quark f g +\end{verbatim} +If you want everything from the frame ``{\bf quark}'', issue +\begin{verbatim} +)frame import quark +\end{verbatim} +You will be asked to verify that you really want everything. + +There are two {\tt )set} flags +\index{set message frame} +to make it easier to tell where you are. +\begin{verbatim} +)set message frame on | off +\end{verbatim} +will print more messages about frames when it is set on. +By default, it is off. +\begin{verbatim} +)set message prompt frame +\end{verbatim} +will give a prompt +\index{set message prompt frame} +that looks like +\begin{verbatim} +initial (1) -> +\end{verbatim} +\index{prompt!with frame name} +when you start up. In this case, the frame name and step make up the +prompt. + +\par\noindent{\bf Also See:} +{\tt )history} \index{)history} and +{\tt )set} \index{)set}. + +\section{)hd} +\index{)hd} +\index{hd} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )hd} +\end{list} + +\par\noindent{\bf Command Description:} + +This command will start the HyperDoc facility if it is not running. +Note that if it issues the message: +\begin{verbatim} +binding UNIX server socket: Address already in use +(HyperDoc) Warning: Not connected to AXIOM Server! +\end{verbatim} +then you probably already had HyperDoc running and the new +copy cannot connect. In this circumstance HyperDoc will still +work but certain interactions with Axiom will not, such as the +Basic Commands facility. + +\section{)help} +\index{)help} +\index{help} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )help} +\item{\tt )help} {\it commandName} +\end{list} + +\par\noindent{\bf Command Description:} + +This command displays help information about system commands. +If you issue +\begin{verbatim} +)help +\end{verbatim} +then this very text will be shown. +You can also give the name or abbreviation of a system command +to display information about it. +For example, +\begin{verbatim} +)help clear +\end{verbatim} +will display the description of the {\tt )clear} system command. + +All this material is available in the Axiom User Guide +and in HyperDoc. +In HyperDoc, choose the {\bf Commands} item from the +{\bf Reference} menu. + + + +\section{)history} +\index{)history} +\index{history} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )history )on} + +\item{\tt )history )off} + +\item{{\tt )history )write} {\it historyInputFileName}} + +\item{\tt )history )show [{\it n}] [both]} + +\item{{\tt )history )save} {\it savedHistoryName}} + +\item{{\tt )history )restore} [{\it savedHistoryName}]} + +\item{\tt )history )reset} + +\item{{\tt )history )change} {\it n}} + +\item{\tt )history )memory} + +\item{\tt )history )file} + +\item{\tt \%} + +\item{\tt \%\%({\it n})} + +\item{\tt )set history on | off} +\end{list} + +\par\noindent{\bf Command Description:} + +The {\it history} facility within Axiom allows you to restore your +environment to that of another session and recall previous +computational results. +Additional commands allow you to review previous +input lines and to create an {\bf .input} file of the lines typed to +\index{file!input} +Axiom. + +Axiom saves your input and output if the history facility is +turned on (which is the default). +This information is saved if either of +\begin{verbatim} +)set history on +)history )on +\end{verbatim} +has been issued. +Issuing either +\begin{verbatim} +)set history off +)history )off +\end{verbatim} +will discontinue the recording of information. +\index{history )on} +\index{set history on} +\index{set history off} +\index{history )off} + +Whether the facility is disabled or not, +the value of {\tt \%} in Axiom always +refers to the result of the last computation. +If you have not yet entered anything, +{\tt \%} evaluates to an object of type +{\tt Variable('\%)}. +The function {\tt \%\%} may be used to refer +to other previous results if the history facility is enabled. +In that case, +{\tt \%\%(n)} is the output from step {\tt n} if {\tt n > 0}. +If {\tt n < 0}, the step is computed relative to the current step. +Thus {\tt \%\%(-1)} is also the previous step, +{\tt \%\%(-2)}, is the step before that, and so on. +If an invalid step number is given, Axiom will signal an error. + +The {\it environment} information can either be saved in a file or entirely in +memory (the default). +Each frame has its own history database. +When it is kept in a file, some of it may also be kept in memory for +efficiency. +When the information is saved in a file, the name of the file is +of the form {\bf FRAME.axh} where ``{\bf FRAME}'' is the name of the +current frame. +The history file is placed in the current working directory. +Note that these history database files are not text files (in fact, +they are directories themselves), and so are not in human-readable +format. + +The options to the {\tt )history} command are as follows: + +\begin{description} +\item[{\tt )change} {\it n}] +will set the number of steps that are saved in memory to {\it n}. +This option only has effect when the history data is maintained in a +file. +If you have issued {\tt )history )memory} (or not changed the default) +there is no need to use {\tt )history )change}. +\index{history )change} + +\item[{\tt )on}] +will start the recording of information. +If the workspace is not empty, you will be asked to confirm this +request. +If you do so, the workspace will be cleared and history data will begin +being saved. +You can also turn the facility on by issuing {\tt )set history on}. + +\item[{\tt )off}] +will stop the recording of information. +The {\tt )history )show} command will not work after issuing this +command. +Note that this command may be issued to save time, as there is some +performance penalty paid for saving the environment data. +You can also turn the facility off by issuing {\tt )set history off}. + +\item[{\tt )file}] +indicates that history data should be saved in an external file on disk. + +\item[{\tt )memory}] +indicates that all history data should be kept in memory rather than +saved in a file. +Note that if you are computing with very large objects it may not be +practical to kept this data in memory. + +\item[{\tt )reset}] +will flush the internal list of the most recent workspace calculations +so that the data structures may be garbage collected by the underlying +Common Lisp system. +Like {\tt )history )change}, this option only has real effect when +history data is being saved in a file. + +\item[{\tt )restore} [{\it savedHistoryName}]] +completely clears the environment and restores it to a saved session, if +possible. +The {\tt )save} option below allows you to save a session to a file +with a given name. If you had issued +{\tt )history )save jacobi} +the command +{\tt )history )restore jacobi} +would clear the current workspace and load the contents of the named +saved session. If no saved session name is specified, the system looks +for a file called {\bf last.axh}. + +\item[{\tt )save} {\it savedHistoryName}] +is used to save a snapshot of the environment in a file. +This file is placed in the current working directory. +Use {\tt )history )restore} to restore the environment to the state +preserved in the file. +This option also creates an input file containing all the lines of input +since you created the workspace frame (for example, by starting your +Axiom session) or last did a {\tt )clear all} or +{\tt )clear completely}. + +\item[{\tt )show} [{\it n}] [{\tt both}]] +can show previous input lines and output results. +{\tt )show} will display up to twenty of the last input lines +(fewer if you haven't typed in twenty lines). +{\tt )show} {\it n} will display up to {\it n} of the last input lines. +{\tt )show both} will display up to five of the last input lines and +output results. +{\tt )show} {\it n} {\tt both} will display up to {\it n} of the last +input lines and output results. + +\item[{\tt )write} {\it historyInputFile}] +creates an {\bf .input} file with the input lines typed since the start +of the session/frame or the last {\tt )clear all} or {\tt )clear +completely}. +If {\it historyInputFileName} does not contain a period (``.'') in the filename, +{\bf .input} is appended to it. +For example, +{\tt )history )write chaos} +and +{\tt )history )write chaos.input} +both write the input lines to a file called {\bf chaos.input} in your +current working directory. +If you issued one or more {\tt )undo} commands, +{\tt )history )write} +eliminates all +input lines backtracked over as a result of {\tt )undo}. +You can edit this file and then use {\tt )read} to have Axiom process +the contents. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )frame} \index{)frame}, +{\tt )read} \index{)read}, +{\tt )set} \index{)set}, and +{\tt )undo} \index{)undo}. + +\section{)library} +\index{)library} +\index{library} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )library {\it libName1 {\tt [}libName2 ...{\tt ]}}} + +\item{\tt )library )dir {\it dirName}} + +\item{\tt )library )only {\it objName1 {\tt [}objlib2 ...{\tt ]}}} + +\item{\tt )library )noexpose} +\end{list} + +\par\noindent{\bf Command Description:} + +This command replaces the {\tt )load} system command that +was available in Axiom releases before version 2.0. +The {\tt )library} command makes available to Axiom the compiled +objects in the libraries listed. + +For example, if you {\tt )compile dopler.as} in your home +directory, issue {\tt )library dopler} to have Axiom look +at the library, determine the category and domain constructors present, +update the internal database with various properties of the +constructors, and arrange for the constructors to be +automatically loaded when needed. +If the {\tt )noexpose} option has not been given, the +constructors will be exposed (that is, available) in the current +frame. + +If you compiled a file with the Spad compiler, you will +have an {\it nrlib} present, for example, {\it DOPLER.nrlib,} +where {\tt DOPLER} is a constructor abbreviation. +The command {\tt )library DOPLER} will then do the analysis and +database updates as above. + +To tell the system about all libraries in a directory, use +{\tt )library )dir dirName} where {\tt dirName} is an explicit +directory. +You may specify ``.'' as the directory, which means the current +directory from which you started the system or the one you set +via the {\tt )cd} command. The directory name is required. + +You may only want to tell the system about particular +constructors within a library. In this case, use the {\tt )only} +option. The command {\tt )library dopler )only Test1} will only +cause the {\sf Test1} constructor to be analyzed, autoloaded, +etc.. + +Finally, each constructor in a library are usually automatically exposed when the +{\tt )library} command is used. Use the {\tt )noexpose} +option if you not want them exposed. At a later time you can use +{\tt )set expose add constructor} to expose any hidden +constructors. + +\par\noindent{\bf Also See:} +{\tt )cd} \index{)cd}, +{\tt )compile} \index{)compile}, +{\tt )frame} \index{)frame}, and +{\tt )set} \index{)set}. + +\section{)lisp} +\index{)lisp} +\index{lisp} + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {{\tt )lisp} [{\it lispExpression}]} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to have single +expressions evaluated by the Common Lisp system on which +Axiom is built. +The {\it lispExpression} is read by the Common Lisp reader and +evaluated. +If this expression is not complete (unbalanced parentheses, say), the reader +will wait until a complete expression is entered. + +Since this command is only useful for evaluating single expressions, the +{\tt )fin} command may be used to drop out of Axiom into Common Lisp. + +\par\noindent{\bf Also See:} +{\tt )system} \index{)system}, +{\tt )boot} \index{)boot}, and +{\tt )fin} \index{)fin}. + +\section{)ltrace} +\index{)ltrace} +\index{ltrace} + +\par\noindent{\bf User Level Required:} development + +\par\noindent{\bf Command Syntax:} + +This command has the same arguments as options as the +{\tt )trace} command. + +\par\noindent{\bf Command Description:} + +This command is used by Axiom system developers to trace +Common Lisp or +BOOT functions. +It is not supported for general use. + +\par\noindent{\bf Also See:} +{\tt )boot} \index{)boot}, +{\tt )lisp} \index{)lisp}, and +{\tt )trace} \index{)trace}. + +\section{)pquit} +\index{)pquit} +\index{pquit} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )pquit} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to terminate Axiom and return to the +operating system. +Other than by redoing all your computations or by +using the {\tt )history )restore} +command to try to restore your working environment, +you cannot return to Axiom in the same state. + +{\tt )pquit} differs from the {\tt )quit} in that it always asks for +confirmation that you want to terminate Axiom (the ``p'' is for +``protected''). +\index{quit} +When you enter the {\tt )pquit} command, Axiom responds +% +\begin{center} + Please enter y or yes if you really want to leave the interactive \\ + environment and return to the operating system: +\end{center} +% +If you respond with {\tt y} or {\tt yes}, +Axiom will terminate and return you to the operating +system (or the environment from which you invoked the system). +If you responded with something other than {\tt y} or {\tt yes}, then +the message +% +\begin{center} +You have chosen to remain in the Axiom interactive environment. +\end{center} +% +will be displayed and, indeed, Axiom would still be running. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{)fin}, +{\tt )history} \index{)history}, +{\tt )close} \index{)close}, +{\tt )quit} \index{)quit}, and +{\tt )system} \index{)system}. + +\section{)quit} +\index{)quit} +\index{quit} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )quit} +\item{\tt )set quit protected | unprotected} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to terminate Axiom and return to the +operating system. +Other than by redoing all your computations or by +using the {\tt )history )restore} +command to try to restore your working environment, +you cannot return to Axiom in the same state. + +{\tt )quit} differs from the {\tt )pquit} in that it asks for +\index{pquit} +confirmation only if the command +\begin{verbatim} +)set quit protected +\end{verbatim} +has been issued. +\index{set quit protected} +Otherwise, {\tt )quit} will make Axiom terminate and return you +to the operating system (or the environment from which you invoked the +system). + +The default setting is {\tt )set quit protected} so that {\tt )quit} +and {\tt )pquit} behave in the same way. +If you do issue +\begin{verbatim} +)set quit unprotected +\end{verbatim} +we +\index{set quit unprotected} +suggest that you do not (somehow) assign {\tt )quit} to be +executed when you press, say, a function key. + +\par\noindent{\bf Also See:} +{\tt )fin} \index{)fin}, +{\tt )history} \index{)history}, +{\tt )close} \index{)close}, +{\tt )pquit} \index{)pquit}, and +{\tt )system} \index{)system}. + +\section{)read} +\index{)read} +\index{read} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {{\tt )read} [{\it fileName}]} + +\item {{\tt )read} [{\it fileName}] [{\tt )quiet}] [{\tt )ifthere}]} +\end{list} +\par\noindent{\bf Command Description:} + +This command is used to read {\bf .input} files into Axiom. +\index{file!input} +The command +\begin{verbatim} +)read matrix.input +\end{verbatim} +will read the contents of the file {\bf matrix.input} into +Axiom. +The ``.input'' file extension is optional. + +This command remembers the previous file you edited, read or compiled. +If you do not specify a file name, the previous file will be read. + +The {\tt )ifthere} option checks to see whether the {\bf .input} file +exists. +If it does not, the {\tt )read} command does nothing. +If you do not use this option and the file does not exist, +you are asked to give the name of an existing {\bf .input} file. + +The {\tt )quiet} option suppresses output while the file is being read. + +\par\noindent{\bf Also See:} +{\tt )compile} \index{)compile}, +{\tt )edit} \index{)edit}, and +{\tt )history} \index{)history}. + +\section{)set} +\index{)set} +\index{set} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )set} +\item {\tt )set} {\it label1 {\tt [}... labelN{\tt ]}} +\item {\tt )set} {\it label1 {\tt [}... labelN{\tt ]} newValue} +\end{list} +\par\noindent{\bf Command Description:} + +The {\tt )set} command is used to view or set system variables that +control what messages are displayed, the type of output desired, the +status of the history facility, the way Axiom user functions are +cached, and so on. +Since this collection is very large, we will not discuss them here. +Rather, we will show how the facility is used. +We urge you to explore the {\tt )set} options to familiarize yourself +with how you can modify your Axiom working environment. +There is a HyperDoc version of this same facility available from the +main HyperDoc menu. + + +The {\tt )set} command is command-driven with a menu display. +It is tree-structured. +To see all top-level nodes, issue {\tt )set} by itself. +\begin{verbatim} +)set +\end{verbatim} +Variables with values have them displayed near the right margin. +Subtrees of selections have ``{\tt ...}'' +displayed in the value field. +For example, there are many kinds of messages, so issue +{\tt )set message} to see the choices. +\begin{verbatim} +)set message +\end{verbatim} +The current setting for the variable that displays +\index{computation timings!displaying} +whether computation times +\index{timings!displaying} +are displayed is visible in the menu displayed by the last command. +To see more information, issue +\begin{verbatim} +)set message time +\end{verbatim} +This shows that time printing is on now. +To turn it off, issue +\begin{verbatim} +)set message time off +\end{verbatim} +\index{set message time} + +As noted above, not all settings have so many qualifiers. +For example, to change the {\tt )quit} command to being unprotected +(that is, you will not be prompted for verification), you need only issue +\begin{verbatim} +)set quit unprotected +\end{verbatim} +\index{set quit unprotected} + +\par\noindent{\bf Also See:} +{\tt )quit} \index{)quit}. + +\section{)show} +\index{)show} +\index{show} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item {\tt )show {\it nameOrAbbrev}} + +\item {\tt )show {\it nameOrAbbrev} )operations} + +\item {\tt )show {\it nameOrAbbrev} )attributes} +\end{list} + +\par\noindent{\bf Command Description:} +This command displays information about Axiom +domain, package and category {\it constructors}. +If no options are given, then the {\tt )operations} option is assumed. +For example, +\begin{verbatim} +)show POLY +)show POLY )operations +)show Polynomial +)show Polynomial )operations +\end{verbatim} +each display basic information about the +{\tt Polynomial} domain constructor and then provide a +listing of operations. +Since {\tt Polynomial} requires a {\tt Ring} (for example, +{\tt Integer}) as argument, the above commands all refer +to a unspecified ring {\tt R}. +In the list of operations, {\tt \$} means +{\tt Polynomial(R)}. + +The basic information displayed includes the {\it signature} +of the constructor (the name and arguments), the constructor +{\it abbreviation}, the {\it exposure status} of the constructor, and the +name of the {\it library source file} for the constructor. + +If operation information about a specific domain is wanted, +the full or abbreviated domain name may be used. +For example, +\begin{verbatim} +)show POLY INT +)show POLY INT )operations +)show Polynomial Integer +)show Polynomial Integer )operations +\end{verbatim} +are among the combinations that will +display the operations exported by the +domain {\tt Polynomial(Integer)} (as opposed to the general +{\it domain constructor} {\tt Polynomial}). +Attributes may be listed by using the {\tt )attributes} option. + +\par\noindent{\bf Also See:} +{\tt )display} \index{)display}, +{\tt )set} \index{)set}, and +{\tt )what} \index{)what}. + +\section{)spool} +\index{)spool} +\index{spool} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{{\tt )spool} [{\it fileName}]} +\item{\tt )spool} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to save {\it (spool)} all Axiom input and output +\index{file!spool} +into a file, called a {\it spool file.} +You can only have one spool file active at a time. +To start spool, issue this command with a filename. For example, +\begin{verbatim} +)spool integrate.out +\end{verbatim} +To stop spooling, issue {\tt )spool} with no filename. + +If the filename is qualified with a directory, then the output will +be placed in that directory. +If no directory information is given, the spool file will be placed in the +\index{directory!for spool files} +{\it current directory.} +The current directory is the directory from which you started +Axiom or is the directory you specified using the +{\tt )cd} command. +\index{cd} + +\par\noindent{\bf Also See:} +{\tt )cd} \index{)cd}. + +\section{)synonym} +\index{)synonym} +\index{synonym} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )synonym} + +\item{{\tt )synonym} {\it synonym fullCommand}} + +\item{\tt )what synonyms} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to create short synonyms for system command expressions. +For example, the following synonyms might simplify commands you often +use. +\begin{verbatim} +)synonym save history )save +)synonym restore history )restore +)synonym mail system mail +)synonym ls system ls +)synonym fortran set output fortran +\end{verbatim} +Once defined, synonyms can be +used in place of the longer command expressions. +Thus +\begin{verbatim} +)fortran on +\end{verbatim} +is the same as the longer +\begin{verbatim} +)set fortran output on +\end{verbatim} +To list all defined synonyms, issue either of +\begin{verbatim} +)synonyms +)what synonyms +\end{verbatim} +To list, say, all synonyms that contain the substring +``{\tt ap}'', issue +\begin{verbatim} +)what synonyms ap +\end{verbatim} + +\par\noindent{\bf Also See:} +{\tt )set} \index{)set} and +{\tt )what} \index{)what}. + +\section{)system} +\index{)system} +\index{system} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{{\tt )system} {\it cmdExpression}} +\end{list} + +\par\noindent{\bf Command Description:} + +This command may be used to issue commands to the operating system while +remaining in Axiom. +The {\it cmdExpression} is passed to the operating system for +execution. + +To get an operating system shell, issue, for example, +{\tt )system sh}. +When you enter the key combination, +\fbox{\bf Ctrl}--\fbox{\bf D} +(pressing and holding the +\fbox{\bf Ctrl} key and then pressing the +\fbox{\bf D} key) +the shell will terminate and you will return to Axiom. +We do not recommend this way of creating a shell because +Common Lisp may field some interrupts instead of the shell. +If possible, use a shell running in another window. + +If you execute programs that misbehave you may not be able to return to +Axiom. +If this happens, you may have no other choice than to restart +Axiom and restore the environment via {\tt )history )restore}, if +possible. + +\par\noindent{\bf Also See:} +{\tt )boot} \index{)boot}, +{\tt )fin} \index{)fin}, +{\tt )lisp} \index{)lisp}, +{\tt )pquit} \index{)pquit}, and +{\tt )quit} \index{)quit}. + +\section{)trace} +\index{)trace} +\index{trace} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )trace} + +\item{\tt )trace )off} + +\item{{\tt )trace} {\it function {\tt [}options{\tt ]}}} + +\item{{\tt )trace} {\it constructor {\tt [}options {\tt ]}}} + +\item{{\tt )trace} {\it domainOrPackage {\tt [}options {\tt ]}}} +\end{list} +% +where options can be one or more of +% +\begin{list}{} +\item{{\tt )after} {\it S-expression}} + +\item{{\tt )before} {\it S-expression}} + +\item{\tt )break after} + +\item{\tt )break before} + +\item{{\tt )cond} {\it S-expression}} + +\item{\tt )count} + +\item{{\tt )count} {\it n}} + +\item{{\tt )depth} {\it n}} + +\item{{\tt )local} {\it op1 {\tt [}... opN {\tt ]}}} + +\item{\tt )nonquietly} + +\item{\tt )nt} + +\item{\tt )off} + +\item{{\tt )only} {\it listOfDataToDisplay}} + +\item{\tt )ops} + +\item{{\tt )ops} {\it op1 {\tt [}... opN {\tt ]}}} + +\item{\tt )restore} + +\item{\tt )stats} + +\item{\tt )stats reset} + +\item{\tt )timer} + +\item{\tt )varbreak} + +\item{{\tt )varbreak} {\it var1 {\tt [}... varN {\tt ]}}} + +\item{\tt )vars} + +\item{{\tt )vars} {\it var1 {\tt [}... varN {\tt ]}}} + +\item{\tt )within} {\it executingFunction} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to trace the execution of functions that make +up the Axiom system, functions defined by users, +and functions from the system library. +Almost all options are available for each type of function but +exceptions will be noted below. + +To list all functions, constructors, domains and packages that are +traced, )issue +\begin{verbatim} +)trace +\end{verbatim} +To untrace everything that is traced, issue +\begin{verbatim} +)trace )off +\end{verbatim} +When a function is traced, the default system action is to display +the arguments to the function and the return value when the +function is exited. +Note that if a function is left via an action such as a {\tt THROW}, no +return value will be displayed. +Also, optimization of tail recursion may decrease the number of +times a function is actually invoked and so may cause less trace +information to be displayed. +Other information can be displayed or collected when a function is +traced and this is controlled by the various options. +Most options will be of interest only to Axiom system +developers. +If a domain or package is traced, the default action is to trace +all functions exported. + +Individual interpreter, lisp or boot +functions can be traced by listing their names after +{\tt )trace}. +Any options that are present must follow the functions to be +traced. +\begin{verbatim} +)trace f +\end{verbatim} +traces the function {\tt f}. +To untrace {\tt f}, issue +\begin{verbatim} +)trace f )off +\end{verbatim} +Note that if a function name contains a special character, it will +be necessary to escape the character with an underscore +% +\begin{verbatim} +)trace _/D_,1 +\end{verbatim} +% +To trace all domains or packages that are or will be created from a particular +constructor, give the constructor name or abbreviation after +{\tt )trace}. +% +\begin{verbatim} +)trace MATRIX +)trace List Integer +\end{verbatim} +% +The first command traces all domains currently instantiated with +{\tt Matrix}. +If additional domains are instantiated with this constructor +(for example, if you have used {\tt Matrix(Integer)} and +{\tt Matrix(Float)}), they will be automatically traced. +The second command traces {\tt List(Integer)}. +It is possible to trace individual functions in a domain or +package. +See the {\tt )ops} option below. + +The following are the general options for the {\tt )trace} +command. + +%!! system command parser doesn't treat general s-expressions correctly, +%!! I recommand not documenting )after )before and )cond +\begin{description} +%\item[{\tt )after} {\it S-expression}] +%causes the given Common Lisp {\it S-expression} to be +%executed after exiting the traced function. + +%\item[{\tt )before} {\it S-expression}] +%causes the given Common Lisp {\it S-expression} to be +%executed before entering the traced function. + +\item[{\tt )break after}] +causes a Common Lisp break loop to be entered after +exiting the traced function. + +\item[{\tt )break before}] +causes a Common Lisp break loop to be entered before +entering the traced function. + +\item[{\tt )break}] +is the same as {\tt )break before}. + +%\item[{\tt )cond} {\it S-expression}] +%causes trace information to be shown only if the given +%Common Lisp {\it S-expression} evaluates to non-NIL. For +%example, the following command causes the system function +%{\tt resolveTT} to be traced but to have the information +%displayed only if the value of the variable +%{\tt \$reportBottomUpFlag} is non-NIL. +%\begin{verbatim} +%)trace resolveTT )cond \_\$reportBottomUpFlag} +%\end{verbatim} + +\item[{\tt )count}] +causes the system to keep a count of the number of times the +traced function is entered. The total can be displayed with +{\tt )trace )stats} and cleared with {\tt )trace )stats reset}. + +\item[{\tt )count} {\it n}] +causes information about the traced function to be displayed for +the first {\it n} executions. After the {\it n}-th execution, the +function is untraced. + +\item[{\tt )depth} {\it n}] +causes trace information to be shown for only {\it n} levels of +recursion of the traced function. The command +\begin{verbatim} +)trace fib )depth 10 +\end{verbatim} +will cause the display of only 10 levels of trace information for +the recursive execution of a user function {\bf fib}. + +\item[{\tt )math}] +causes the function arguments and return value to be displayed in the +Axiom monospace two-dimensional math format. + +\item[{\tt )nonquietly}] +causes the display of additional messages when a function is +traced. + +\item[{\tt )nt}] +This suppresses all normal trace information. This option is +useful if the {\tt )count} or {\tt )timer} options are used and +you are interested in the statistics but not the function calling +information. + +\item[{\tt )off}] +causes untracing of all or specific functions. Without an +argument, all functions, constructors, domains and packages are +untraced. Otherwise, the given functions and other objects +are untraced. To +immediately retrace the untraced functions, issue {\tt )trace +)restore}. + +\item[{\tt )only} {\it listOfDataToDisplay}] +causes only specific trace information to be shown. The items are +listed by using the following abbreviations: +\begin{description} +\item[a] display all arguments +\item[v] display return value +\item[1] display first argument +\item[2] display second argument +\item[15] display the 15th argument, and so on +\end{description} +\end{description} +\begin{description} + +\item[{\tt )restore}] +causes the last untraced functions to be retraced. If additional +options are present, they are added to those previously in effect. + +\item[{\tt )stats}] +causes the display of statistics collected by the use of the +{\tt )count} and {\tt )timer} options. + +\item[{\tt )stats reset}] +resets to 0 the statistics collected by the use of the +{\tt )count} and {\tt )timer} options. + +\item[{\tt )timer}] +causes the system to keep a count of execution times for the +traced function. The total can be displayed with {\tt )trace +)stats} and cleared with {\tt )trace )stats reset}. + +%!! only for lisp, boot, may not work in any case, recommend removing +%\item[{\tt )varbreak}] +%causes a Common Lisp break loop to be entered after +%the assignment to any variable in the traced function. + +\item[{\tt )varbreak} {\it var1 {\tt [}... varN {\tt ]}}] +causes a Common Lisp break loop to be entered after +the assignment to any of the listed variables in the traced +function. + +\item[{\tt )vars}] +causes the display of the value of any variable after it is +assigned in the traced function. +Note that library code must +have been compiled using the {\tt )vartrace} option in order +to support this option. + +\item[{\tt )vars} {\it var1 {\tt [}... varN {\tt ]}}] +causes the display of the value of any of the specified variables +after they are assigned in the traced function. +Note that library code must +have been compiled using the {\tt )vartrace} option in order +to support this option. + +\item[{\tt )within} {\it executingFunction}] +causes the display of trace information only if the traced +function is called when the given {\it executingFunction} is running. +\end{description} + +The following are the options for tracing constructors, domains +and packages. + +\begin{description} +\item[{\tt )local} {\it {\tt [} op1 {\tt [}\ldots opN{\tt ]]}}] +causes local functions of the constructor to be traced. Note that +to untrace an individual local function, you must use the fully +qualified internal name, using the escape character +{\tt \_} before the semicolon. +\begin{verbatim} +)trace FRAC )local +)trace FRAC_;cancelGcd )off +\end{verbatim} + +\item[{\tt )ops} {\it op1 {\tt [}... opN {\tt ]}}] +By default, all operations from a domain or package are traced +when the domain or package is traced. This option allows you to +specify that only particular operations should be traced. The +command +% +\begin{verbatim} +)trace Integer )ops min max _+ _- +\end{verbatim} +% +traces four operations from the domain {\tt Integer}. Since +{\tt +} and {\tt -} are special +characters, it is necessary +to escape them with an underscore. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )boot} \index{)boot}, +{\tt )lisp} \index{)lisp}, and +{\tt )ltrace} \index{)ltrace}. + +\section{)undo} +\index{)undo} +\index{undo} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )undo} +\item{\tt )undo} {\it integer} +\item{\tt )undo} {\it integer {\tt [}option {\tt ]}} +\item{\tt )undo} {\tt )redo} +\end{list} +% +where {\it option} is one of +% +\begin{list}{} +\item{\tt )after} + +\item{\tt )before} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to +restore the state of the user environment to an earlier +point in the interactive session. +The argument of an {\tt )undo} is an integer which must designate some +step number in the interactive session. + +\begin{verbatim} +)undo n +)undo n )after +\end{verbatim} +These commands return the state of the interactive +environment to that immediately after step {\tt n}. +If {\tt n} is a positive number, then {\tt n} refers to step nummber +{\tt n}. If {\tt n} is a negative number, it refers to the {\tt n}-th +previous command (that is, undoes the effects of the last $-n$ +commands). + +A {\tt )clear all} resets the {\tt )undo} facility. +Otherwise, an {\tt )undo} undoes the effect of {\tt )clear} with +options {\tt properties}, {\tt value}, and {\tt mode}, and +that of a previous {\tt undo}. +If any such system commands are given between steps $n$ and +$n + 1$ ($n > 0$), their effect is undone +for {\tt )undo m} for any $0 < m \leq n$.. + +The command {\tt )undo} is equivalent to {\tt )undo -1} (it undoes +the effect of the previous user expression). +The command {\tt )undo 0} undoes any of the above system commands +issued since the last user expression. + +\begin{verbatim} +)undo n )before +\end{verbatim} +This command returns the state of the interactive +environment to that immediately before step {\tt n}. +Any {\tt )undo} or {\tt )clear} system commands +given before step {\tt n} will not be undone. + +\begin{verbatim} +)undo )redo +\end{verbatim} +This command reads the file {\tt redo.input}. +created by the last {\tt )undo} command. +This file consists of all user input lines, excluding those +backtracked over due to a previous {\tt )undo}. + +\par\noindent{\bf Also See:} +{\tt )history} \index{)history}. +The command {\tt )history )write} will eliminate the ``undone'' command +lines of your program. + +\section{)what} +\index{)what} +\index{what} + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{{\tt )what categories} {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what commands } {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what domains } {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what operations} {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what packages } {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what synonym } {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )what things } {\it pattern1} [{\it pattern2 ...}]} + +\item{{\tt )apropos } {\it pattern1} [{\it pattern2 ...}]} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to display lists of things in the system. The +patterns are all strings and, if present, restrict the contents of the +lists. Only those items that contain one or more of the strings as +substrings are displayed. For example, +\begin{verbatim} +)what synonym +\end{verbatim} +displays all command synonyms, +\begin{verbatim} +)what synonym ver +\end{verbatim} +displays all command synonyms containing the substring ``{\tt ver}'', +\begin{verbatim} +)what synonym ver pr +\end{verbatim} +displays all command synonyms +containing the substring ``{\tt ver}'' or the substring +``{\tt pr}''. +Output similar to the following will be displayed +\begin{verbatim} +---------------- System Command Synonyms ----------------- + +user-defined synonyms satisfying patterns: + ver pr + + )apr ........................... )what things + )apropos ....................... )what things + )prompt ........................ )set message prompt + )version ....................... )lisp *yearweek* +\end{verbatim} + +Several other things can be listed with the {\tt )what} command: + +\begin{description} +\item[{\tt categories}] displays a list of category constructors. +\index{what categories} +\item[{\tt commands}] displays a list of system commands available at your +user-level. +\index{what commands} +Your user-level +\index{user-level} +is set via the {\tt )set userlevel} command. +\index{set userlevel} +To get a description of a particular command, such as ``{\tt )what}'', issue +{\tt )help what}. +\item[{\tt domains}] displays a list of domain constructors. +\index{what domains} +\item[{\tt operations}] displays a list of operations in the system library. +\index{what operations} +It is recommended that you qualify this command with one or +more patterns, as there are thousands of operations available. For +example, say you are looking for functions that involve computation of +eigenvalues. To find their names, try {\tt )what operations eig}. +A rather large list of operations is loaded into the workspace when +this command is first issued. This list will be deleted when you +clear the workspace via {\tt )clear all} or {\tt )clear completely}. +It will be re-created if it is needed again. +\item[{\tt packages}] displays a list of package constructors. +\index{what packages} +\item[{\tt synonym}] lists system command synonyms. +\index{what synonym} +\item[{\tt things}] displays all of the above types for items containing +\index{what things} +the pattern strings as substrings. +The command synonym {\tt )apropos} is equivalent to +\index{apropos} +{\tt )what things}. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )display} \index{)display}, +{\tt )set} \index{)set}, and +{\tt )show} \index{)show}. + +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol1 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol10.pamphlet b/books/bookvol10.pamphlet new file mode 100644 index 0000000..089b5b2 --- /dev/null +++ b/books/bookvol10.pamphlet @@ -0,0 +1,333 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 10: Axiom Algebra}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol10 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol11.pamphlet b/books/bookvol11.pamphlet new file mode 100644 index 0000000..4213c45 --- /dev/null +++ b/books/bookvol11.pamphlet @@ -0,0 +1,50494 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 11: Axiom Browser}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2007 Timothy Daly +Portions Copyright (c) 2007 Alfredo Portes +Portions Copyright (c) 2007 Arthur Ralfs + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +This book is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Overview} +This book contains the Firefox browser AJAX routines. + +\section{Build Instructions} +\begin{verbatim} + mkdir -p /home/silver/bitmaps + cp bookvol11.pamphlet /home/silver + cd /home/silver + export AXIOM=(where) + export PATH=$AXIOM/bin/lib:$AXIOM/bin:$PATH + notangle -t8 bookvol11.pamphlet > Makefile + make -j 10 + axiom -nox + -> )set mes auto off + -> )set out mathml on + -> axServer(8085,multiServ)$AXSERV + + Now start your browser and go to: + file:///home/silver/rootpage.xhtml + and then do: + Basic Commands -> Calculus -> Differentiate -> Continue + Basic Commands -> Matrix -> Continue + + You should see the result of the differentiate appear inline in the + page. You can change the values in the text areas, click continue, + and see the new result. +\end{verbatim} + +\section{The Makefile} +<<*>>= +TANGLE=${AXIOM}/bin/lib/notangle + +%.xhtml: bookvol11.pamphlet + @ echo making $*.xhtml + @ ${TANGLE} -R"$*.xhtml" bookvol11.pamphlet > $*.xhtml + +<> + +all: ${PAGES} + @ mkdir -p bitmaps + @ ${TANGLE} -R"axiom1.bitmap" bookvol11.pamphlet >bitmaps/axiom1.bitmap + @ ${TANGLE} -R"rcm3720.input" bookvol11.pamphlet >rcm3720.input + @ ${TANGLE} -R"strang.input" bookvol11.pamphlet >strang.input + @ ${TANGLE} -R"signatures.txt" bookvol11.pamphlet >signatures.txt + +clean: + @ rm -f *.xhtml + @ rm -f rcm3720.input + @ rm -f signatures.txt +@ + +\section{Building new pages} +To add a new page you need to create a page with the default layout below +and add the name of the page to the PAGES variable below. + +Most of the pages have a default layout of the form: +\begin{verbatim} +\subsection{pagename.xhtml} +<>= +<> + + +<> +
Page subtitle goes here
+
+your basic page text goes here. +<> +@ +\end{verbatim} +There are several things to observe here: +
    +
  1. Each page lives in its own subsection and its own chunk.
  2. +
  3. The pagename and the chunkname are the same
  4. +
  5. The chunk includes the @<>
  6. +
  7. The chunk includes the @<>
  8. +
  9. The chunk includes the @<>
  10. +
+The default page layout cannot communicate with Axiom. + +\subsection{Communicating with Axiom} +If your page needs to communicate with Axiom you need to add some +information in the header of the page. +The default page that talks to Axiom has the form: +\begin{verbatim} +\subsection{pagename.xhtml} +@<>= +@<> + + + +@<> +
Page subtitle goes here
+
+your text goes here +your communication blocks go here +@<> +\end{verbatim} + +\subsection{Handling statements with no free variables} +Use a makeRequest call with a parameter of the id. +Note that the div with id of ``ansXX'' will get replaced +automatically and the ``ans'' prefix is required. +\begin{verbatim} +
  • + +
    +
  • +\end{verbatim} + +\subsection{Handling statements with free variables} +Free variables exist are used in statements but they are defined in +other statements. To make sure the free variables have the correct +values you need to include an explicit list of the other ids that +need to be executed {\sl before} this statement. You do this with +a call to ``handleFree''. It expects a list, enclosed in brackets, +of the ids to execute in order. Be certain that the current id is +at the end of the list. +\begin{verbatim} +
  • + +
    +
  • +\end{verbatim} + +\subsection{Handling domain database lookups} +Use an anchor tag of the form: +\begin{verbatim} +Vector +\end{verbatim} +This will be interpreted by Axiom to mean that you want to do a +lookup on a domain, category, or package whose name follows the +question mark. Note that the domain name should NOT be an abbreviation. + +\subsection{Handling )show domain} +Use a block containing a showcall of the form: +\begin{verbatim} +
  • + +
    +
  • +\end{verbatim} +Note that the ``)show'' must be at the beginning of the line +and that there can only be one space between the word show and +the following argument. + +\subsection{Handling lisp expressions} +Use a block containing a lispcall of the form: +
  • + +
    +
  • +Note that this works but you can easily blow away your Axiom +session with random statements. Let the coder beware. + +\subsection{Handling expressions that have no output} +Use the CSS class=``noresult'' tag on the input form. +This causes the item to show up in black text. It is +still executable and is generally executed by handleFree +calls because it contains definitions. However, things like +function definitions in Axiom return no interesting output +so there is no point in clicking on them. +\begin{verbatim} +
  • + +
    +
  • +\end{verbatim} + + +\section{Defined Pages} +Every page in this file is extracted by the Makefile. This is the list +of pages that will be extracted. It is organized roughly in the hierarchy +that you see in the browser pages. 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\ + pageset.xhtml \ + pagesquarematrix.xhtml \ + pagetable.xhtml \ + pagetwodimensionalarray.xhtml \ + pagevector.xhtml + +@ + +\section{The Standard Layout} +Generally a page has a standard layout using a couple of chunks +to minimize the typing. The defined chunks are: +\begin{itemize} +\item ``standard head'' which includes the head element, xmlns, meta, and +title element. It also contains the ``style'' element for CSS information. +\item ``page head'' contains the banner information +\item ``page foot'' contains the trailing page information and the +body-end and html-end tags +\end{itemize} +So the basic layout looks like +\begin{verbatim} +@<> + (local and general javascript goes here) + + +@<> + (local page definition goes here) +@<> +\end{verbatim} +So all you need to worry about are the actual page forms and the +javascript to fetch those forms. + +For ``active pages'', that is those that communicate with Axiom +they generally define a javascript function called ``commandline'' +which formats the request to be sent to the host. You also need to +include the ``axiom talker'' chunk. Note that ``axiom talker'' +expects the ``commandline'' function to exist and calls it. Thus, +for the page that handles differentiation calls to Axiom we add +the local javascript: +\begin{verbatim} + + + +\end{verbatim} +This defined the ``commandline'' function and embeds the ``axiom talker''. +The ``commandline'' function knows how to fetch fields from the rest of +the page and format them into a single Axiom string. This is page +specific code. For example, this shows a single input line which +will be sent to the host when the ``Continue'' is pressed: +\begin{verbatim} +
    +

    + Type an input command line to Axiom:
    + + @<> +

    +
    + @<> +\end{verbatim} + +Note that the commandline function takes an argument which it gets +from the caller, makeRequest. This argument can be used to distinguish +which button was pressed. + +The {\bf div} section with {\bf id=``mathAns''} is replaced by the +result sent from the server. + +\section{Cascading Style Sheet} +\subsection{Standard Style Sheet} +This is the standard CSS style section that gets included with every +page. We do this here but it could be a separate style sheet. It +hardly matters either way as the style sheet is trivial. +\begin{verbatim} +\end{verbatim} +< +@ +\subsection{Menu style sheet} +<>= + +@ +\section{standard head} +This is the standard head section. It is used on pages that do not +include javascript. Note that it does NOT include the so +the javascript can be added easily. +<>= + + + + + Axiom Documentation +< + + +<> + +<> +@ + +\subsection{graphexamples.xhtml} +<>= +<> + + +<> +
    Graphics Examples
    +
    +Here are some examples of Axiom graphics. + +<> +@ + +\subsection{graphexamplesassorted.xhtml} +<>= +<> + + + +<> +
    Assorted Graphics Examples
    +
    +Function of two variables: z=f(x,y) +
      +
    • + +
      +
    • +
    +Function of one variable: y=f(x) +
      +
    • + +
      +
    • +
    +Plane parametric curve: x=f(t),y=g(t) +
      +
    • + +
      +
    • +
    +Space parametric curve: x=f(t),y=g(t),z=h(t) +
      +
    • + +
      +
    • +
    +Polar coordinates: r=f(θ) +
      +
    • + +
      +
    • +
    +Implicit curves: p(x,y)=0 +
      +
    • + +
      +
    • +
    +<> +@ + + +\subsection{graphexamplesimplicit.xhtml} +<>= +<> + + + +<> +
    Implicit Curves
    +
    +A Conic Section (Hyperbola) +
      +
    • + +
      +
    • +
    +An Elliptic Curve +
      +
    • + +
      +
    • +
    +Cartesian Ovals +
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    • + +
      +
    • +
    • + +
      +
    • +
    +Cassinian Ovals: two loops +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + + +\subsection{graphexampleslistofpoints.xhtml} +<>= +<> + + + +<> +
    Plotting Lists of Points
    +
    +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
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    • +
    +The makeViewport2D command takes a +list of options as a parameter in this example. The string "Lines" is +designated as the viewport's title. +<> +@ + + +\subsection{graphexamplesonevariable.xhtml} +<>= +<> + + + +<> +
    Functions of One Variable
    +
    +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + + +\subsection{graphexamplesparametric.xhtml} +<>= +<> + + + +<> +
    Parametric Curves
    +
    +The Lemnicate of Bernoulli +
      +
    • + +
      +
    • +
    +Lissajous curve +
      +
    • + +
      +
    • +
    +A gnarly closed curve +
      +
    • + +
      +
    • +
    +Another closed curve +
      +
    • + +
      +
    • +
    +<> +@ + + +\subsection{graphexamplespolar.xhtml} +<>= +<> + + + +<> +
    Polar Coordinates
    +
    +A circle +
      +
    • + +
      +
    • +
    +A spiral +
      +
    • + +
      +
    • +
    +A Petal Curve +
      +
    • + +
      +
    • +
    +A Limacon +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{graphexamplesthreed.xhtml} +<>= +<> + + + +<> +
    Three Dimensional Graphics
    +
    +Plots of parametric surfaces defined by functions f(u,v), g(u,v) and h(u,v). + +Pear Surface +
      +
    • + +
      +
    • +
    + +Trigonometric Screw +
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    • + +
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    • +
    + +Etruscan Venus +
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      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    + +Banchoff Klein Bottle +
      +
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      +
    • +
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      +
    • +
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    • +
    + +<> +@ + +\subsection{graphicspage.xhtml} +<>= +<> + + +<> +Axiom can plot curves and surfaces of various types, as well as +lists of points in the plane. + + + + + + + + + + + + + + + + + +
    + Examples + + See examples of Axiom graphics +
    + 2D Graphics + + Graphics in the real and complex plane +
    + 3D Graphics + + Plot surfaces, curves, or tubes around curves +
    + Viewports + + Customize graphics using Viewports +
    +<> +@ + +\subsection{graphviewports.xhtml} +<>= +<> + + + +<> +
    Stand-alone Viewport
    +
    +To get a viewport on a page, you first need to create on in Axiom and +write it out to a file that can be called up. For example, we draw a +saddle function and assign the result to the variable v. +
      +
    • + +
      +
    • +
    +Now that we've created the viewport, we want to write the data out to +a file. To do this, we use the write command +which takes as arguments the viewport to write out, the title of the file +to be written to, and an optiona argument telling the write command what +type (or types) of data you want to write (in addition to the ones that +Axiom writes). The optional argument could be a string, like "pixmap", or +a list of strings, like ["postscript", "pixmap"]. We need a "bitmap" data +type to include a graph in a page so in this case, we write the viewport +and tell it to also write a "pixmap" file: +
      +
    • + +
      +
    • +
    +Currently supported file formats are "pixmap", "bitmap", "postscript" +and "image". + +Axiom automatically adds ".view" at the end of the viewport data file +to specify the file type. The ".view" is actually a directory and +contains a bitmap file, usually called image.bm.Z, which is a compressed +bitmap. Firefox can display bitmap files, as shown here. Clicking on the +image should start a "live graphics copy" so you can manipulate the image. + +
    +
    + + + +<> +@ + +\subsection{graph2d.xhtml} +<>= +<> + + + +<> +
    Two Dimensional Graphics
    +
    + +<> +@ + +\subsection{graph2dimplicit.xhtml} +<>= +<> + + + +<> +
    Implicit Curves
    +
    +Axiom has facilities for graphing a non-singular algebraic curve in a +rectangular region of the plane. An algebraic curve is a curve defined by +a polynomial equation p(x,y)=0. Non-singular means that the curve is +"smooth" in that it does not cross itself or come to a point (cusp). +Algebraically, this means that for any point (a,b) on the curve (i.e. +a point such that p(a,b)=0), the partial derivatives dp/dx(a,b) and +dp/dy(a,b) are not both zero. We require that the polynomial have rational +or integral coefficients. Here is a Cartesian ovals algebraic curve +example: +
      +
    • + +
      +
    • +
    • + +
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    • +
    +A range must be declared for each variable specified in the algebraic +curve equation. +<> +@ + +\subsection{graph2dlistsofpoints.xhtml} +<>= +<> + + + +<> +
    List of Points
    +
    +Axiom has the ability to create lists of points in a two dimensional +graphics viewport. This is done by utilizing the +GraphImage and +TwoDimensionalViewport +domain facilities. + +In this example, the makeGraphImage +takes a list of lists of points parameter, a list of colors for each point +in the graph, a list of colors for each line in the graph, and a list of +numbers which indicate the size of each point in the graph. The following +lines create list of lists of points which can be read be made into two +dimensional graph images. + +First we make a list of points +
      +
    • + +
      +
    • +
    +Then we select pairs of these points which represent the endpoints of lines. +
      +
    • + +
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    • +
    +Next we set the point color and size, and the line color for all components +of the graph. +
      +
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    • +
    • + +
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    • +
    +Now the graph image is created and named according to the component +specifications indicated above. The +makeViewport2D command then +creates a two dimensional viewport for this graph according to the list +of options specified within the brackets. +
      +
    • + +
      +
    • +
    +The makeViewport2D command takes a +list of options as a parameter. In this example the string "Lines" is +designated as the viewport's title. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{graph2donevariable.xhtml} +<>= +<> + + + +<> +
    Functions of One Variable
    +
    +Here we wish to plot a function y=f(x) on the interval [a,b]. As an +example, let's take the function y=sin(tan(x))-tan(sin(x)) on the interval +[0,6]. Here is the simplest command that will do this: +
      +
    • + +
      +
    • +
    +Notice that Axiom compiled a function before the graph was put on the +screen. The expression sin(tan(x))-tan(sin(x)) was converted to a compiled +function so that its value for various values of x could be computed +quickly and efficiently. Let's graph the same function on a different +interval and this time we'll give the graph a title. The title is a string, +which is an optional argument of the command 'draw'. +
      +
    • + +
      +
    • +
    +Once again the expression sin(tan(x))-tan(sin(x)) was converted to a +compiled function before any points were computed. If you want to graph the +same function on a number of intervals, it's a good idea to write down a +function definition so that the function only has to be compiled once. +Here's an example: +
      +
    • + +
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    • + +
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    • + +
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    • +
    +Notice that our titles can be whatever we want, as long as they are +encloded by double quotes. However, a title which is too long to fit +within the viewport title window will be clipped. +<> +@ + +\subsection{graph2dparametric.xhtml} +<>= +<> + + + +<> +
    Parametric Curves
    +
    +One way of producing interesting curves is by using parametric equations. +Let x=f(t) and y=g(t) for two functions f and g as the parameter t ranges +over an interval [a,b]. Here's an example: +
      +
    • + +
      +
    • +
    +Here 0..2*%pi represents the interval over which the variable t ranges. +In the case of parametric curves, Axiom will compile two functions, one for +each of the functions f and g. You may also put a title on a graph. The +title may be an arbitrary string and is an optional argument to the +command 'draw'. For example: +
      +
    • + +
      +
    • +
    +If you plan on plotting x=f(t), y=g(t) as t ranges over several intervals, +you may want to define functions f and g, so that they need not be +recompiled every time you create a new graph. Here's an example: +
      +
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    • + +
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    • + +
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    • +
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    +These examples show how the curve changes as the range of the parameter +t varies. +<> +@ + +\subsection{graph2dpolar.xhtml} +<>= +<> + + + +<> +
    Polar Coordinates
    +
    +Graphs in polar coordinates are given by an equation r=f(θ) as +theta ranges over an interval. This is equivalent to the parametric curve +x=f(θ)*cos(θ),y=f(θ)*sin(θ) as θ +ranges over the same interval. You may create such curves using the +command 'draw', with the optional argument 'coordinates==polar'. +Here are some examples: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +You may also define your own functions, when you plan on plotting the same +curve as θ varies over several intervals. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +For information on plotting graphs in other coordinate systems see the +pages for the CoordinateSystems +domain. +<> +@ + + +\subsection{graph3d.xhtml} +<>= +<> + + +<> +
    Three Dimensional Graphing
    +
    + +<> +@ + +\subsection{graph3dobjects.xhtml} +<>= +<> + + + +<> +
    Building Three Dimensional Objects from Primatives
    +
    +Rather than using the draw and +makeObject commands, you can create +three-dimensional graphs from primitives. Operation +create3Space creates a three-space +object to which points, curves, and polygons can be added using the +operations from the ThreeSpace domain. +The resulting object can then be displayed in a viewport using +makeViewport3D. + +Create the empty three-space object space. +
      +
    • + +
      +
    • +
    +Objects can be sent to this space using the operations exported by the +ThreeSpace domain. The following examples +place curves into space. + +Add these three curves to the space. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Create and display the viewport using +makeViewport3D. Options may also be +given but here are displayed as a list with values enclosed in parentheses. +
      +
    • + +
      +
    • +
    +
    +
    +Cube Example +
    +As a second example of the use of primitives, we generate a cube using +a polygon mesh. It is important to use a consistent orientation of the +polygons for correct generation of three-dimensional objects. + +Again start with an empty three-space object. +
      +
    • + +
      +
    • +
    +For convenience, give the DoubleFloat +values +1 and -1 names. +
      +
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    +Define the vertices of the cube. +
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    +Add the faces of the cube as polygons to the space using a consistent +orientation. +
      +
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    +Create and display the viewport. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{graph3dparametric.xhtml} +<>= +<> + + + +<> +
    Parametric Space Curves
    +
    +This page describes the plotting in three dimensional space of a curve +defined by the parametric equations x=f(t), y=g(t), z=h(t), where f,g, and +h are functions of the parameter t which ranges over a specified interval. +The basic draw command for this function utilizes either the uncompiled +functions or compiled functions format and uses the +curve command to specify the three +functions for the x, y, and z components of the curve. The general format +for uncompiled functions is: +
    +   draw(curve(f(t),g(t),h(t)), t=a..b)
    +
    +where a..b is the segment defining the interval [a,b] over which the +parameter t ranges. In this case the functions are not compiled until +the draw command is executed. Here is an example: +
      +
    • + +
      +
    • +
    +In the case of compiled functions, the functions are named and compiled +independently. This is useful if you intend to use the functions often, +or if the functions are long and complex. The following lines show +functions whose parameters are of the type SmallFloat. The functions +are compiled and stored by Axiom when entered. +
      +
    • + +
      +
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    • + +
      +
    • +
    • + +
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    +Once the functions are compiled the draw command only needs the names of +the functions to execute. Here is a compiled functions example: +
      +
    • + +
      +
    • +
    +Note that the parameter range does not take the variable name as in the +case of uncompiled functions. It is understood that the indicated range +applies to the parameter of the functions, which in this case is t. +<> +@ + +\subsection{graph3dsurfaces.xhtml} +<>= +<> + + + +<> +
    Parametric Surfaces
    +
    +Graphing a surface defined by x=f(u,v), y=g(u,v), z=h(u,v). This page +describes plotting of surfaces defined by the parametric equations of two +variables, x=f(u,v), y=g(u,v), and z=h(u,v), for which the ranges of u and +v are explicitly defined. The basic draw command for this function utilizes +either the uncompiled function or compiled function format and uses the +surface command to specify the three +functions for the x, y, and z components of the surface. The general +format for uncompiled functions is: +
    +    draw(surface(f(u,v),g(u,v),h(u,v)), u=a..b, v=c..d)
    +
    +where a..b and c..d are segments defining the intervals [a,b] and [c,d] +over which the parameters u and v span. In this case the functions are +not compiled until the draw command is executed. Here is an example of a +surface plotted using the parabolic cylindrical coordinate system option: +
      +
    • + +
      +
    • +
    +In the case of compiled functions, the functions are named and compiled +independentlyh. This is useful if you intend to use the functions often, +or if the functions are long and complex. The following lines show functions +whose parameters are of the type SmallFloat. The functions are compiled and +stored by Axiom when entered. +
      +
    • + +
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    • + +
      +
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    • + +
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    +Once the function is compiled the draw command only needs the names of +the functions to execute. Here is a compiled functions example plotted +using the toroidal coordinate system option: +
      +
    • + +
      +
    • +
    +Note that the parameter ranges do not take the variable names as in the case +of uncompiled functions. The variables are entered in the order in which +they are defined in the function specification. In this case the first +range specifies the u-variable and the second range specifies the v-variable. +<> +@ + +\subsection{graph3dtubeplots.xhtml} +<>= +<> + + + +<> +
    Parametric Tube Plots
    +
    +This page describes the plotting in three dimensional space of a tube +around a parametric space curve defined by the parametric equations +x=f(t), y=g(t), z=h(t), where f, g, and h are functions of the parameter t +which ranges over a specified interval. The basic draw command for this +function utilizes either the uncompiled functions or compiled functions +format and uses the curve command to specify +the three functions for the x, y, and z components of the curve. This uses +the same format as that for space curves except that it requires a +specification for the radius of the tube. If the radius of the tube is 0, +then the result is the space curve itself. The general format for +uncompiled functions is: +
    +    draw(curve(f(t),g(t),h(t)),t=a..b,tubeRadius==r)
    +
    +where a..b is the segment defining the interval [a,b] over which the +parameter t ranges, and the tubeRadius is indicated by the variable r. +In this case the functions are not compiled until the draw command is +executed. Here is an example: +
      +
    • + +
      +
    • +
    +In the case of compiled functions, the functions are named and compiled +independently. This is useful if you intend to use the functions often, +or if the functions are long and complex. The following lines show +functions whose parameters are of the type SmallFloat. The functions are +compiled and stored by Axiom when entered. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Once the function is compiled the draw command only needs the names of +the functions to execute. Here is a compiled functions example of a trefoil +knot: +
      +
    • + +
      +
    • +
    +Note that the parameter range does not take the variable name as in the +case of uncompiled functions. It is understood that the indicated range +applies to the parameter of the functions, which in this case is t. +Typically, the radius of the tube should be set between 0 and 1. A radius +of less than 0 results in it's positive counterpart and a radius of greater +than one cause self-intersection. +<> +@ + +\subsection{graph3dtwovariables.xhtml} +<>= +<> + + + +<> +
    Functions of Two Variables
    +
    +This page describes the plotting of surfaces defined by an equation of +two variables, z=f(x,y), for which the ranges of x and y are explicitly +defined. The basic draw command for this function utilizes either the +uncompiled function or compiled function format. The general format for an +uncompiled function is: +
    +   draw(f(x,y), x=a..b, y=c..d)
    +
    +where a..b and c..d are segments defining the intervals [a,b] and [c,d] +over which the variables x and y span. In this case, the function is not +compiled until the draw command is executed. Here is an example: +
      +
    • + +
      +
    • +
    +In the case of a compiled function, the function is named and compiled +independently. This is useful if you intend to use a function often, or +if the function is long and complex. The following line shows a function +whose parameters are of the type SmallFloat. The function is compiled and +stored by Axiom when it is entered. +
      +
    • + +
      +
    • +
    +Once the function is compiled the draw command only needs the name of the +function to execute. Here is a compiled function example: +
      +
    • + +
      +
    • +
    +Note that the parameter ranges do not take the variable names as in the +case of uncompiled functions. The variables are entered in the order in +which they are defined in the function specificationl. In this case the +first range specifies the x-variable and the second range specifies the +y-variable. +<> +@ + +%%H +\subsection{htxtoppage.xhtml} +<>= +<> + + +<> + htxtoppage not implemented +<> +@ + + +%%I +\subsection{indefiniteintegral.xhtml} +<>= +<> + + + +<> +
    + Enter the function you want to integrate:
    +
    + Enter the variable of integration: +
    +
    +<> +<> +<> +@ + +\subsection{introtofloat.xhtml} +<>= +<> + + + +<> +
    Introduction to Float
    +
    +Scientific notation is supported for input and output of floating +point numbers. A floating point number is written as a string of +digits containing a decimal point optionally followed by the letter "E", +and then the exponent. We begin by doing some calculations using +arbitrary precision floats. The default precision is twenty decimal +digits. +
      +
    • + +
      +
    • +
    +A decimal base for the exponent is assumed, so the number 1.234E2 denotes +1.234*10**2 +
      +
    • + +
      +
    • +
    +The normal arithmetic operations are available for floating point numbers. +
      +
    • + +
      +
    • +
    +<> +@ + +%%J +\subsection{jenks.xhtml} +<>= +<> + + +<> +
    + + + +
    +
    +

    + + AXIOM -- Richard D. Jenks and Robert S. Sutor + +

    +
    +
    +

    + + The Scientific Computation System + +

    +
    +
    +

    + + Volume 0 -- The Textbook + +

    +
    + + Chapter 0: Introduction to Axiom +
    + + Chapter 1: An Overview of Axiom +
    + + Chapter 2: Using Types and Modes +
    + + Chapter 3: Using HyperDoc +
    + + Chapter 4: Input Files and Output Styles +
    + + Chapter 5: Overview of Interactive Language +
    + + Chapter 6: User-Defined Functions, Macros and Rules +
    + + Chapter 7: Graphics +
    + + Chapter 8: Advanced Problem Solving +
    + + Chapter 9: Some Examples of Domains and Packages +
    + + Chapter 10: Interactive Programming +
    + + Chapter 11: Packages +
    + + Chapter 12: Categories +
    + + Chapter 13: Domains +
    + + Chapter 14: Browse +
    + + Chapter 15: What's New in Axiom Version 2.0 +
    + + Chapter 17: Categories +
    + + Chapter 18: Domains +
    + + Chapter 19: Packages +
    + + Chapter 21: Programs for AXIOM Images +
    +<> +@ + +%%K +%%L + +\subsection{laurentseries.xhtml} +<>= +<> + + + +<> + + + + + + + + + + + + + + + + +
    + Enter the formula for the general coefficient of the series: +
    + +
    + Enter the index variable for your formula: + +
    + Enter the power series variable: + +
    + Enter the point about which to expand: + +
    +For Laurent Series, the exponent of the power series variable ranges +from an initial value, an arbitrary integer value, to plus +infinity; the step size is any positive integer. + + + + + + + +
    + Enter the initial value of the index (an integer): + +
    + Enter the step size (a positive integer): + +
    +<> +<> +<> + +@ + +\subsection{linalgpage.xhtml} +<>= +<> + + +<> +
    Linear Algebra
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + +
    + Introduction +
    + Create and manipulate matrices. Work with the entries of a + matrix. Perform matrix arithmetic. +
    + Creating Matrices +
    + Create matrices from scratch and from other matrices +
    + Operations on Matrices +
    + Algebraic manipulations with matrices. Compute the inverse, + determinant, and trace of a matrix. Find the rank, nullspace, + and row echelon form of a matrix. +
    + Eigenvalues and Eigenvectors +
    + How to compute eigenvalues and eigenvectors +
    +
    + +<> +@ + +\subsection{linconversion.xhtml} +<>= +<> + + + +<> +
    Conversion
    +
    +Conversion is the process of changing an object of one type into an +object of another type. The syntax for conversion is object::newType. +
    +By default, 3 has the type +PositiveInteger +
      +
    • + +
      +
    • +
    +We can change thisinto an object of type +Fraction Integer by using "::". +
      +
    • + +
      +
    • +
    +A coercion is a special kind of conversion that Axiom is allowed to do +automatically when you enter an expression. Coercions are usually +somewhat safer than more general conversions. The Axiom library contains +operations called +coerce and +convert. Only the +coerce operations can be used by the +interpreter to change an object into an object of another type unless +you explicitly use a "::". + +By now you will be quite familiar with what types and modes look like. +It is useful to think of a type or mode as a pattern for what you want +the result to be. Let's start with a square matrix of polynomials with +complex rational number coefficients. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +We first want to interchange the Complex +and Fraction layers. We do the conversion +by doing the interchange in the type expression. +
      +
    • + +
      +
    • +
    +Interchange the Polynomial and the +Fraction levels. +
      +
    • + +
      +
    • +
    +Interchange the Polynomial and the +Complex levels. +
      +
    • + +
      +
    • +
    +All the entries have changed types, although in comparing the last two +results only the entry in the lower left corner looks different. We did +all the intermediate steps to show you what Axiom can do. + +In fact, we could have combined all these into one conversion. +
      +
    • + +
      +
    • +
    + +There are times when Axiom is not able to do the conversion in one step. +You may need to break up the transformation into several conversions in +order to get an object of the desired type. + +We cannot move either the Fraction or +Complex above (or to the left of, +depending on how you look at it) +SquareMatrix because each of these +levels requires that its argument type have commutative multiplication, +whereas SquareMatrix does not. +(Fraction requires that its argument +belong to the category +IntegralDomain and +Complex requires that its argument belongs to +CommutativeRing. See the +Jenks section 2.1 for a brief +discussion of categories. The Integer level +did not move anywhere because it does not allow any arguments. We also did +not move the SquareMatrix part +anywhere, but we could have. Recall that m looks like this: +
      +
    • + +
      +
    • +
    +If we want a polynomial with matrxi coefficients rather than a matrix with +polynomial entries, we can just do the conversion. +
      +
    • + +
      +
    • +
    +We have not yet used modes for any conversions. Modes are a great +shorthand for indicating the type of the object you want. Instead of +using the long type expression in the last example we could have +simply said this: +
      +
    • + +
      +
    • +
    +We can also indicate more structure if we want the entries of the matrices +to be fractions. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{lincreate.xhtml} +<>= +<> + + + +<> +
    Creating Matrices
    +
    +There are many ways to create a matrix from a collection of values or +from existing matrices. + +If the matrix has almost all items equal to the same value, use +new to create a matrix filled with that value +and then reset the entries that are different. +
      +
    • + +
      +
    • +
    +To change the entry in the second row, third column to 5, use +setelt. +
      +
    • + +
      +
    • +
    +An alternative syntax is to use assignment. +
      +
    • + +
      +
    • +
    +The matrix was destructively modified. +
      +
    • + +
      +
    • +
    +If you already have the matrix entries as a list of lists, use +matrix. +
      +
    • + +
      +
    • +
    +If the matrix is diagonal, use +diagonalMatrix +
      +
    • + +
      +
    • +
    +Use setRow! and +setColumn! +to change a row or column of a matrix. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use copy to make a copy of a matrix. +
      +
    • + +
      +
    • +
    +This is useful if you intend to modify a matrix destructively but want a +copy of the original. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use subMatrix(dm,2,3,2,4) to extract +part of an existing matrix. The syntax is +
    +  subMatrix(m,firstrow,lastrow,firstcol,lastcol)
    +
    +
      +
    • + +
      +
    • +
    +To change a submatrix, use +setsubMatrix!. +
      +
    • + +
      +
    • +
    +If e is too big to fit where you specify, an error message is displayed. Use +subMatrix. +
      +
    • + +
      +
    • +
    +This changes the submatrix of d whose upper left corner is at the first row +and second column and whose size is that of e. +
      +
    • + +
      +
    • +
    +
      +
    • + +
      +
    • +
    +Matrices can be joined either horizontally or vertically to make new +matrices. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use horizConcat to append them side to +side. The two matrices must have the same number of rows. +
      +
    • + +
      +
    • +
    +Use vertConcat to stack one upon the +other. The two matrices must have the same number of columns. +
      +
    • + +
      +
    • +
    +The operation transpose is used to create +a new matrix by reflection across the main diagonal. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{lineigen.xhtml} +<>= +<> + + + +<> +
    Computation of Eigenvalues and Eigenvectors
    +
    +In this section we show you some of Axiom's facilities for computing and +manipulating eigenvalues and eigenvectors, also called characteristic +values and characteristic vectors, respectively. + +Let's first create a matrix with integer entries. +
      +
    • + +
      +
    • +
    +To get a list of the rational eigenvalues, use the operation +eigenvalues. +
      +
    • + +
      +
    • +
    +Given an explicit eigenvalue, +eigenvector computes the eigenvectors +corresponding to it. +
      +
    • + +
      +
    • +
    +The operation eigenvectors returns a +list of pairs of values and vectors. When an eigenvalue is rational, Axiom +gives you the value explicitly; otherwise, its minimal polynomial is given, +(the polynomial of lowest degree with the eigenvalues as roots), together +with a parametric representation of the eigenvector using the eigenvalue. +This means that if you ask Axiom to solve +the minimal polynomial, then you can substitute these roots into the +parametric form of the corresponding eigenvectors. + +You must be aware that unless an exact eigenvalue has been computed, +the eigenvector may be badly in error. +
      +
    • + +
      +
    • +
    +Another possibility is to use the operation +radicalEigenvectors tries to +compute explicitly the eignevectors in terms of radicals. +
      +
    • + +
      +
    • +
    +Alternatively, Axiom can compute real or complex approximations to the +eigenvectors and eigenvalues using the operations +realEigenvectors or +complexEigenvectors. They +each take an additional argument epsilon to specify the "precision" +required. In the real case, this means that each approximation will be +within plus or minus epsilon of the actual result. In the complex case, this +means that each approximation will be within plus or minus epsilon of the +actual result in each of the real and imaginary parts. + +The precision can be specified as a Float if +the results are desired in floating-point notation, or as +Fraction Integer if the results are +to be expressed using rational (or complex rational) numbers. +
      +
    • + +
      +
    • +
    +If an n by n matrix has n distinct eigenvalues (and therefore n eigenvectors) +the operation eigenMatrix gives you a +matrix of the eigenvectors. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +If a symmetric matrix has a basis of orthonormal eigenvectors, then +orthonormalBasis computes a list +of these vectors. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{linhilbert.xhtml} +<>= +<> + + + +<> +
    An Example: Determinant of a Hilbert Matrix
    +
    +Consider the problem of computing the determinant of a 10 by 10 Hilbert +matrix. The (i,j)-th entry of a Hilbert matrix is given by 1/(i+j+1). + +First do the computation using rational numbers to obtain the exact result. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The result given by hardware floats is correct only to four significant digits +of precision. In the jargon of numerical analysis, the Hilbert matrix is said +to be "ill-conditioned". +
      +
    • + +
      +
    • +
    +Now repeat the computation at a higher precision using Float. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Reset digits to its default value. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{linintro.xhtml} +<>= +<> + + + +<> +
    Expanding to Higher Dimensions
    +
    +To get higher dimensional aggregates, you can create one-dimensional +aggregates with elements that are themselves aggregates, for example, +lists of list, one-dimensional arrays of list of multisets, and so on. For +applications requiring two-dimensional homogeneous aggregates, you will +likely find two-dimensional arrays and matrices useful. + +The entries in TwoDimensionalArray +and Matrix objects are all the same type, +except that those for Matrix must belong to a +Ring. You create and access elements in roughly +the same way. Since matrices have an understood algebraic structure, certain +algebraic operations are available for matrices but not for arrays. Because +of this, we limit our discussion here to Matrix, +that can be regarded as an extension of +TwoDimensionalArray. See +TwoDimensionalArray +For more +information about Axiom's linear algebra facilities see +Matrix, +Permanent, +SquareMatrix, +Vector, + +Computation of Eigenvalues and Eigenvectors, and + +Solution of Linear and Polynomial Equations. + +You can create a matrix from a list of lists, where each of the inner +lists represents a row of the matrix. +
      +
    • + +
      +
    • +
    +The "collections" construct (see + +Creating Lists and Streams with Iterators) +is useful for creating matrices whose entries are given by formulas. +
      +
    • + +
      +
    • +
    +Let vm denote the three by three Vandermonde matrix. +
      +
    • + +
      +
    • +
    +Use this syntax to extract an entry in the matrix. +
      +
    • + +
      +
    • +
    +You can also pull out a row or a column. +
      +
    • + +
      +
    • +
    +You can do arithmetic. +
      +
    • + +
      +
    • +
    +You can perform operations such as +transpose, +trace, and +determinant +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{linoperations.xhtml} +<>= +<> + + + +<> +
    Operations on Matrices
    +
    +Axiom provides both left and right scalar multiplication. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +You can add, subtract, and multiply matrices provided, of course, that the +matrices have compatible dimensions. If not, an error message is displayed. +
      +
    • + +
      +
    • +
    +This following product is defined but n*m is not. +
      +
    • + +
      +
    • +
    +The operations nrows and +ncols return the number of rows and +columns of a matrix. You can extract a row or a column of a matrix using +the operations row and +column. The object returned ia a +Vector. Here is the third column of the matrix n. +
      +
    • + +
      +
    • +
    +You can multiply a matrix on the left by a "row vector" and on the right by +a "column vector". +
      +
    • + +
      +
    • +
    +The operation inverse computes the inverse +of a matrix if the matrix is invertible, and returns "failed" if not. This +Hilbert matrix invertible. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +This matrix is not invertible. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The operation determinant computes the +determinant of a matrix provided that the entries of the matrix belong to a +CommutativeRing. The above matrix mm +is not invertible and, hence, must have determinant 0. +
      +
    • + +
      +
    • +
    +The operation trace computes the trace of a +square matrix. +
      +
    • + +
      +
    • +
    +The operation rank computes the rank of a matrix: +the maximal number of linearly independent rows or columns. +
      +
    • + +
      +
    • +
    +The operation nullity computes the nullity +of a matrix: the dimension of its null space. +
      +
    • + +
      +
    • +
    +The operation nullSpace returns a list +containing a basis for the null space of a matrix. Note that the nullity is +the number of elements in a basis for the null space. +
      +
    • + +
      +
    • +
    +The operation rowEchelon returns the row +echelon form of a matrix. It is easy to see that the rank of this matrix is +two and that its nullity is also two. +
      +
    • + +
      +
    • +
    +For more information see +Expanding to Higher Dimensions, + +Computation of Eigenvalues and Eigenvectors, and + +An Example: Determinant of a Hilbert Matrix. Also see +Permanent, +Vector, +OneDimensionalArray, and +TwoDimensionalArray. Issue the +system command +
      +
    • + +
      +
    • +
    +to display the full ist of operations defined by +Matrix. +<> +@ + +\subsection{linpermaent.xhtml} +<>= +<> + + + +<> +
    Permanent
    +
    +The package Permanent provides the function +permanent for square matrices. The +permanent of a square matrix can be +computed in the same way as the determinant by expansion of minors except +that for the permanent the sign for each element is 1, rather than being 1 +if the row plus column indices is positive and -1 otherwise. This function +is much more difficult to compute efficiently than the +determinant. An example of the use of +permanent is the calculation of the nth +derangement number, defined to be the number of different possibilities +for n couples to dance but never with their own spouse. Consider an n by x +matrix with entries 0 on the diagonal and 1 elsewhere. Think of the rows as +one-half of each couple (for example, the males) and the columns the other +half. The permanent of such a matrix gives the desired derangement number. +
      +
    • + +
      +
    • +
    +Here are some derangement numbers, which you see grow quite fast. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{linsquarematrices.xhtml} +<>= +<> + + + +<> +
    SquareMatrix
    +
    +The top level matrix type in Axiom is +Matrix, see +(Matrix), which provides basic arithmetic +and linear algebra functions. However, since the matrices can be of any +size it is not true that any pair can be added or multiplied. Thus +Matrix has little algebraic structure. + +Sometimes you want to use matrices as coefficients for polynomials or in +other algebraic contexts. In this case, +SquareMatrix should be used. The +domain SquareMatrix(n,R) gives the +ring of n by n square matrices over R. +
      +
    • + +
      +
    • +
    +The usual arithmetic operations are available. +
      +
    • + +
      +
    • +
    +Square matrices can be used where ring elements are required. For example, +here is a matrix with matrix entries. +
      +
    • + +
      +
    • +
    +Or you can construct a polynomial with square matrix coefficients. +
      +
    • + +
      +
    • +
    +This value can be converted to a square matrix with polynomial coefficients. +
      +
    • + +
      +
    • +
    +For more information on related topics see +Modes and +Matrix. Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +SquareMatrix. +<> +@ + +\subsection{linvectors.xhtml} +<>= +<> + + + +<> +
    Vector
    +
    +The Vector domain is used for storing +data in a one-dimensonal indexed data structure. A vector is a +homogeneous data structure in that all the components of the vector +must belong to the same Axiom domain. Each vector has a fixed length +specified by the user; vectors are not extensible. This domain is +similar to the +OneDimensionalArray domain, +except that when the components of a +Vector belong to a +Ring, arithmetic operations are provided. +For more examples of operations that are defined for both +Vector and +OneDimensionalArray, see +OneDimensionalArray. + +As with the OneDimensionalArray +domain, a +Vector can be created by calling the operation +new, its components can be accessed by calling +the operations elt and +qelt, and its components can be reset by +calling the operations +setelt and +setelt!. This creates a vector of +integers of length 5 all of whose components are 12. +
      +
    • + +
      +
    • +
    +This is how you create a vector from a list of its components. +
      +
    • + +
      +
    • +
    +Indexing for vectors begins at 1. The last element has index equal to +the length of the vector, which is computed by +#. +
      +
    • + +
      +
    • +
    +This is the standard way to use elt to extract +an element. +
      +
    • + +
      +
    • +
    +This is the standard way to use setelt to change an element. It is the +same as if you had typed setelt(v,3,99). +
      +
    • + +
      +
    • +
    +Now look at v to see the change. You can use +qelt and +qsetelt! (instead of +elt and +setelt, respectively) but only when you +know that the indexis within the valid range. +
      +
    • + +
      +
    • +
    +When the components belong to a +Ring, +Axiom provides arithmetic operations for +Vector. These include left and right +scalar multiplication. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Addition and subtraction are also available +
      +
    • + +
      +
    • +
    +Of course, when adding or subtracting, the two vectors must have the +same length or an error message is displayed. +
      +
    • + +
      +
    • +
    +For more information about other aggregate domains, see +List, +Matrix, +OneDimensionalArray. +Set, +Table, and +TwoDimensionalArray. +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +Vector. +<> +@ + +\subsection{lin1darrays.xhtml} +<>= +<> + + + +<> +
    One Dimensional Array
    +
    +The OneDimensionalArray is used +for storing data in a one-dimensional indexed data structure. Such an array +is a homogeneous data structure in that all the entries of the array must +belong to the same Axiom domain. Each array has a fixed length specified +by the user and arrays are not extensible. The indexing of +one-dimensional arrays is one-based. This means that the "first" element of +an array is given the index 1. See also +Vector and +FlexibleArray. To create a +one-dimensional array, apply the operation +oneDimensionalArray to a list. +
      +
    • + +
      +
    • +
    +Another approach is to first create a, a one-dimensional array of 10 0's. +OneDimensionalArray has a +convenient abbreviation +ARRAY1. +
      +
    • + +
      +
    • +
    +Set each ith element to i, then display the result. +
      +
    • + +
      +
    • +
    +Square each element by mapping the function i+->i**2 onto each element. +
      +
    • + +
      +
    • +
    +Reverse the elements in place. +
      +
    • + +
      +
    • +
    +Swap the 4th and 5th element. +
      +
    • + +
      +
    • +
    +Sort the elements in place. +
      +
    • + +
      +
    • +
    +Create a new one-dimensional array b containing the last 5 elements of a. +
      +
    • + +
      +
    • +
    +Replace the first 5 elements of a with those of b. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{lin2darrays.xhtml} +<>= +<> + + + +<> +
    Two Dimensional Array
    +
    +The TwoDimensionalArray is used +for storing data in a two-dimensional data structure indexed by row and by +column. Such an array is a homogeneous data structure in that all the +entries of the array must belog to the same Axiom domain (although see +the Any domain). Each array has a +fixed number of rows and columns specified by the user and arrays are not +extensible. In Axiom, the indexing of two-dimensional arrays is one-based. +This means that both the "first" row of an array and the "first" column of +an array are given the index 1. Thus, the entry in the upper left corner +of an array is in position (1,1). + +The operation new creates an array with a +specified number of rows and columns and fills the components of that +array with a specified entry. The arguments of this operation specify the +number of rows, the number of columns, and the entry. This creates a +five-by-four array of integers, all of which are zero. +
      +
    • + +
      +
    • +
    +The entries of this array can be set to other integers using the +operation setelt. + +Issue this to set the element in the upper left corner of this array to 17. +
      +
    • + +
      +
    • +
    +Now the first element of the array is 17. +
      +
    • + +
      +
    • +
    +Likewise, elements of an array are extracted using the operation +elt. +
      +
    • + +
      +
    • +
    +Another way to use these two operations is as follows. This sets the +element in position (3,2) of the array to 15. +
      +
    • + +
      +
    • +
    +This extracts the element in position (,32) of the array. +
      +
    • + +
      +
    • +
    +The operations elt and +setelt come equipped with an error check which +verifies that the indices are in the proper ranges. For example, the +above array has five rows and four columns, so if you ask for the entry +in position (6,2) with arr(6,2) Axiom displays an error message. If there +is no need for an error check, you can call the operations +qelt and +qsetelt! which provide the same +functionality but without the error check. Typically, these operations +are called in well-tested programs. + +The operations row and +column extract rows and columns, +respectively, and return objects of +OneDimensionalArray with the +same underlying element type. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    + +You can determine the dimensions of an array by calling the operations +nrows and ncols, +which return the number of rows and columns, respectively. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To apply an operation to every element of an array, use +map. This creates a new array. This +expression negates every element. +
      +
    • + +
      +
    • +
    +This creates an array where all the elements are doubled. +
      +
    • + +
      +
    • +
    +To change the array destructively, use +map! instead of +map. If you need to make a copy of any array, +use copy. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use member? to see if a given element +is in an array. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To see how many times an element appears in an array, use +count. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    + +For more information about the operations available for +TwoDimensionalArray, issue +
      +
    • + +
      +
    • +
    +For more information on related topics, see +Matrix and +OneDimensionalArray +<> +@ + +%%M +\subsection{man0page.xhtml} +<>= +<> + + +<> +Enter search string (use * for wild card unless counter-indicated): +
    + +
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + + + + + + + +
    + + Constructors + + + Search for + + categories + , + + domains + , + or + + packages + +
    + + Operations + + Search for operations.
    + + Attributes + + Search for attributes.
    + + General + + Search for all three of the above.
    + + Documentation + + Search library documentation. +
    + + Complete + + All of the above. +
    + + Selectable + + Detailed search with selectable options. +
    + + Reference + + Search Reference documentation (* wild card is not accepted). +
    + + Commands + + View system command documentation. +
    +<> +@ + +\subsection{menualgebraadjointmatrix.xhtml} +<>= +<> + + +<> +menualgebraadjointmatrix not implemented +<> +@ + +\subsection{menualgebraapplytolist.xhtml} +<>= +<> + + +<> +menualgebraapplytolist not implemented +<> +@ + +\subsection{menualgebracharacteristicpolynomial.xhtml} +<>= +<> + + +<> +menualgebracharacteristicpolynomial not implemented +<> +@ + +\subsection{menualgebradeterminant.xhtml} +<>= +<> + + +<> +menualgebradeterminant not implemented +<> +@ + +\subsection{menualgebraeigenvalues.xhtml} +<>= +<> + + +<> +menualgebraeigenvalues not implemented +<> +@ + +\subsection{menualgebraeigenvectors.xhtml} +<>= +<> + + +<> +menualgebraeigenvectors not implemented +<> +@ + +\subsection{menualgebraentermatrix.xhtml} +<>= +<> + + +<> +menualgebraentermatrix not implemented +<> +@ + +\subsection{menualgebrainvertmatrix.xhtml} +<>= +<> + + +<> +menualgebrainvertmatrix not implemented +<> +@ + +\subsection{menualgebrageneratematrix.xhtml} +<>= +<> + + +<> 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implemented +<> +@ + +\subsection{menuaxiominterrupt.xhtml} +<>= +<> + + +<> +menuaxiominterrupt not implemented +<> +@ + +\subsection{menuaxiomrestart.xhtml} +<>= +<> + + +<> +menuaxiomrestart not implemented +<> +@ + +\subsection{menuaxiomshowdefinition.xhtml} +<>= +<> + + +<> +menuaxiomshowdefinition not implemented +<> +@ + +\subsection{menuaxiomdisplay.xhtml} +<>= +<> + + +<> +menuaxiomdisplay not implemented +<> +@ + +\subsection{menuaxiomset.xhtml} +<>= +<> + + +<> +menuaxiomset not implemented +<> +@ + +\subsection{menuaxiomshowfunctions.xhtml} +<>= +<> + + +<> +menuaxiomshowfunctions not implemented +<> +@ + +\subsection{menuaxiomshowvariables.xhtml} +<>= +<> + + +<> +menuaxiomshowvariables not implemented +<> +@ + +\subsection{menuaxiomtoggletimedisplay.xhtml} +<>= +<> + + +<> +menuaxiomtoggletimedisplay not implemented +<> +@ + +\subsection{menucalculuscalculussum.xhtml} +<>= +<> + + +<> +menucalculuscalculussum not implemented +<> +@ + +\subsection{menucalculuscalculusproduct.xhtml} +<>= +<> + + +<> +menucalculuscalculusproduct not implemented +<> +@ + +\subsection{menucalculuschangevariable.xhtml} +<>= +<> + + +<> +menucalculuschangevariable not implemented +<> +@ + +\subsection{menucalculuscontinuedfractions.xhtml} +<>= +<> + + +<> +menucalculuscontinuedfractions not implemented +<> +@ + +\subsection{menucalculusdifferentiate.xhtml} +<>= +<> + + +<> +menucalculusdifferentiate not implemented +<> +@ + +\subsection{menucalculusdividepolynomials.xhtml} +<>= +<> + + +<> +menucalculusdividepolynomials not implemented +<> +@ + +\subsection{menucalculusfindlimit.xhtml} +<>= +<> + + +<> +menucalculusfindlimit not implemented +<> +@ + +\subsection{menucalculusgetseries.xhtml} +<>= +<> + + +<> +menucalculusgetseries not implemented +<> +@ + +\subsection{menucalculusgreatestcommondivisor.xhtml} +<>= +<> + + +<> +menucalculusgreatestcommondivisor not implemented +<> +@ + +\subsection{menucalculusleastcommonmultiple.xhtml} +<>= +<> + + +<> +menucalculusleastcommonmultiple not implemented +<> +@ + +\subsection{menucalculusintegrate.xhtml} +<>= +<> + + +<> +menucalculusintegrate not implemented +<> +@ + +\subsection{menucalculusinverselaplacetransform.xhtml} +<>= +<> + + +<> +menucalculusinverselaplacetransform not implemented +<> +@ + +\subsection{menucalculuslaplacetransform.xhtml} +<>= +<> + + +<> +menucalculuslaplacetransform not implemented +<> +@ + +\subsection{menucalculuslevel3.xhtml} +<>= +<> + + +<> +menucalculuslevel3 not implemented +<> +@ + +\subsection{menucalculuslevel3a.xhtml} +<>= +<> + + +<> +menucalculuslevel3a not implemented +<> +@ + +\subsection{menucalculuslevel3b.xhtml} +<>= +<> + + +<> +menucalculuslevel3b not implemented +<> +@ + +\subsection{menucalculuslevel3c.xhtml} +<>= +<> + + +<> +menucalculuslevel3c not implemented +<> +@ + +\subsection{menucalculuspadeapproximation.xhtml} +<>= +<> + + +<> +menucalculuspadeapproximation not implemented +<> +@ + +\subsection{menucalculuspartialfractions.xhtml} +<>= +<> + + +<> +menucalculuspartialfractions not implemented +<> +@ + +\subsection{menucalculusrischintegrate.xhtml} +<>= +<> + + +<> +menucalculusrischintegrate not implemented +<> +@ + +\subsection{menueditcopy.xhtml} +<>= +<> + + +<> +menueditcopy not implemented +<> +@ + +\subsection{menueditcopyasimage.xhtml} +<>= +<> + + +<> +menueditcopyasimage not implemented +<> +@ + +\subsection{menueditcopytex.xhtml} +<>= +<> + + +<> +menueditcopytex not implemented +<> +@ + +\subsection{menueditcopytext.xhtml} +<>= +<> + + +<> +menueditcopytext not implemented +<> +@ + +\subsection{menueditcut.xhtml} +<>= +<> + + +<> +menueditcut not implemented +<> +@ + +\subsection{menueditpaste.xhtml} +<>= +<> + + +<> +menueditpaste not implemented +<> +@ + +\subsection{menueditdeleteselection.xhtml} +<>= +<> + + +<> +menueditdeleteselection not implemented +<> +@ + +\subsection{menueditselectiontoimage.xhtml} +<>= +<> + + +<> +menueditselectiontoimage not implemented +<> +@ + +\subsection{menueditselectiontoinput.xhtml} +<>= +<> + + +<> +menueditselectiontoinput not implemented +<> +@ + +\subsection{menuequationsrealrootsofpolynmial.xhtml} +<>= +<> + + +<> +menuequationsrealrootsofpolynmial not implemented +<> +@ + +\subsection{menuequationsatvalue.xhtml} +<>= +<> + + +<> +menuequationsatvalue not implemented +<> +@ + +\subsection{menuequationsboundaryvalueproblem.xhtml} +<>= +<> + + +<> +menuequationsboundaryvalueproblem not implemented +<> +@ + +\subsection{menuequationsinitialvalueproblem1.xhtml} +<>= +<> + + +<> +menuequationsinitialvalueproblem1 not implemented +<> +@ + +\subsection{menuequationsinitialvalueproblem2.xhtml} +<>= +<> + + +<> +menuequationsinitialvalueproblem2 not implemented +<> +@ + +\subsection{menuequationssolvealgebraicsystem.xhtml} +<>= +<> + + +<> +menuequationssolvealgebraicsystem not implemented +<> +@ + +\subsection{menuequationseliminatevariable.xhtml} +<>= +<> + + +<> +menuequationseliminatevariable not implemented +<> +@ + +\subsection{menuequationssolvelinearsystem.xhtml} +<>= +<> + + +<> +menuequationssolvelinearsystem not implemented +<> +@ + +\subsection{menuequationssolveode.xhtml} +<>= +<> + + +<> +menuequationssolveode not implemented +<> +@ + +\subsection{menuequationssolveodewithlaplace.xhtml} +<>= +<> + + +<> +menuequationssolveodewithlaplace not implemented +<> +@ + +\subsection{menuequationsrootsofpolynomial.xhtml} +<>= +<> + + +<> +menuequationsrootsofpolynomial not implemented +<> +@ + +\subsection{menuequationssolve.xhtml} +<>= +<> + + +<> +menuequationssolve not implemented +<> +@ + +\subsection{menuequationssolvenumerically.xhtml} +<>= +<> + + +<> +menuequationssolvenumerically not implemented +<> +@ + +\subsection{menufileexit.xhtml} +<>= +<> + + +<> +menufileexit not implemented +<> +@ + +\subsection{menufileinputfile.xhtml} +<>= +<> + + +<> +menufileinputfile not implemented +<> +@ + +\subsection{menufileloadlibrary.xhtml} +<>= +<> + + +<> +menufileloadlibrary not implemented +<> +@ + +\subsection{menufileopen.xhtml} +<>= +<> + + +<> +menufileopen not implemented +<> +@ + +\subsection{menufileprint.xhtml} +<>= +<> + + +<> +menufileprint not implemented +<> +@ + +\subsection{menufileread.xhtml} +<>= +<> + + +<> +menufileread not implemented +<> +@ + +\subsection{menufilesave.xhtml} +<>= +<> + + +<> +menufilesave not implemented +<> +@ + +\subsection{menufilesaveas.xhtml} +<>= +<> + + +<> +menufilesaveas not implemented +<> +@ + +\subsection{menufiletogglespool.xhtml} +<>= +<> + + +<> +menufiletogglespool not implemented +<> +@ + +\subsection{menunumericsetprecision.xhtml} +<>= +<> + + +<> +menunumericsetprecision not implemented +<> +@ + +\subsection{menunumerictobigfloat.xhtml} +<>= +<> + + +<> +menunumerictobigfloat not implemented +<> +@ + +\subsection{menunumerictofloat.xhtml} +<>= +<> + + +<> +menunumerictofloat not implemented +<> +@ + +\subsection{menunumerictogglenumericoutput.xhtml} +<>= +<> + + +<> +menunumerictogglenumericoutput not implemented +<> +@ + +\subsection{menusimplifyaddalgebraicequality.xhtml} +<>= +<> + + +<> +menusimplifyaddalgebraicequality not implemented +<> +@ + +\subsection{menusimplifycomplexsimplification.xhtml} +<>= +<> + + +<> +menusimplifycomplexsimplification not implemented +<> +@ + +\subsection{menusimplifycontractlogarithms.xhtml} +<>= +<> + + +<> +menusimplifycontractlogarithms not implemented +<> +@ + +\subsection{menusimplifyevaluatenounform.xhtml} +<>= +<> + + +<> +menusimplifyevaluatenounform not implemented +<> +@ + +\subsection{menusimplifyexpandexpression.xhtml} +<>= +<> + + +<> +menusimplifyexpandexpression not implemented +<> +@ + +\subsection{menusimplifyexpandlogarithms.xhtml} +<>= +<> + + +<> +menusimplifyexpandlogarithms not implemented +<> +@ + +\subsection{menusimplifyfactorialsandgamma.xhtml} +<>= +<> + + +<> +menusimplifyfactorialsandgamma not implemented +<> +@ + +\subsection{menusimplifyfactorcomplex.xhtml} +<>= +<> + + +<> +menusimplifyfactorcomplex not implemented +<> +@ + +\subsection{menusimplifyfactorexpression.xhtml} +<>= +<> + + +<> +menusimplifyfactorexpression not implemented +<> +@ + +\subsection{menusimplifymoduluscomputation.xhtml} +<>= +<> + + +<> +menusimplifymoduluscomputation not implemented +<> +@ + +\subsection{menusimplifysimplifyexpression.xhtml} +<>= +<> + + +<> +menusimplifysimplifyexpression not implemented +<> +@ + +\subsection{menusimplifysubtitute.xhtml} +<>= +<> + + +<> +menusimplifysubtitute not implemented +<> +@ + +\subsection{menusimplifysimplifyradicals.xhtml} +<>= +<> + + +<> +menusimplifysimplifyradicals not implemented +<> +@ + +\subsection{menusimplifytogglealgebraicflag.xhtml} +<>= +<> + + +<> +menusimplifytogglealgebraicflag not implemented +<> +@ + +\subsection{menusimplifytrigsimplification.xhtml} +<>= +<> + + +<> +menusimplifytrigsimplification not implemented +<> +@ + +%%N +\subsection{numbasicfunctions.xhtml} +<>= +<> + + + +<> +
    Basic Functions
    +
    +The size of an integer in Axiom is only limited by the amount of computer +storage you have available. The usual arithmetic operations are available. +
      +
    • + +
      +
    • +
    +There are a number of ways of working with the sign of an integer. Let's +use the x as an example. +
      +
    • + +
      +
    • +
    +First of all, there is the absolute value function. +
      +
    • + +
      +
    • +
    +The sign operation returns -1 if its argument +is negative, 0 if zero and 1 if positive. +
      +
    • + +
      +
    • +
    +You can determine if an integer is negative in several other ways. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Similarly, you can find out if it is positive. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +This is the recommended way of determining whether an integer is zero. +
      +
    • + +
      +
    • +
    +
    +Use the zero? whenever you are testing any +mathematical object for equality with zero. This is usually more efficient +than using = (think of matrices: it is easier +to tell if a matrix is zero by just checking term by term than constructing +another "zero" amtrix and comparing the two matrices term by term) and also +avoids the problem that = is usually used +for creating equations. +
    +This is the recommended way of determining whether an integer is equal to one. +
      +
    • + +
      +
    • +
    +This syntax is used to test equality using =. +It says that you want a Boolean (true or false) +answer rather than an equation. +
      +
    • + +
      +
    • +
    +The operations odd? and +even? determine whether an integer is odd +or even, respectively. They each return a +Boolean +object. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The operation gcd computes the greatest common +divisor of two integers. +
      +
    • + +
      +
    • +
    +The operation lcm computes their least common +multiple. +
      +
    • + +
      +
    • +
    +To determine the maximum of two integers, use max. +
      +
    • + +
      +
    • +
    +To determine the minimum, use min. +
      +
    • + +
      +
    • +
    +The reduce operation is used to extend +binary operations to more than two arguments. For example, you can use +reduce to find the maximum integer in a +list or compute the least common multiple of all integers in a list. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The infix operator "/" is not used to compute the quotient of integers. +Rather , it is used to create rational numbers as described in +Fractions. +
      +
    • + +
      +
    • +
    +The infix operator quo computes the integer +quotient. +
      +
    • + +
      +
    • +
    +The infix operation rem computes the integer +remainder. +
      +
    • + +
      +
    • +
    +One integer is evenly divisible by another if the remainder is zero. +The operation exquo can also be used. See +Unions for an example. +
      +
    • + +
      +
    • +
    +The operation divide returns a record of +the quotient and remainder and thus is more efficient when both are needed. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Records are discussed in detail in +Records. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numberspage.xhtml} +<>= +<> + + +<> +The following types of numbers are among those available in Axiom + + + + + + + + + + + + + + + + + + + + + + + + + +
    + Integers + + Arithmetic with arbitrarily large integers +
    + Fractions + + Rational numbers and general fractions +
    + Machine Floats + + Fixed precision machine floating point +
    + Real Numbers + + Arbitrary precision decimal arithmetic +
    + Complex Numbers + + Complex numbers in general +
    + Finite Fields + + Arithmetic in characteristic p +
    +
    +Addtional topics + +<> +@ + +\subsection{numcardinalnumbers.xhtml} +<>= +<> + + + +<> +
    Cardinal Numbers
    +
    +The CardinalNumber can be used for +values indicating the cardinality of sets, both finite and infinite. For +example, the dimension operation in the +category VectorSpace returns a cardinal +number. + +The non-negative integers have a natural construction as cardinals +
    +0=#{ }, 1={0}, 2={0,1}, ..., n={i | 0 <= i < n}
    +
    +The fact that 0 acts as a zero for the multiplication of cardinals is +equivalent to the axiom of choice. + +Cardinal numbers can be created by conversion from non-negative integers. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The can also be obtained as the named cardinal Aleph(n) +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The finite? operation tests whether a value +is a finite cardinal, that is, a non-negative integer. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Similarly, the countable? operation +determines whether a value is a countable cardinal, that is, finite or +Aleph(0). +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Arithmetic operations are defined on cardinal numbers as follows: + + + + + + + + + + + + + + + + + +
    + x+y = #(X+Y) + + cardinality of the disjoint union +
    + x-y = #(X-Y) + + cardinality of the relative complement +
    + x*y = #(X*Y) + + cardinality of the Cartesian product +
    + x+*y = #(X**Y) + + cardinality of the set of maps from Y to X +
    +Here are some arithmetic examples: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Subtraction is a partial operation; it is not defined when subtracting +a larger cardinal from a smaller one, nor when subtracting two equal +infinite cardinals. +
      +
    • + +
      +
    • +
    +The generalized continuum hypothesis asserts that +
    + 2**Aleph i = Aleph(i+1)
    +
    +and is independent of the axioms of set theory. (Goedel, The consistency +of the continuum hypothesis, Ann. Math. Studies, Princeton Univ. Press, +1940) The CardinalNumber domain +provides an operation to assert whether the hypothesis is to be assumed. +
      +
    • + +
      +
    • +
    +When the generalized continuum hypothesis is assumed, exponentiation to +a transfinite power is allowed. +
      +
    • + +
      +
    • +
    +Three commonly encountered cardinal numbers are +
    +  a = #Z                 countable infinity
    +  c = #R                 the continuum
    +  f = #{g|g: [0,1]->R}
    +
    +In this domain, these values are obtained under the generalized continuum +hypothesis in this way: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{numcomplexnumbers.xhtml} +<>= +<> + + + +<> +
    Complex Numbers
    +
    +The Complex constructor implements +complex objects over a commutative ring R. Typically, the ring R is +Integer, +Fraction Integer, +Float, +DoubleFloat, +R can also be a symbolic type, like +Polynomial Integer. +For more information about the numerical and graphical aspects of +complex numbers, see +Numeric Functions +in section 8.1. + +Complex objects are created by the +complex operation +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The standard arithmetic operations are available. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +If R is a field, you can also divide the complex objects. +
      +
    • + +
      +
    • +
    +Use a conversion +(see Conversion in +section 2.7) to view the last object as a fraction of complex +integers. +
      +
    • + +
      +
    • +
    +The predefined macro %i is defined to be complex(0,1). +
      +
    • + +
      +
    • +
    +You can also compute the +conjugate and +norm of a complex number. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The real and +imag operations are provided to +extract the real and imaginary parts, respectively. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The domain +Complex Integer +is also called the Gaussian integers. If R is the integers (or, more +generally, a +Euclidean Domain), +you can compute greatest common divisors. +
      +
    • + +
      +
    • +
    +You can also compute least common multiples +
      +
    • + +
      +
    • +
    +You can factor Gaussian integers. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{numcontinuedfractions.xhtml} +<>= +<> + + + +<> +
    Continued Fractions
    +
    +Continued fractions have been a fascinating and useful tool in mathematics +for well over three hundred years. Axiom implements continued fractions +for fractions of any Euclidean domain. In practice, this usually means +rational numbers. In this section we demonstrate some of the operations +available for manipulating both finite and infinite continued fractions. +It may be helpful if you review +Stream to remind yourself of some of the +operations with streams. + +The ContinuedFraction domain is a +field and therefore you can add, subtract, multiply, and divide the +fractions. The +continuedFraction operation +converts its fractional argument to a continued fraction. +
      +
    • + +
      +
    • +
    +This display is the compact form of the bulkier +
    +  3 +             1
    +     ---------------------------
    +     7 +            1
    +         -----------------------
    +         15 +         1
    +              ------------------
    +              1 +        1
    +                  --------------
    +                  25 +     1
    +                       ---------
    +                       1 +   1
    +                           -----
    +                           7 + 1
    +                               -
    +                               4
    +
    +You can write any rational number in a similar form. The fraction will +be finite and you can always take the "numerators" to be 1. That is, any +rational number can be written as a simple, finite continued fraction of +the form +
    +a(1) +            1
    +     ---------------------------
    +  a(2) +            1
    +         -----------------------
    +       a(3) +         1
    +                        .
    +                         .
    +                          .
    +                           1
    +
    +              -----------------
    +              a(n-1) +     1
    +                       ---------
    +                          a(n)
    +
    +The a(i) are called partial quotients and the operation +partialQuotients creates a +stream of them. +
      +
    • + +
      +
    • +
    +By considering more and more of the fraction, you get the +convergents. For example, the +first convergent is a(1), the second is a(1)+1/a(2) and so on. +
      +
    • + +
      +
    • +
    +Since this ia a finite continued fraction, the last convergent is the +original rational number, in reduced form. The result of +approximants is always an infinite +stream, though it may just repeat the "last" value. +
      +
    • + +
      +
    • +
    +Inverting c only changes the partial quotients of its fraction by +inserting a 0 at the beginning of the list. +
      +
    • + +
      +
    • +
    +Do this to recover the original continued fraction from this list of +partial quotients. The three argument form of the +continuedFraction operation takes +an element which is the whole part of the fraction, a stream of elements +which are the denominators of the fraction. +
      +
    • + +
      +
    • +
    +The streams need not be finite for +continuedFraction. Can you guess +which irrational number has the following continued fraction? See the end +of this section for the answer. +
      +
    • + +
      +
    • +
    +In 1737 Euler discovered the infinite continued fraction expansion +
    + e - 1                 1
    + ----- =  ---------------------------
    +p          2 +            1
    +              -----------------------
    +              6  +         1
    +                   ------------------
    +                  10 +        1
    +                       --------------
    +                       14 +  ... 
    +
    +We use this expansion to compute rational and floating point +approximations of e. (For this and other interesting expansions, +see C. D. Olds, Continued Fractions, New Mathematical Library, +Random House, New York, 1963 pp.134-139). + +By looking at the above expansion, we see that the whole part is 0 +and the numerators are all equal to 1. This constructs the stream of +denominators. +
      +
    • + +
      +
    • +
    +Therefore this is the continued fraction expansion for (e-1)/2. +
      +
    • + +
      +
    • +
    +These are the rational number convergents. +
      +
    • + +
      +
    • +
    +You can get rational convergents for e by multiplying by 2 and adding 1. +
      +
    • + +
      +
    • +
    +You can also compute the floating point approximations to these convergents. +
      +
    • + +
      +
    • +
    +Compare this to the value of e computed by the +exp operation in +Float. +
      +
    • + +
      +
    • +
    +In about 1658, Lord Brouncker established the following expansion for 4/pi. +
    +  1 +             1
    +     ---------------------------
    +     2 +            9
    +         -----------------------
    +         2  +         25
    +              ------------------
    +              2 +        49
    +                  --------------
    +                  2  +     81
    +                       ---------
    +                       2 +   ...
    +
    +Let's use this expansion to compute rational and floating point +approximations for pi. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +As you can see, the values are converging to +
    +  pi = 3.14159265358979323846..., but not very quickly.
    +
    +
      +
    • + +
      +
    • +
    +You need not restrict yourself to continued fractions of integers. Here is +an expansion for a quotient of Gaussian integers. +
      +
    • + +
      +
    • +
    +This is an expansion for a quotient of polynomials in one variable with +rational number coefficients. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +To conclude this section, we give you evidence that +
    +  z =  3 +             1
    +          ---------------------------
    +          3 +            1
    +              -----------------------
    +              6 +          1
    +                  -------------------
    +                   3 +        1
    +                       --------------
    +                       6  +     1
    +                            ---------
    +                            3 + ...
    +
    +is the expansion of the square root of 11. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numexamples.xhtml} +<>= +<> + + + +<> +
    Examples
    +
    +One can show that if an integer of the form 2**k+1 is prime, then k +must be a power of two. + +Pierre Fermat conjectured that every integer of the form 2**(2**n)+1 +is prime. Let's look for a counterexample. First define a function: +
      +
    • + +
      +
    • +
    +Now try commands like: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +until you find an integer of this form which is composite. You can also +try the following command: +
      +
    • + +
      +
    • +
    +Obviously, Fermat didn't have access to Axiom. +<> +@ + +\subsection{numfactorization.xhtml} +<>= +<> + + + +<> +
    Primes and Factorization
    +
    +Use the operation factor to factor integers. +It returns an object of type +Factored Integer. See Factored for a +discussion of the manipulation of factored objects. +
      +
    • + +
      +
    • +
    +The operation prime? returns true or false +depending on whether its argument is a prime. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The operation nextPrime returns the +least prime number greater than its argument. +
      +
    • + +
      +
    • +
    +The operation prevPrime returns the +greatest prime number less than its argument. +
      +
    • + +
      +
    • +
    +To compute all primes between two integers (inclusively), use the +operation primes. +
      +
    • + +
      +
    • +
    +You might sometimes want to see the factorization of an integer +when it is considered a Gaussian (that is, complex) integer. See +Complex for more details. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numfinitefields.xhtml} +<>= +<> + + +<> +
    Finite Fields
    +
    +A finite field (also called a Galois field) is a finite +algebraic structure where on can add, multiply, and divide under the same +laws (for example, commutativity, associativity, or distributivity) as +apply to the rational, real, or complex numbers. Unlike those three fields, +for any finite field there exists a positive prime integer p, called the +characteristic, such that p*x=0 for +any element x in the finite field. In fact, the number of elements in a +finite filed is a power of the characteristic and for each prime p and +positive integer n there exists exactly one finite field with p**n elements, +up to an isomorphism. (For more information about the algebraic structure and +properties of finite fields, see for example, S. Lang Algebr, +Second Edition, New York, Addison-Wesley Publishing Company, Inc. 1984, +ISBN 0 201 05476 6; or R. Lidl, H. Niederreiter, Finite Fields, +Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge. +Cambridge Univ. Press, 1983, ISBN 0 521 30240 4) + +When n=1, the field has p elements and is called a prime field, +discussed in + +Modular Arithmetic and Prime Fields +in section 8.11.1. There are several ways of implementing extensions of +finite fields, and Axiom provides quite a bit of freedom to allow you to +choose the one that is best for your application. Moreover, we provide +operations for converting among the different representations of extensions +and different extensions of a single field. Finally, note that you usually +need to package call operations from finite fields if the operations do not +take as an argument an object of the field. See +Package Calling and Target Types +in section 2.9 for more information on package calling. + +<> +@ + +\subsection{numfloat.xhtml} +<>= +<> + + +<> +
    Real Numbers
    +
    +Axiom provides two kinds of floating point numbers. The domain +Float +(abbreviation FLOAT) +implements a model of arbitrary precisions floating point numbers. The +domain +DoubleFloat +(abbreviation DFLOAT) +is intended to make available hardware floating point arithmetic in Axiom. +The actual model of floating point +DoubleFloat that Axiom +provides is system dependent. For example, on the IBM System 370, Axiom +uses IBM double precision which has fourteen hexadecimal digits of +precision or roughly sixteen decimal digits. Arbitrary precision floats +allow the user to specify the precision at which arithmetic operations +are computed. Although this is an attractive facility, it comes at a cost. +Arbitrary precision floating point arithmetic typically takes twenty to +two hundred times more time than hardware floating point. + +For more information about Axiom's numeric and graphic facilities +see Graphics in section 7, +Numeric Functions +in section 8.1, and DoubleFloat + +<> +@ + +\subsection{numfractions.xhtml} +<>= +<> + + + +<> +
    Fractions
    +
    +Axiom handles fractions in many different contexts and will +automatically simplify fractions whenever possible. Here are +some examples: +
      +
    • +
      +
    • +
    • +
      +
    • +
    • +
      +
    • +
    • +
      +
    • +
    +
    +Additional Topics: + + + + + + + + + +
    + Rational Numbers + + Quotients of integers +
    + Quotient Fields + + Quotients over an arbitrary integral domain +
    +<> +@ + +\subsection{numfunctions.xhtml} +<>= +<> + + + +<> +
    Integer Number Theory Functions
    +
    +The +IntegerNumberTheoryFunctions package contains a variety of +operations of interest to number theorists. Many of these operations +deal with divisibility properties of integers (Recall that an integer +a divides an integer b if there is an integer c such that b=a*c.) + +The operation divisors returns a list +of the divisors of an integer +
      +
    • + +
      +
    • +
    +You can now compute the number of divisors of 144 and the sum of the +divisors of 144 by counting and summing the elements of the list we +just created. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Of course, you can compute the number of divisors of an integer n, +usually denoted d(n), and the sum of the divisors of an integer n, +usually denoted ς(n), without ever listing the divisors of n. + +In Axiom, you can simply call the operations +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The key is that d(n) and ς(n) are "multiplicative functions". +This means that when n and m are relatively prime, that is, when n and +m have no factors in common, then d(nm)=d(n)d(m) and ς(nm)= +ς(n)ς(m). Note that these functions are trivial to +compute when n is a prime power and are computed for general n from +the prime factorization of n. Other examples of multiplicative functions +are ς_k(n), the sum of the k-th powers of the divisors of n and +φ(n), the number of integers between 1 and n which are prime to n. +The corresponding Axiom operations are called +sumOfKthPowerDivisors and +eulerPhi. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    + +An interesting function is called μ(n), the Moebius mu function, +defined as +
    +        0 if n has a repeated prime factor 
    +          (i.e. is divisible by a square)
    +  μ(n)= 1 if n is 1
    +       (-1)^k if n is the product of k distinct primes
    +
    +The corresponding Axiom operation is +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +This function occurs in the following theorem: +
    +Theorem(Moebius Inversion Formula):
    +Let f(n) be a function on the positive integers and let F(n) be defined +by F(n)=sum of f(n) over d | n where the sum is taken over the positive +divisors of n. Then the values of f(n) can be recovered from the values +of F(n):f(n) = sum of μF(n/d) over d|n, where the sum is taken +over the positive divisors of n. + +When f(n)=1, the F(n)=d(n). Thus, if you sum μ(d)*d(n/d) over +the positive divisors of d of n, you should always get 1. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Similarly, when f(n)=n, then F(n)=ς(n). Thus, if you sum +μ(d)*ς(n/d) over the positive divisors d of n, you +should always get n. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The Fibonacci numbers are defined by +
    +  F(1)=1
    +  F(2)=1
    +  F(n)=F(n-1)+F(n-2) for n=3,4,...
    +
    +The operation fibonacci computes the +nth Fibonacci number. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Fibonacci numbers can also be expressed as sums of binomial +coefficients. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    + +Quadratic symbols can be computed with the operations +legendre and +jacobi. The Legendre symbol (a/p) is +defined for integers a and p with p an odd prime number. By definition, +
    +        = -1 when a is not a square (mod p)
    +  (a/p) =  0 when a is divisible by p
    +        = +1 when a is a square (mod p)
    +
    +You compute (a/p) via the command legendre(a,p) +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    + +The Jacobi symbol (a/n) is the usual extension of the Legendre symbol, +where n is an arbitrary integer. The most important property of the +Jacobi symbol is the following: if K is a quadratic field with +discriminant d and quadratic character χ, the χ(n)=(d/n). +Thus, you can use the Jacobi symbol to compute, say, the class numbers +of imaginary quadratic fields from a standard class number formula. This +function computes the class number of the imaginary quadratic field with +discriminant d. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    + +<> +@ + +\subsection{numgeneralinfo.xhtml} +<>= +<> + + +<> +
    General Integer Information
    +
    +Axiom provides many operations for manipulating arbitrary precision integers. +In this section we will show some of those that come from +Integer itself plus some that are implemented +in other packages. More examples of integers are in the following sections: +Numbers. +IntegerNumberTheoryFunctions, +DecimalExpansion, +BinaryExpansion, +HexadecimalExpansion, and +RadixExpansion + +<> +@ + +\subsection{numintegerfractions.xhtml} +<>= +<> + + +<> + numintegerfractions not implemented +<> +@ + +\subsection{numintegers.xhtml} +<>= +<> + + + +<> +
    Integers
    +
    +In Axiom, integers can be as large as you like. Try the following +examples. +
      +
    • +
      +
    • +
    • +
      +
    • +
    +Of course, you can now do arithmetic as usual on these (very) large +integers: +
      +
    • +
      +
    • +
    • +
      +
    • +
    • +
      +
    • +
    +Axiom can factor integers, but numbers with small prime factors +
      +
    • +
      +
    • +
    +will factor more rapidly than numbers with large prime factors. +
      +
    • +
      +
    • +
    +
    +Additional topics + + + + + + + + + + + + + + + + + + + + + +
    + General Info + + General information and examples of integers +
    + Factorization + + Primes and factorization +
    + Functions + + Number theoretic functions +
    + Examples + + Examples from number theory +
    + Problems + + Problems from number theory +
    +<> +@ + +\subsection{nummachinefloats.xhtml} +<>= +<> + + + +<> +
    Machine Floats
    +
    +Axiom provides two kinds of floating point numbers. The domain +Float +(abbreviation FLOAT) +implements a model of arbitrary precisions floating point numbers. The +domain +DoubleFloat +(abbreviation DFLOAT) +is intended to make available hardware floating point arithmetic in Axiom. +The actual model of floating point +DoubleFloat that Axiom +provides is system dependent. For example, on the IBM System 370, Axiom +uses IBM double precision which has fourteen hexadecimal digits of +precision or roughly sixteen decimal digits. Arbitrary precision floats +allow the user to specify the precision at which arithmetic operations +are computed. Although this is an attractive facility, it comes at a cost. +Arbitrary precision floating point arithmetic typically takes twenty to +two hundred times more time than hardware floating point. + +By default, floating point numbers that you enter into Axiom are of type +Float. +
      +
    • + +
      +
    • +
    +You must therefore tell Axiom that you want to use +DoubleFloat values and operations. The +following are some conservative guidelines for getting Axiom to use +DoubleFloat. + +To get a value of type DoubleFloat., +use a target with + "@", ... +
      +
    • + +
      +
    • +
    +a conversion,... +
      +
    • + +
      +
    • +
    +or an assignment to a declared variable. It is more efficient if you +use a target rather than an explicit or implicit conversion. +
      +
    • + +
      +
    • +
    +You also need to declare functions that work with +DoubleFloat. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use package calling for operations from +DoubleFloat +unless the arguments themselves are already of type +DoubleFloat. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +By far, the most common usage of +DoubleFloat +is for functions to be graphied. For more information about Axiom's +numerical and graphical facilities, see +Graphics +in section 7, +Numeric Functions +in section 8.1, and +Float + +The usual arithmetic and elementary functions are available for +DoubleFloat. Use +
      +
    • + +
      +
    • +
    +to get a list of operations. +<> +@ + +\subsection{nummachinesizedintegers.xhtml} +<>= +<> + + + +<> +
    Machine-sized Integers
    +
    +The SingleInteger is intended to +provide support in Axiom for machine integer arithmetic. It is generally +much faster than (bignum) Integer arithmetic +but suffers from a limited range of values. Since Axiom can be implemented +on top of various dialects of Lisp, the actual representation of small +integers may not correspond exactly to the host machines integer +representation. + +You can discover the minimum and maximum values in your implementation by +using min and max +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To avoid confusion with Integer, which is +the default type for integers, you usually need to work with declared +variables (see Declarations). +
      +
    • + +
      +
    • +
    +or use package calling (see +Package Calling and Target Types). +
      +
    • + +
      +
    • +
    +You can add, multiply, and subtract +SingleInteger objects, and ask for the +greatest common divisor +(gcd). +
      +
    • + +
      +
    • +
    +The least common multiple +(lcm) is also available. +
      +
    • + +
      +
    • +
    +Operations +mulmod, +addmod, +submod, and +invmod +are similar -- they provide arithmetic modulo a given small integer. +Here is 5*6 mod 13. +
      +
    • + +
      +
    • +
    +To reduce a small integer modulo a prime, use +positiveRemainder +
      +
    • + +
      +
    • +
    +Operations And, +Or, +xor, +and Not +provide bit level operations on small integers. +
      +
    • + +
      +
    • +
    +Use shift(int,numToShift) to shift bits, where int is shifted left if +numToShift is positive, right if negative. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Many other operations are available for small integers, including many of +those provided for Integer. +To see other operations use the system command +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numnumbertheoreticfunctions.xhtml} +<>= +<> + + + +<> +
    Some Number Theoretic Functions
    +
    +Axiom provides several number theoretic operations for integers. +More examples are in +IntegerNumberTheoryFunctions, + +The operation fibonacci computes the +Fibonacci numbers. The algorithm has a running time O(log(n)**3) for +argument n. +
      +
    • + +
      +
    • +
    +The operation legendre computes the +Legendre symbol for its two integer arguments where the second one is +prime. If you know the second argument to be prime, use +jacobi instead where no check is made. +
      +
    • + +
      +
    • +
    +The operation jacobi computes the Jacobi symbol +for its two integer arguments. By convention, 0 is returned if the greatest +common divisor of the numerator and denominator is not 1. +
      +
    • + +
      +
    • +
    +The operation eulerPhi compute the values +of Euler's φ-function where φ(n) equals the number of +positive integers less than or equal to n that are relatively prime to the +positive integer n. +
      +
    • + +
      +
    • +
    +The operation moebiusMu computes the +Moebius μ function. +
      +
    • + +
      +
    • +
    +Although they have somewhat limited utility, Axiom provides Roman numerals. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{numnumericfunctions.xhtml} +<>= +<> + + + +<> +
    Numeric Functions
    +
    +Axiom provides two basic floating point types: +Float and +DoubleFloat. This section +describes how to use numerical operations defined on these types and +the related complex types. As we mentioned in +An Overview of Axiom +in chapter 1., the +Float type is a software implementation of +floating point numbers in which the exponent and the significand may have +any number of digits. See +Float for detailed information about this +domain. The +DoubleFloat is usually a hardware +implementation of floating point numbers, corresponding to machine double +precision. The types +Complex Float and +Complex DoubleFloat are the +corresponding software implementations of complex floating point numbers. +In this section the term floating point type means any of these four +types. The floating point types immplement the basic elementary functions. +These include (where $ means +DoubleFloat, +Float, +Complex Float, +Complex DoubleFloat):
    +exp, +log: $ -> $
    +sin, +cos, +tan, +cot, +sec, +csc: $ -> $
    +asin, +acos, +atan, +acot, +asec, +acsc: $ -> $
    +sinh, +cosh, +tanh, +coth, +sech, +csch: $ -> $
    +asinh, +acosh, +atanh, +acoth, +asech, +acsch: $ -> $
    +pi: () -> $
    +sqrt: $ -> $
    +nthRoot: ($,Integer) -> $
    +**: ($,Fraction Integer) -> $
    +**: ($,$) -> $
    +The handling of roots depends on whether the floating point type is +real or complex: for the real floating point types, +DoubleFloat and +Float, if a real root exists the one with +the same sign as the radicand is returned; for the complex floating +point types, the principal value is returned. Also, for real floating +point types the inverse functions produce errors if the results are not +real. This includes cases such as asin(1.2), log(-3.2), sqrt(-1,1). +The default floating point type is Float +or Complex Float, just use normal +decimal notation. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To evaluate functions using +DoubleFloat or +Complex DoubleFloat, a +declaration or conversion is required. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +A number of special functions are provided by the package +DoubleFloatSpecialFunctions +for the machine precision floating point types. The special functions +provided are listed below, where F stands for the types +Float +or Complex Float. The real versions +of the functions yield an error if the result is not real. +
      +
    • + Gamma: F -> F
      + Gamma(z) is the Euler gamma + function, Gamma(Z), defined by
      + Gamma(z) = integrate(t^(z-1)*exp(-t),t=0..%infinity) +
    • +
    • + Beta: F -> F
      + Beta(u,v) is the Euler Beta + function B(u,v), defined by
      + Beta(u,v)=integrate(t^(u-1)*(1-t)^(b-1),t=0..1)
      + This is related to Gamma(z) by
      + Beta(u,v)=Gamma(u)*Gamma(v)/Gamma(u+v) +
    • +
    • + logGamma: F -> F
      + logGamma(z) is the natural logarithm of Gamma(z). This can often be + computed even if Gamma(z) cannot. +
    • +
    • + digamma: F -> F
      + digamma(z), also called psi(z), is the function psi(z), defined by
      + psi(z)=Gamma'(z)/Gamma(z) +
    • +
    • + polygamma: (NonNegativeInteger, F) -> F
      + polygamma(n,z) is the n-th derivative of digamma(z) +
    • +
    • + besselJ: (F, F) -> F
      + besselJ(v,z) is the Bessel function of the first kind, J(v,z). This + function satisfies the differential equation
      + z^(2w)''(z)+zw'(z)+(z^2-v^2)w(z)=0 +
    • +
    • + besselY: (F, F) -> F
      + besselY(v,z) is the Bessel function of the second kind, Y(v,z). This + function satisfies the same differential equation as + besselJ. The implementation simply + uses the relation
      + Y(v,z)=(J(v,z)cos(v*%pi)-J(-v,z))/sin(v*%pi) +
    • +
    • + besselI: (F, F) -> F
      + besselI(v,z) if the modifed Bessel function of the first kind, I(v,z). + This function satisfies the differential equation
      + z^2w''(z)+zw'(z)-(z^2+v^2)w(z)=0 +
    • +
    • + besselK: (F, F) -> F
      + besselK(v,z) is the modifed Bessel function of the second kind, K(v,z). + This function satisfies the same differential equation as + besselI. The implementation simply uses + the relation
      + K(v,z)=%pi*(I(v,z)-I(-v,z))/(2sin(v*%pi)) +
    • +
    • + airyAi: F -> F
      + airyAi(z) is the Airy function Ai(z). This function satisfies the + differential equation
      + w''(z)-zw(z)=0
      + The implementation simply uses the relation
      + Ai(-z)=1/3*sqrt(z)*(J(-1/3,2/3*z^(3/2))+J(1/3,2/3*z^(3/2))) +
    • +
    • + airyBi: F -> F
      + airyBi(z) is the Airy function Bi(z). This function satisfies the + same differential equation as airyAi. + The implementation simply uses the relation
      + Bi(-z)=1/3*sqrt(3*z)*(J(-1/3,2/3*z^(3/2))-J(1/3,2/3*z^(3/2))) +
    • +
    • + hypergeometric0F1: (F, F) -> F
      + hypergeometric0F1(c,z) is the hypergeometric function 0F1(;c;z). The above + special functions are defined only for small floating point types. If you + give Float arguments, they are converted to + DoubleFloat by Axiom. +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +A number of additional operations may be used to compute numerical +values. These are special polynomial functions that can be evaluated +for values in any commutative ring R, and in particular for values in +any floating-point type. The following operations are provided by the +package +OrthogonalPolynomialFunctions: +
      +
    • chebyshevT: + (nonNegativeInteger,R) -> R +
      + chebyshevT(n,z) is the nth Chebyshev polynomial of the first kind, + T[n](z). These are defined by +
      + (1-t*z)/(1-2*t*z*t**2)=sum(T[n](z)*t**n,n=0..) +
    • +
    • chebyshevU: + (nonNegativeInteger,R) -> R +
      + chebyshevU(n,z) is the nth Chebyshev polynomial of the second kind, + U[n](z). These are defined by +
      + 1/(1-2*t*z+t**2)=sum(U[n](z)*t**n,n=0..) +
    • +
    • hermiteH: + (NonNegativeInteger,R) -> R +
      + hermiteH(n,z) is the nth Hermite polynomial, H[n](z). These are + defined by +
      + exp(2*t*z-t**2)=sum(H[n](z)*t**n/n!,n=0..) +
    • +
    • laguerreL: + (NonNegativeInteger,R) -> R +
      + laguerreL(n,z) is the nth Laguerre polynomial, L[n](z). These are + defined by +
      + (exp(-t*z/(1-t))/(1-t)=sum(L[n](z)*t**n/n!,n=0..) +
    • +
    • laguerreL: + (NonNegativeInteger,NonNegativeInteger,R) -> R +
      + labuerreL(m,n,2) is the associated Laguerre polynomial, L<m>[n](z). + This is the nth derivative of L[n](z). +
    • +
    • legendreP: + (NonNegativeInteger,R) -> R +
      + legendreP(n,z) is the nth Legendre polynomial, P[n](z). These are + defined by +
      + 1/sqrt(1-2*z*t+t**2)=sum(P[n](z)*t**n,n=0..) +
    • +
    +
    +
    +These operations require non-negative integers for the indices, +but otherwise the argument can be given as desired. +
      +
    • + +
      +
    • +
    +The expression chebyshevT(n,z) evaluates to the nth Chebyshev polynomial +of the first kind. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression chebyshevU(n,z) evaluates to the nth Chebyshev polynomial +of the second kind. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression hermiteH(n,z) evaluates to the nth Hermite polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression laguerreL(n,z) evaluates to the nth Laguerre polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression legendreP(n,z) evaluates to the nth Legendre polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +
    +
    +Finally, three number-theoretic polynomial operations may be evaluated. +The following operations are provided by the package + +NumberTheoreticPolynomialFunctions. +
      +
    • bernoulliB: + (NonNegativeInteger,R) -> R +
      + bernoulliB(n,z) is the nth Bernoulli polynomial, B[n](z). These are + defined by +
      + t*exp(z*t)/(exp t - 1)=sum(B[n](z)*t**n/n! for n=0..) +
    • +
    • eulerE: + (NonNegativeInteger,R) -> R +
      + eulerE(n,z) is the nth Euler polynomial, E[n](z). These are defined by +
      + 2*exp(z*t)/(exp t + 1)=sum(E[n](z)*t**n/n! for n=0..) +
    • +
    • cyclotomic: + (NonNegativeInteger,R) -> R +
      + cyclotomic(n,z) is the nth cyclotomic polynomial φ(n,z). + This is the polynomial whose roots are precisely the primitive nth + roots of unity. This polynomial has degree given by the Euler + totient function φ(n). +
    • +
    + +The expression bernoulliB(n,z) evaluates to the nth Bernoulli polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression eulerE(n,z) evaluates to the nth Euler polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The expression cyclotomic(n,z) evaluates to the nth cyclotomic polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +
    +
    +Drawing complex functions in Axiom is presently somewhat awkward compared +to drawing real functions. It is necessary to use the +draw operations that operate on functions +rather than expressions. + +This is the complex exponential function. When this is displayed in color, +the height is the value of the real part of the function and the color is +the imaginary part. Red indicates large negative imaginary values, green +indicates imaginary values near zero and blue/violet indicates large +positive imaginary values. +
      +
    • + +
      +
    • +
    +This is the complex arctangent function. Again, the height is the real part +of the function value but here the color indicates the function value's phase. +The position of the branch cuts are clearly visible and one can see that the +function is real only for a real argument. +
      +
    • + +
      +
    • +
    +This is the complex Gamma function. +
      +
    • + +
      +
    • +
    +This shows the real Beta function near the origin. +
      +
    • + +
      +
    • +
    +This is the Bessel function J(alpha,x) for index alpha in the range -6..4 and +argument x in the range 2..14. +
      +
    • + +
      +
    • +
    +This is the modified Bessel function I(alpha,x) evaluated for various real +values of the index alpha and fixed argument x=5. +
      +
    • + +
      +
    • +
    +This is similar to the last example except the index alpha takes on complex +values in a 6x6 rectangle centered on the origin. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numoctonions.xhtml} +<>= +<> + + + +<> +
    Octonions
    +
    +The Octonions, also called the Cayley-Dixon algebra, defined over a +commutative ring are an eight-dimensional non-associative algebra. Their +construction from quaternions is similar to the construction of quaternions +from complex numbers (see Quaternion). +As Octonion creates an eight-dimensional +algebra, you have to give eight components to construct an octonion. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Or you can use two quaternions to create an octonion. +
      +
    • + +
      +
    • +
    +You can easily demonstrate the non-associativity of multiplication. +
      +
    • + +
      +
    • +
    +As with the quaternions, we have a real part, the imaginary parts i, j, +k, and four additional imaginary parts E, I, J, and K. These parts +correspond to the canonical basis (1,i,j,k,E,I,J,K). For each basis +element there is a component operation to extract the coefficient of +the basis element for a given octonion. +
      +
    • + +
      +
    • +
    +A basis with respect to the quaternions is given by (1,E). However, you +might ask, what then are the commuting rules? To answer this, we create +some generic elements. We do this in Axim by simply changing the ground +ring from +Integer to +Polynomial Integer. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Note that quaternions are automatically converted to octonions in the +obvious way. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Finally, we check that the norm, defined as +the sum of the squares of the coefficients, is a multiplicative map. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Since the result is 0, the norm is multiplicative +
      +
    • + +
      +
    • +
    +Issue the system command +
      +
    • + +
      +
    • +
    +to display the list of operations defined by +Octonion. +<> +@ + +\subsection{numotherbases.xhtml} +<>= +<> + + + +<> +
    Expansions in other Bases
    +
    +It is possible to expand numbers in general bases. Here we expand +111 in base 5. This means +
    +    2   1   0      2    1  -
    +  10 +10 +10  = 4*5 +2*5 +5
    +
    +
      +
    • + +
      +
    • +
    +You can expand fractions to form repeating expansions. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +For bases from 11 to 36 the letters A through Z are used. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +For bases greater than 36, the ragits are separated by blanks. +
      +
    • + +
      +
    • +
    +The RadixExpansion type provides +operations to obtain the individual ragits. Here is a rational number +in base 8. +
      +
    • + +
      +
    • +
    +The operation wholeRagits returns +a list of the ragits for the integral part of the number. +
      +
    • + +
      +
    • +
    +The operations prefixRagits and +cycleRagits returns lists of the +initial and repeating ragist in the fractional part of the number. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +You can construct any radix expansion by giving the whole, prefix, and +cycle parts. The declaration is necessary to let Axiom know the base +of the ragits. +
      +
    • + +
      +
    • +
    +If there is no repeating part, then the list [0] should be used. +
      +
    • + +
      +
    • +
    +If you are not interested in the repeating nature of the expansion, +an infinite stream of ragits can be obtained using +fractRagits +
      +
    • + +
      +
    • +
    +Of course, it's possible to recover the fraction representation:n +
      +
    • + +
      +
    • +
    +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +RadixExpansion. More examples of +expansions are available in +DecimalExpansion, +BinaryExpansion, and +HexadecimalExpansion +<> +@ + +\subsection{numpartialfractions.xhtml} +<>= +<> + + + +<> +
    Partial Fractions
    +
    +A partial fraction is a decomposition of a quotient into a sum of quotients +where the denominators of the summand are powers of primes. (Most people +first encounter partial fractions when they are learning integral calculus. +For a technical discussion of partial fractions see, for example, Lang's +Algebra.) For example, the rational number 1/6 is decomposed into 1/2-1/3. +You can compute partial fractions of quotients of objects from domains +belonging to the category +EuclideanDomain. For example, +Integer, +Complex Integer, and + +UnivariatePolynomial(x,Fraction Integer) +all belong to +EuclideanDomain. +In the examples following, we demonstrate how to decompose quotients of +each of these kinds of objects into partial fractions. + +It is necessary that we know how to factor the denominator when we want to +compute a partial fraction. Although the interpreter can often do this +automatically, it may be necessary for you to include a call to +factor. In these examples, it is not +necessary to factor the denominators explicitly. The main operation for +computing partial fractions is called +partialFraction and we use this +to compute a decomposition of 1/10!. The first argument top +partialFraction is the numerator +of the quotient and the second argument is the factored denominator. +
      +
    • + +
      +
    • +
    +Since the denominators are powers of primes, it may be possible to expand +the numerators further with respect to those primes. Use the operation +padicFraction to do this. +
      +
    • + +
      +
    • +
    +The operation compactFraction +returns an expanded fraction into the usual form. The compacted version +is used internally for computational efficiency. +
      +
    • + +
      +
    • +
    +You can add, subtract, multiply, and divide partial fractions. In addition, +you can extract the parts of the decomposition. +numberOfFractionalTerms +computes the number of terms in the fractional part. This does not include +the whole part of the fraction, which you get by calling +wholePart. In this example, the whole part +is 0. +
      +
    • + +
      +
    • +
    +The operation +nthFractionalTerm +returns the individual terms in the decomposition. Notice that the object +returned is a partial fraction itself. +firstNumer and +firstDenom extract the numerator and +denominator of the first term of the fraction. +
      +
    • + +
      +
    • +
    +Given two gaussian integers (see Complex), +you can decompose their quotient into a partial fraction. +
      +
    • + +
      +
    • +
    +To convert back to a quotient, simply use the conversion +
      +
    • + +
      +
    • +
    +To conclude this section, we compute the decomposition of +
    +                   1
    +     -------------------------------
    +                   2       3       4
    +     (x + 1)(x + 2) (a + 3) (x + 4)
    +
    +The polynomials in this object have type + +UnivariatePolynomial(x,Fraction Integer). +We use the primeFactor operation +(see Factored) to create the denominator +in factored form directly. +
      +
    • + +
      +
    • +
    +These are the compact and expanded partial fractions for the quotient. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Also see + +FullPartialFractionExpansion for examples of factor-free conversion of +quotients to full partial fractions. + +Issue the system +command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +PartialFraction. + +<> +@ + +\subsection{numproblems.xhtml} +<>= +<> + + + +<> +
    Problems
    +
    +One can show that if an integer of the form 2**k-1 is prime then +k must be prime. +
    +Proof +Suppose that k=m*n is a non-trivial factorization. Then +
    +       2^m = 1 (mod (2^m-1))
    +   2^(m*n) = 1 (mod (2^m-1))
    +  so 2^m-1 is a non-trivial factor of 2^k-1
    +
    + +Problem Find the smallest prime p such that 2**p-1 is not prime +
    +Answer +
    +First, define a function: +
      +
    • + +
      +
    • +
    +You can try factoring f(p) as p ranges through the set of primes. +For example, +
      +
    • + +
      +
    • +
    +This gets tedious after a while, so let's use Axiom's stream facility. +A streamm is essentially an infinite sequence. First, we create a stream +consisting of the positive integers: +
      +
    • + +
      +
    • +
    +Now, we create a stream consisting of the primes: +
      +
    • + +
      +
    • +
    +Here is the 25th prime: +
      +
    • + +
      +
    • +
    +Next, create the stream of numbers of the form 2**p-1 with p prime: +
      +
    • + +
      +
    • +
    +Finally, form the stream of factorizations of the elements of numbers: +
      +
    • + +
      +
    • +
    +You can see that the fifth number in the stream (2047=23*89) is the first +one that has a non-trivial factorization. Since 2**11=2048, the solution +to the problem is 11. + +Here is another way to see that 2047 is the first number in the stream +that is composite: +
      +
    • + +
      +
    • +
    +

    +Problem: Find the smallest positive integer n such that +n**2-n+41 is not prime. +
    +Answer: When n=41, n**2-n+41=41**2, which certainly isn't prime. +Is there any smaller integer that works? Here are the first 40 values: +
      +
    • + +
      +
    • +
    +Now have Axiom factor the numbers on this list: +
      +
    • + +
      +
    • +
    +You can see that 41 is the smallest positive integer n such that +n**2-n+41 is not prime. +<> +@ + +\subsection{numquaternions.xhtml} +<>= +<> + + + +<> +
    Quaternions
    +
    +The domain contructor Quaternion +implements quaternions over commutative rings. + +The basic operation for creating quaternions is +quatern. This is a quaternion +over the rational numbers. +
      +
    • + +
      +
    • +
    +The four arguments are the real part, the i imaginary part, +the j imaginary part, and the k imaginary part, respectively. +
      +
    • + +
      +
    • +
    +Because q is over the rationals (and nonzero), you can invert it. +
      +
    • + +
      +
    • +
    +The usual arithmetic (ring) operations are available. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +In general, multiplication is not commutative. +
      +
    • + +
      +
    • +
    +There are no predefined constants for the imaginary i, j, and k parts, +but you can easily define them +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +These satisfy the normal identities. +
      +
    • + +
      +
    • +
    +The norm is the quaternion times its conjugate. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +For information on +related topics, see Complex and +Octonion. You can also issue the +system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +Quaternion. +<> +@ + +\subsection{numquotientfields.xhtml} +<>= +<> + + + +<> +
    Quotient Fields
    +
    +The Fraction domain implements quotients. +The elements must belong to a domain of category +IntegralDomain: multiplication +must be commutative and the product of two non-zero elements must not be +zero. This allows you to make fractions of most things you would think of, +but don't expect to create a fraction of two matrices. The abbreviation for +Fraction is +FRAC. + +Use / to create a fraction. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The standard arithmetic operations are available. +
      +
    • + +
      +
    • +
    +Extract the numerator and denominator by using +numer and denom, +respectively. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Operations like +max, min, +negative?, +positive?, and +zero? are all available if they are +provided for the numerators and denominators. See +Integer for examples. + +Don't expect a useful answer from +factor, +gcd, or +lcm if you apply them to fractions. +
      +
    • + +
      +
    • +
    +Since all non-zero fractions are invertible, these operations have +trivial definitions. +
      +
    • + +
      +
    • +
    +Use map to apply +factor to the numerator and denominator, +which is probably what you mean. +
      +
    • + +
      +
    • +
    + +Other forms of fractions are available, Use +continuedFraction +to create a continued fraction. +
      +
    • + +
      +
    • +
    +Use partialFraction to create a +partial fraction. +See continuedFraction +and PartialFraction for +additional information and examples. +
      +
    • + +
      +
    • +
    +Use conversion to create alternative views of fractions with objects +moved in and out of the numerator and denominator. +
      +
    • + +
      +
    • +
    +Conversion is discussed in detail in +Conversion. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numrationalnumbers.xhtml} +<>= +<> + + + +<> +
    Rational Numbers
    +
    +Like integers, rational numbers can be arbitrarily large. For example: +
      +
    • + +
      +
    • +
    +Rational numbers will not be converted to decimals unless you explicitly +ask Axiom to do so. To convert a rational number to a decimal, use the +function numeric. Here's an example: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +You can find the numerator and denominator of rational numbers using the +functions numer and +denom, respectively. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To factor the numerator and denominator of a fraction, use the following +command: +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{numrepeatingbinaryexpansions.xhtml} +<>= +<> + + + +<> +
    Repeating Binary Expansions
    +
    +All rational numbers have repeating binary expansions. Operations to +access the individual bits of a binary expansion can be obtained by +converting the value to +RadixExpansion(2). More examples +of expansions are available in +DecimalExpansion, +HexadecimalExpansion, and +RadixExpansion. + +The expansion (of type +BinaryExpansion) +of a rational number is returned by the +binary operation. +
      +
    • + +
      +
    • +
    +Arithmetic is exact. +
      +
    • + +
      +
    • +
    +The period of the expansion can be short or long... +
      +
    • + +
      +
    • +
    +or very long +
      +
    • + +
      +
    • +
    +These numbers are bona fide algebraic objects. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{numrepeatingdecimals.xhtml} +<>= +<> + + + +<> +
    Repeating Decimals
    +
    +All rationals have repeating decimal expansions. Operations to access +the individual digits of a decimal expansion can be obtained by converting +the value to RadixExpansion(10). + +The operation decimal is used to create +this expansion of type +DecimalExpansion. +
      +
    • + +
      +
    • +
    +Arithmetic is exact. +
      +
    • + +
      +
    • +
    +The period of the expansion can be short or long... +
      +
    • + +
      +
    • +
    +or very long +
      +
    • + +
      +
    • +
    +These numbers are bona fide algebraic objects. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +More examples of expansions are available in +BinaryExpansion, +HexadecimalExpansion, and +RadixExpansion. Issue the system +command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +RadixExpansion. +<> +@ + +\subsection{numrepeatinghexexpansions.xhtml} +<>= +<> + + + +<> +
    Repeating Hexadecimal Expansions
    +
    +All rationals have repeating hexadecimals expansions. The operation +hex returns these expansions of type +HexadecimalExpansion. +Operations to access the individual numerals of a hexadecimal expansion +can be obtained by converting the value to +RadixExpansion(16). More examples of +expansions are available in +DecimalExpansion, +BinaryExpansion, and +RadixExpansion. + +This is a hexadecimal expansion of a rational number. +
      +
    • + +
      +
    • +
    +Arithmetic is exact. +
      +
    • + +
      +
    • +
    +The period of the expansion can be short or long... +
      +
    • + +
      +
    • +
    +or very long. +.
      +
    • + +
      +
    • +
    +These numbers are bona fide algebraic objects. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +HexadecimalExpansion. + +<> +@ + +\subsection{numromannumerals.xhtml} +<>= +<> + + + +<> +
    Roman Numerals
    +
    +The Roman numeral package was added to Axiom in MCMLXXXVI for use in +denoting higher order derivatives. + +For example, let f be a symbolic operator. +
      +
    • + +
      +
    • +
    +This is the seventh derivative of f with respect to x +
      +
    • + +
      +
    • +
    +You can have integers printed as Roman numerals by declaring variables +to be of type +RomanNumeral +(abbreviation ROMAN). +
      +
    • + +
      +
    • +
    +This package now has a small but devoted group of followers that claim +this domain has shown its efficacy in many other contexts. They claim +that Roman numerals are every bit as useful as ordinary integers. +In a sense, they are correct, because Roman numerals form a ring and +you can therefore construct polynomials with Roman numeral +coefficients, matrices over Roman numerals,etc.. +
      +
    • + +
      +
    • +
    +Was Fibonacci Italian or ROMAN? +
      +
    • + +
      +
    • +
    +You can also construct fractions with Roman numeral numerators and +denominators, as this matrix Hilberticus illustrates. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Note that the inverse of the matrix has integral +ROMAN entries. +
      +
    • + +
      +
    • +
    +Unfortunately, the spoil-sports say that the fun stops when the +numbers get big -- mostly because the Romans didn't establish +conventions about representing very large numbers. +
      +
    • + +
      +
    • +
    +You work it out! +
      +
    • + +
      +
    • +
    +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +RomanNumeral). +<> +@ + +%%O +\subsection{ocwmit18085.xhtml} +<>= +<> + + +<> +18.085 Mathematical Methods for Engineers I Course Notes +
    +These are course notes based on the + + M.I.T. Open Courseware lectures by Gilbert Strang. + + +<> +@ + +\subsection{ocwmit18085lecture1.xhtml} +<>= +<> + + +<> + Positive Definite Matrices K=A'CA +
    +In applied mathematics we have 2 basic tasks: +
      +
    • Find the equations
    • +
    • Solve the equations
    • +
    +

    Positive Definite Matrices

    +Certain matrices occur frequently in applied math. These three +matrices (K,T,and M) are canonical examples. +We have 3 3x3 matrices, +
    +K:Matrix(Integer):=[[2,-1,0],[-1,2,-1],[0,-1,2]]
    +
    +        + 2   - 1   0 +
    +        |             |
    +        |- 1   2   - 1|
    +        |             |
    +        + 0   - 1   2 +
    +               Type: Matrix Integer
    +T:Matrix(Integer):=[[1,-1,0],[-1,2,-1],[0,-1,2]]
    +
    +        + 1   - 1   0 +
    +        |             |
    +        |- 1   2   - 1|
    +        |             |
    +        + 0   - 1   2 +
    +               Type: Matrix Integer
    +B:Matrix(Integer):=[[1,-1,0],[-1,2,-1],[0,-1,1]]
    +
    +        + 1   - 1   0 +
    +        |             |
    +        |- 1   2   - 1|
    +        |             |
    +        + 0   - 1   1 +
    +               Type: Matrix Integer
    +
    +These matrices are similar and can be generalized to square matrices +of order N, with n x n elements. All of these matrices have the same +element along the diagonal. T (aka Top) differs from K in the first row. +B (aka Both) differs from K in the first and last row. These represent +different boundary conditions in the problem. + +We can create K(n), T(n) and B(n) with the following commands: +
    +k(n) == 
    + M := diagonalMatrix([2 for i in 1..n]) 
    + for i in 1..n-1 repeat M(i,i+1):=-1 
    + for i in 1..n-1 repeat M(i+1,i):=-1 
    + M::SquareMatrix(n,Fraction(Integer))
    +
    +
    +t(n) == 
    + M:=k(n)
    + N:=M::Matrix(Fraction(Integer)) 
    + qsetelt!(N,1,1,1) 
    + N::SquareMatrix(n,Fraction(Integer))
    +
    +
    +b(n) == 
    + M:=k(n)
    + N:=M::Matrix(Fraction(Integer)) 
    + qsetelt!(N,1,1,1) 
    + qsetelt!(N,n,n,1)
    + N::SquareMatrix(n,Fraction(Integer))
    +
    + +K:=k(n) has a few key properties: +
      +
    • K is symmetric, that is K=K^T
    • +
    • K might be nonsingular, that is, it is invertible
    • +
    • K has a non-zero determinant
    • +
    • K is banded (main diagonal and neighbors)
    • +
    • K is tri-diagonal (main diagonal and nearest neighbors
    • +
    • K is extremely sparse
    • +
    • K has constant diagonals, (shift invariant, time invariant)
    • +
    • K is Toeplitz (constant diagonal, shows up in filters)
    • +
    • K is good for Fourier analysis
    • +
    + +
    The inverse of T
    +If we look at the inverse of the T matrix we see: +
    +T^-1
    +
    +        +3  2  1+
    +        |       |
    +        |2  2  1|
    +        |       |
    +        +1  1  1+
    +               Type: Matrix Fraction Integer
    +
    +Notice that these are all integers because the determinant of +this matrix is 1 +
    +determinant T
    +
    +     1
    +               Type: Fraction Integer
    +
    +
    +We can check that this matrix is the inverse of T. + +When computing the inverse the row pattern [-1 2 -1] is a +``second difference''. The first column of the inverse matrix +is [3 2 1] which is linear. When we take the second difference +of a linear object we should get 0. Thus, +
    +[[-1,2,-1]]::MATRIX(INT)*[[3],[2],[1]]
    +
    +     [0]
    +               Type: Matrix Integer
    +
    +
    +The third column of the T matrix is linear and constant. If we +take the second difference of that we also find it is zero: +
    + [[-1,2,-1]]::MATRIX(INT)*[[1],[1],[1]]
    +
    +    [0]
    +               Type: Matrix Integer
    +
    +and the diagonal element of the unit matrix must be one. So +the second difference of the second column is: +
    + [[-1,2,-1]]::MATRIX(INT)*[[2],[2],[1]]
    +
    +    [1]
    +               Type: Matrix Integer
    +
    +So these simple checks show that we're getting the correct +row and column values for the identity matrix by multiplying +T times its inverse. + +
    +
    The inverse of B
    +If we look for the inverse of the B matrix we can observe +that the rows sum to zero which implies that it is not +invertible. Thus it is singular. + +K and T are positive definite. B is only positive semi-definite. + +If we can find a vector that it takes to zero, that is if we can +solve for x,y,z in: +
    +        + 1   - 1   0 + + x +    + 0 +
    +        |             | |   |    |   |
    +        |- 1   2   - 1| | y | =  | 0 |
    +        |             | |   |    |   |
    +        + 0   - 1   1 + + z +    + 0 +
    +
    +
    +The constant vector [1 1 1] solves this equation. When +the rows sum to zero we are adding each row by a constant +and thus we add each row times the constant one and we +get zeros. If the matrix takes some vector to zero it +cannot have an inverse since if +
    +   B x = 0
    +
    +and x is not zero. If B had an inverse only x=0 would +solve the equation. Since x=1 solves the equation B has +no inverse. The vector x is in the nullspace of B. In +fact any constant vector, e.g. [3 3 3] is in the nullspace. +Thus the nullspace of B is cx for any constant c. + +When doing matrix multiplication one way to think about the +work is to consider the problem by columns. Thus in the +multiplication +
    +        + 1   - 1   0 + + x +    + 0 +
    +        |             | |   |    |   |
    +        |- 1   2   - 1| | y | =  | 0 |
    +        |             | |   |    |   |
    +        + 0   - 1   1 + + z +    + 0 +
    +
    +
    +we can think about this as +
    +x*(first column) + y*(second column) + z*(third column).
    +
    +and for the constant vector [1 1 1] this means that we +just need to sum the columns. + +Alternatively this can be computed by thinking of the +multiplication as +
    + (first row)*(vector)
    + (second row)*(vector)
    + (third row)*(vector)
    +
    + +
    +
    The inverse of K
    +Now we consider the K matrix we see the inverse +
    +K
    +
    +         + 2   - 1   0 +
    +         |             |
    +         |- 1   2   - 1|
    +         |             |
    +         + 0   - 1   2 +
    +               Type: SquareMatrix(3,Fraction Integer)
    +kinv:=K^-1
    +
    +         +3  1  1+
    +         |-  -  -|
    +         |4  2  4|
    +         |       |
    +         |1     1|
    +         |-  1  -|
    +         |2     2|
    +         |       |
    +         |1  1  3|
    +         |-  -  -|
    +         +4  2  4+
    +               Type: SquareMatrix(3,Fraction Integer)
    +
    +We can take the determinant of k +
    +determinant K
    +
    +    4
    +               Type: Fraction Integer
    +
    +Thus there is a constant 1/4 which can be factored out +
    +4*kinv
    +
    +         +3  2  1+
    +         |       |
    +         |2  4  2|
    +         |       |
    +         +1  2  3+
    +               Type: SquareMatrix(3,Fraction Integer)
    +
    +Notice that the inverse is a symmetric matrix but not tri-diagonal. +The inverse is not a sparse matrix so much more computation would +be involved when using the inverse. + +In order to solve the system +
    + K u = f
    +
    +by elimination which implies multiplying and subtracting rows. +
    +       K    u  =  f    ==>   U     u  =    f
    +
    +For the 2x2 case we see: +
    +                             +2  -1+        +  f1  +
    +    +2  -1+  +x+   +f1+      |     |  +x+   |      |
    +    |     |  | | = |  |  ==> |    3|  | | = |   1  |
    +    +-1  2+  +y+   +f2+      |0   -|  +y+   |f2+-f1|
    +                             +    2+        +   2  +
    +
    +
    +
    +By multiplying row1 by 1/2 and adding it to row2 we create an +upper triangular matrix U. Since we chose K(1,1), the number 2 +is called the first pivot. K(2,2), the number 3/2, is called +the second pivot. + +For K 2x2 above is symmetric and invertible (since the pivots +are all non-zero). + +For the K 3x3 case the pivots are 2, 3/2, and 4/3. (The next pivots +would be 5/4, 6/5, etc. for larger matrices). + +For the T 3x3 case the pivots are 1, 1, and 1. + +For the B 3x3 case the third pivot would be zero. + +
    +
    Generalizing the matrix pivot operations
    +For the 2x2 case we see contruct an elimination matrix E which we can use +to pre-multipy by K to give us the upper triangular matrix U +
    +      E     K    =   U
    +
    +In detail we see +
    +
    +    +1  0+            +2  -1+
    +    |    |  +2  -1+   |     |
    +    |1   |  |     | = |    3|
    +    |-  1|  +-1  2+   |0   -|
    +    +2   +            +    2+
    +
    +
    +We wish to rewrite this as +
    +       K = L U 
    +
    + +
    +
    The big 4 solve operations in Linear Algebra
    +
      +
    1. Elimination
    2. +
    3. Gram-Schmidt Orthoginalization
    4. +
    5. Eigenvalues
    6. +
    7. Singular Value Decomposition
    8. +
    +Each of these operations is described by a factorization of K. +Elimination is written +
    +  K = L U
    +
    +where L is lower triangular and U is upper triangular. +Thus we need a matrix L which when multiplied by U gives K. +The required matrix is the inverse of the E matrix above since +
    +
    +1)      E K =     U
    +
    +     -1        -1
    +2)  E   E K = E   U
    +
    +               -1
    +3)      I K = E   U
    +
    +               -1
    +4)  but   L = E
    +
    +5)  so    K = L U
    +
    +Given the matrix operations above we had +
    +      E       K   =   U
    +
    +    +1  0+            +2  -1+
    +    |    |  +2  -1+   |     |
    +    |1   |  |     | = |    3|
    +    |-  1|  +-1  2+   |0   -|
    +    +2   +            +    2+
    +
    +
    +and the inverse of E is the same matrix with a minus sign in +the second row, thus: +
    +        +  1  0+ 
    +   -1   |      | 
    +  E   = |  1   | = L 
    +        |- -  1| 
    +        +  2   + 
    +
    +
    + +
    +
    Making the matrices symmetric
    +We would like to preserve the symmetry property which we can +do with a further decomposition of LU as follows: +
    +      L        U     =     L        D       U'
    +
    +  +  1  0+  +2  -1+    +  1  0+  +2  0+  +1   1+
    +  |      |  |     |    |      |  |    |  |  - -|
    +  |  1   |  |    3|  = |  1   |  |   3|  |    2|
    +  |- -  1|  |0   -|    |- -  1|  |0  -|  |     |
    +  +  2   +  +    2+    +  2   +  +   2+  +0   1+
    +
    +
    +So now we have 3 matrices; L is the lower triangular, +D is symmetric and contains the pivots, and U' is upper triangular and +is the transpose of the lower. So the real form we have is +
    +           T
    +    L  D  L
    +
    +This result will always be symmetric. We can check this by taking +its transpose. If we get the same matrix we must have a symmetric +matrix. So the transpose of +
    +            T  T     TT  T   T        T T        T
    +  (  L  D  L  )   = L   D   L   =  L D L  = L D L
    +
    +
    +
    Positive Definite Matrices
    +There are several ways to recognize a positive definite matrix. +First, it must be symmetric. The "positive" aspect comes from +the pivots, all of which must be positive. Note that T is also +positive definite. B is positive semi-definite because one of +the pivots is zero. So +
    +   positive definite      == all pivots >  0
    +   positive semi-definite == all pivots >= 0
    +
    +When all the pivots are positive then all the eigenvalues are positive. + +So a positive definite matrix K and any non-zero vector X +
    +    T
    +   X  K X  > 0
    +
    +X transpose is just a row and X is just a column. + +<> +@ + +\subsection{ocwmit18085lecture2.xhtml} +<>= +<> + + +<> + One-dimensional Applications: A = Difference Matrix +
    +
    Difference Matrices
    +
    +
    Second Differences
    +
    +
    Stiffness Matrix
    +
    +
    Boundary Conditions
    +<> +@ + +\subsection{operations.xhtml} +<>= +<> + + +<> + operations not implemented +<> +@ + +\subsection{outputfunctions.xhtml} +<>= +<> + + + +<> +
    Output Functions
    +
    +A number of operations exist for specifying how numbers of type +Float are to be displayed. By default, +spaces are inserted every ten digits in the output for readability. +(Not that you cannot include spaces in the input form of a floating +point number, though you can use underscores.) + +Output spacing can be modified with the +outputSpacing operation. This +inserts no spaces and then displays the value of x. +
      +
    • + +
      +
    • +
    +Issue this to have the spaces inserted every 5 digits. +
      +
    • + +
      +
    • +
    +By default, the system displays floats in either fixed format or +scientific format, depending on the magnitude of the number. +
      +
    • + +
      +
    • +
    +A particular format may be requested with the operations +outputFloating and +outputFixed. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Additionally, you can ask for n digits to be displayed after the +decimal point. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The outputGeneral function resets +the output printing to the default behavior. +
      +
    • + +
      +
    • +
    +<> +@ + +%%P +\subsection{pagelist.xhtml} +<>= +<> + + +<> + pagelist not implemented +<> +@ + +\subsection{pagematrix.xhtml} +<>= +<> + + +<> + pagematrix not implemented +<> +@ + +\subsection{pageonedimensionalarray.xhtml} +<>= +<> + + +<> + pageonedimensionalarray not implemented +<> +@ + +\subsection{pageset.xhtml} +<>= +<> + + +<> + pageset not implemented +<> +@ + +\subsection{pagetable.xhtml} +<>= +<> + + +<> + pagetable not implemented +<> +@ + +\subsection{pagepermanent.xhtml} +<>= +<> + + +<> + pagepermanent not implemented +<> +@ + +\subsection{pagesquarematrix.xhtml} +<>= +<> + + +<> + pagesquarematrix not implemented +<> +@ + +\subsection{pagetwodimensionalarray.xhtml} +<>= +<> + + + +<> +
    TwoDimensionalArray
    +
    +The TwoDimensionalArray is used for +storing data in a two-dimensional data structure indexed by row and column. +Such an array is a homogeneous data structure in that all the entries of the +array must belong to the same Axiom domain (although see +The Any Domain). Each array has a fixed +number of rows and columns specified by the user and arrays are not +extensible. In Axiom, the indexing of two-dimensional arrays is one-based. +This means that both the "first" row of an array and the "first" column of +an array are given the index 1. Thus, the entry in the upper left corner +of an array is in position (1,1). + +The operation new creates an array with a +specified number of rows and columns and fills the components of that array +with a specified entry. The arguments of this operation specify the number +of rows, the number of columns, and the entry. This creates a five-by-four +array of integers, all of whose entries are zero. +
      +
    • + +
      +
    • +
    +The entries of this array can be set to other integers using the operation +setelt. + +Issue this to set the element in the upper left corner of this array to 17. +
      +
    • + +
      +
    • +
    +Now the first element of the array is 17. +
      +
    • + +
      +
    • +
    +Likewise, elements of an array are extracted using the operation +elt. +
      +
    • + +
      +
    • +
    +Another way to use these two operations is as follows. This sets the +element in position (3,2) of the array to 15. +
      +
    • + +
      +
    • +
    +This extracts the element in position (3,2) of the array. +
      +
    • + +
      +
    • +
    +The operation elt and +setelt come equipped with an error check +which verifies that the indices are in the proper ranges. For example, +the above array has five rows and four columns, so if you ask for the +entry in position (6,2) with arr(6,2) Axiom displays an error message. +If there is no need for an error check, you can call the operations +qelt and +qsetelt! +which provide the same functionality +but without the error check. Typically, these operations are called in +well-tested programs. + +The operations row and +column extract rows and columns, respectively, +and return objects of +OneDimensionalArray with the +same underlying element type. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +You can determine the dimensions of an array by calling the operations +nrows and +ncols, which return the number of rows +and columns, respectively. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To apply an operation to every element of an array, use +map. This creates a new array. This +expression negates every element. +
      +
    • + +
      +
    • +
    +This creates an array where all the elements are doubled. +
      +
    • + +
      +
    • +
    +To change the array destructively, use +map! instead of +map. +If you need to make a copy of an array, +use copy. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use member? to see if a given element is in +an array. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To see how many times an element appears in an array, use +count. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +For more information about the operations available for +TwoDimensionalArray, issue +
      +
    • + +
      +
    • +
    +For more information on related topics, see +Matrix and +OneDimensionalArray. +<> +@ + +\subsection{pagevector.xhtml} +<>= +<> + + +<> + pagevector not implemented +<> +@ + + +\subsection{polybasicfunctions.xhtml} +<>= +<> + + + +<> +
    Basic Operations on Polynomials
    +
    +You create polynomials using the usual operations of ++, +-, +* +(for multiplication), and +** (or +^. Here are two examples: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +These operations can also be used to combine polynomials. Try the following: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +As you can see from the above examples, the variables are ordered by defaults +
    +  z > y > x > c > b > a
    +
    +That is, z is the main variable, then y and so on in reverse alphabetical +order. You can redefine this ordering (for display purposes) with the +setVariableOrder. For example, the +following makes a the main variable, then b, and so on: +
      +
    • + +
      +
    • +
    +Now compare the way polynomials are displayed: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +To return to the system's default ordering, use +resetVariableOrder. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Polynomial coefficients can be pulled out using the function +coefficient. For example: +
      +
    • + +
      +
    • +
    +will give you the coefficient of x**2 in the polynomial q. Try these +commands: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Coefficients of monomials can be obtained as follows: +
      +
    • + +
      +
    • +
    +This will return the coefficient of x**2*z in the polynomial q**2. Also, +
      +
    • + +
      +
    • +
    +will return the coefficient of x**2*y**2 in the polynomial r(x,y). +<> +@ + +\subsection{polyfactorization.xhtml} +<>= +<> + + +<> +
    Polynomial Factorization
    +
    +The Axiom polynomial factorization facilities are available for all +polynomial types and a wide variety of coefficient domains. Here are +some examples. + +<> +@ + +\subsection{polyfactorization1.xhtml} +<>= +<> + + + +<> +
    Integer and Rational Number Coefficients
    +
    +Polynomials with integer coefficients can be factored. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Also, Axiom can factor polynomials with rational number coefficients +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyfactorization2.xhtml} +<>= +<> + + + +<> +
    Finite Field Coefficients
    +
    +Polynomials with coefficients in a finite filed can also be factored. +
      +
    • + +
      +
    • +
    +These include the integers mod p, where p is prime, and extensions of these +fields. +
      +
    • + +
      +
    • +
    +Convert this to have coefficients in the finite field with +19**3 elements. See +FiniteFields for more information +about finite fields. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyfactorization3.xhtml} +<>= +<> + + + +<> +
    Simple Algebraic Extension Field Coefficients
    +
    +Polynomials with coefficients in simple algebraic extensions of the +rational numbers can be factored. + +Here, aa and bb are symbolic roots of polynomials. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Note that the second argument to factor can be a list of algebraic +extensions to factor over. +
      +
    • + +
      +
    • +
    +This factors x^2+3 over the integers. +
      +
    • + +
      +
    • +
    +Factor the same polynomial over the field obtained by adjoining aa to the +rational numbers. +
      +
    • + +
      +
    • +
    +Factor x^6+108 over the same field. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Factor again over the field obtained by adjoining both aa and bb to the +rational numbers. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyfactorization4.xhtml} +<>= +<> + + + +<> +
    Factoring Rational Functions
    +
    +Since fractions of polynomials form a field, every element (other than zero) +divides any other, so there is no useful notion of irreducible factors. +Thus the factor operation is not very useful +for fractions of polynomials. + +Instead, there is a specific operation +factorFraction that separately +factors the numerator and denominator and returns a fraction of the +factored results. +
      +
    • + +
      +
    • +
    +You can also use map. This expression applies +the factor operation to the numerator and +denominator. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polygcdandfriends.xhtml} +<>= +<> + + + +<> +
    + Greatest Common Divisors, Resultants, and Discriminants +
    +
    +You can compute the greatest common divisor of two polynomials using the +function gcd. Here's an example: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +You could also see that p and q have a factor in common by using the +function resultant: +
      +
    • + +
      +
    • +
    +The resultant of two polynomials vanishes precisely when they have a +factor in common. (In the example above we specified the variable with which +we wanted to compute the resultant because the polynomials could have +involved variables other than x.) +<> +@ + +\subsection{polynomialpage.xhtml} +<>= +<> + + +<> +
    Polynomials
    +
    + + + + + + + + + + + + + + + + + + + + + + + + + +
    + Basic Functions + + Create and manipulate polynomials +
    + Substitutions + + Evaluate Polynomials +
    + Factorization + + Factor in different contexts +
    + GCD and Friends + + Greatest Common Divisors, Resultants, and Discriminants +
    + Roots + + Work with and solve for roots +
    + Specific Types + + More specific information +
    +<> +@ + +\subsection{polyroots.xhtml} +<>= +<> + + +<> +
    Roots of Polynomials
    +
    + + + + + + + + + + + + + + + + + +
    + + Using a Single Root of a Polynomial + + + Working with a single root of a polynomial +
    + + Using All Roots of a Polynomial + + + Working with all the roots of a polynomial +
    + + Solution of a Single Polynomial Equation + + + Finding the roots of one polynomial +
    + + Solution of Systems of Polynomial Equations + + + Finding the roots of a system of polynomials +
    +<> +@ + +\subsection{polyroots1.xhtml} +<>= +<> + + + +<> +
    Using a Single Root of a Polynomial
    +
    +Use rootOf to get a symbolic root of a +polynomial. The call rootOf(p,x) returns a root of p(x). + +This creates an algebraic number a, which is a root of the polynomial +returned in symbolic form. +
      +
    • + +
      +
    • +
    +To find the algebraic relation that defines a, use +definingPolynomial +
      +
    • + +
      +
    • +
    +You can use a in any further expression, including a nested +rootOf. +
      +
    • + +
      +
    • +
    +Higher powers of the roots are automatically reduced during calculations. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The operation zeroOf is similar to +rootOf, except that it may express the +root using radicals in some cases. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyroots2.xhtml} +<>= +<> + + + +<> +
    Using All Roots of a Polynomial
    +
    +Use rootsOf to get all symbolic roots +of a polynomial. The call rootsOf(p,x) returns a list of all the roots +of p(x). If p(x) has a multiple root of order n, then that root appears +n times in the list. + +Compute all the roots of x^4+1. +
      +
    • + +
      +
    • +
    +As a side effect, the variables %x0, %x1, and %x2 are bound to the first +three roots of x^4+1. +
      +
    • + +
      +
    • +
    +Although they all satisfy x^4+1=0, %x0, %x1, and %x2 are different +algebraic numbers. To find the algebraic relation that defines each of +them, use definingPolynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +We can check that the sum and product of the roots of x^4+1 are its +trace and norm. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Corresponding to the pair of operations +rootOf and +zeroOf in + +Solution of a Single Polynomial Equation +there is an operations zerosOf that, like +rootsOf, computes all the roots of a given +polynomial, but which expresses some of them in terms of radicals. +
      +
    • + +
      +
    • +
    +As you see, only one implicit algebraic number was created (%y1), and its +defining equation is this. The other three roots are expressed in radicals. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyroots3.xhtml} +<>= +<> + + + +<> +
    Solution of a Single Polynomial Equation
    +
    +Axiom can solve polynomial equations producing either approximate or exact +solutions. Exact solutions are either members of the ground field or can +be presented symbolically as roots of irreducible polynomials. + +This returns one rational root along with an irreducible polynomial +describing the other solutions +
      +
    • + +
      +
    • +
    +If you want solutions expressed in terms of radicals you would use this +instead. +
      +
    • + +
      +
    • +
    +The solve command always returns a value but +radicalSolve returns only the solutions +that it is able to express in terms of radicals. + +If the polynomial equation has rational coefficients you can ask for +approximations to its real roots by calling solve with a second argument +that specifies the "precision" epsilon. This means that each approximation +will be within plus or minus epsilon of the actual result. + +Notice that the type of second argument controls the type of the result. +
      +
    • + +
      +
    • +
    +If you give a floating point precision you get a floating point result. +If you give the precision as a ration number you get a rational result. +
      +
    • + +
      +
    • +
    +If you want approximate complex results you should use the command +complexSolve that takes the same +precision argument epsilon. +
      +
    • + +
      +
    • +
    +Each approximation will be within plus or minus epsilon of the actual result +in each of the real and imaginary parts. +
      +
    • + +
      +
    • +
    +Note that if you omit the = from the first argument Axiom generates +an equation by equating the first argument to zero. Also, when only one +variable is present in the equation, you do not need to specify the +variable to be solved for, that is, you can omit the second argument. + +Axiom can also solve equations involving rational functions. Solutions +where the denominator vanishes are discarded. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyroots4.xhtml} +<>= +<> + + + +<> +
    Solution of Systems of Polynomial Equations
    +
    +Given a system of equations of rational functions with exact coefficients +
    +     p1(x1,...,xn)
    +         .
    +         .
    +     pm(x1,...,xn)
    +
    +Axiom can find numeric or symbolic solutions. The system is first split +into irreducible components, then for each component, a triangular system +of equations is found that reduces the problem to sequential solutions of +univariate polynomials resulting from substitution of partial solutions +from the previous stage. +
    +     q1(x1,...,xn)
    +         .
    +         .
    +     qm(xn)
    +
    +Symbolic solutions can be presented using "implicit" algebraic numbers +defined as roots of irreducible polynomials or in terms of radicals. Axiom +can also find approximations to the real or complex roots of a system of +polynomial equations to any user specified accuracy. + +The operation solve for systems is used in +a way similar to solve for single equations. +Instead of a polynomial equation, one has to give a list of equations and +instead of a single variable to solve for, a list of variables. For +solutions of single equations see + +Solution of a Single Polynomial Equation + +Use the operation solve if you want +implicitly presented solutions. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Use radicalSolve if you want your +solutions expressed in terms of radicals. +
      +
    • + +
      +
    • +
    +To get numeric solutions you only need to give the list of equations and +the precision desired. The list of variables would be redundant information +since there can be no parameters for the numerical solver. + +If the precision is expressed as a floating point number you get results +expressed as floats. +
      +
    • + +
      +
    • +
    +To get complex numeric solutions, use the operation +complexSolve, which takes the same +arguments as in the real case. +
      +
    • + +
      +
    • +
    +It is also possible to solve systems of equations in rational functions +over the rational numbers. Note that [x=0.0,a=0.0] is not returned as +a solution since the denominator vanishes there. +
      +
    • + +
      +
    • +
    +When solving equations with denominators, all solutions where the +denominator vanishes are discarded. +
      +
    • + +
      +
    • +
    +<> +@ + +\subsection{polyspecifictypes.xhtml} +<>= +<> + + +<> +
    The Specific Polynomial Types
    +
    + + + + + + + + + + + + + + + + Multiple variable polynomials, non-recursive structure + + +
    + + Polynomial + + + The general type +
    + + UnivariatePolynomial + + + One variable polynomials +
    + + MultivariatePolynomial + + + Multiple variable polynomials, recursive structure +
    + + DistributedMultivariatePolynomial + + +
    +<> +@ + +\subsection{polyspecifictypes1.xhtml} +<>= +<> + + + +<> +
    Polynomial
    +
    +The domain constructor Polynomial +(abbreviation: POLY) provides polynomials +with an arbitrary number of unspecified variables. + +It is used to create the default polynomial domains in Axiom. Here the +coefficients are integers. +
      +
    • + +
      +
    • +
    +Here the coefficients have type Float. +
      +
    • + +
      +
    • +
    +And here we have a polynomial in two variables with coefficients which +have type Fraction Integer +
      +
    • + +
      +
    • +
    +The representation of objects of domains created by +Polynomial is that of recursive univariate +polynomials. (The term univariate means "one variable". The term +multivariate means "possibly more than one variable".) This recursive +structure is sometimes obvious from the display of a polynomial. +
      +
    • + +
      +
    • +
    +In this example, you see that the polynomial is stored as a polynomial in y +with coefficients that are polynomials in x with integer coefficients. In +fact, you really don't need to worry about the representation unless you are +working on an advanced application where it is critical. The polynomial +types created from + +DistributedMultivariatePolynomial and +XDistributedPolynomial +(discussed in +"DistributedMultivariatePolynomial" +are stored and displayed in a +non-recursive manner. You see a "flat" display of the above polynomial by +converting to one of those types. +
      +
    • + +
      +
    • +
    +We will demonstrate many of the polynomial facilities by using two +polynomials with integer coefficients. By default, the interpreter +expands polynomial expressions, even if they are written in a factored +format. +
      +
    • + +
      +
    • +
    +See Factored +to see how to create objects in factored form directly. +
      +
    • + +
      +
    • +
    +The fully factored form can be recovered by using +factor +
      +
    • + +
      +
    • +
    +This is the same name used for the operation to factor integer. +Such reuse of names is called +overloading and makes it much +easier to think of solving problems in general ways. Axiom facilities +for factoring polynomials created with +Polynomial +are currently restricted to the integer and rational number coefficients +cases. There are more complete facilities for factoring univariate +polynomials (see +Polynomial Factorization) + +The standard arithmetic operations are available for polynomials. +
      +
    • + +
      +
    • +
    +The operation gcd is used to compute the +greated common divisor of two polynomials. +
      +
    • + +
      +
    • +
    +In the case of p and q, the gcd is obvious from their definitions. +We factor the gcd to show this relationship better. +
      +
    • + +
      +
    • +
    +The least common multiple is computed by using +lcm. +
      +
    • + +
      +
    • +
    +Use content to compute the greatest common +divisor of the coefficients of the polynomial. +
      +
    • + +
      +
    • +
    +Many of the operations on polynomials require you to specify a variable. +For example, resultant requires you to +give the variable in which the polynomials should be expressed. This +computes the resultant of the values of p and q, considering them as +polynomials in the variable z. They do not share a root when thought +of as polynomials in z. +
      +
    • + +
      +
    • +
    +This value is 0 because as polynomials in x the polynomials have a +common root. +
      +
    • + +
      +
    • +
    +The data type used for the variables created by +Polynomial is +Symbol. As mentioned above, the representation +used by Polynomial is recursive and so +there is a main variable for nonconstant polynomials. The operation +makeVariable returns this variable. +The return type is actually a union of Symbol +and "failed". +
      +
    • + +
      +
    • +
    +The latter branch of the union is used if the polynomial has no +variables, that is, is a constant. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +The complete list of variables actually used in a particular polynomial +is returned by variables. For constant +polynomials, this list is empty. +
      +
    • + +
      +
    • +
    +The degree operation returns the degree +of a polynomial in a specific variable. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +If you give a list of variables for the second argument, a list of the +degrees in those variables is returned. +
      +
    • + +
      +
    • +
    +The minimum degree of a variable in a polynomial is computed using +minimumDegree. +
      +
    • + +
      +
    • +
    +The total degree of a polynomial is returned by +totalDegree. +
      +
    • + +
      +
    • +
    +It is often convenient to think of a polynomial as a leading monomial +plus the remaining terms, using the operation +leadingMonomial +
      +
    • + +
      +
    • +
    +The reductum operation returns a polynomial +consisting of the sum of the monomials after the first. +
      +
    • + +
      +
    • +
    +These have the obvious relationship that the original polynomial is equal +to the leading monomial plus the reductum. +
      +
    • + +
      +
    • +
    +The value returned by leadingMonomial +includes the coefficient of that term. This is extracted by using +leadingCoefficient on the +original polynomial. +
      +
    • + +
      +
    • +
    +The operation eval is used to substitute a +value for a varialbe in a polynomial. +
      +
    • + +
      +
    • +
    +This value may be another variable, a constant or a polynomial. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Actually, all the things being substituted are just polynomials, some +more trivial than others. +
      +
    • + +
      +
    • +
    +Derivatives are computed using the D operation. +
      +
    • + +
      +
    • +
    +The first argument is the polynomial and the second is the variable. +
      +
    • + +
      +
    • +
    +Even if the polynomial has only one variable, you must specify it. +
      +
    • + +
      +
    • +
    +Integration of polynomials is similar and the +integrate operation is used. + +Integration requires that the coefficients support division. +Consequently, Axiom converts polynomials over the integers to polynomials +over the rational numbers before integrating them. +
      +
    • + +
      +
    • +
    +It is not possible, in general, to divide two polynomials. In our example +using polynomials over the integers, the operation +monicDivide divides a polynomial by a +monic polynomial (that is, a polynomial with leading coefficient equal to +1). The result is a record of the quotient and remainder of the division. +You must specify the variable in which to express the polynomial. +
      +
    • + +
      +
    • +
    +The selectors of the components of the record are quotient and +remainder. Issue this to extract the remainder: +
      +
    • + +
      +
    • +
    +Now that we can extract the components, we can demonstrate the +relationship among them and the arguments to our original expression +
    +  qr:=monicDivide(p,x+1,x)
    +
    +
      +
    • + +
      +
    • +
    +If the / operator is used with polynomials, +a fraction object is created. In this example, the result is an object of +type +Fraction Polynomial Integer. +
      +
    • + +
      +
    • +
    +If you use rational numbers as polynomial coefficients, the resulting +object is of type +Polynomial Fraction Integer +
      +
    • + +
      +
    • +
    +This can be converted to a fraction of polynomials and back again, if +required. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +To convert the coefficients to floating point, map the +numeric operation on the coefficients +of the polynomial. +
      +
    • + +
      +
    • +
    +For more information on related topcis, see +UnivariatePolynomial, +MultivariatePolynomial, and +DistributedMultivariatePolynomial. +You can also issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +Polynomial. +<> +@ + +\subsection{polyspecifictypes2.xhtml} +<>= +<> + + + +<> +
    UnivariatePolynomial
    +
    +The domain constructor +UnivariatePolynomial +(abbreviated UP) +creates domains of univariate polynomials in a specified variable. +For example, the domain UP(a1,POLY FRAC INT) provides polynomials in +the single variable a1 whose coefficients are general polynomials with +rational number coefficients. +
    +Restriction:
    +Axiom does not allow you to create types where +UnivariatePolynomial +is contained in the coefficient type of +Polynomial. +Therefore, UP(x,POLY INT) is legal but POLY UP(x,INT) is not. +
    +UP(x,INT) is the domain of polynomials in the single variable x with +integer coefficients. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +The usual arithmetic operations are available for univariate polynomials. +
      +
    • + +
      +
    • +
    +The operation +leadingCoefficient +extracts the coefficient of the term of highest degree. +
      +
    • + +
      +
    • +
    +The operation degree returns the degree of +the polynomial. Since the polynomial has only one variable, the variable +is not supplied to operations like degree. +
      +
    • + +
      +
    • +
    +The reductum of the polynomial, the polynomial obtained by subtracting +the term of highest order, is returned by +reductum. +
      +
    • + +
      +
    • +
    +The operation gcd computes the greatest common +divisor of two polynomials. +
      +
    • + +
      +
    • +
    +The operation lcm computes the least common +multiple. +
      +
    • + +
      +
    • +
    +The operation resultant computes the +resultant of two univariate polynomials. In the case of p and q, the +resultant is 0 because they share a common root. +
      +
    • + +
      +
    • +
    +To compute the derivative of a univariate polynomial with respect to +its variable, use D. +
      +
    • + +
      +
    • +
    +Univariate polynomials can also be used as if they were functions. +To evaluate a univariate polynomial at some point, apply the polynomial +to the point. +
      +
    • + +
      +
    • +
    +The same syntax is used for composing two univariate polynomials, i.e. +substituting one polynomial for the variable in another. This substitutes q +for the variable in p. +
      +
    • + +
      +
    • +
    +This substitutes p for the variable in q. +
      +
    • + +
      +
    • +
    +To obtain a list of coefficients of the polynomial, use +coefficients. +
      +
    • + +
      +
    • +
    +From this you can use gcd and +reduce to compute the contents of the +polynomial. +
      +
    • + +
      +
    • +
    +Alternatively (and more easily), you can just call +content +
      +
    • + +
      +
    • +
    +Note that the operation coefficients +omits the zero coefficients from the list. Sometimes it is useful to +convert a univariate polynomial to a vector whose i-th position contains +the degree i-1 coefficient of the polynomial. +
      +
    • + +
      +
    • +
    +To get a complete vector of coefficients, use the operation +vectorise, which takes a univariate +polynomial and an integer denoting the length of the desired vector. +
      +
    • + +
      +
    • +
    +It is common to want to do something to every term of a polynomial, +creating a new polynomial in the process. This is a function for +iterating across the terms of a polynomial, squaring each term. +
      +
    • + +
      +
    • +
    +Recall what p looked like. +
      +
    • + +
      +
    • +
    +We can demonstrate squareTerms on p. +
      +
    • + +
      +
    • +
    +When the coefficients of the univariate polynomial belong to a field, +(for example, when the coefficients are rational numbers, as opposed to +integers. The important property of a field is that non-zero elements can +be divided and produce another element. The quotient of the integers 2 and 3 +is not another integer.) It is possible to compute quotients and remainders. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +When the coefficients are rational numbers or rational expressions, the +operation quo computes the quotient of two +polynomials. +
      +
    • + +
      +
    • +
    +The operation rem computes the remainder. +
      +
    • + +
      +
    • +
    +The operation divide can be used to return +a record of both components. +
      +
    • + +
      +
    • +
    +Now we check the arithmetic. +
      +
    • + +
      +
    • +
    +It is also possible to integrate univariate polynomials when the +coefficients belong to a field. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +One application of univariate polynomials is to see expressions in terms of +a specific variable. We start with a polynomial in a1 whose coefficients are +quotients of polynomials in b1 and b2. +
      +
    • + +
      +
    • +
    +Since in this case we are not talking about using multivariate polynomials +in only two variables, we use Polynomial. +We also use Fraction because we want fractions. +
      +
    • + +
      +
    • +
    +We push all the variables into a single quotient of polynomials. +
      +
    • + +
      +
    • +
    +Alternatively, we can view this as a polynomial in the variable. This is a +mode-directed conversion: You indicate as much of the structure as you care +about and let Axiom decide on the full type and how to do the transformation. +
      +
    • + +
      +
    • +
    +See Polynomial Factorization for a +discussion of the factorization facilities in Axiom for univariate +polynomials. For more information on related topics, see +Polynomials, +Conversion, +Polynomial, +MultivariatePolynomial, and +DistributedMultivariatePolynomial. +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +UnivariatePolynomial. +<> +@ + +\subsection{polyspecifictypes3.xhtml} +<>= +<> + + + +<> +
    MultivariatePolynomial
    +
    +The domain constructor +MultivariatePolynomial is +similar to Polynomial except that it +specifies the variables to be used. +Polynomial are available for +MultivariatePolynomial. +The abbreviation for +MultivariatePolynomial is +MPOLY. The type expressions +
    +   MultivariatePolynomial([x,y],Integer)
    +
    +and +
    +         MPOLY([x,y],INT)
    +
    +refer to the domain of multivariate polynomials in the variables x and y +where the coefficients are restricted to be integers. The first variable +specified is the main variable and the display of the polynomial reflects +this. This polynomial appears with terms in descending powers of the +variable x. +
      +
    • + +
      +
    • +
    +It is easy to see a different variable ordering by doing a conversion. +
      +
    • + +
      +
    • +
    +You can use other, unspecified variables, by using +Polynomial in the coefficient type of +MPOLY. +
      +
    • + +
      +
    • +
    +Conversions can be used to re-express such polynomials in terms of the +other variables. For example, you can first push all the variables into a +polynomial with integer coefficients. +
      +
    • + +
      +
    • +
    +Now pull out the variables of interest. +
      +
    • + +
      +
    • +
    +
    +Restriction: Axiom does not allow you to create types where +MultivariatePolynomial is +contained in the coefficient type of +Polynomial. Therefore, +
    +     MPOLY([x,y],POLY INT)
    +
    +is legal but this is not: +
    +     POLY MPOLY([x,y],INT)n
    +
    +
    +Multivariate polynomials may be combined with univariate polynomials to +create types with special structures. +
      +
    • + +
      +
    • +
    +This is a polynomial in x whose coefficients are quotients of polynomials +in y and z. Use conversions for the structural rearrangements. z does not +appear in a denominator and so it can be made the main variable. +
      +
    • + +
      +
    • +
    +Or you can make a multivariate polynomial in x and z whose coefficients +are fractions in polynomials in y +
      +
    • + +
      +
    • +
    +A conversion like +
    +  q::MPOLY([x,y],FRAC UP(z,INT))
    +
    +is not possible in this example because y appears in the denominator of +a fraction. As you can see, Axiom provides extraordinary flexibility in +the manipulation and display of expressions via its conversion facility. + +For more information on related topics, see +Polynomial, +UnivariatePolynomial, and +DistributedMultivariatePolynomial. +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by +MultivariatePolynomial. +<> +@ + +\subsection{polyspecifictypes4.xhtml} +<>= +<> + + + +<> +
    DistributedMultivariatePolynomial
    +
    + +DistributedMultivariatePolynomial and + +HomogeneousDistributedMultivariatePolynomial, abbreviated +DMP and +HDMP +repspectively, are very similar to +MultivariatePolynomial +except that they are represented and displayed in a non-recursive manner. +
      +
    • + +
      +
    • +
    +The construction +DMP orders its +monomials lexicographically while +HDMP +orders them by total order refined by reverse lexicographic order. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +These constructors are mostly used in Groebner basis calculations. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    • + +
      +
    • +
    +Note that we get a different Groebner basis when we use the +HDMP +polynomials, as expected. +
      +
    • + +
      +
    • +
    + +GeneralDistributedMultivariatePolynomial is somewhat more flexible in +the sense that as well as accepting a list of variables to specify the +variable ordering, it also takes a predicate on exponent vectors to specify +the term ordering. With this polynomial type the user can experiment with +the effect of using completely arbitrary term orderings. This flexibility +is mostly important for algorithms such as Groebner basis calculations +which can be very sensitive to term orderings. + +For more information on related topics, see +Polynomials, +Conversion, +Polynomial, +UnivariatePolynomial. and +MultivariatePolynomial, +Issue the system command +
      +
    • + +
      +
    • +
    +to display the full list of operations defined by + +DistributedMultivariatePolynomial and +<> +@ + +\subsection{polysubstitutions.xhtml} +<>= +<> + + + +<> +
    Polynomial Evaluation and Substitution
    +
    +The function eval is used to substitute values +into polynomials. Here's an example of how to use it: +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +This example would give you the value of the polynomial p at 5. You can +also substitute into polynomials with several variables. First, specify +the polynomial, then give a list of the bindings of the form +
    +  variable = value
    +
    +For examples: +
      +
    • + +
      +
    • +
    +Here x was replaced by a+b, and y was replaced by c+d. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +Substitution is done "in parallel". That is, Axiom takes q(x,y) and +returns q(y,x). + +You can also substitute numerical values for some or all of the variables. +
      +
    • + +
      +
    • +
    • + +
      +
    • +
    +<> +@ + + +\subsection{puiseuxseries.xhtml} +<>= +<> + + + +<> + + + + + + + + + + + + + + + + +
    + Enter the formula for the general coefficient of the series: +
    + +
    + Enter the index variable for your formula: + +
    + Enter the power series variable: + +
    + Enter the point about which to expand: + +
    +For Puiseux Series, the exponent of the power series variable ranges +from an initial value, an arbitrary rational number, to plus +infinity; the step size is any positive rational number. + + + + + + + +
    + Enter the initial value of the index (a rational number): + +
    + Enter the step size (a positive rational number): + +
    +<> +<> +<> + +@ + +%%Q +%%R +\subsection{reallimit.xhtml} +<>= +<> + + + +<> +
    + Enter the function you want to compute the limit of:
    +
    + Enter the name of the variable:
    +
    + + A finite point +
    + + %plusInfinity
    + + %minusInfinity


    + Compute the limit from:
    + + both directions
    + + the right
    + + the left
    +
    +<> +<> +<> + +@ +\subsection{refsearchpage.xhtml} +<>= +<> + + +<> + refsearchpage not implemented +<> +@ + +\subsection{releasenotes.xhtml} +<>= +<> + + +<> +The November 2007 release of Axiom contains +
      +
    • + New MathML output mode. This mode allows Axiom to output expressions + using standard MathML format. This complements the existing ability + to output Fortran, IBM script, Latex, OpenMath, and algebra formats. +
    • +
    • + Ninety-five domains have been documented for the )help command. + Type )help to see the list. +
    • +
    • + New regression tests were added to improve the release testing. +
    • +
    • + Hyperdoc can now be restarted. Type )hd +
    • +
    • + Testing has begun against Spiegel's Mathematical Handbook from the + Schaum's Outline Series. These tests include Axiom's solutions and + have uncovered mistakes in the published text. +
    • +
    +Bug fixes +
      +
    • + Bug100 integrate((z^a+1)^b,z) no longer loops infinitely. +
    • +
    • + Bug101 laplace(log(z),z,w) returns "failed" instead of crashing. +
    • +
    • + Bug103 solve(z=z,z) returns the correct answer +
    • +
    +Additional information sources: + + + + + + + +
    + + Online information is available here + +
    + + The changelog file contains specific file-by-file changes. + +
    +<> +@ + +\subsection{rootpage.xhtml} +\begin{verbatim} + notangle -R"rootpage.xhtml" bookvol11.pamphlet > rootpage.xhtml +\end{verbatim} +<>= +<> + + + +
    + What would you like to do?
    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + Any Command + + Try command line input
    + + Basic Commands + + Solve problems by filling in templates
    + + Axiom Textbook + + Read Volume 0 -- The Jenks/Sutor Book
    + + Axiom Tutorial + + Read Volume 1 -- The Tutorial
    + + Reference + + Scan on-line documentation for AXIOM
    + + Topics + + Learn how to use Axiom, by topic
    + + Browser + + Browse through the AXIOM library
    + + Examples + + See examples of use of the library
    + + Settings + + Display and change the system environment
    + + What's New + + Enhancements in this version of Axiom
    + + Fonts + + Test Axiom Fonts in your Browser
    +<> +@ + +%%S +\subsection{series.xhtml} +<>= +<> + + +<> + Create a series by + + + + + + + + + + + + + + + + + +
    + + Expansion + + + Expand a function in a series around a point +
    + + Taylor Series + +
    + Series where the exponent ranges over the integers from a + non-negative integer value to plus infinity by an arbitrary + positive integer step size. +
    + + Laurent Series + +
    + Series where the exponent ranges from an arbitrary integer value + to plus infinity by an arbitrary positive integer step size. +
    + + Puiseux Series + +
    + Series where the exponent ranges from an arbitrary rational value + to plus infinity by an arbitrary positive rational number step size. +
    +<> +@ + +\subsection{seriesexpand.xhtml} +<>= +<> + + + +<> + + + + + + + + + + + + + +
    + What function would you like to expand in a power series? +
    + +
    + Enter the power series variable: + +
    + Expand around the point: + +
    +<> +<> +<> + +@ + +\subsection{solve.xhtml} +<>= +<> + + +<> + What do you want to solve? + + + + + + + + + + + + + +
    + + A System of Linear Equations in equation form + +
    + + A System of Linear Equations in matrix form + +
    + + A System of Polynomial Equations + +
    + + A Single Polynomial Equation + +
    +<> +@ + + +\subsection{solvelinearequations.xhtml} +<>= +<> + + + +<> + + + + +
    + Enter the number of equations: + +
    +
    +
    +
    + +
    +
    +
    +<> +<> +@ + +\subsection{solvelinearmatrix.xhtml} +<>= +<> + + + +<> +Enter the size of the matrix: + + + + + + + + + +
    Rows
    Columns
    +
    +
    + How would you like to enter the matrix elements? +
    + + +
    +
    +
    +<> +<> +@ + + +@ + +\subsection{solvesinglepolynomial.xhtml} +<>= +<> + + +<> + solvesinglepolynomial.xhtml not implemented +<> + +@ + +\subsection{solvesystempolynomials.xhtml} +<>= +<> + + +<> + solvesystempolynomials.xhtml not implemented +<> + +@ + +\subsection{summation.xhtml} +<>= +<> + + + +<> +
    + Enter the function you want to sum:
    +
    + Enter the summation index: +
    + Enter the limits of the sum: From: + + To: +
    +
    +<> +<> +<> +@ + +\subsection{systemvariables.xhtml} +<>= +<> + + +<> + systemvariables not implemented +<> +@ + + +%%T + +\subsection{taylorseries.xhtml} +<>= +<> + + + +<> + + + + + + + + + + + + + + + + +
    + Enter the formula for the general coefficient of the series: +
    + +
    + Enter the index variable for your formula: + +
    + Enter the power series variable: + +
    + Enter the point about which to expand: + +
    +For Taylor Series, the exponent of the power series variable ranges +from an initial value, an arbitrary non-negative integer, to plus +infinity; the step size is any positive integer. + + + + + + + +
    + Enter the initial value of the index (an integer): + +
    + Enter the step size (a positive integer): + +
    +<> +<> +<> + +@ + +\subsection{topexamplepage.xhtml} +<>= +<> + + +<> + + + + + + + + + + + + + +
    GraphicsExamples of Axiom Graphics
    DomainsExamples of use of Axiom domains and packages
    OperationsExamples of Axiom Operations, by topic
    +<> +@ + +\subsection{topicspage.xhtml} +<>= +<> + + +<> + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    NumbersA look at different types of numbers
    PolynomialsPolynomials in Axiom
    FunctionsBuilt-in and user-defined functions
    Solving EquationsFacilities for solving equations
    CalculusUsing Axiom to do calculus
    Linear AlgebraAxiom's linear algebra facilities
    GraphicsAxiom's graphics facilities
    AlgebraAxiom's abstract algebra facilities
    CryptographyAlasdair McAndrew's Crytography Course Notes
    Mathematical MethodsMIT 18-08 Mathematical Methods for Engineers Course Notes
    CATSComputer Algebra Test Suite
    +<> +@ + +\subsection{topreferencepage.xhtml} +<>= +<> + + +<> + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    AXIOM BookThe on-line version of the Jenks/Sutor book.
    Aldor GuideThe on-line Aldor Users Guide.
    NAG LibraryThe on-line NAG Library documentation.
    TopicsLearn how to use Axiom, by topic.
    LanguageIntroduction to the Axiom language.
    ExamplesExamples for exposed domains and packages
    CommandsSystem commands that control your workspace.
    OperationsA guide to useful operations
    System VariablesView and change a system-defined variable
    GlossaryA glossary of Axiom terms.
    HyperDocHow to write your own HyperDoc pages.
    SearchReference pages for occurrences of a string.
    +<> +@ + +\subsection{topsettingspage.xhtml} +<>= +<> + + +<> +System commands are used to perform Axiom environment +management and change Axiom system variables. +
    + + + + + + + + + +
    CommandsSystem commands that control your environment.
    SettingsChange an Axiom variable.
    +<> +@ + +\subsection{tutorial.xhtml} +<>= +<> + + +<> + tutorial not implemented +<> +@ + +%%U +\subsection{uglangpage.xhtml} +<>= +<> + + +<> + uglangpage not implemented +<> +@ + +\subsection{ugsyscmdpage.xhtml} +<>= +<> + + +<> + ugsyscmdpage not implemented +<> +@ + + +\subsection{usersguidepage.xhtml} +<>= +<> + + +<> + usersguidepage not implemented +<> +@ + + +%%V +%%W +%%X +%%Y +%%Z +\subsection{rcm3720.input} +<>= +str2lst(str) == [ord(str.i)-65 for i in 1..#str] + +lst2str(lst) == concat [char(lst.i+65)::String for i in 1..#lst] + +str2num(str) == + local strlst + strlst:=[ord(str.i) for i in 1..#str] + return wholeRadix(strlst)$RadixExpansion(256)::INT + +num2str(n) == + local tmp + tmp:=wholeRagits(n::RadixExpansion(256)) + return concat [char(tmp.i)::String for i in 1..#tmp] + +superIncreasing?(lst) == + reduce(/\,[lst.i>reduce(+,[lst.j for j in 1..i-1]) for i in 2..#lst]) + +siSolve(lst,n) == + local res,m,i + if not superIncreasing?(lst) then error "The list is not super-increasing" + m := n + res := [0 for i in 1..#lst] + for i in #lst..1 by -1 repeat + if lst.i <= m then + res.i := 1 + m := m - lst.i + if m = 0 then return res + error "Unsolvable" + +subsetsum(L:List(INT),N:INT):List(INT) == + local x,Y + if N=0 then return([]) + if N<0 or #L=0 then return([-1]) + for x in L repeat + Y:=subsetsum(remove(x,L),N) + if Y~=[-1] then return(Y) + Y:=subsetsum(remove(x,L),N-x) + if Y~=[-1] then return(cons(x,Y)) + return([-1]) +@ + +\subsection{signatures.txt} +<>= +RSA --- +n = 2^137-1 e = 17 +message = "This is my text." +signature = 68767027465671577191073128495082795700768 +n = (6^67-1)/5 e = 17 +message = "Please feed my dog!" +signature = 1703215098456351993605104919259566435843590978852633 + +Rabin ----- +n = (3^59-1)/2 +message = "Leave now." +signature = +n = (7^47-1)/6 +message = "Arrive Thursday." +signature = 189479723122534414019783447271411895509 + +El Gamal -------- +p = next prime after 2^150 +a = 2 +B = 1369851585774063312693119161120024351761244461 +message = "Leave AT ONCE!" +signature r = 1389080525305754392111976715361069425353578198 +s = 1141326468070168229982976133801721430306004477 + +DSS --- +p = next prime after 2^170 +q = 143441505468590696209 +g = 672396402136852996799074813867123583326389281120278 +B = 1394256880659595564848116770226045673904445792389839 +message = "Now's your chance!" +signature r = 64609209464638355801 +s = 13824808741200493330 +@ + +\subsection{strang.input} +<>= +rowmatrix(r:List(Fraction(Integer))):Matrix(Fraction(Integer)) == + [r]::Matrix(Fraction(Integer)) + +columnmatrix(c:List(Fraction(Integer))):Matrix(Fraction(Integer)) == + [[i] for i in c]::Matrix(Fraction(Integer)) + +k(n) == + M := diagonalMatrix([2 for i in 1..n]) + for i in 1..n-1 repeat M(i,i+1):=-1 + for i in 1..n-1 repeat M(i+1,i):=-1 + M::SquareMatrix(n,Fraction(Integer)) + +t(n) == + M:=k(n) + N:=M::Matrix(Fraction(Integer)) + qsetelt!(N,1,1,1) + N::SquareMatrix(n,Fraction(Integer)) + +b(n) == + M:=k(n) + N:=M::Matrix(Fraction(Integer)) + qsetelt!(N,1,1,1) + qsetelt!(N,n,n,1) + N::SquareMatrix(n,Fraction(Integer)) + +K:=k(3) +T:=t(3) +B:=b(3) + + +@ +\subsection{bitmaps/axiom1.bitmap} +<>= +#define axiom_width 270 +#define axiom_height 100 +static char axiom_bits[] = { + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 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0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0xe0, 0xff, 0x07, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0xc0, 0xff, 0x03, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0xff, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x7e, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x10, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0x00, 0x00, 0x00, 0x00}; +@ + +\section{License} +<>= +--Copyright (c) 2007 Arthur C. Ralfs +--All rights reserved. +-- +--Redistribution and use in source and binary forms, with or without +--modification, are permitted provided that the following conditions are +--met: +-- +-- - Redistributions of source code must retain the above copyright +-- notice, this list of conditions and the following disclaimer. +-- +-- - Redistributions in binary form must reproduce the above copyright +-- notice, this list of conditions and the following disclaimer in +-- the documentation and/or other materials provided with the +-- distribution. +-- +-- - Neither the name of Arthur C. Ralfs nor the +-- names of its contributors may be used to endorse or promote products +-- derived from this software without specific prior written permission. +-- +--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} diff --git a/books/bookvol12.pamphlet b/books/bookvol12.pamphlet new file mode 100644 index 0000000..7c3c8cc --- /dev/null +++ b/books/bookvol12.pamphlet @@ -0,0 +1,483 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 12: Axiom Crystal}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Axiom Crystal Design} +\chapter{Experiments} +\section{Hide/Show a div element} +Here we demonstrate the ability to hide or show a named div element. +<>= + + + + + + + +it works + + +
    + Hide + Show + +@ +\section{Hide/Show a nested div element} +Now that we can hide or show a div element we demonstrate the ability +to hide or show a nested div element. +<>= + + + + + + + +it works + + +
    + Hide + Show + +@ +\section{Hide/Show a ring of elements} +Now that we can hide or show a div element we demonstrate the ability +to hide or show a ring of div elements. There are 3 elements in the +ring, 'facet1', 'facet2', and 'facet3'. Each facet can open or close +the associated 'answer' sub-div element. +<>= + + + + + + + +it works + + + + +
    + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\bibitem{7} Winograd, Terry ``Bringing Design to Software", +ACM Press Books, Addison-Wesley Publishing, 1996 +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol2.pamphlet b/books/bookvol2.pamphlet new file mode 100644 index 0000000..1dc6019 --- /dev/null +++ b/books/bookvol2.pamphlet @@ -0,0 +1,333 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 2: Axiom Users Guide}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol2 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol3.pamphlet b/books/bookvol3.pamphlet new file mode 100644 index 0000000..f8ad6f3 --- /dev/null +++ b/books/bookvol3.pamphlet @@ -0,0 +1,333 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 3: Axiom Programmers Guide}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol3 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol4.pamphlet b/books/bookvol4.pamphlet new file mode 100644 index 0000000..32abce4 --- /dev/null +++ b/books/bookvol4.pamphlet @@ -0,0 +1,4424 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 4: Axiom Developers Guide}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\section{How Axiom Works} +\subsection{Input and Type Selection} +First we change the default setting for autoload messages to +turn off the noise of file loading from the library: +\begin{verbatim} +(1) -> )set mes auto off +\end{verbatim} + +Next we tell the interpreter to show us the modemaps used to +classify input and select types. This is known as ``bottomup'' +messages. We can watch the interpreter ponder the input. +\begin{verbatim} +(1) -> )set mes bot on +\end{verbatim} + +Now we give it something nontrivial to ponder. +\begin{verbatim} +(1) -> f:=1/(a*x+b) +\end{verbatim} + +After parsing the input Axiom begins to figure out the type +of the expression. In this case it starts with the multiply +operator in the denominator. + +Axiom has determined that ``a'' is of type VARIABLE and ``x'' +is of type VARIABLE. It is looking for function of the form +\begin{verbatim} + VARIABLE * VARIABLE +\end{verbatim} +so it looks in the domain of the left argument ``a'' which is +VARIABLE and does not find the required function. Similarly +it looks in the domain of the right argument ``x'' which is +VARIABLE and, not surprisingly, does not find the required function. + +It tried to promote each VARIABLE to SYMBOL and looks for a way +to mulitply VARIABLES and SYMBOLS or SYMBOLS and SYMBOLS. Neither +succeeds. +\begin{verbatim} + + Function Selection for * + Arguments: (VARIABLE a,VARIABLE x) + -> no appropriate * found in Variable a + -> no appropriate * found in Variable x + -> no appropriate * found in Symbol + -> no appropriate * found in Variable a + -> no appropriate * found in Variable x + -> no appropriate * found in Symbol + + Modemaps from Associated Packages + no modemaps +\end{verbatim} + +Since it cannot find a specific modemap that uses the exact types +it now expands the search to look for the general modemaps. It +searches these modemaps in order to try to find one that fits. +\begin{verbatim} + + Remaining General Modemaps + [1] (D,D1) -> D from D + if D has XFALG(D2,D1) and D2 has ORDSET and D1 has RING +\end{verbatim} + +The first match will fail because Symbol does not have RING. We can +determine this by asking the interpreter: +\begin{verbatim} +SYMBOL has RING + + (1) false + Type: Boolean +\end{verbatim} + +The following modemaps will fail for various similar reasons: +\begin{verbatim} + [2] (D1,D) -> D from D + if D has XFALG(D1,D2) and D1 has ORDSET and D2 has RING + [3] (Integer,D) -> D from D + if D has VECTCAT D2 and D2 has TYPE and D2 has ABELGRP + [4] (D1,D) -> D from D + if D has VECTCAT D1 and D1 has TYPE and D1 has MONOID + [5] (D,D1) -> D from D + if D has VECTCAT D1 and D1 has TYPE and D1 has MONOID + [6] (D,D1) -> D1 from D + if D has SMATCAT(D2,D3,D4,D1) and D3 has RING and D4 has + DIRPCAT(D2,D3) and D1 has DIRPCAT(D2,D3) + [7] (D1,D) -> D1 from D + if D has SMATCAT(D2,D3,D1,D4) and D3 has RING and D1 has + DIRPCAT(D2,D3) and D4 has DIRPCAT(D2,D3) + [8] (D,D) -> D from D if D has SGROUP + [9] (D,D1) -> D from D if D has RMODULE D1 and D1 has RNG + [10] (D,D) -> D from D if D has MONAD + [11] (D,D) -> D from D + if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG + D1 and D3 has FLAGG D1 + [12] (D1,D) -> D from D + if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG + D1 and D3 has FLAGG D1 + [13] (D,D1) -> D from D + if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG + D1 and D3 has FLAGG D1 + [14] (Integer,D) -> D from D + if D has MATCAT(D2,D3,D4) and D2 has RING and D3 has FLAGG + D2 and D4 has FLAGG D2 + [15] (D,D1) -> D1 from D + if D has MATCAT(D2,D3,D1) and D2 has RING and D3 has FLAGG + D2 and D1 has FLAGG D2 + [16] (D1,D) -> D1 from D + if D has MATCAT(D2,D1,D3) and D2 has RING and D1 has FLAGG + D2 and D3 has FLAGG D2 + [17] ((D5 -> D6),(D4 -> D5)) -> (D4 -> D6) from MappingPackage3(D4, + D5,D6) + if D4 has SETCAT and D5 has SETCAT and D6 has SETCAT + [18] (D1,D) -> D from D if D has LMODULE D1 and D1 has RNG + [19] (PolynomialIdeals(D1,D2,D3,D4),PolynomialIdeals(D1,D2,D3,D4)) + -> PolynomialIdeals(D1,D2,D3,D4) + from PolynomialIdeals(D1,D2,D3,D4) + if D1 has FIELD and D2 has OAMONS and D3 has ORDSET and D4 + has POLYCAT(D1,D2,D3) + [20] (D1,D) -> D from D + if D has GRMOD(D1,D2) and D1 has COMRING and D2 has ABELMON + + [21] (D,D1) -> D from D + if D has GRMOD(D1,D2) and D1 has COMRING and D2 has ABELMON + + [22] (D1,D2) -> D from D + if D has FMCAT(D1,D2) and D1 has RING and D2 has SETCAT + [23] (D1,D2) -> D from D + if D has FAMONC(D2,D1) and D2 has SETCAT and D1 has CABMON + + [24] (Equation D1,D1) -> Equation D1 from Equation D1 + if D1 has SGROUP and D1 has TYPE + [25] (D1,Equation D1) -> Equation D1 from Equation D1 + if D1 has SGROUP and D1 has TYPE + [26] (D,D1) -> D from D + if D has DIRPCAT(D2,D1) and D1 has TYPE and D1 has MONOID + + [27] (D1,D) -> D from D + if D has DIRPCAT(D2,D1) and D1 has TYPE and D1 has MONOID + + [28] (DenavitHartenbergMatrix D2,Point D2) -> Point D2 + from DenavitHartenbergMatrix D2 + if D2 has Join(Field,TranscendentalFunctionCategory) + [29] (PositiveInteger,Color) -> Color from Color + [30] (DoubleFloat,Color) -> Color from Color + [31] (CartesianTensor(D1,D2,D3),CartesianTensor(D1,D2,D3)) -> + CartesianTensor(D1,D2,D3) + from CartesianTensor(D1,D2,D3) + if D1: INT and D2: NNI and D3 has COMRING + [32] (PositiveInteger,D) -> D from D if D has ABELSG + [33] (NonNegativeInteger,D) -> D from D if D has ABELMON + [34] (Integer,D) -> D from D if D has ABELGRP + +\end{verbatim} + +Eventually the interpreter decides that it can coerce Symbol to +Polynomial(Integer). We can do this in the interpreter also: +\begin{verbatim} +a::Symbol::POLY(INT) + + (1) a + Type: Polynomial Integer +\end{verbatim} + +And the interpreter can find multiply in POLY(INT): +\begin{verbatim} + + [1] signature: (POLY INT,POLY INT) -> POLY INT + implemented: slot $$$ from POLY INT + [2] signature: (POLY INT,POLY INT) -> POLY INT + implemented: slot $$$ from POLY INT +\end{verbatim} + +We can see this signature exists by asking the interpreter to show +us the domain POLY(INT) (truncated here for brevity): +\begin{verbatim} + +)show POLY(INT) + Polynomial Integer is a domain constructor. + Abbreviation for Polynomial is POLY + This constructor is exposed in this frame. + Issue )edit src/algebra/POLY.spad to see algebra source code for POLY + +------------------------------- Operations -------------------------------- + + ?*? : (Fraction Integer,%) -> % ?*? : (Integer,%) -> % + ?*? : (PositiveInteger,%) -> % ?*? : (%,Fraction Integer) -> % + ?*? : (%,Integer) -> % ?*? : (%,%) -> % + +\end{verbatim} + +Having found multipy the interpreter now starts a search for the operation +\begin{verbatim} + (POLY(INT)) + (VARIABLE) +\end{verbatim} + +It cannot find this modemap +\begin{verbatim} + + Function Selection for + + Arguments: (POLY INT,VARIABLE b) + -> no appropriate + found in Polynomial Integer + -> no appropriate + found in Variable b + -> no appropriate + found in Variable b + +\end{verbatim} +so it promotes VARIABLE to POLY(INT) and finds the operation: +\begin{verbatim} + (POLY(INT)) + (POLY(INT)) +\end{verbatim} + +\begin{verbatim} + [1] signature: (POLY INT,POLY INT) -> POLY INT + implemented: slot $$$ from POLY INT + +\end{verbatim} + +Next it tackles the division operation where the numerator is +PI (PositiveInteger) and the denominator is POLY(INT). It tries +to find +\begin{verbatim} + (PI) / (POLY(INT)) +\end{verbatim} + +in PositiveInteger, Polynomial Integer and Integer. All attempts fail. +\begin{verbatim} + + Function Selection for / + Arguments: (PI,POLY INT) + -> no appropriate / found in PositiveInteger + -> no appropriate / found in Polynomial Integer + -> no appropriate / found in Integer + -> no appropriate / found in PositiveInteger + -> no appropriate / found in Polynomial Integer + -> no appropriate / found in Integer + + Modemaps from Associated Packages + no modemaps + +\end{verbatim} + +So now it turns to the general modemaps: +\begin{verbatim} + Remaining General Modemaps + [1] (D,D1) -> D from D if D has VSPACE D1 and D1 has FIELD + [2] (D,D1) -> D from D + if D has RMATCAT(D2,D3,D1,D4,D5) and D1 has RING and D4 has + DIRPCAT(D3,D1) and D5 has DIRPCAT(D2,D1) and D1 has FIELD + + [3] (D1,D1) -> D from D if D has QFCAT D1 and D1 has INTDOM + [4] (D,D1) -> D from D + if D has MATCAT(D1,D2,D3) and D1 has RING and D2 has FLAGG + D1 and D3 has FLAGG D1 and D1 has FIELD + [5] (D,D1) -> D from D + if D has LIECAT D1 and D1 has COMRING and D1 has FIELD + [6] (D,D) -> D from D if D has GROUP + [7] (SparseMultivariatePolynomial(D2,Kernel D), + SparseMultivariatePolynomial(D2,Kernel D)) -> D + from D if D2 has INTDOM and D2 has ORDSET and D has FS D2 + + [8] (Float,Integer) -> Float from Float + [9] (D,D) -> D from D if D has FIELD + [10] (D,D) -> D from D + if D = EQ D1 and D1 has FIELD and D1 has TYPE or D = EQ D1 + and D1 has GROUP and D1 has TYPE + [11] (DoubleFloat,Integer) -> DoubleFloat from DoubleFloat + [12] (D,D1) -> D from D + if D has AMR(D1,D2) and D1 has RING and D2 has OAMON and D1 + has FIELD + +\end{verbatim} +and it eventually promotes PI to FRAC(POLY(INT)) and +POLY(INT) to FRAC(POLY(INT)) and finds the match: +\begin{verbatim} + (FRAC(POLY(INT))) / (FRAC(POLY(INT))) +\end{verbatim} + +We can ask the intepreter to show us this operation (again, the +output is truncated for brevity): +\begin{verbatim} + )show FRAC(POLY(INT)) + Fraction Polynomial Integer is a domain constructor. + Abbreviation for Fraction is FRAC + This constructor is exposed in this frame. + Issue )edit src/algebra/FRAC.spad to see algebra source code for FRAC + +------------------------------- Operations -------------------------------- + + ?*? : (Fraction Integer,%) -> % ?*? : (Integer,%) -> % + ?*? : (PositiveInteger,%) -> % ?*? : (%,Fraction Integer) -> % + ?*? : (%,%) -> % ?**? : (%,Integer) -> % + ?**? : (%,PositiveInteger) -> % ?+? : (%,%) -> % + ?-? : (%,%) -> % -? : % -> % + ?/? : (%,%) -> % ? Boolean +\end{verbatim} + +\begin{verbatim} + [1] signature: (FRAC POLY INT,FRAC POLY INT) -> FRAC POLY INT + implemented: slot $$$ from FRAC POLY INT + +\end{verbatim} + +At this point the interpreter has succeeded in finding a type for +the expression and eventually returns the result badged with the +appropriate type: +\begin{verbatim} + + 1 + (1) ------- + a x + b + Type: Fraction Polynomial Integer + +\subsection{A simple integral} + +Now we will show an integration with successive levels of expansion +of explanation. We will use the expression above: +\begin{verbatim} +(1) -> f:=1/(a*x+b) + + 1 + (1) ------- + a x + b + Type: Fraction Polynomial Integer +(2) -> integrate(f,x) + + log(a x + b) + (2) ------------ + a + Type: Union(Expression Integer,...) + +\end{verbatim} +\subsection{A simple integral, expansion 1 interpreter} + +\begin{verbatim} +(2) -> integrate(f,x) +\end{verbatim} + +Here we assume the previous discussion of modemap handling for the +expression f and we only look at the modemap handling for the integrate +function. We are looking for a modemap of the form: + +\begin{verbatim} + integrate(FRAC(POLY(INT)),VARIABLE x) +\end{verbatim} + +So first we look in the domains of the arguments, that is, in +Fraction Polynomial Integer, and Variable. Neither one succeeds: +\begin{verbatim} + + Function Selection for integrate + Arguments: (FRAC POLY INT,VARIABLE x) + -> no appropriate integrate found in Fraction Polynomial Integer + -> no appropriate integrate found in Variable x + -> no appropriate integrate found in Fraction Polynomial Integer + -> no appropriate integrate found in Variable x + + Modemaps from Associated Packages + no modemaps + +\end{verbatim} + +Next we look at the general modemaps to find one that might work: +\begin{verbatim} + Remaining General Modemaps + [1] (D,D1) -> D from D + if D1 = SYMBOL and D has UTSCAT D2 and D2 has RING and D2 + has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 + has ALGEBRA FRAC INT or D1 = SYMBOL and D has UTSCAT D2 and + D2 has RING and D2 has variables: D2 -> List D1 and D2 has + integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT + [2] (D,D1) -> D from D + if D1 = SYMBOL and D has UPXSCAT D2 and D2 has RING and D2 + has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 + has ALGEBRA FRAC INT or D1 = SYMBOL and D has UPXSCAT D2 + and D2 has RING and D2 has variables: D2 -> List D1 and D2 + has integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT + + [3] (D,D1) -> D from D + if D1 = SYMBOL and D has ULSCAT D2 and D2 has RING and D2 + has ACFS INT and D2 has PRIMCAT and D2 has TRANFUN and D2 + has ALGEBRA FRAC INT or D1 = SYMBOL and D has ULSCAT D2 and + D2 has RING and D2 has variables: D2 -> List D1 and D2 has + integrate: (D2,D1) -> D2 and D2 has ALGEBRA FRAC INT + [4] (Polynomial D2,Symbol) -> Polynomial D2 from Polynomial D2 + if D2 has ALGEBRA FRAC INT and D2 has RING + [5] (D,D1) -> D from D + if D has MTSCAT(D2,D1) and D2 has RING and D1 has ORDSET + and D2 has ALGEBRA FRAC INT + [6] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List + Expression D4) + from IntegrationResultRFToFunction D4 + if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo + Integer,OrderedSet,LinearlyExplicitRingOver Integer) + [7] (Expression Float,List Segment OrderedCompletion Float) -> + Result + from AnnaNumericalIntegrationPackage + [8] (Expression Float,Segment OrderedCompletion Float) -> Result + from AnnaNumericalIntegrationPackage + [9] (GeneralUnivariatePowerSeries(D2,D3,D4),Variable D3) -> + GeneralUnivariatePowerSeries(D2,D3,D4) + from GeneralUnivariatePowerSeries(D2,D3,D4) + if D3: SYMBOL and D2 has ALGEBRA FRAC INT and D2 has RING + and D4: D2 + [10] (D2,Symbol) -> Union(D2,List D2) from FunctionSpaceIntegration( + D4,D2) + if D4 has Join(EuclideanDomain,OrderedSet, + CharacteristicZero,RetractableTo Integer, + LinearlyExplicitRingOver Integer) and D2 has Join( + TranscendentalFunctionCategory,PrimitiveFunctionCategory, + AlgebraicallyClosedFunctionSpace D4) + [11] (Fraction Polynomial D4,SegmentBinding OrderedCompletion + Fraction Polynomial D4) -> Union(f1: OrderedCompletion Expression + D4,f2: List OrderedCompletion Expression D4,fail: failed, + pole: potentialPole) + from RationalFunctionDefiniteIntegration D4 + if D4 has Join(EuclideanDomain,OrderedSet, + CharacteristicZero,RetractableTo Integer, + LinearlyExplicitRingOver Integer) + [12] (Fraction Polynomial D4,SegmentBinding OrderedCompletion + Expression D4) -> Union(f1: OrderedCompletion Expression D4,f2: + List OrderedCompletion Expression D4,fail: failed,pole: + potentialPole) + from RationalFunctionDefiniteIntegration D4 + if D4 has Join(EuclideanDomain,OrderedSet, + CharacteristicZero,RetractableTo Integer, + LinearlyExplicitRingOver Integer) + [13] (D2,SegmentBinding OrderedCompletion D2) -> Union(f1: + OrderedCompletion D2,f2: List OrderedCompletion D2,fail: failed, + pole: potentialPole) + from ElementaryFunctionDefiniteIntegration(D4,D2) + if D2 has Join(TranscendentalFunctionCategory, + PrimitiveFunctionCategory,AlgebraicallyClosedFunctionSpace + D4) and D4 has Join(EuclideanDomain,OrderedSet, + CharacteristicZero,RetractableTo Integer, + LinearlyExplicitRingOver Integer) + +\end{verbatim} +Modemap [6] wins because we can construct the first argument +by matching +\begin{verbatim} + Fraction Polynomial Integer +\end{verbatim} +to +\begin{verbatim} + Fraction Polynomial D4 +\end{verbatim} +so we can infer that D4 == Integer +\begin{verbatim} + + [6] (Fraction Polynomial D4,Symbol) -> Union(Expression D4,List + Expression D4) + from IntegrationResultRFToFunction D4 + if D4 has CHARZ and D4 has Join(GcdDomain,RetractableTo + Integer,OrderedSet,LinearlyExplicitRingOver Integer) + +\end{verbatim} + +Given that match we have two requirements on Integer, both of which +we can check with the interpreter: + +\begin{verbatim} +INT has CHARZ + + (3) true + Type: Boolean +(4) -> INT has Join(GcdDomain,RetractableTo Integer,OrderedSet,_ + LinearlyExplicitRingOver Integer) + + (4) true + Type: Boolean +\end{verbatim} + +So we have a match +\begin{verbatim} + + [1] signature: (FRAC POLY INT,SYMBOL) -> Union(EXPR INT,LIST EXPR INT) + implemented: slot (Union (Expression (Integer)) (List (Expression (Integer))))(Fraction (Polynomial (Integer)))(Symbol) from IRRF2F INT + [2] signature: (EXPR INT,SYMBOL) -> Union(EXPR INT,LIST EXPR INT) + implemented: slot (Union (Expression (Integer)) (List (Expression (Integer))))(Expression (Integer))(Symbol) from FSINT(INT,EXPR INT) + +\end{verbatim} + +Now we invoke +\begin{verbatim} + integrate(FRAC(POLY(INT)),SYMBOL) -> Union(EXPR INT,LIST EXPR INT) + from IRRF2F(INT) + + integrate(1/(a*x+b),x) + +\end{verbatim} +can print the result: +\begin{verbatim} + + log(a x + b) + (2) ------------ + a + Type: Union(Expression Integer,...) + +\end{verbatim} +\subsection{A simple integral, expansion 2 integrate} +Now that we know how the interpreter has matched the input and called +the function we need to follow the first level call into the function. + +Axiom provides a trace tool that will allow us to walk into the function +invocation and watch what happens. We will follow this same invocation +path many times, each time we will descend another layer, repeating the +information as we do. + +For now, we look at the domain IRRF2F from irexpand.spad. +The categorical definition of this domain reads (we remove parts +of the definition for brevity): +\begin{verbatim} +IntegrationResultRFToFunction(R): Exports == Implementation where + R: Join(GcdDomain, RetractableTo Integer, OrderedSet, + LinearlyExplicitRingOver Integer) + + RF ==> Fraction Polynomial R + F ==> Expression R + IR ==> IntegrationResult RF + OF ==> OutputForm + + Exports ==> with + expand : IR -> List F + ++ expand(i) returns the list of possible real functions + ++ corresponding to i. + if R has CharacteristicZero then + integrate : (RF, Symbol) -> Union(F, List F) + ++ integrate(f, x) returns the integral of \spad{f(x)dx} + ++ where x is viewed as a real variable.. + + Implementation ==> add + import IntegrationTools(R, F) + import TrigonometricManipulations(R, F) + import IntegrationResultToFunction(R, F) + + toEF: IR -> IntegrationResult F + + toEF i == map(#1::F, i)$IntegrationResultFunctions2(RF, F) + expand i == expand toEF i + complexExpand i == complexExpand toEF i + + if R has CharacteristicZero then + import RationalFunctionIntegration(R) + + if R has imaginary: () -> R then + integrate(f, x) == complexIntegrate(f, x) + else + integrate(f, x) == + l := [mkPrim(real g, x) for g in expand internalIntegrate(f, x)] + empty? rest l => first l + l + +@ +\end{verbatim} + +We can see that this domain constructor takes one argument which, in +this case, is Integer. We've already determined that Integer has the +required Joins: +\begin{verbatim} +(4) -> INT has Join(GcdDomain,RetractableTo Integer,OrderedSet,_ + LinearlyExplicitRingOver Integer) + + (4) true + Type: Boolean +\end{verbatim} +and we can see that: +\begin{verbatim} +(5) -> INT has CharacteristicZero + + (5) true + Type: Boolean +\end{verbatim} +so we can match the signature of integrate: +\begin{verbatim} + integrate(Fraction Polynomial Integer, Symbol) -> + Union(Expression Integer, List Expression Integer) +\end{verbatim} + +We can trace this domain and ask to see the output in math form: +\begin{verbatim} +(6) -> )trace IRRF2F )math + + Packages traced: + IntegrationResultRFToFunction Integer + Parameterized constructors traced: + IRRF2F +\end{verbatim} +and now, when we do the integration, we see the output of the trace: +\begin{verbatim} +integrate(1/(a*x+b),x) +1exit IntegrationResultRFToFunction.expand,18 : + a x + b + log(-------) + a + [------------] + a +1>exit IntegrationResultRFToFunction.integrate,32 : + log(a x + b) + ------------ + a + + log(a x + b) + (6) ------------ + a + Type: Union(Expression Integer,...) +\end{verbatim} + +From this we learn that the arguments to integrate are exactly the +arguments we supplied and we know the exact types of the arguments +because they have to match the signature of the function: +\begin{verbatim} +1 + 1 + arg1= ------- <== Fraction Polynomial Integer + a x + b + arg2= x <== Symbol + +\end{verbatim} +and returns the result +\begin{verbatim} +1>exit IntegrationResultRFToFunction.integrate,32 : + log(a x + b) + ------------ <== Union(Expression Integer, List Expression Integer) + a +\end{verbatim} +\subsection{A simple integral, expansion 2 internalIntegrate} +If we look at the function definition for integrate: +\begin{verbatim} + integrate(f, x) == + l := [mkPrim(real g, x) for g in expand internalIntegrate(f, x)] + empty? rest l => first l + l +\end{verbatim} +we can see that there is a call to the function +\begin{verbatim} + internalIntegrate(f, x) +\end{verbatim} +and we can compute the types of the arguments since they are exactly +the types of the integrate function itself: +\begin{verbatim} + internalIntegrate(Fraction Polynomial Integer, Symbol) +\end{verbatim} +and since the return value will be fed to the expand function we +can look at the signature of expand: +\begin{verbatim} + expand: IntegrationResult Fraction Polynomial Integer -> + List Expression Integer +\end{verbatim} +and we can get the full signature for internalIntegrate: +\begin{verbatim} + internalIntegrate(Fraction Polynomial Integer, Symbol) -> + IntegrationResult Fraction Polynomial Integer +\end{verbatim} + +This comes from the domain +\begin{verbatim} + RationalFunctionIntegration(F): Exports == Implementation where + F: Join(IntegralDomain, RetractableTo Integer, CharacteristicZero) +\end{verbatim} +where F is Integer. +\begin{verbatim} + SE ==> Symbol + P ==> Polynomial F + Q ==> Fraction P + UP ==> SparseUnivariatePolynomial Q + QF ==> Fraction UP + LGQ ==> List Record(coeff:Q, logand:Q) + UQ ==> Union(Record(ratpart:Q, coeff:Q), "failed") + ULQ ==> Union(Record(mainpart:Q, limitedlogs:LGQ), "failed") + + Exports ==> with + internalIntegrate: (Q, SE) -> IntegrationResult Q + ++ internalIntegrate(f, x) returns g such that \spad{dg/dx = f}. + Implementation ==> add + import RationalIntegration(Q, UP) + import IntegrationResultFunctions2(QF, Q) + import PolynomialCategoryQuotientFunctions(IndexedExponents SE, + SE, F, P, Q) + internalIntegrate(f, x) == + map(multivariate(#1, x), integrate univariate(f, x)) + +\end{verbatim} +If we look the signature for internalIntegrate and expand it we see: +\begin{verbatim} + internalIntegrate: (Q, SE) -> IntegrationResult Q + + internalIntegrate: ( Fraction Polynomial Integer, Symbol) -> + IntegrationResult Fraction Polynomial Integer + +\end{verbatim} +which is exactly what we need. When we look at the function we see: +\begin{verbatim} + internalIntegrate(f, x) == + map(multivariate(#1, x), integrate univariate(f, x)) +\end{verbatim} +We can watch the function call by tracing INTRF: +\begin{verbatim} +(7) -> )trace INTRF )math + + Packages traced: + IntegrationResultRFToFunction Integer, + RationalFunctionIntegration Integer + Parameterized constructors traced: + IRRF2F, INTRF +\end{verbatim} +and we see: +\begin{verbatim} +(7) -> integrate(1/(a*x+b),x) +1exit RationalFunctionIntegration.internalIntegrate,25 : + 1 a x + b + - log(-------) + a a + 1exit IntegrationResultRFToFunction.expand,18 : + a x + b + log(-------) + a + [------------] + a +1>exit IntegrationResultRFToFunction.integrate,32 : + log(a x + b) + ------------ + a + + log(a x + b) + (7) ------------ + a + Type: Union(Expression Integer,...) +\end{verbatim} +Now we see that internalIntegrate was called with the arguments +\begin{verbatim} + 1exit RationalFunctionIntegration.internalIntegrate,25 : + 1 a x + b + - log(-------) <== IntegrationResult Fraction Polynomial Integer + a a +\end{verbatim} +\subsection{A simple integral, expansion 3 univariate} +But the internalIntegrate function does its work by calling yet +other functions, the deepest of which is univariate: +\begin{verbatim} + internalIntegrate(f, x) == + map(multivariate(#1, x), integrate univariate(f, x)) +\end{verbatim} +Since univariate uses the arguments to the internalIntegrate function +which has the signature: +\begin{verbatim} + internalIntegrate: ( Fraction Polynomial Integer, Symbol) -> +\end{verbatim} +we can determine that we need a univariate function with the signature: +\begin{verbatim} + univariate: ( Fraction Polynomial Integer, Symbol) -> +\end{verbatim} +This function is found in PolynomialCategoryQuotientFunctions, POLYCATQ +which has the form: +\begin{verbatim} +PolynomialCategoryQuotientFunctions(E, V, R, P, F): + Exports == Implementation where + E: OrderedAbelianMonoidSup + V: OrderedSet + R: Ring + P: PolynomialCategory(R, E, V) + F: Field with + coerce: P -> % + numer : % -> P + denom : % -> P + + UP ==> SparseUnivariatePolynomial F + RF ==> Fraction UP + + Exports ==> with + variables : F -> List V + ++ variables(f) returns the list of variables appearing + ++ in the numerator or the denominator of f. + mainVariable: F -> Union(V, "failed") + ++ mainVariable(f) returns the highest variable appearing + ++ in the numerator or the denominator of f, "failed" if + ++ f has no variables. + univariate : (F, V) -> RF + ++ univariate(f, v) returns f viewed as a univariate + ++ rational function in v. + Implementation ==> add + P2UP: (P, V) -> UP + + univariate(f, x) == P2UP(numer f, x) / P2UP(denom f, x) + + P2UP(p, x) == + map(#1::F, + univariate(p, x))$SparseUnivariatePolynomialFunctions2(P, F) +\end{verbatim} + +So we are calling the function: +\begin{verbatim} + univariate: ( Fraction Polynomial Integer, Symbol) -> + Fraction SparseUnivariatePolynomial Field with + coerce: PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet) -> % + numer: % -> PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet) + denom: % -> PolynomialCategory(Ring, OrderedAbelianMonoidSup, OrderedSet) +\end{verbatim} +which we can see by tracing that domain: +\begin{verbatim} +(8) -> )trace POLYCATQ )math + + Packages traced: + IntegrationResultRFToFunction Integer, + RationalFunctionIntegration Integer, + PolynomialCategoryQuotientFunctions(IndexedExponents + Kernel Expression Integer,Kernel Expression Integer, + Integer,SparseMultivariatePolynomial(Integer,Kernel + Expression Integer),Expression Integer), + PolynomialCategoryQuotientFunctions(IndexedExponents + Symbol,Symbol,Integer,Polynomial Integer,Fraction + Polynomial Integer) + Parameterized constructors traced: + IRRF2F, INTRF, POLYCATQ +\end{verbatim} +which gives the input: +\begin{verbatim} + 1exit PolynomialCategoryQuotientFunctions.univariate,16 : + 1 + - + a + ----- <== Fraction SparseUnivariatePolynomial Field with + b coerce: P -> % + ? + - numer: % -> P + a denom: % -> P +\end{verbatim} + +It should be clear that univariate divided the numerator and +denominator by the leading coefficient of the polynomial in +the denominator. It also replaced ``x'' with the variable ``?''. +\subsection{A simple integral, expansion 4 integrate} +When univariate returns, the results are fed to another integrate, +this time from RationalIntegration (INTRAT). This domain looks like: +\begin{verbatim} +RationalIntegration(F, UP): Exports == Implementation where + F : Join(Field, CharacteristicZero, RetractableTo Integer) + UP: UnivariatePolynomialCategory F + + RF ==> Fraction UP + IR ==> IntegrationResult RF + LLG ==> List Record(coeff:RF, logand:RF) + URF ==> Union(Record(ratpart:RF, coeff:RF), "failed") + U ==> Union(Record(mainpart:RF, limitedlogs:LLG), "failed") + OF ==> OutputForm + + Exports ==> with + integrate : RF -> IR + ++ integrate(f) returns g such that \spad{g' = f}. + + Implementation ==> add + import TranscendentalIntegration(F, UP) + + integrate f == + rec := monomialIntegrate(f, differentiate) + integrate(rec.polypart)::RF::IR + rec.ir + +\end{verbatim} + +This domain was constructed and "brought into scope" in +RationalFunctionIntegration(F) with the statement +\begin{verbatim} + import RationalIntegration(Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial Integer) +\end{verbatim} +and the function has the signature +\begin{verbatim} + integrate: + Fraction SparseUnivariatePolynomial Fraction Polynomial Integer -> + IntegrationResult Fraction + Fraction Polynomial Integer +\end{verbatim} +\begin{verbatim} + 1exit RationalIntegration.integrate,32 : + 1 b + - log(? + -) <== IntegrationResult Fraction SparseUnivariatePolynomial + a a Fraction Polynomial Integer +\end{verbatim} + +\subsection{A simple integral, expansion 5 monomialIntegrate} +The integrate function is defined as: +\begin{verbatim} + integrate f == + print(outputForm("tpdhere INTRAT 1")@OF)$OF + rec := monomialIntegrate(f, differentiate) + integrate(rec.polypart)::RF::IR + rec.ir +\end{verbatim} +Notice that while ``f'' is an argument to integrate, the ``differentiate'' +function is a free variable. The Axiom compiler will look at all of the +symbols "in scope" to find its meaning. This code does an import: +\begin{verbatim} + import TranscendentalIntegration(Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial Integer) +\end{verbatim} +which exports monomialIntegrate +\begin{verbatim} +TranscendentalIntegration(F, UP): Exports == Implementation where + F : Field + UP : UnivariatePolynomialCategory F + + RF ==> Fraction UP + FF ==> Record(ratpart:F, coeff:F) + UF ==> Union(FF, "failed") + IR ==> IntegrationResult RF + REC ==> Record(ir:IR, specpart:RF, polypart:UP) + + Exports ==> with + monomialIntegrate : (RF, UP -> UP) -> REC + ++ monomialIntegrate(f, ') returns \spad{[ir, s, p]} such that + ++ \spad{f = ir' + s + p} and all the squarefree factors of the + ++ denominator of s are special w.r.t the derivation '. + + Implementation ==> add + import SubResultantPackage(UP, UP2) + import MonomialExtensionTools(F, UP) + import TranscendentalHermiteIntegration(F, UP) + import CommuteUnivariatePolynomialCategory(F, UP, UP2) + + monomialIntegrate(f, derivation) == + zero? f => [0, 0, 0] + r := HermiteIntegrate(f, derivation) + zero?(inum := numer(r.logpart)) => + [r.answer::IR, r.specpart, r.polypart] + iden := denom(r.logpart) + x := monomial(1, 1)$UP + resultvec := subresultantVector(UP2UP2 inum - + (x::UP2) * UP2UP2 derivation iden, UP2UP2 iden) + respoly := primitivePart leadingCoefficient resultvec 0 + rec := splitSquarefree(respoly, kappa(#1, derivation)) + logs:List(LOG) := [ + [1, UP2UPR(term.factor), + UP22UPR swap primitivePart(resultvec(term.exponent),term.factor)] + for term in factors(rec.special)] + dlog := + ((derivation x) = 1) => r.logpart + differentiate(mkAnswer(0, logs, empty()), + differentiate(#1, derivation)) + (u := retractIfCan(p := r.logpart - dlog)@Union(UP, "failed")) case UP => + [mkAnswer(r.answer, logs, empty), r.specpart, r.polypart + u::UP] + [mkAnswer(r.answer, logs, [[p, dummy]]), r.specpart, r.polypart] + +\end{verbatim} +which expands into the type signature: +\begin{verbatim} + + monomialIntegrate: + (Fraction SparseUnivariatePolynomial Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial Integer -> + SparseUnivariatePolynomial Fraction Polynomial Integer) -> + Record(ir: IntegrationResult Fraction + SparseUnivariatePolynomial Fraction Polynomial Integer, + specpart: Fraction + SparseUnivariatePolynomial Fraction Polynomial Integer, + polypart: SparseUnivariatePolynomial Fraction Polynomial Integer) + ++ monomialIntegrate(f, ') returns \spad{[ir, s, p]} such that + ++ \spad{f = ir' + s + p} and all the squarefree factors of the + ++ denominator of s are special w.r.t the derivation '. +\end{verbatim} +we can watch this happen: +\begin{verbatim} +)trace INTTR )math + + Function traced: UnivariatePolynomialCategory + Packages traced: + IntegrationResultRFToFunction Integer, + RationalFunctionIntegration Integer, + RationalIntegration(Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial + Integer), PolynomialCategoryQuotientFunctions( + IndexedExponents Kernel Expression Integer,Kernel + Expression Integer,Integer, + SparseMultivariatePolynomial(Integer,Kernel + Expression Integer),Expression Integer), + PolynomialCategoryQuotientFunctions(IndexedExponents + Symbol,Symbol,Integer,Polynomial Integer,Fraction + Polynomial Integer), TranscendentalIntegration( + Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial + Integer) + Parameterized constructors traced: + IRRF2F, INTRF, INTRAT, POLYCATQ, INTTR +\end{verbatim} +and we can watch the monomialIntegrate function call +\begin{verbatim} +(34) -> integrate(1/(a*x+b),x) +1exit PolynomialCategoryQuotientFunctions.univariate,16 : + 1 + - + a + ----- + b + ? + - + a + 1exit TranscendentalIntegration.monomialIntegrate,81 : + 1 b + [ir= - log(? + -),specpart= 0,polypart= 0] + a a + 1>exit RationalIntegration.integrate,32 : + 1 b + - log(? + -) + a a + 1>exit RationalFunctionIntegration.internalIntegrate,25 : + 1 a x + b + - log(-------) + a a +1>exit IntegrationResultRFToFunction.integrate,32 : + log(a x + b) + ------------ + a + + log(a x + b) + (34) ------------ + a + Type: Union(Expression Integer,...) +(35) -> +\end{verbatim} +\subsection{A simple integral, expansion 6 HermiteIntegrate} +Since ``f'' is not zero we invoke HermiteIntegrate from the domain +TranscendentalHermiteIntegration which looks like: +\begin{verbatim} +TranscendentalHermiteIntegration(F, UP): Exports == Implementation where + F : Field + UP : UnivariatePolynomialCategory F + + N ==> NonNegativeInteger + RF ==> Fraction UP + REC ==> Record(answer:RF, lognum:UP, logden:UP) + HER ==> Record(answer:RF, logpart:RF, specpart:RF, polypart:UP) + + Exports ==> with + HermiteIntegrate: (RF, UP -> UP) -> HER + ++ HermiteIntegrate(f, D) returns \spad{[g, h, s, p]} + ++ such that \spad{f = Dg + h + s + p}, + ++ h has a squarefree denominator normal w.r.t. D, + ++ and all the squarefree factors of the denominator of s are + ++ special w.r.t. D. Furthermore, h and s have no polynomial parts. + ++ D is the derivation to use on \spadtype{UP}. + + Implementation ==> add + import MonomialExtensionTools(F, UP) + + HermiteIntegrate(f, derivation) == + rec := decompose(f, derivation) + hi := normalHermiteIntegrate(rec.normal, derivation) + qr := divide(hi.lognum, hi.logden) + [hi.answer, qr.remainder / hi.logden, rec.special, qr.quotient + rec.poly] +\end{verbatim} +The function has the same input signature as monomialIntegrate but a +different return signature. +\begin{verbatim} + HermiteIntegrate: + (Fraction SparseUnivariatePolynomial Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial Integer -> + SparseUnivariatePolynomial Fraction Polynomial Integer) -> + Record(answer:Fraction SparseUnivariatePolynomial + Fraction Polynomial Integer, + logpart:Fraction SparseUnivariatePolynomial + Fraction Polynomial Integer, + specpart:Fraction SparseUnivariatePolynomial + Fraction Polynomial Integer, + polypart:SparseUnivariatePolynomial Fraction Polynomial Integer) +\end{verbatim} +so we trace this domain +\begin{verbatim} +(37) -> )trace INTHERTR )math + + Function traced: UnivariatePolynomialCategory + Packages traced: + IntegrationResultRFToFunction Integer, + RationalFunctionIntegration Integer, + RationalIntegration(Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial + Integer), PolynomialCategoryQuotientFunctions( + IndexedExponents Kernel Expression Integer,Kernel + Expression Integer,Integer, + SparseMultivariatePolynomial(Integer,Kernel + Expression Integer),Expression Integer), + PolynomialCategoryQuotientFunctions(IndexedExponents + Symbol,Symbol,Integer,Polynomial Integer,Fraction + Polynomial Integer), TranscendentalIntegration( + Fraction Polynomial Integer, + SparseUnivariatePolynomial Fraction Polynomial + Integer), TranscendentalHermiteIntegration(Fraction + Polynomial Integer,SparseUnivariatePolynomial + Fraction Polynomial Integer) + Parameterized constructors traced: + IRRF2F, INTRF, INTRAT, POLYCATQ, INTTR, INTHERTR +\end{verbatim} +and now we see +\begin{verbatim} +(38) -> integrate(1/(a*x+b),x) +1exit TranscendentalHermiteIntegration.HermiteIntegrate,18 : + 1 + - + a + [answer= 0,logpart= -----,specpart= 0,polypart= 0] + b + ? + - + a + 1>exit TranscendentalIntegration.monomialIntegrate,81 : + 1 b + [ir= - log(? + -),specpart= 0,polypart= 0] + a a + "tpdhere UPOLYC 1" + 1>exit RationalIntegration.integrate,32 : + 1 b + - log(? + -) + a a + 1>exit RationalFunctionIntegration.internalIntegrate,25 : + 1 a x + b + - log(-------) + a a + 1exit IntegrationResultRFToFunction.expand,18 : + a x + b + log(-------) + a + [------------] + a +1>exit IntegrationResultRFToFunction.integrate,32 : + log(a x + b) + ------------ + a + + log(a x + b) + (38) ------------ + a + Type: Union(Expression Integer,...) +\end{verbatim} +so HermiteIntegrate did nothing to the input. +Next we call normalHermiteIntegrate which is a local function + +\section{Tools} +\subsection{svn} +SVN is a source control system on all platforms. Axiom 'silver' is +maintained in an SVN archive on sourceforge. This can be pulled from: +\begin{verbatim} +svn co https://axiom.svn.sf.net/svnroot/axiom/trunk/axiom axiom +\end{verbatim} +\subsection{git} +Git is a unix-based source code control system. Axiom 'silver' is +maintained in a git archive. This can be pulled from: +\begin{verbatim} +git-clone ssh://git@axiom-developer.org/home/git/silver +\end{verbatim} +the password for the userid git is linus. + +\subsection{cvs} +This assumes that you have set up ssh on the Savannah site. +CVS does not use a password. You have to log onto the Savannah +site and set up a public key. This requires you to: +\begin{itemize} +\item set up a local public key: +ssh-keygen -b 1024 -t rsa1 +\item open a browser +\item nagivate to the savannah page that has your personal keys +\item open .ssh/identity.pub +\item cut .ssh/identity.pub +\item paste it into your personal key list on savannah +\item go have a beer (the page takes an hour or two to update) +\end{itemize} + +Once you have a working key you can do the cvs login. +If it prompts you for a password then the key is not working. +If it prompts you to ``Enter the passphrase for RSA key'' then +cvs login will work. + +I maintain a directory where I work (call this WORK) +\begin{verbatim} + /home/axiomgnu/new +\end{verbatim} +and a directory for CVS (call this GOLD) +\begin{verbatim} + /axiom +\end{verbatim} + +When I want to export a set of changes I do the following steps: + +\noindent +0) MAKE SURE THE ~/.ssh/config FILE IS CORRECT: +\begin{verbatim} + (you should only need to do this once. + you need to change the User= field) + +Host *.gnu.org + Protocol=1 + Compression=yes + CompressionLevel=3 + User=axiom + StrictHostKeyChecking=no + PreferredAuthentications=publickey,password + NumberOfPasswordPrompts=2 +\end{verbatim} + +\noindent +1) MAKE SURE THE SHELL VARIABLES ARE OK: +\begin{verbatim} + (normally set in .bashrc) + +export CVS_RSH=ssh +export CVSROOT=:pserver:axiom@subversions.gnu.org:/cvsroot/axiom + ^^^^^ + change this to your id +\end{verbatim} + +\noindent +2) MAKE SURE YOU'RE LOGGED IN: +\begin{verbatim} + (I keep a session open all the time but it doesn't seem to care + if you login again. i'm not sure what login does, actually) + +cvs login +\end{verbatim} + +\noindent +3) GET A FRESH COPY FOR THE FIRST TIME OR AT ANY TIME: +\begin{verbatim} + (you only need to do this the first time but you can erase + your whole axiom subtree and refresh it again doing this. + + note that i work as root so i can update /. Most rational + people are smarter than me and work as a regular user so + you have to change the instructions for cd. But you knew that) + +cd / +cvs co axiom +\end{verbatim} + +\noindent +4) MAKE SURE THAT GOLD, MY LOCAL CVS COPY, IS UP TO DATE: +\begin{verbatim} + (I maintain an exact copy of the CVS repository and only make + changes to it when i want to export the changes. that way I + won't export my working tree by accident. my working tree is + normally badly broken. + + The update command makes sure that you have all of the changes + other people might have made and checked in. you have to merge + your changes so you don't step on other people's work. + So be sure to run update BEFORE you copy files to GOLD) + +cd /axiom +cvs update +\end{verbatim} + +\noindent +5) COPY CHANGED FILES FROM WORK TO THE GOLD TREE: +\begin{verbatim} + (This is an example for updating the *.daase files. + You basically are changing your GOLD tree to reflect the + way you want CVS to look once you check in all of the files.) + +cd /home/axiomgnu/new +cp src/share/algebra/*.daase /axiom/src/share/algebra +\end{verbatim} + +\noindent +6) IF A FILE IS NEW (e.g. src/interp/foo.lisp.pamphlet) THEN: +\begin{verbatim} + (If you create a file you need to "put it under CVS control" + CVS only cares about files you explicitly add or delete. + If you make a new file and copy it to GOLD you need to do this. + + Don't do the "cvs add" in your WORK directory. The cvs add + command updates the files in the CVS directory and you won't + have them in your WORK directory. + + Notice that you do the "cvs add" in the directory where the + file was added (hence, the cd commands). + +cd /axiom/src/interp +cvs add -m"some pithy comment" foo.lisp.pamphlet +cd /axiom +\end{verbatim} + +\noindent +7) IF A FILE IS DELETED (e.g. src/interp/foo.lisp.pamphlet) THEN: +\begin{verbatim} + (you have to delete the file from the GOLD directory BEFORE you + do a "cvs remove". The "cvs remove" will update the files in + the CVS directory + + Notice that you do the "cvs remove" in the directory where the + file was deleted (hence, the cd commands). + +cd /axiom/src/interp +rm foo.lisp.pamphlet +cvs remove foo.lisp.pamphlet +cd /axiom +\end{verbatim} + +\noindent +8) IF A DIRECTORY IS NEW (e.g. foodir) THEN: +\begin{verbatim} + (this will put "foodir" under CVS control. It will also create + foodir/CVS as a directory with a bunch of control files in the + foodir/CVS directory. Don't mess with the control files. + + (there are a bunch of special rules about directories. + empty directories are not downloaded by update.) + + (NOTE: THERE IS NO WAY TO DELETE A DIRECTORY) + +cd /axiom/src +mkdir foodir +cvs add -m "pithy comment" foodir +cd /axiom +\end{verbatim} + +\noindent +9) EDIT CHANGELOG: +\begin{verbatim} + changelog is already under CVS control so it will get uploaded + when you do the checkin.) + +cd /axiom +emacs -nw changelog + (add a date, initials, and pithy comment, save it, and exit) +\end{verbatim} + +\noindent +10) CHECK IN THE CHANGES +\begin{verbatim} + (This will actually change the savannah CVS repository. + + The "cvs ci" command will recurse thru all of the lower + subdirectories and look for changed files. It will change + the host versions of those files to agree with your copy. + If somebody else has changed a file while you were busy + developing code then the checkin MAY complain (if it can't + merge the changes) + +cd /axiom +cvs ci -m"pithy comment" +\end{verbatim} + +Congrats. You've now done your first change to the production +image. Please be very careful as this is a world readable copy. +We don't want to ship nonsense. Test everything. Even trivial +changes before you upload. + +\section{Common Lisps} +\subsection{GCL} +Axiom was ported to run under AKCL which was a common lisp +developed by Bill Schelter. He started with KCL (Kyoto Common +Lisp) and, since he lived and worked in Austin, Texas, named +his version AKCL (Austin-Kyoto Common Lisp). Bill worked under +contract to the Scratchpad group at IBM Research. I was the +primary developer for system internals so Bill and I worked +closely together on a lot of issues. After Axiom was sold to +NAG Bill continued to develop AKCL and it eventually became +GCL (Gnu Common Lisp). + +In order to port Axiom to run on GCL we need to do several things. +First, we need to apply a few patches. These patches enlarge the +default stack size, remove the startup banner, link with Axiom's +socket library, and rename collectfn. + +The issue with the stack size is probably bogus. At one point the +system was running out of stack space but the problem was due to +a recursive expansion of a macro and no amount of stack space +would be sufficient. This patch remains at the moment but should +probably be removed and tested. + +The startup banner is an issue because we plan to run under +various frontend programs like Texmacs and the Magnus ZLC. +We need to just output a single prompt. + +Axiom has a socket library because at the time it was developed under +AKCL there was no socket code in Lisp. There is still not a standard +common lisp socket library but I believe all common lisps have a way +to manipulate sockets. This code should be rewritten in lisp +and \verb|#+| for each common lisp. + +The collectfn file is a major optimization under GCL. When collectfn +is loaded and the lisp compiler is run then collectfn will output +a .fn file. The second time the compiler is invoked the .fn +file is consulted to determine the actual types of arguments used. +Function calling is highly optimized using this type information +so that fast function calling occurs. Axiom should be built one +time to create the int/*/*.fn files. It should then be rebuilt +using the cached .fn files. I will automate this process into +the Makefiles in the future. + +GCL implementation will have a major porting problem to brand new +platforms. The compiler strategy is to output C code, compile it +using GCC, and dynamically link the machine code to the running +image. This requires deep knowledge of the symbol tables used +by the native linker for each system. In general this is a hard +problem that requires a lot of expertise. Bill Schelter and I +spent a lot of time and effort making this work for each port. +The magic knowledge is not written down anywhere and I no longer +remember the details. + +\subsection{CCL} +When Axiom was sold to NAG it was ported to CCL (Codemist Common +Lisp) which is not, strictly speaking, a common lisp implementation. +It contains just enough common lisp to support Axiom and, as I'm +a great believer in simple code, it only needed a small subset of +a full common lisp. + +CCL can be considered the best way to get Axiom running on a new +architecture as the porting issues are minimal. + +CCL is a byte-interpreter implementation and has both the +positive and negative aspects of that design choice. The positive +aspect is that porting the code to run on new architectures is +very simple. Once the CCL byte-code interpreter is running +then Axiom is running. The saved-system image is pure byte-codes +and is completely system independent. + +The negative aspects are that it is slow and the garbage collector +appears broken. Compiling the Axiom library files on an file-by-file +basis takes about 1 hour on GCL and about 12 hours on CCL. Compiling +all of the Axiom library files in the same image (as opposed to +starting a new image per file) still takes about 1 hour on GCL. +It never finishes in CCL. Indeed it stops doing useful work after +about the 40th file (out of several hundred). + +When Axiom became open source I moved the system back to GCL +because I could not understand how to build a CCL system. I plan +to revisit this in the future and document the process so others +can follow it as well as build Makefiles to automate it. + +\subsection{CMU CL} +CMU CL grew out of the Carnegie-Mellon University SPICE project. +That project studied the issues involved in building an optimizing +compiler for common lisp. Axiom, back when it was Scratchpad at IBM, +ran on CMU CL. Indeed, a lot of the lisp-level optimizations are +due to use of the CMU CL compiler and the disassemble function. + +\subsection{Franz Lisp} +Axiom, as Scratchpad, ran on Franz Lisp. + +\subsection{Lucid Common Lisp} +Axiom, as Scratchpad, ran on Lucid Common Lisp. + +\subsection{Symbolics Common Lisp} +Axiom, as Scratchpad, ran on Symbolics Common Lisp. + +\subsection{Golden Common Lisp} +Axiom, as Scratchpad, ran on Golden Common Lisp. +This was a PC version of Common Lisp which appears to have died. + +\subsection{VM/LISP 370} +Axiom, as Scratchpad, ran on VM/Lisp 370. This was an IBM version +of lisp and was not a common lisp. The .daase random access +file format is an artifact of running on this lisp. + +\subsection{Maclisp} +Axiom, as Scratchpad, ran on Maclisp. This was an early MIT version +of lisp and is not common lisp. Many of the funny function names +that have slightly different semantics than their common lisp +counterparts still exist in the system as macros due to this +lisp. + +\section{Literate Programming} +The Axiom source code was originally developed at IBM Research. +It was sold to The Numerical Algorithms Group (NAG) and was on +the market as a commercial competitor to Mathematica and Maple. + +Axiom was withdrawn from the market in 2000 and released as free +and open source software in 2001. When the Axiom project was +started on savannah, the GNU Free Software Foundation site the +source code had been rewritten into ``pamphlet'' files. The reasons +for this are twofold. + +\subsection{Pamphlet files} + +When the Axiom code was released it contained few comments. That made +it very difficult to understand what the code actually did. Unlike +commercial software there would be no group of individuals who would +work on the project for its lifetime. Thus there needed to be a way to +capture the expertise and understanding underlying ongoing +development. + +Unlike any other piece of free and open source software Axiom will +still give useful answers 30 years from now. Thus it is important, +and worthwhile, to invest a large amount of effort into documenting +how these answers are arrived at and why the algorithms are written +the way they are. + +The pamphlet file format follows Knuth's idea of literate +programming. Knuth made the observation that a program should be a +work designed to be read by humans. Making the program readable by +machine was a secondary consideration. Making documentation primary +and code secondary was a dramatic shift for a programmer. + +Knuth created a file format that combined documentation and code. He +created a tool called ``Web'' which had two basic command, tangle and +weave. The tangle command would be run against a literate document and +extract the source code, the weave command would be run against the +literate document and extract the TeX. + +\subsection{noweb} + +Knuth's Web tool was specifically designed to use Pascal code. The +``tangle'' operation would prettyprint the output according to the +style rules of Pascal. + +Axiom was written in a variety of languages, such as C and Lisp, +and used tools such as Makefiles which have their own syntax. Thus +Web could not be used directly. + +Norman Ramsey had the insight to realize that there was no reason +why the ``tangle'' command had to know anything about the programming +language. If you remove the prettyprinting feature but kept the code +extraction idea then ``tangle'' could be generic. + +Ramsey wrote a program called ``noweb'' that was similar in spirit +to Knuth's Web. It has two commands ``notangle'' and ``noweave'' +which perform the extractions but do so in a language neutral manner. + +The language neutral feature of noweb made it ideal for Axiom. Using +noweb every file in the system could be rewritten into literate form. + +The noweb program is not a perfect match, however. Axiom does not +``understand'' pamphlet files using noweb so it cannot extract the +code directly using the )compile command. This could be addressed by +writing lisp code which would be able to extract the code chunks +from a pamphlet and collect them into a file. The src/interp/gclweb.lisp +program does this. + +Axiom pamphlets need an additional step to weave them into standard latex. +This step is just a syntax question. The noweb program used a syntax +similar to Knuth and defined chunks between delimiters, thus: +\begin{verbatim} + <>= + your code + @ +\end{verbatim} +but this is just a syntactic convention. If the syntax followed the +rules of latex then the pamphlet files would be pure latex files. + +The alternate syntax defines a new latex environment called chunk. +This chunk environment makes the pamphlet file a pure latex file. +This eliminates the need for the weave operation. The tangle +operation only needs to occur while manipulating code, either during +system build or end user interaction. At both of these times the +tangle operation can be built into the system and hidden. + +The latest changeset introduces two related changes, gclweb and +axiom.sty. Together these changes allow optional syntactic changes to +pamphlets. These changes will completely eliminate the need to weave +files since now a pamphlet file can be a valid latex file. Tangle is +the only remaining command and it will eventually be an option on +)compile, etc. + +The src/interp/gclweb.lisp file introduces the ability to extract code +from pamphlet files while inside Axiom. The short description is that +gclweb will now automatically distinguish the type of chunk style +(latex or noweb) based on the chunk name. It is a first step to a native +understanding of pamphlet files. Future work involves integrating it +into commands like )compile and adding commands like )tangle. + +To tangle a file from within Axiom: +\begin{verbatim} + )lisp (tangle "filename.pamphlet" "<>") +\end{verbatim} +which is noweb syntax. Output goes to the console. You can direct the +output to a file with the optional third argument: +\begin{verbatim} + )lisp (tangle "filename.pamphlet" "<>" "filename.spad") +\end{verbatim} + +If you use the new latex chunk environment the syntax is: +\begin{verbatim} + )lisp (tangle "filename.pamphlet" "chunkname") + )lisp (tangle "filename.pamphlet" "chunkname" "filename.spad") +\end{verbatim} + +gclweb distinguishes the input syntax by looking at the first character +of the chunkname. If it is a '$<$' then noweb is used, otherwise latex. + +The src/doc/axiom.sty.pamphlet introduces the new chunk environment. +This is a completely compatible change and has no impact on existing +pamphlets. The new syntax makes pamphlet files = tex files so there is +no need to use weave. The gclweb change has a compatible tangle +function which can be invoked from inside Axiom. + +Noweb syntax of: +\begin{verbatim} + <>= + your code goes here + @ +\end{verbatim} + +can also be written as: +\begin{verbatim} + \begin{chunk}{chunkname} + your code goes here + \end{chunk} +\end{verbatim} + +One new feature of the latex chunk style is that latex commands +work within the chunk. To get typeset mathematics use +\verb|\(| and \verb|\)| + +\begin{verbatim} +-- This will typeset in a chunk \( x^2+\epsilon \) +-- And you can format things {\bf bold} +\end{verbatim} + +\section{Databases} +\subsection{libcheck} +The databases are built from the .kaf files in the .nrlib directories. +(.kaf files are random access files). + +interp.exposed is a file that names all of the CDPs (Category, Domain, +and Packages) and classifies them. Only some CDPs are exposed because +most are used to implement algebra and are not intended to be user level +functions. Exposing all of the functions causes much ambiguity. + +There is a function called libcheck (see src/interp/util.lisp.pamphlet) +that will check nrlibs vs interp.exposed. This is only partially +functional as I see that changes were made to the system which broke +this function. + +The libcheck function requires an absolute pathname to the int directory +so call it thus: +\begin{verbatim} + --> )lisp (libcheck "/axiom/int/algebra") +\end{verbatim} +The main reason this function is broken is that the system now gets +exposure information from src/algebra/exposed.lsp.pamphlet. It appears +that interp.exposed.pamphlet is no longer used (although I made sure +that both files have the same information). I'm going to modify libcheck +to use exposed.lsp in the future and eliminate all references in the +system to interp.exposed. + +For the moment, however, the libcheck function is quite useful. +It used to be run during system build because I frequently ran into +database problems and this function would alert me to that fact. +I'll add it back into the Makefile once I elide interp.exposed. + +\subsection{asq} +Axiom has several databases which contain information about domains, +categories, and packages. The databases are in a compressed format and +are organized as random-access files using numeric index values so it +is hard to get at the stored information. However, there is a +command-line query function called asq (pronounced ask) that knows the +format of the files and can be used for stand-alone queries. For +instance, if you know the abbreviation for a domain but want to know +what source file generated that domain you can say: +\begin{verbatim} +asq -so FOOBAR +\end{verbatim} +and it will tell you the name of the algebra source file that +defines FOOBAR. + +\section{Axiom internal representations} +\begin{verbatim} +PRIMITIVE REPRESENTATIONS OF AXIOM OBJECTS + + +There are several primitive representations in axiom. These are: + +boolean + this is represented as a lisp boolean + +integer + this is represented as a lisp integer + +small integer + this is represented as a lisp integer + +small float + this is represented as a lisp float + +list + this is represented as a lisp list + +vector + this is represented as a lisp vector + +record + there are 3 cases: + records of 1 element are a pair (element . nil) + records of 2 element are a pair (element1 . element2) + records of 3 or more are a vectors # + +mapping + mappings are a spadcall objects. they are represented as a pair + (lispfn . env) + where the env is usually a type object. A spadcall rips this + pair open and applies the lispfn to its args with env as the + last arg. + +union + there are 2 cases + if the object can be determined by a lisp predicate + (eg integer) then the union is just the object (eg 3) + itself since we can use lisp to decide which branch of + the union the object belongs to. that is, 3 is of the + integer branch in union(list,integer) + + if the object cannot be determined then the object is + wrapped into a pair where the car of the pair is the + union branch name and the cdr of the pair is the object. + that is, given union(a:SUP,b:POLY(INT)) x might be (a . x) + + note: if no tags are given in the union the system uses + consecutive integers, thus union(SUP,POLY(INT)) will give + a pair of (1 . x) or (2 . x) depending on the type of x + +other types are built up of compositions of these primitive +types. a sparse univariate polynomial (SUP) over the integers + + x**2+1 + +is represented as + +Term := Record(k:NonNegativeInteger,c:R) +Rep := List Term + +that is, the representation is a list of terms where each term +is a record whose first field is a nonnegative integer (the +exponent) and the second field is a member of the coefficient +ring. since this is a record of length 2 it is represented as +a pair. thus, the internal form of this polynomial is: + + ((2 . 1) (0 . 1)) + +a more complex object (recursively defined) is POLY(INT). given + + x**2+1 + +as a POLY(INT) we look at its representation and see: + +D := SparseUnivariatePolynomial($) +VPoly := Record(v:VarSet,ts:D) +Rep := Union(R,VPoly) + +so first we find that we are a member of the second form of the +union and since this is an untagged union the system uses 2 as +the tag. thus the first level of internal representation is: + + ( 2 . ) + +next we need to define the VPoly object. VPolys are records of +length 2 so we know they are represented by a pair. the car of +the pair is a VarSet. the cdr is a D which is a +SparseUnivariatePolynomial. Thus we consider this to be a poly +in x (at the top level) and we get: + + ( 2 . ( x . )) + +the SUP is over the SparseMultivariatePolynomials (SMP) so the +representation is recursive. Since an SUP is represented as a +list of + + (non-negative int . coefficient) + +one per term and we have 2 terms we know the next level of +structure is: + + ( 2 . ( x . (( 2 . ) ( 0 . )))) + +the SMP is just the integers so it fits into the first branch +of the union and each SMP looks like: + + ( uniontag . value ) + +in this case, being the first branch we get + + ( 2 . ( x . (( 2 . ( 1 . 1 )) ( 0 . ( 1 . 1 ))))) + +as the internal representation of + + x**2 + 1 + +what could be easier? +\end{verbatim} + +\section{axiom command} +The axiom command will eventually be a shell script. At the moment +it is just a copy of the interpsys image. However the whole Axiom +system consists of several processes and the axiom command starts +these processes. The shell script will transparently replace the +axiom executable image which will be renamed to spadsys. + +\section{help command documentation} +Axiom supports a )help command that takes a single argument. +This argument is interpreted as the name of a flat ascii file +which should live in \$AXIOM/doc/src/spadhelp. + +\subsection{help documentation for algebra} +The help documentation for algebra files lives within the algebra +pamphlet. The help chunk contains the name of the domain, thus: + +\begin{verbatim} +<>= +==================================================================== +thisdomain examples +==================================================================== + + (documentation for this domain) + + examplefunction foo + output + Type: thetype + +See Also: +o )show thisdomain +o $AXIOM/bin/src/doc/algebra/thisfile.spad.dvi + +@ +\end{verbatim} + +The documentation starts off with the domain enclosed in two lines +of equal signs. The documentation is free format. Generally the +functions are indented two spaces, the output is indented 3 spaces, +and the Type field has been moved toward the center of the line. + +The ``See Also:'' section lists the domain with the ``show'' command +and the path to the source file in dvi format. + +\subsection{adding help documentation in Makefile} + +There is a section in the src/algebra/Makefile.pamphlet that reads: +\begin{verbatim} +SPADHELP=\ + ${HELP}/AssociationList.help ${HELP}/BalancedBinaryTree.help \ +\end{verbatim} +which is essentially a list of all of the algebra help files. Each item +in this list refers to a stanza that looks like: + +\begin{verbatim} +${HELP}/AssociationList.help: ${IN}/list.spad.pamphlet + @echo 7000 create AssociationList.help from ${IN}/list.spad.pamphlet + @${TANGLE} -R"AssociationList.help" ${IN}/list.spad.pamphlet \ + >${HELP}/AssociationList.help + @cp ${HELP}/AssociationList.help ${HELP}/ALIST.help + @${TANGLE} -R"AssociationList.input" ${IN}/list.spad.pamphlet \ + >${INPUT}/AssociationList.input +\end{verbatim} + +Notice that the first line has an connection between the help file and +the spad file that contains it. The second line gives debugging output +containing a unique number. The third line extracts the help file. The +fourth line copies the file with the long name of the domain to a file +with the abbreviation of the domain so the user can query the domain +with either form. + +\subsection{Using help documentation for regression testing} + +The last line extracts an input test file for the algebra. In general +each help file is used to create an input test file for regression testing. + +There is a Makefile variable called REGRESS in the algebra Makefile: +\begin{verbatim} +REGRESS=\ + AssociationList.regress BalancedBinaryTree.regress \ +\end{verbatim} + +This is part of a Makefile that structure within the algebra Makefile. +This Makefile gets extracted by the Makefile in the input subdirectory. +Thus there is a connection between the two Makefiles (algebra and input). +This algebra regression Makefile goes by the chunk name +{\bf algebra.regress}. It contains a list of regression files and a +single stanza: +\begin{verbatim} +%.regress: %.input + @ echo algebra regression testing $* + @ rm -f $*.output + @ echo ')read $*.input' | ${TESTSYS} + @ echo ')lisp (regress "$*.output")' | ${TESTSYS} \ + | egrep -v '(Timestamp|Version)' | tee $*.regress +\end{verbatim} +The input Makefile extracts {\bf algebra.regress} and then calls +make to process this file. + +This keeps the regression test list in the algebra Makefile. + +\subsection{help documentation as algebra test files} + + +\section{debugsys} +The "debugsys" executable is the "interpsys" image but it is +built using the interpreted lisp code rather than using compiled +lisp code. This will make it slower but may, in certain cases, +give much better feedback in case of errors. If you find you need +to use debugsys you're really doing deep debugging. It isn't useful +for much else. It can be started by typing: +\begin{verbatim} +export AXIOM=/home/axiomgnu/new/mnt/linux +/home/axiomgnu/new/obj/linux/bin/debugpsys +\end{verbatim} +Notice that this image lives in the "obj" subtree. It is not +shipped with the "final" system image as only developers could +find it useful. +\subsection{debugging hyperdoc} +Hyperdoc will sometimes exit and also kill the AXIOMsys image with +no error message. One way to get around this is to replace the +AXIOMsys image with the debugsys image: +\begin{enumerate} +\item mv \$AXIOM/bin/AXIOMsys \$AXIOM/bin/AXIOMsys.backup\\ +This keeps the failing axiomsys image around for later restoration. +\item cp obj/sys/bin/debugsys \$AXIOM/bin/AXIOMsys\\ +This puts an interpreted version of axiom in place of the compiled form +\item axiom\\ +Now we are running a fully interpreted form and the error messages are +much more informative. +\end{enumerate} +\section{Understanding a compiled function} +Suppose we stop a program at a function call to some low level lisp function, +say ONEP. We can do that by entering +\begin{verbatim} +)trace ONEP )break +\end{verbatim} +at the Axiom command prompt. Or at the lisp prompt: +\begin{verbatim} +(trace (ONEP :entry (break))) +\end{verbatim} +Next we execute some function that will +eventually call ONEP thus: +\begin{verbatim} +p := numeric %pi + +Break: onep +Broken at ONEP. Type :H for Help. +BOOT>> +\end{verbatim} +We have stopped and entered a lisp command prompt. We can enter any +lisp expression here and there are commands that begin with a ``:'' +character. ``:b'' requests a backtrace of the call stack, thus: +\begin{verbatim} +BOOT>>:b +Backtrace: funcall > system:top-level > restart > /read > + |upLET| > eval > |Pi| > |newGoGet| > |newGoGet| > ONEP +\end{verbatim} +Here we see that the function ONEP was called by the function newGoGet. +Notice that the name is surrounded by vertical bars. Vertical bars are +a common lisp escape sequence used to allow non-standard characters to +occur in symbol names. Common lisp is not case sensitive. Boot code is +case sensitive. Thus symbol names that were written in Boot tend to have +escape sequence characters around the name. + +Now that we see the simple backtrace we can ask for a more complex one. +The command is ``:bt''. It shows more detail about each level of call +on the invocation history stack (ihs) including the function name, its +arguments and the depth of the invocation history stack ([ihs=13]): +\begin{verbatim} +BOOT>>:bt + +#0 ONEP {1=nil,} [ihs=13] +#1 newGoGet {g3629=("0" (# 45 . |char|)), + loc1=# 0 . |coerce|)), + loc1=(# 0 . |c...} [ihs=11] +#3 Pi {g109299=nil,loc1=nil,loc2=#, + loc3=|Pi|,loc4=15,loc5=#} [ihs=9] +#5 upLET {t=(# # + (# (#} [ihs=4] +BOOT>>:bl +>> (LAMBDA-BLOCK ONEP (&REST X) ...)(): +X : (1) +NIL +\end{verbatim} +We can ask to see the local variables that are used at the current +level of the invocation history stack. The command is ``:bl'' thus: +\begin{verbatim} +BOOT>>:bl +>> (LAMBDA-BLOCK ONEP (&REST X) ...)(): +X : (1) +NIL +\end{verbatim} +We can move up the stack one level at a time looking at the function +that called the current function (the previous function) using ``:p'' thus: +\begin{verbatim} +BOOT>>:p +Broken at |NEWGOGET|. +\end{verbatim} +And again, we can look at the variables that can be accessed locally: +\begin{verbatim} +BOOT>>:bl +>> newGoGet(): +Local0(G3629): (0 (# 45 . char)) +Local(1): # +Local(2): 0 +Local(3): # +Local(4): 1 +NIL +\end{verbatim} +Here we see that the function newGoGet is calling +CHAR;char;S\$;20 which is a mangled form of +the name of the original spad function. To decode this name +we can see that the CHAR portion is used to identify the +domain where the function lives. This domain, CHAR, comes +from the source file ``string.spad'' which ultimately lives in +``src/algebra/string.spad.pamphlet''. To discover this we use +the Axiom ``asq'' command with the ``-so'' (sourcefile) option +at a standard shell prompt (NOT in the lisp prompt) thus: +\begin{verbatim} +asq -so CHAR +string.spad +\end{verbatim} +If we look at the code in the string.spad.pamphlet file we find +the following code signature: +\begin{verbatim} + char: String -> % + ++ char(s) provides a character from a string s of length one. +\end{verbatim} +and it's implementation code: +\begin{verbatim} + char(s:String) == + (#s) = 1 => s(minIndex s) pretend % + error "String is not a single character" +\end{verbatim} +The string.spad file can be compiled at the command prompt. +In particular, we can compile only the CHAR domain out of this +file thus: +\begin{verbatim} +)co string.spad )con CHAR +\end{verbatim} +This will produce a directory called CHAR.NRLIB containing 3 files: +\begin{verbatim} +ls CHAR.NRLIB +code.lsp index.kaf info +\end{verbatim} +The info file contains information used by the spad compiler. We can +ignore it for now. + +The index.kaf file contains information that will go into the various +Axiom database (.daase) files. The kaf file format is a random access +file. The first entry is an integer that will be an index into the +file that can be used in an operating system call to seek. In this +case it will be an index which is the last used byte in the file. +Go to the last expression in the file and we find: +\begin{verbatim} +( + ("slot1Info" 0 11302) + ("documentation" 0 9179) + ("ancestors" 0 9036) + ("parents" 0 9010) + ("abbreviation" 0 9005) + ("predicates" 0 NIL) + ("attributes" 0 NIL) + ("signaturesAndLocals" 0 8156) + ("superDomain" 0 NIL) + ("operationAlist" 0 7207) + ("modemaps" 0 6037) + ("sourceFile" 0 5994) + ("constructorCategory" 0 5434) + ("constructorModemap" 0 4840) + ("constructorKind" 0 4831) + ("constructorForm" 0 4817) + ("NILADIC" 0 4768) + ("compilerInfo" 0 2093) + ("loadTimeStuff" 0 20)) +\end{verbatim} +This is a list of triples. Each triple has two interesting parts, the +name of the data and the seek index of the data in the index.kaf file. +So, for instance, if you want to know what source file contains this +domain you can start at the top of the index.kaf file, move ahead 5994 +bytes and you will be at the start of the string: +\begin{verbatim} +"/usr/local/axiom/src/algebra/string.spad" +\end{verbatim} +The information in the index.kaf files are collected into the +special databases (the .daase files). The stand-alone ``asq'' function +can query these databases and answer questions. The kind of questions +you can ask are the names in the list above. + +The third file in the CHAR.NRLIB directory is the code.lsp file. This +is the actual common lisp code that will be executed as a result of +calling the various spad functions. The spad code from the char command +was: +\begin{verbatim} + char(s:String) == + (#s) = 1 => s(minIndex s) pretend % + error "String is not a single character" +\end{verbatim} +which got compiled into the common lisp code: +\begin{verbatim} +(DEFUN |CHAR;char;S$;20| (|s| |$|) + (COND + ((EQL (QCSIZE |s|) 1) + (SPADCALL |s| + (SPADCALL |s| (QREFELT |$| 47)) + (QREFELT |$| 48))) + ((QUOTE T) + (|error| "String is not a single character")))) +\end{verbatim} +To understand what is going on here we need to understand the low level +details of Axiom's interface to Common Lisp. The ``Q'' functions are +strongly typed (Quick) versions of standard common lisp functions. +QCSIZE is defined in src/interp/vmlisp.lisp.pamphlet thus: +\begin{verbatim} +(defmacro qcsize (x) + `(the fixnum (length (the simple-string ,x)))) +\end{verbatim} +This macro will compute the length of a string. + +QREFELT is defined in the same file as: +\begin{verbatim} +(defmacro qrefelt (vec ind) + `(svref ,vec ,ind)) +\end{verbatim} +This macro will return the element of a vector. + +SPADCALL is defined in src/interp/macros.lisp.pamphlet as: +\begin{verbatim} +(defmacro SPADCALL (&rest L) + (let ((args (butlast l)) (fn (car (last l))) (gi (gensym))) + `(let ((,gi ,fn)) + (the (values t) (funcall (car ,gi) ,@args (cdr ,gi)))) + )) +\end{verbatim} +This macro will call the last value of the argument list as a function +and give it everything but the last argument as arguments to the function. +There are confusing historical reasons for this I won't go into here. + +So you can see that these are simply macros that will expand into +highly optimizable (the optimizations depend on the abilities of the +common lisp compiler) common lisp code. + +The common lisp code computes the length of the string s. If the length +is 1 then we call the minIndex function from string on s. The minIndex +function is found by looking "in the domain". The compiler changes the +minIndex function call into a reference into a vector. The 47th element +of the vector contains the function minIndex. +\begin{verbatim} + (SPADCALL |s| (QREFELT |$| 47)) +\end{verbatim} +This code is equivalent (ignoring the gensyms) to the call +\begin{verbatim} +(minIndex s) +\end{verbatim} +The \$ symbol refers to the domain. At runtime this amounts to a lookup +of the ``infovec''. The compile-time infovec shown here: +\begin{verbatim} +(MAKEPROP + (QUOTE |Character|) + (QUOTE |infovec|) + (LIST + (QUOTE + #(NIL + NIL + NIL + NIL + NIL + NIL + (QUOTE |Rep|) + (|List| 28) + (|PrimitiveArray| 28) + (0 . |construct|) + (QUOTE |OutChars|) + (QUOTE |minChar|) + (|Boolean|) + |CHAR;=;2$B;1| + |CHAR;<;2$B;2| + (|NonNegativeInteger|) + |CHAR;size;Nni;3| + (|Integer|) + |CHAR;char;I$;6| + (|PositiveInteger|) + |CHAR;index;Pi$;4| + |CHAR;ord;$I;7| + |CHAR;lookup;$Pi;5| + (5 . |coerce|) + |CHAR;random;$;8| + |CHAR;space;$;9| + |CHAR;quote;$;10| + |CHAR;escape;$;11| + (|OutputForm|) + |CHAR;coerce;$Of;12| + (|CharacterClass|) + (10 . |digit|) + (|Character|) + (14 . |member?|) + |CHAR;digit?;$B;13| + (20 . |hexDigit|) + |CHAR;hexDigit?;$B;14| + (24 . |upperCase|) + |CHAR;upperCase?;$B;15| + (28 . |lowerCase|) + |CHAR;lowerCase?;$B;16| + (32 . |alphabetic|) + |CHAR;alphabetic?;$B;17| + (36 . |alphanumeric|) + |CHAR;alphanumeric?;$B;18| + (|String|) + |CHAR;latex;$S;19| + (40 . |minIndex|) + (45 . |elt|) + |CHAR;char;S$;20| + |CHAR;upperCase;2$;21| + |CHAR;lowerCase;2$;22| + (|SingleInteger|))) + (QUOTE + #(|~=| 51 |upperCase?| 57 |upperCase| 62 |space| 67 + |size| 71 |random| 75 |quote| 79 |ord| 83 |min| 88 + |max| 94 |lowerCase?| 100 |lowerCase| 105 |lookup| 110 + |latex| 115 |index| 120 |hexDigit?| 125 |hash| 130 + |escape| 135 |digit?| 139 |coerce| 144 |char| 149 + |alphanumeric?| 159 |alphabetic?| 164 |>=| 169 |>| 175 + |=| 181 |<=| 187 |<| 193)) + (QUOTE NIL) + (CONS + (|makeByteWordVec2| 1 (QUOTE (0 0 0 0 0 0))) + (CONS + (QUOTE #(NIL |OrderedSet&| NIL |SetCategory&| + |BasicType&| NIL)) + (CONS + (QUOTE + #((|OrderedFinite|) + (|OrderedSet|) + (|Finite|) + (|SetCategory|) + (|BasicType|) + (|CoercibleTo| 28))) + (|makeByteWordVec2| 52 + (QUOTE + (1 8 0 7 9 1 6 0 17 23 0 30 0 31 2 30 12 32 0 33 + 0 30 0 35 0 30 0 37 0 30 0 39 0 30 0 41 0 30 0 + 43 1 45 17 0 47 2 45 32 0 17 48 2 0 12 0 0 1 1 + 0 12 0 38 1 0 0 0 50 0 0 0 25 0 0 15 16 0 0 0 24 + 0 0 0 26 1 0 17 0 21 2 0 0 0 0 1 2 0 0 0 0 1 1 0 + 12 0 40 1 0 0 0 51 1 0 19 0 22 1 0 45 0 46 1 0 0 + 19 20 1 0 12 0 36 1 0 52 0 1 0 0 0 27 1 0 12 0 34 + 1 0 28 0 29 1 0 0 45 49 1 0 0 17 18 1 0 12 0 44 1 + 0 12 0 42 2 0 12 0 0 1 2 0 12 0 0 1 2 0 12 0 0 13 + 2 0 12 0 0 1 2 0 12 0 0 14)))))) + (QUOTE |lookupComplete|))) +\end{verbatim} +Which is a 5 element list. This contains all kinds of information used at +runtime by the compiled routines. In particular, functions are looked up +at runtime in the first element of the infovec list. This first element +contains 53 items (in this domain). Item 47 is +\begin{verbatim} + (40 . |minIndex|) +\end{verbatim} +which is the minIndex function we seek. + +At runtime this infovec lives on the property list of the domain name. +The domain name of CHAR is Character. So we look on the property list +(a lisp a-list) thus: +\begin{verbatim} +BOOT>>(symbol-plist '|Character|) + +(SYSTEM:DEBUG (#:G85875) + |infovec| (# + # + NIL + (# + # + # . #) + |lookupComplete|) + LOADED "/home/axiomgnu/new/mnt/linux/algebra/CHAR.o" + NILADIC T + PNAME "Character" + DATABASE #S(DATABASE + ABBREVIATION CHAR + ANCESTORS NIL + CONSTRUCTOR NIL + CONSTRUCTORCATEGORY 228064 + CONSTRUCTORKIND |domain| + CONSTRUCTORMODEMAP 227069 + COSIG (NIL) + DEFAULTDOMAIN NIL + MODEMAPS 227404 + NILADIC T + OBJECT "CHAR" + OPERATIONALIST 226402 + DOCUMENTATION 152634 + CONSTRUCTORFORM 152626 + ATTRIBUTES 154726 + PREDICATES 154731 + SOURCEFILE "string.spad" + PARENTS NIL + USERS NIL + DEPENDENTS NIL + SPARE NIL)) +\end{verbatim} +This list is organized contains many runtime lookup items (notice the +PNAME entry is ``Character'', the LOADED entry says where the file came +from, the DATABASE structure entry has database indicies +(see daase.lisp.pamphlet for the structure definition), etc). + +Lets get the property list +\begin{verbatim} +BOOT>>(setq a (symbol-plist '|Character|)) + +(SYSTEM:DEBUG (#:G85875) + |infovec| (# + # + NIL + (# + # + # . #) + |lookupComplete|) + LOADED "/home/axiomgnu/new/mnt/linux/algebra/CHAR.o" + NILADIC T + PNAME "Character" + DATABASE #S(DATABASE + ABBREVIATION CHAR + ANCESTORS NIL + CONSTRUCTOR NIL + CONSTRUCTORCATEGORY 228064 + CONSTRUCTORKIND |domain| + CONSTRUCTORMODEMAP 227069 + COSIG (NIL) + DEFAULTDOMAIN NIL + MODEMAPS 227404 + NILADIC T + OBJECT "CHAR" + OPERATIONALIST 226402 + DOCUMENTATION 152634 + CONSTRUCTORFORM 152626 + ATTRIBUTES 154726 + PREDICATES 154731 + SOURCEFILE "string.spad" + PARENTS NIL + USERS NIL + DEPENDENTS NIL + SPARE NIL)) +\end{verbatim} +Next we get the infovec value +\begin{verbatim} +BOOT>>(setq b (fourth a)) + +(# + # + NIL + (# + # + # . #) + |lookupComplete|) +\end{verbatim} +Then we get the function table +\begin{verbatim} +BOOT>>(setq c (car b)) + +# +\end{verbatim} +In this common lisp (GCL) the array is identified by it's memory address. + +Notice that it has the right number of entries: +\begin{verbatim} +BOOT>>(length c) + +53 +\end{verbatim} +And we can ask for the 47th entry thus: +\begin{verbatim} +BOOT>>(elt c 47) + +(40 . |minIndex|) +\end{verbatim} +Later we end up calling the 48th function (which is elt and +returns the actual character in the string). We ask for it: +\begin{verbatim} +BOOT>>(elt c 48) + +(45 . |elt|) +\end{verbatim} +At this point we've reached the metal. Common lisp will evaluate +the macro-expanded functions and execute the proper code. Essentially +the compiler has changed all of our spad code into runtime table lookups. + +\section{The axiom.input startup file} +If you add a file in your home directory called ``.axiom.input'' it will +be read and executed when Axiom starts. This is useful for various +reasons including setting various switches. Mine reads: +\begin{verbatim} +)lisp (pprint ``running /root/.axiom.input'') +)set quit unprotected +)set message autoload off +)set message startup off +\end{verbatim} +You can execute any command in .axiom.input. Be aware that this will +ALSO be run while you are doing a ``make'' so be careful what you ask to do. + +\section{Where are Axiom symbols stored?} + +You'd think that your question about where the symbol is interned +would be easy to answer but it is not. The top level loop uses Bill +Burge's dreaded zipper parser. You can see it in action by executing +the following sequence: +\begin{verbatim} + +)lisp (setq $DALYMODE t) + ; this is a special mode of the top level interpreter. If + ; $DALYMODE is true then any top-level form that begins + ; with an open-paren is considered a lisp expression. + ; For almost everything I ever do I end up peeking at the + ; lisp so this bit of magic helps. +(trace |intloopProcessString|) + ; from int-top.boot.pamphlet +(trace |intloopProcess|) + ; the third argument is the "zippered" input +(trace |intloopSpadProcess|) + ; now it is all clear, no? sigh. +(trace |phInterpret|) + ; from int-top.boot.pamphlet +(trace |intInterpretPform|) + ; from intint.lisp.pamphlet +(trace |processInteractive|) + ; from i-toplev.boot.pamphlet +(setq |$reportInstantiations| t) + ; shows what domains were created +(setq |$monitorNewWorld| t) + ; watch the interpreter resolve operations +(trace |processInteractive1|) + ; from i-toplev.boot.pamphlet + +\end{verbatim} +ah HA! I remember now. There is the notion of a "frame" which is +basically a namespace in Axiom or an alist in Common Lisp. It is +possible to maintain different "frames" and move among them. There +is the notion of the current frame and it contains all the defined +variables. At any given time the current frame is available as +\$InteractiveFrame. This variable is used in processInteractive1. +If you do: +\begin{verbatim} + +a:=7 +(pprint |$InteractiveFrame|) + +\end{verbatim} +you'll see |a| show up on the alist. When you do the +\begin{verbatim} + +pgr:=MonoidRing(Polynomial PrimeField 5, Permutation Integer) +p:pgr:=1 + +\end{verbatim} +you'll see |p| show up with 2 other things: (|p| mode value) +where mode is the "type" of the variable. The value is the +internal value. In this case MonoidRing has an internal +representation. You can find out what the internal representation +of a MonoidRing is by first asking where the source file is: +\begin{verbatim} + +(do this at a shell prompt, not in axiom) +asq -so MonoidRing ==> mring.spad + + -- or -- in Axiom type: + +)show MonoidRing + +\end{verbatim} +and you'll see a line that reads: +\begin{verbatim} + +Issue )edit (yourpath)/../../src/algebra/mring.spad + +\end{verbatim} + +If you look in mring.spad.pamphlet you'll see line 91 that reads: +\begin{verbatim} + + Rep := List Term + +\end{verbatim} +which says that we will store elements of type MonoidRing as a list +of Term objects. Term is defined in the same file (as a macro, which +is what '$==>$' means in spad files) on line 43: +\begin{verbatim} + + Term ==> Record(coef: R, monom: M) + +\end{verbatim} +which means that elements of a MonoidRing are Lists of Records. +The 'R' is defined on line 42 as the first argument to MonoidRing +which in this case is ``Polynomial PrimeField 5''. The ``M'' is also +defined on line 42 as the second argument to MonoidRing and in this +case is ``Permutation Integer''. So the real representation is +\begin{verbatim} + + List Record(coef: Polynomial PrimeField 5, + monom: Permutation Integer) + +\end{verbatim} +In the \$InteractiveFrame we printed out you can see in the value +field that the value is: +\begin{verbatim} + +(|value| + (|MonoidRing| (|Polynomial| (|PrimeField| 5)) + (|Permutation| (|Integer|))) + WRAPPED ((0 . 1) . #)) + +\end{verbatim} +which basically means that we know how the MonoidRing was constructed and +what it's current value is. The (0 . 1) likely means that this is the +zeroth (constant) term with a leading coefficient of 1. This is just a +guess as I haven't decoded the representation of either Polynomial PrimeField +or Permutation Integer. You can do the same deconstruction of these two +domains by setting +\begin{verbatim} + +pi:=Permutation Integer +z:pi:=1 + +pp5:=Polynomial PrimeField 5 +w:pp5:=1 + +and following the same steps as above: + (pprint |$InteractiveFrame|) + )show pi + (find the source file) + (find the representation and decode it) + + (pprint |$InteractiveFrame|) + )show pp5 + (find the source file) + (find the representation and decode it) + +\end{verbatim} +Be sure to set \$DALYMODE to nil if you plan to use Axiom for any +real computation. Otherwise every expression that begins with an +open-paren will go directly to lisp. +\section{Translating individual boot files to common lisp} + +If you are making changes to boot code it is sometimes helpful to +check the generated lisp code to ensure it does what you want. +You can convert an individual boot file to common lisp using the +boottran::boottocl function: +\begin{verbatim} +)fin -- drop into common lisp +(boottran::boottocl "foo.boot") +\end{verbatim} +when you do this it creates a foo.clisp file in ../../int/interp + +Alternatively if you work from the pamphlet file the process is +more painful as you have to do +\begin{verbatim} +)cd (yourpath)/int/interp +)sys tangle ../../src/interp/foo.boot.pamphlet >foo.boot +)fin +(boottran::boottocl "foo.boot") +(restart) +\end{verbatim} +The )cd step tells axiom to cd to the int/interp subdirectory. +The )sys tangle... extracts the boot file from the pamphlet file +The )fin step drops into common lisp +The (bootran... converts the foo.boot file to foo.clisp +The (restart) re-enters the top level loop + +\section{Directories} +For this discussion I assume that you have your system rooted +at /spad and was build to run on linux. These directories +may not yet be in the CVS tree but are documented here so they +make sense when the show up. + +\vskip .25in +\noindent +The AXIOM variable + +The usual setting of the AXIOM variable is /spad/mnt/linux. The +name is composed of three parts, the rooted path, in this case +/spad, ``mnt'', and the system you are running, in this case +linux. Builds for other systems will have other system names. + +\vskip .25in +\noindent +/spad + +This is the usual root directory of the Axiom system. The name is +historical, a contraction of Scratchpad. This name can be anything +provided the shell variable AXIOM contains the new prefix. + +\vskip .25in +\noindent +/spad/mnt + +This is a directory which contains files which are specific to a +given platform. At a site that contains multiple platforms this +directory will contain a subdirectory for each type of platform +(e.g. linux, rios, ps2, rt, sun, etc). + +\vskip .25in +\noindent +/spad/mnt/linux + +This directory contains the complete copy of the Axiom system for +the linux system. This is the 'mount point' of the system. +Executable systems (for RedHat) are shipped relative to this +point. In what follows, the ./ refers to /spad/mnt/linux. + +\begin{verbatim} +******************************************************** +There are several directories explained below. They are: + +./bin -- user executables +./doc -- system documentation +./algebra -- algebra libraries +./lib -- system executables +./etc -- I haven't a clue.... +******************************************************** +\end{verbatim} + +\subsection{The mnt/linux/bin directory} +\vskip .25in +\noindent +./bin + +This is a directory of user executable commands, either at the top +level or thru certain Axiom system calls. Support executables live +in ./lib + +\vskip .25in +\noindent +./bin/htadd + +This function adds pages to the Hyperdoc database (ht.db, which lives +in ./doc/hypertex/pages; hypertex, since we have a penchant for these +things, is an historical name for Hyperdoc. The single word 'lawyers' +will probably explain away a lot of name changes.) + +\vskip .25in +\noindent +./bin/spadsys + +This is the Axiom interpreter. It is one of the functions started when +the user invokes the system using the spadsys command. Normally this +command is run under the control of sman (./lib/sman) and the console +is under the control of clef (./bin/clef), the wonderous command-line +editor. It is possible to start spadsys standalone but it will not +talk to Hyperdoc or graphics. Users who rlogin or use an ascii-only +terminal (for historical reasons, no doubt) can profit by invoking +spadsys directly rather than using ./bin/axiom + +\vskip .25in +\noindent +./bin/axiom + +This is a shell script that spins the world. It kicks off a whole tree +of processes necessary to perform the X-related magic we do. It expects +the shell variable AXIOM to be set to the 'mount point' (usually to +/spad/mnt/linux). + +\vskip .25in +\noindent +./bin/clef + +This is the wonderous command-line editor used by Axiom. It can be used +in a stand-alone fashion if you wish. + +\vskip .25in +\noindent +./bin/SPADEDFN + +This script is invoked by the spad )fe command. It can be changed to +invoke your favorite editor. While you may invoke your editor, it may +not run (as in, yes, I can invoke the devil but will he come when I +call?) + +\vskip .25in +\noindent +./bin/viewalone + +This is a function to run the graphics in a stand-alone fashion. The +Graphics package (an amazing contribution by several very talented +people, most notably Jim Wen and Jon Steinbach) is a C program that +communicates with Axiom thru sockets. It will, however, perform its +miracles unaided if invoked by the sufficiently chaste... + +\vskip .25in +\noindent +./bin/hypertex + +This is a function to run Hyperdoc (remember the penchant!) stand-alone. +The Hyperdoc package owes its existence to the efforts of J.M. Wiley and +Scott Morrison. This function works off 'pages' that live in +hypertex pages directory and are referenced in the ``hyperdoc database'' +called ht.db (for historical reasons, but you +knew that). It is possible for creative plagerists to figure out how +to write their own pages and add them to the database (see htadd above), +thus gaining fame far and wide... + +\vskip .25in +\noindent +./bin/sys-init.lsp + +This is a file of lisp code that gets loaded +before Axiom starts. Thus, we distribute patches by adding lisp +(load ...) commands to this file. The sufficiently clever should +have a field day with this one. (All others should worship the +sufficiently clever and send them money, eh?) + +\vskip .25in +\noindent +./bin/init.lsp + +This is a file of lisp code loaded if and only if +you start spadsys in this directory. The user can put a file of this +name in her home directory and it will get loaded at startup with +the probable effect of injecting luser errors into the running system. +sigh. + +\subsection{The mnt/linux/doc directory} +\vskip .25in +\noindent +./doc + +The doc subdirectory contains system documentation. + +\vskip .25in +\noindent +./doc/command.list + +This is a file of command completions used by clef when you hit the +tab key. This is a little known feature that will surprise someone +someday (hopefully pleasantly). + +\vskip .25in +\noindent +./doc/book + +This is an attempt at a book describing Axiom. It represents a +combination of fantasy, describing what never will be and history +(remember the penchant?) describing what was. Any description matching +what is may be regarded as failure of the imagination and ignored. + +\vskip .25in +\noindent +./doc/compguide + +This is an attempt to describe a compiler that doesn't exist, never +did exist, and never will exist. It makes for entertaining reading +so we included it. + +\vskip .25in +\noindent +./doc/hypertex + +This is the fabled Hyperdoc subdirectory where all of the pages and +the database live, along with several other obscure files needed to +make the wizards look good. + +\vskip .25in +\noindent +./doc/hypertex/pages + +This is where the 'pages' live. Each file ending in .ht contains +several pages related, if only by chance, to the same topic. You +may find it instructive to try to read some of these files. Hyperdoc +was learned by the 'campfire' method (sitting around the fire +passing along historical facts by word of mouth) and will probably +continue to propagate by the same method. Ye may become th' local +scribe and soothsayer if ye study the writings here below.... + +\vskip .25in +\noindent +./doc/hypertex/pages/rootpage.ht + +This file is the magic 'first page' that gets displayed when +Hyperdoc starts. There is a macro (see ./doc/hypertex/pages/util.ht) +called /localinfo which is intended to allow the luser to add +her own pages without modifying the system copies. How this is +done was lost when the campfire got rained out. + +\vskip .25in +\noindent +./doc/hypertex/pages/util.ht + +This file contains the macros used to extend the system commands. +The syntax is hard to learn (it was hard to write, it ought to +be hard to learn, eh?). + +\vskip .25in +\noindent +./doc/hypertex/pages/ht.db + +This is the Hyperdoc database. It is updated using ./bin/htadd +which must be run whenever a page in this directory gets changed. +The necessary arguments to htadd are obvious to those in the know. + +\vskip .25in +\noindent +./doc/hypertex/bitmaps + +There are several pretty bitmaps used as cursors, buttons and +general decorations that hide in this directory. + +\vskip .25in +\noindent +./doc/hypertex/ht.files + +This is a list of some Hyperdoc files. It seems to have no +purpose in life but it is useful as a koan, as in, What is +the length of half a list? + +\vskip .25in +\noindent +./doc/hypertex/ht.db + +Another copy of the Hyperdoc database. It isn't clear which +one is the real one so I guess we keep both. Maybe we'll +figure it out at the friday night campfire provided we don't +get too lit. + +\vskip .25in +\noindent +./doc/hypertex/gloss.text + +The text used in the glossary. Many magic words lie herein. +Some are spoken only by campfire gurus. + +\vskip .25in +\noindent +./doc/library + +This is a directory of Hyperdoc pages that can be freely smashed, +trashed and generally played with. It uses the /localinfo connection +to set up a 'library' containing Hyperdoc pages keyed to your +favorite textbook. It is interesting to set the shell variable\\ +HTPATH=/spad/mnt/linux/doc/library:\\ + /spad/mnt/linux/doc/hypertex/pages\\ +and then start Hyperdoc. See the file ./doc/library/macros.ht + +\vskip .25in +\noindent +./doc/msgs + +This directory contains several 'message databases'; the only one +of which we seem to care about being s2-us.msgs but I can't swear +to it. + +\vskip .25in +\noindent +./doc/spadhelp + +This is a directory containing help information for a copy of the +system that once ran long ago and far away. It is kept for +historical reasons (programmers NEVER throw anything away). + +\vskip .25in +\noindent +./doc/viewports + +There are several dozen truly fine pictures in Axiom. We have +created them and hidden them here. Hyperdoc will insert them +at various places (where the text gets too boring, hopefully) +and you can click on them there. They get snarfed from here. +It is possible to view them with stand-alone graphics but don't +ask me how. I missed that campfire due to poisoned marshmellows. + +\vskip .25in +\noindent +./doc/complang + +This directory contains fantasy from the past as opposed to facts +from the future. Ignore it. + +\vskip .25in +\noindent +./doc/ug + +This directory left intentionally blank :-) (an old IBM joke). + +\vskip .25in +\noindent +./doc/tex + +These are the files necessary to create the famous goertler document. +If you figure hou how to use these please send us the instructions +and we will add a log to the campfire with your name on it (a rare +honor indeed as luser's names rarely reach the inner circle). + +\vskip .25in +\noindent +./doc/htex + +This directory contains the original tex-like source for the luser's +guide. There are many functions that munch on these between here and +paper but this is approximately where they start. If you do your own +algebra perchance you might document it like this. Figuring out the +syntax will also get your name into the inner circle (probably +connnected with a smirk :-) ) + +\vskip .25in +\noindent +./doc/newug + +Please don't ask me. I couldn't begin to guess. You wouldn't believe +how many 'new' things there are that really aren't. We have more NEW +things than Madison Avenue has NEW laundry soap. + +\vskip .25in +\noindent +./doc/gloss.text + +This one is here because it is here. Existentially speaking, of course. + +\vskip .25in +\noindent +./doc/submitted + +This was what the htex files said before history was rewritten... +(and renamed?) + +\subsection{The mnt/linux/algebra directory} +\vskip .25in +\noindent +./algebra + +This is where all of the interesting action lives. +Each .NRLIB directory contains 2 files, a code.o and an index.kaf* +file. The code.o contains the executable algebra that gets loaded +into the system. The index.kaf* file contains all kinds of things +like signatures, source paths, properties and dried bat droppings. +The documentation for each of these can be reached by using the +BROWSE feature of Hyperdoc. + +\vskip .25in +\noindent +./algebra/MODEMAP.daase + +This is an inverted database that contains information gleaned from +the index.kaf* files. Without this there is no way to figure out +which .NRLIB file to load. This database is opened on startup +and kept open. + +\vskip .25in +\noindent +./algebra/interp.exposed + +This is a control file for the interpeter that limits the number +of places to search for function names. + +********************************************* + +\subsection{The mnt/linux/lib directory} +\vskip .25in +\noindent +./lib + +This directory contains functions that get loaded by the system. +Nothing in here is executable by the user but the system needs +these functions to run. + +\vskip .25in +\noindent +./lib/htrefs\\ +./lib/htsearch\\ +./lib/hthits + +These three functions are used to search the Hyperdoc pages. +There is no way in the current system to request a search of +those pages so these files are fascinating examples of history +in the making... + +\vskip .25in +\noindent +./lib/hypertex + +This is Hyperdoc. What is in a name? + +\vskip .25in +\noindent +./lib/sman + +This is sman, which comes before all. Methinks the name originated +as a contraction of superman, the name of a stack frame in a system +long ago and far away (VMLisp) chosen because a certain programmer had a +penchant for comic books when he was young. + +\vskip .25in +\noindent +./lib/session\\ +./lib/spadclient + +These two files are processes started by sman for some reason or other. +I can never remember what they do or why. However, the campfire fails +to smoke if they don't work. + +\vskip .25in +\noindent +./lib/viewman + +This is the controlling function for the graphics. + +\vskip .25in +\noindent +./lib/view2d + +This is invoked when a 2 dimensional window is requested. +This is provided mostly for those math majors who never got over +the insights from flatland. + +\vskip .25in +\noindent +./lib/view3d + +This is invoked when a 3 dimensional window is requested. +Option IBM3634-A is required to convert your 2 dimensional +screen to 3 dimensions for realistic viewing. A mathematically +accurate, if somewhat more achievable, rendering can be had +on a color or monochrome crt without this upgrade. + +\vskip .25in +\noindent +./lib/gloss.text\\ +./lib/glosskey.text\\ +./lib/glossdef.text + +These are three files related to the glossary. The first (gloss.text) +is the original glossary text. The second (glosskey.text) is a list +of terms and pointers into glossdef.text. The third (glossdef.text for +those math majors who can't count) is a list of definitions and pointers +back into the second (guess). These files are used by Hyperdoc. + +\vskip .25in +\noindent +./lib/browsedb.lisp + +This is the original file that creates an in-memory hash table used by +browse. It is used during system build time. We keep it here to ensure +that the bytes on this section of the disk have a well-defined orientation, +allowing us to compute the spin vectors of the individual magnetic domains. +This allows us to give Heisenburg a sense of direction (at least over the +long run). + +\vskip .25in +\noindent +./lib/comdb.text\\ +./lib/libdb.text + +The first file (comdb.text) contains the so-called $++$ (plus +plus) comments from the algebra files. It contains pointers into +the second file. The second file (libdb.text) contains flags +(constructor, operation, attribute) and pointers into the first file. +These files are used by browse in Hyperdoc. + +\vskip .25in +\noindent +./lib/loadmprotect\\ +./lib/mprotect + +This set of two files has been mercifully de-installed from the system. +They will, if used and despite the meaning behind the name, cause +random system reboots (yeah, HARDWARE reboots. don't ask me how, I'm +just the historian). + +\vskip .25in +\noindent +./lib/SPADEDIT\\ +./lib/fc\\ +./lib/spadbuf\\ +./lib/SPADEDFN\\ +./lib/obey\\ +./lib/ex2ht + +I've drawn a blank; intentionally. + +\subsection{The mnt/linux/lib directory} +\vskip .25in +\noindent +./etc + +This directory intentionally left blank. +We just can't figure out WHY we intended to leave it blank. +Historical reasons, no doubt. + +\section{The )set command} +The {\bf )set} command contains many possible options such as: +\begin{verbatim} + Current Values of )set Variables + +Variable Description Current Value +---------------------------------------------------------------- +breakmode execute break processing on error break +compiler Library compiler options ... +expose control interpreter constructor exposure ... +functions some interpreter function options ... +fortran view and set options for FORTRAN output ... +kernel library functions built into the kernel for + efficiency ... +hyperdoc options in using HyperDoc ... +help view and set some help options ... +history save workspace values in a history file on +messages show messages for various system features ... +naglink options for NAGLink ... +output view and set some output options ... +quit protected or unprotected quit unprotected +streams set some options for working with streams ... +system set some system development variables ... +userlevel operation access level of system user development + +Variables with current values of ... have further +sub-options. For example, +issue )set system to see what the options are for system . +For more information, issue )help set . +\end{verbatim} +The table that contains these options lives in setvart.boot.pamphlet. +The actual code that implements these options is sprinkled around +but most of the first-level calls resolve to functions in +setvars.boot.pamphlet. Thus if you plan to add a new output +style to the system, or figure out where a current style is +broken, these two files are the place to start. + +\section{Special Output Formats} +The first level of special output formatting is handled by +functions in setvart.boot.pamphlet. This handles the options +given to the )set command. + +\section{Low Level Debugging Techniques} +It should be observed that Axiom is basically Common Lisp and +some very low level techniques can be used to find where problems +occur in algebra code. This section walks thru a small problem and +illustrates some techniques that can be used to find bugs. The +point of this exercise is to show a few techniques, not to show a +general method. +\subsection{The example bug} +Axiom can generate TeX output by typing: +\begin{verbatim} +)set output tex on +\end{verbatim} + +Here we give an example of TeX output that contains a bug: +\begin{verbatim} +(1) -> )set output tex on +(1) -> radix(10**10,32) + Loading /axiom/mnt/linux/algebra/RADUTIL.o for package + RadixUtilities + Loading /axiom/mnt/linux/algebra/RADIX.o + for domain RadixExpansion + Loading /axiom/mnt/linux/algebra/ANY1.o + for package AnyFunctions1 + Loading /axiom/mnt/linux/algebra/NONE1.o + for package NoneFunctions1 + Loading /axiom/mnt/linux/algebra/ANY.o + for domain Any + Loading /axiom/mnt/linux/algebra/SEX.o + for domain SExpression + + (1) 9A0NP00 + Loading /axiom/mnt/linux/algebra/TEX.o + for domain TexFormat + Loading /axiom/mnt/linux/algebra/CCLASS.o + for domain CharacterClass + Loading /axiom/mnt/linux/algebra/IBITS.o + for domain IndexedBits + Loading /axiom/mnt/linux/algebra/UNISEG.o + for domain UniversalSegment +$$ +9#\A0#\N#\P00 +\leqno(1) +$$ + Loading /axiom/mnt/linux/algebra/VOID.o for domain Void + + Type: RadixExpansion 32 +\end{verbatim} +The correct output should be: +\begin{verbatim} +$$ +9A0NP00 +\leqno(1) +$$ +\end{verbatim} +So we need to figure out where the \# prefixes are being generated. +In the above code we can see various domains being loaded. These domains +are lisp code. Each domain lives in a subdirectory of its own. For example, +the ANY domain lives in ANY.NRLIB. The ANY.NRLIB directory contains a +common lisp file named code.lsp. The compiled form of this code ANY.o +is loaded whenever the domain Any is referenced. We can look at the lisp +code: +\begin{verbatim} + +(/VERSIONCHECK 2) + +(PUT (QUOTE |ANY;obj;$N;1|) + (QUOTE |SPADreplace|) + (QUOTE QCDR)) + +(DEFUN |ANY;obj;$N;1| (|x| $) (QCDR |x|)) + +(PUT (QUOTE |ANY;dom;$Se;2|) + (QUOTE |SPADreplace|) + (QUOTE QCAR)) + +(DEFUN |ANY;dom;$Se;2| (|x| $) (QCAR |x|)) + +(PUT (QUOTE |ANY;domainOf;$Of;3|) + (QUOTE |SPADreplace|) + (QUOTE QCAR)) + +(DEFUN |ANY;domainOf;$Of;3| (|x| $) (QCAR |x|)) + +(DEFUN |ANY;=;2$B;4| (|x| |y| $) + (COND + ((SPADCALL (QCAR |x|) (QCAR |y|) (QREFELT $ 17)) + (EQ (QCDR |x|) (QCDR |y|))) + ((QUOTE T) (QUOTE NIL)))) + +(DEFUN |ANY;objectOf;$Of;5| (|x| $) + (|spad2BootCoerce| + (QCDR |x|) + (QCAR |x|) + (SPADCALL + (SPADCALL "OutputForm" (QREFELT $ 21)) + (QREFELT $ 23)))) + +(DEFUN |ANY;showTypeInOutput;BS;6| (|b| $) + (SEQ + (SETELT $ 10 (SPADCALL |b| (QREFELT $ 9))) + (EXIT + (COND + (|b| "Type of object will be displayed in + output of a member of Any") + ((QUOTE T) "Type of object will not be displayed in + output of a member of Any"))))) + +(DEFUN |ANY;coerce;$Of;7| (|x| $) + (PROG (|obj1| |p| |dom1| #0=#:G1426 |a| #1=#:G1427) + (RETURN + (SEQ + (LETT |obj1| + (SPADCALL |x| (QREFELT $ 24)) + |ANY;coerce;$Of;7|) + (COND + ((NULL (SPADCALL (QREFELT $ 10) (QREFELT $ 26))) + (EXIT |obj1|))) + (LETT |dom1| + (SEQ + (LETT |p| + (|prefix2String| (|devaluate| (QCAR |x|))) + |ANY;coerce;$Of;7|) + (EXIT + (COND + ((SPADCALL |p| (QREFELT $ 27)) + (SPADCALL |p| (QREFELT $ 23))) + ((QUOTE T) (SPADCALL |p| (QREFELT $ 29)))))) + |ANY;coerce;$Of;7|) + (EXIT + (SPADCALL + (CONS |obj1| + (CONS ":" + (PROGN + (LETT #0# NIL |ANY;coerce;$Of;7|) + (SEQ + (LETT |a| NIL |ANY;coerce;$Of;7|) + (LETT #1# |dom1| |ANY;coerce;$Of;7|) + G190 + (COND + ((OR (ATOM #1#) + (PROGN + (LETT |a| (CAR #1#) |ANY;coerce;$Of;7|) + NIL)) + (GO G191))) + (SEQ + (EXIT + (LETT #0# + (CONS + (SPADCALL |a| (QREFELT $ 30)) + #0#) + |ANY;coerce;$Of;7|))) + (LETT #1# (CDR #1#) |ANY;coerce;$Of;7|) + (GO G190) + G191 + (EXIT (NREVERSE0 #0#)))))) + (QREFELT $ 31))))))) + +(DEFUN |ANY;any;SeN$;8| (|domain| |object| $) + (SEQ + (COND + ((|isValidType| |domain|) (CONS |domain| |object|)) + ((QUOTE T) + (SEQ + (LETT |domain| (|devaluate| |domain|) |ANY;any;SeN$;8|) + (EXIT + (COND + ((|isValidType| |domain|) (CONS |domain| |object|)) + ((QUOTE T) + (|error| +"function any must have a domain as first argument"))))))))) + +(DEFUN |Any| NIL + (PROG NIL + (RETURN + (PROG (#0=#:G1432) + (RETURN + (COND + ((LETT #0# + (HGET |$ConstructorCache| (QUOTE |Any|)) + |Any|) + (|CDRwithIncrement| (CDAR #0#))) + ((QUOTE T) + (UNWIND-PROTECT + (PROG1 + (CDDAR + (HPUT |$ConstructorCache| + (QUOTE |Any|) + (LIST (CONS NIL (CONS 1 (|Any;|)))))) + (LETT #0# T |Any|)) + (COND + ((NOT #0#) + (HREM |$ConstructorCache| (QUOTE |Any|)))))))))))) + +(DEFUN |Any;| NIL + (PROG (|dv$| $ |pv$|) + (RETURN + (PROGN + (LETT |dv$| (QUOTE (|Any|)) . #0=(|Any|)) + (LETT $ (GETREFV 35) . #0#) + (QSETREFV $ 0 |dv$|) + (QSETREFV $ 3 (LETT |pv$| (|buildPredVector| 0 0 NIL) . #0#)) + (|haddProp| |$ConstructorCache| (QUOTE |Any|) NIL (CONS 1 $)) + (|stuffDomainSlots| $) + (QSETREFV $ 6 + (|Record| (|:| |dm| (|SExpression|)) (|:| |ob| (|None|)))) + (QSETREFV $ 10 (SPADCALL (QUOTE NIL) (QREFELT $ 9))) + $)))) + +(MAKEPROP (QUOTE |Any|) (QUOTE |infovec|) + (LIST + (QUOTE + #(NIL NIL NIL NIL NIL NIL (QUOTE |Rep|) + (|Boolean|) (|Reference| 7) (0 . |ref|) + (QUOTE |printTypeInOutputP|) (|None|) + |ANY;obj;$N;1| (|SExpression|) |ANY;dom;$Se;2| + (|OutputForm|) |ANY;domainOf;$Of;3| (5 . =) + |ANY;=;2$B;4| (|String|) (|Symbol|) (11 . |coerce|) + (|List| 20) (16 . |list|) |ANY;objectOf;$Of;5| + |ANY;showTypeInOutput;BS;6| (21 . |deref|) + (26 . |atom?|) (|List| $) (31 . |list|) + (36 . |coerce|) (41 . |hconcat|) |ANY;coerce;$Of;7| + |ANY;any;SeN$;8| (|SingleInteger|))) + (QUOTE #(~= 46 |showTypeInOutput| 52 |objectOf| 57 |obj| + 62 |latex| 67 |hash| 72 |domainOf| 77 |dom| 82 + |coerce| 87 |any| 92 = 98)) + (QUOTE NIL) + (CONS (|makeByteWordVec2| 1 (QUOTE (0 0 0))) + (CONS (QUOTE #(|SetCategory&| |BasicType&| NIL)) + (CONS + (QUOTE #((|SetCategory|) (|BasicType|) (|CoercibleTo| 15))) + (|makeByteWordVec2| 34 + (QUOTE (1 8 0 7 9 2 13 7 0 0 17 1 20 0 19 21 1 22 0 20 + 23 1 8 7 0 26 1 13 7 0 27 1 20 28 0 29 1 20 15 + 0 30 1 15 0 28 31 2 0 7 0 0 1 1 0 19 7 25 1 0 + 15 0 24 1 0 11 0 12 1 0 19 0 1 1 0 34 0 1 1 0 + 15 0 16 1 0 13 0 14 1 0 15 0 32 2 0 0 13 11 33 + 2 0 7 0 0 18)))))) + (QUOTE |lookupComplete|))) + +(MAKEPROP (QUOTE |Any|) (QUOTE NILADIC) T) +\end{verbatim} +We can ignore this information and focus on the functions that are +defined in this file. These functions can be traced with the usual +common lisp tracing facility. So lets create a file /tmp/debug.lisp +that contains a trace statement for each DEFUN in ANY.NRLIB/code.lsp. +It looks like: +\begin{verbatim} +(trace |ANY1;retractable?;AB;1|) +(trace |ANY1;coerce;SA;2|) +(trace |ANY1;retractIfCan;AU;3|) +(trace |ANY1;retract;AS;4|) +(trace |AnyFunctions1|) +(trace |AnyFunctions1;|) +\end{verbatim} +We can now restart the axiom system, rerun the failing expression +(this will autoload ANY.o; alternatively we could hand-load the +ANY.NRLIB/code.lsp file), and then load our /tmp/debug.lisp file. +Now all of the functions in the ANY domain are traced and we can +watch the trace occur while the expression is evaluated. In this example +I've created a larger file that traces all of the loaded domains: +\begin{verbatim} +(trace |RADUTIL;radix;FIA;1|) +(trace |RadixUtilities|) +(trace |RadixUtilities;|) + +(trace |RADIX;characteristic;Nni;1|) +(trace |RADIX;differentiate;2$;2|) +(trace |RADIX;Zero;$;3|) +(trace |RADIX;One;$;4|) +(trace |RADIX;-;2$;5|) +(trace |RADIX;+;3$;6|) +(trace |RADIX;-;3$;7|) +(trace |RADIX;*;I2$;8|) +(trace |RADIX;*;3$;9|) +(trace |RADIX;/;3$;10|) +(trace |RADIX;/;2I$;11|) +(trace |RADIX;<;2$B;12|) +(trace |RADIX;=;2$B;13|) +(trace |RADIX;numer;$I;14|) +(trace |RADIX;denom;$I;15|) +(trace |RADIX;coerce;$F;16|) +(trace |RADIX;coerce;I$;17|) +(trace |RADIX;coerce;F$;18|) +(trace |RADIX;retractIfCan;$U;19|) +(trace |RADIX;retractIfCan;$U;20|) +(trace |RADIX;ceiling;$I;21|) +(trace |RADIX;floor;$I;22|) +(trace |RADIX;wholePart;$I;23|) +(trace |RADIX;fractionPart;$F;24|) +(trace |RADIX;wholeRagits;$L;25|) +(trace |RADIX;fractRagits;$S;26|) +(trace |RADIX;prefixRagits;$L;27|) +(trace |RADIX;cycleRagits;$L;28|) +(trace |RADIX;wholeRadix;L$;29|) +(trace |RADIX;fractRadix;2L$;30|) +(trace |RADIX;intToExpr|) +(trace |RADIX;exprgroup|) +(trace |RADIX;intgroup|) +(trace |RADIX;overBar|) +(trace |RADIX;coerce;$Of;35|) +(trace |RADIX;checkRagits|) +(trace |RADIX;radixInt|) +(trace |RADIX;radixFrac|) +(trace |RadixExpansion|) +(trace |RadixExpansion;|) + +(trace |ANY1;retractable?;AB;1|) +(trace |ANY1;coerce;SA;2|) +(trace |ANY1;retractIfCan;AU;3|) +(trace |ANY1;retract;AS;4|) +(trace |AnyFunctions1|) +(trace |AnyFunctions1;|) + +(trace |NONE1;coerce;SN;1|) +(trace |NoneFunctions1|) +(trace |NoneFunctions1;|) + +(trace |ANY;obj;$N;1|) +(trace |ANY;dom;$Se;2|) +(trace |ANY;domainOf;$Of;3|) +(trace |ANY;=;2$B;4|) +(trace |ANY;objectOf;$Of;5|) +(trace |ANY;showTypeInOutput;BS;6|) +(trace |ANY;coerce;$Of;7|) +(trace |ANY;any;SeN$;8|) +(trace |Any|) +(trace |Any;|) + +(trace |SExpression|) +(trace |SExpression;|) + +(trace |TEX;new;$;1|) +(trace |TEX;newWithNum|) +(trace |TEX;coerce;Of$;3|) +(trace |TEX;convert;OfI$;4|) +(trace |TEX;display;$IV;5|) +(trace |TEX;display;$V;6|) +(trace |TEX;prologue;$L;7|) +(trace |TEX;tex;$L;8|) +(trace |TEX;epilogue;$L;9|) +(trace |TEX;setPrologue!;$2L;10|) +(trace |TEX;setTex!;$2L;11|) +(trace |TEX;setEpilogue!;$2L;12|) +(trace |TEX;coerce;$Of;13|) +(trace |TEX;ungroup|) +(trace |TEX;postcondition|) +(trace |TEX;stringify|) +(trace |TEX;lineConcat|) +(trace |TEX;splitLong|) +(trace |TEX;splitLong1|) +(trace |TEX;group|) +(trace |TEX;addBraces|) +(trace |TEX;addBrackets|) +(trace |TEX;parenthesize|) +(trace |TEX;precondition|) +(trace |TEX;formatSpecial|) +(trace |TEX;formatPlex|) +(trace |TEX;formatMatrix|) +(trace |TEX;formatFunction|) +(trace |TEX;formatNullary|) +(trace |TEX;formatUnary|) +(trace |TEX;formatBinary|) +(trace |TEX;formatNary|) +(trace |TEX;formatNaryNoGroup|) +(trace |TEX;formatTex|) +(trace |TexFormat|) +(trace |TexFormat;|) + +(trace |CCLASS;digit;$;1|) +(trace |CCLASS;hexDigit;$;2|) +(trace |CCLASS;upperCase;$;3|) +(trace |CCLASS;lowerCase;$;4|) +(trace |CCLASS;alphabetic;$;5|) +(trace |CCLASS;alphanumeric;$;6|) +(trace |CCLASS;=;2$B;7|) +(trace |CCLASS;member?;C$B;8|) +(trace |CCLASS;union;3$;9|) +(trace |CCLASS;intersect;3$;10|) +(trace |CCLASS;difference;3$;11|) +(trace |CCLASS;complement;2$;12|) +(trace |CCLASS;convert;$S;13|) +(trace |CCLASS;convert;$L;14|) +(trace |CCLASS;charClass;S$;15|) +(trace |CCLASS;charClass;L$;16|) +(trace |CCLASS;coerce;$Of;17|) +(trace |CCLASS;#;$Nni;18|) +(trace |CCLASS;empty;$;19|) +(trace |CCLASS;brace;$;20|) +(trace |CCLASS;insert!;C2$;21|) +(trace |CCLASS;remove!;C2$;22|) +(trace |CCLASS;inspect;$C;23|) +(trace |CCLASS;extract!;$C;24|) +(trace |CCLASS;map;M2$;25|) +(trace |CCLASS;map!;M2$;26|) +(trace |CCLASS;parts;$L;27|) +(trace |CharacterClass|) +(trace |CharacterClass;|) + +(trace |IBITS;minIndex;$I;1|) +(trace |IBITS;range|) +(trace |IBITS;coerce;$Of;3|) +(trace |IBITS;new;NniB$;4|) +(trace |IBITS;empty;$;5|) +(trace |IBITS;copy;2$;6|) +(trace |IBITS;#;$Nni;7|) +(trace |IBITS;=;2$B;8|) +(trace |IBITS;<;2$B;9|) +(trace |IBITS;and;3$;10|) +(trace |IBITS;or;3$;11|) +(trace |IBITS;xor;3$;12|) +(trace |IBITS;setelt;$I2B;13|) +(trace |IBITS;elt;$IB;14|) +(trace |IBITS;Not;2$;15|) +(trace |IBITS;And;3$;16|) +(trace |IBITS;Or;3$;17|) +(trace |IndexedBits|) +(trace |IndexedBits;|) + +(trace |UNISEG;segment;S$;1|) +(trace |UNISEG;segment;2S$;2|) +(trace |UNISEG;BY;$I$;3|) +(trace |UNISEG;lo;$S;4|) +(trace |UNISEG;low;$S;5|) +(trace |UNISEG;hasHi;$B;6|) +(trace |UNISEG;hi;$S;7|) +(trace |UNISEG;high;$S;8|) +(trace |UNISEG;incr;$I;9|) +(trace |UNISEG;SEGMENT;S$;10|) +(trace |UNISEG;SEGMENT;2S$;11|) +(trace |UNISEG;coerce;S$;12|) +(trace |UNISEG;convert;S$;13|) +(trace |UNISEG;=;2$B;14|) +(trace |UNISEG;coerce;$Of;15|) +(trace |UNISEG;expand;$S;16|) +(trace |UNISEG;map;M$S;17|) +(trace |UNISEG;plusInc|) +(trace |UNISEG;expand;LS;19|) +(trace |UNISEG;expand;LS;19!0|) +(trace |UniversalSegment|) +(trace |UniversalSegment;|) + +\end{verbatim} +Now we rerun the function and get the trace output +\begin{verbatim} +(2) -> )lisp (load "/axiom/debug.lisp") + +Value = T +(2) -> radix(10**10,32) + + 1> (|RadixUtilities|) + <1 (|RadixUtilities| #) + 1> (|RadixExpansion| 32) + <1 (|RadixExpansion| #) + 1> (|AnyFunctions1| #) + <1 (|AnyFunctions1| #) + 1> (|RadixExpansion| 32) + <1 (|RadixExpansion| #) + 1> (|RADIX;radixInt| 10000000000 32 #) + <1 (|RADIX;radixInt| (9 10 0 23 25 0 0)) + 1> (|RADIX;radixFrac| 0 1 32 #) + <1 (|RADIX;radixFrac| (NIL 0)) + + 1> (|RadixExpansion| 32) + <1 (|RadixExpansion| #) + 1> (|RADIX;intgroup| (9 10 0 23 25 0 0) #) + 2> (|RADIX;intToExpr| 9 #) + <2 (|RADIX;intToExpr| 9) + 2> (|RADIX;intToExpr| 10 #) + <2 (|RADIX;intToExpr| #\A) + 2> (|RADIX;intToExpr| 0 #) + <2 (|RADIX;intToExpr| 0) + 2> (|RADIX;intToExpr| 23 #) + <2 (|RADIX;intToExpr| #\N) + 2> (|RADIX;intToExpr| 25 #) + <2 (|RADIX;intToExpr| #\P) + 2> (|RADIX;intToExpr| 0 #) + <2 (|RADIX;intToExpr| 0) + 2> (|RADIX;intToExpr| 0 #) + <2 (|RADIX;intToExpr| 0) + <1 (|RADIX;intgroup| (CONCAT 9 #\A 0 #\N #\P 0 0)) + 1> (|RADIX;exprgroup| + ((CONCAT 9 #\A 0 #\N #\P 0 0)) #) + <1 (|RADIX;exprgroup| (CONCAT 9 #\A 0 #\N #\P 0 0)) + (2) 9A0NP00 + 1> (|TexFormat|) + <1 (|TexFormat| #) + 1> (|TexFormat|) + <1 (|TexFormat| #) + 1> (|TEX;newWithNum| 2 #) + <1 (|TEX;newWithNum| #) + 1> (|TEX;precondition| + (CONCAT 9 #\A 0 #\N #\P 0 0) #) + <1 (|TEX;precondition| (CONCAT 9 #\A 0 #\N #\P 0 0)) + 1> (|TEX;formatTex| + (CONCAT 9 #\A 0 #\N #\P 0 0) 0 #) + 2> (|TEX;stringify| CONCAT #) + <2 (|TEX;stringify| "CONCAT") + 2> (|TEX;formatSpecial| "CONCAT" + (9 #\A 0 #\N #\P 0 0) 0 #) + 3> (|TEX;formatNary| "" + (9 #\A 0 #\N #\P 0 0) 0 #) + 4> (|TEX;formatNaryNoGroup| "" + (9 #\A 0 #\N #\P 0 0) 0 #) + 5> (|TEX;formatTex| 9 0 #) + 6> (|TEX;stringify| 9 #) + <6 (|TEX;stringify| "9") + <5 (|TEX;formatTex| "9") + 5> (|TEX;formatTex| #\A 0 #) + 6> (|TEX;stringify| #\A #) + <6 (|TEX;stringify| "#\\A") + 6> (|IBITS;range| + # 35 #) + <6 (|IBITS;range| 35) + <5 (|TEX;formatTex| "#\\A") + 5> (|TEX;formatTex| 0 0 #) + 6> (|TEX;stringify| 0 #) + <6 (|TEX;stringify| "0") + <5 (|TEX;formatTex| "0") + 5> (|TEX;formatTex| #\N 0 #) + 6> (|TEX;stringify| #\N #) + <6 (|TEX;stringify| "#\\N") + 6> (|IBITS;range| + # 35 #) + <6 (|IBITS;range| 35) + <5 (|TEX;formatTex| "#\\N") + 5> (|TEX;formatTex| #\P 0 #) + 6> (|TEX;stringify| #\P #) + <6 (|TEX;stringify| "#\\P") + 6> (|IBITS;range| + # 35 #) + <6 (|IBITS;range| 35) + <5 (|TEX;formatTex| "#\\P") + 5> (|TEX;formatTex| 0 0 #) + 6> (|TEX;stringify| 0 #) + <6 (|TEX;stringify| "0") + <5 (|TEX;formatTex| "0") + 5> (|TEX;formatTex| 0 0 #) + 6> (|TEX;stringify| 0 #) + <6 (|TEX;stringify| "0") + <5 (|TEX;formatTex| "0") + <4 (|TEX;formatNaryNoGroup| "9#\\A0#\\N#\\P00") + 4> (|TEX;group| "9#\\A0#\\N#\\P00" #) + <4 (|TEX;group| "{9#\\A0#\\N#\\P00}") + <3 (|TEX;formatNary| "{9#\\A0#\\N#\\P00}") + <2 (|TEX;formatSpecial| "{9#\\A0#\\N#\\P00}") + <1 (|TEX;formatTex| "{9#\\A0#\\N#\\P00}") + 1> (|TEX;postcondition| + "{9#\\A0#\\N#\\P00}" #) + 2> (|TEX;ungroup| "{9#\\A0#\\N#\\P00}" #) + <2 (|TEX;ungroup| "9#\\A0#\\N#\\P00") + <1 (|TEX;postcondition| "9#\\A0#\\N#\\P00") +$$ + 1> (|TEX;splitLong| + "9#\\A0#\\N#\\P00" 77 #) + 2> (|TEX;splitLong1| + "9#\\A0#\\N#\\P00" 77 #) + 3> (|TEX;lineConcat| + "9#\\A0#\\N#\\P00 " NIL #) + <3 (|TEX;lineConcat| ("9#\\A0#\\N#\\P00 ")) + <2 (|TEX;splitLong1| ("9#\\A0#\\N#\\P00 ")) + <1 (|TEX;splitLong| ("9#\\A0#\\N#\\P00 ")) +9#\A0#\N#\P00 +\leqno(2) +$$ + + Type: RadixExpansion 32 +\end{verbatim} +Notice the call that reads: +\begin{verbatim} + 2> (|RADIX;intToExpr| 10 #) + <2 (|RADIX;intToExpr| #\A) +\end{verbatim} +This means that calling |RADIX;intToExpr| with the number 10 +and ``the domain vector'' generates the character \#\\A which fails. +If we had the domain vector in a variable we could hand-execute this +algebra function directly and watch it fail. So we go to the file +RADIX.NRLIB/code.lsp which contains the definition of RADIX;intToExpr. +The definition is: +\begin{verbatim} +(DEFUN |RADIX;intToExpr| (|i| $) + (COND + ((< |i| 10) + (SPADCALL |i| (QREFELT $ 66))) + ((QUOTE T) + (SPADCALL + (SPADCALL + (QREFELT $ 64) + (+ (- |i| 10) (SPADCALL (QREFELT $ 64) (QREFELT $ 68))) + (QREFELT $ 70)) + (QREFELT $ 71))))) +\end{verbatim} +We can put this definition into our /tmp/debug.lisp file and modify +it to capture the domain vector passed in the \$ variable thus: +\begin{verbatim} +(DEFUN |RADIX;intToExpr| (|i| $) + (setq tpd $) + (COND + ((< |i| 10) + (SPADCALL |i| (QREFELT $ 66))) + ((QUOTE T) + (SPADCALL + (SPADCALL + (QREFELT $ 64) + (+ (- |i| 10) (SPADCALL (QREFELT $ 64) (QREFELT $ 68))) + (QREFELT $ 70)) + (QREFELT $ 71))))) +\end{verbatim} +Now when this function is executed the tpd variable will contain the +value of \$, the domain vector. So we load debug.lisp again +to redefine RADIX;intToExpr and re-execute the function. The trace +results will be the same but now the global variable tpd will have +the domain vector: +\begin{verbatim} +(4) -> (identity tpd) + +Value = # +\end{verbatim} +Now we can use common lisp to step the RADIX;intToExpr function: +\begin{verbatim} +(4) -> (step (|RADIX;intToExpr| 10 tpd)) + +Type ? and a newline for help. + (|RADIX;intToExpr| 10 ...) ? + +Stepper commands: +n (or N or Newline): advances to the next form. +s (or S): skips the form. +p (or P): pretty-prints the form. +f (or F) FUNCTION: skips until the FUNCTION is called. +q (or Q): quits. +u (or U): goes up to the enclosing form. +e (or E) FORM: evaluates the FORM and prints the value(s). +r (or R) FORM: evaluates the FORM and returns the value(s). +b (or B): prints backtrace. +?: prints this. + + (|RADIX;intToExpr| 10 ...) +10 +TPD + = # + (SYSTEM::TRACE-CALL (QUOTE #:G1624) ...) +(QUOTE #:G1624) +SYSTEM::ARGS + = (10 #) + (QUOTE T) +(QUOTE T) +(QUOTE (CONS # ...)) +(QUOTE T) +(QUOTE (CONS # ...)) +(LET (#) ...) +(QUOTE (10 #)) +T + = T + 1> (LET (#) ...) +(QUOTE (10 #)) +(CONS (QUOTE |RADIX;intToExpr|) ...) +(QUOTE |RADIX;intToExpr|) +SYSTEM::ARGLIST + = (10 #) + = (|RADIX;intToExpr| 10 ...) + = (|RADIX;intToExpr| 10 ...) +(|RADIX;intToExpr| 10 ...) + (SETQ TPD ...) +$ + = # + = # + (COND (# #) ...) +(< |i| ...) +|i| + = 10 + 10 + = NIL + (QUOTE T) +(SPADCALL (SPADCALL # ...) ...) +(LET (#) ...) +(QREFELT $ ...) +(SVREF $ ...) +$ + = # + 71 + = (# . + #) + = (# . + #) + (THE (VALUES T) ...) +(FUNCALL (CAR #:G1776) ...) +(CAR #:G1776) +#:G1776 + = (# . + #) + = # + (SPADCALL (QREFELT $ ...) ...) +(LET (#) ...) +(QREFELT $ ...) +(SVREF $ ...) +$ + = # + 70 + = (# . + #) + = (# . + #) + (THE (VALUES T) ...) +(FUNCALL (CAR #:G1777) ...) +(CAR #:G1777) +#:G1777 + = (# . + #) + = # + (QREFELT $ ...) +(SVREF $ ...) +$ + = # + 64 + = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + (+ (- |i| ...) ...) +(- |i| ...) +|i| + = 10 + 10 + = 0 + (SPADCALL (QREFELT $ ...) ...) +(LET (#) ...) +(QREFELT $ ...) +(SVREF $ ...) +$ + = # + 68 + = (# . + #) + = (# . + #) + (THE (VALUES T) ...) +(FUNCALL (CAR #:G1778) ...) +(CAR #:G1778) +#:G1778 + = (# . + #) + = # + (QREFELT $ ...) +(SVREF $ ...) +$ + = # + 64 + = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + = "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + (CDR #:G1778) +#:G1778 + = (# . + #) + = # + = 1 + = 1 + = 1 + = 1 + = 1 + (CDR #:G1777) +#:G1777 + = (# . + #) + = # + = 65 + = 65 + = 65 + = 65 + (CDR #:G1776) +#:G1776 + = (# . + #) + = # + = #\A + = #\A + = #\A + = #\A + = #\A + <1 (LET (# #) ...) +(QUOTE (10 #)) +(QUOTE (#\A)) +(CONS (QUOTE |RADIX;intToExpr|) ...) +(QUOTE |RADIX;intToExpr|) +VALUES + = (#\A) + = (|RADIX;intToExpr| #\A) + = (|RADIX;intToExpr| #\A) +(|RADIX;intToExpr| #\A) + = #\A + = #\A +Value = #\A +(4) -> +\end{verbatim} +If we examine the source code for this function in int/algebra/radix.spad +we find: +\begin{verbatim} + + ALPHAS : String := "ABCDEFGHIJKLMNOPQRSTUVWXYZ" + + intToExpr(i:I): OUT == + -- computes a digit for bases between 11 and 36 + i < 10 => i :: OUT + elt(ALPHAS,(i-10) + minIndex(ALPHAS)) :: OUT + +\end{verbatim} +We do some lookups by hand to find out what functions are being called +from the domain vectors thus: +\begin{verbatim} +(4) -> )lisp (qrefelt tpd 68) + +Value = (# . #) + +\end{verbatim} + + + + +The \#\\A value appears as a result of a call to +CHAR;coerce;\$Of;12. We can look in CHAR.NRLIB/code.lsp +for this function and continue our descent into the code. The +function looks like: +\begin{verbatim} +(DEFUN |CHAR;coerce;$Of;12| (|c| $) + (ELT (QREFELT $ 10) + (+ (QREFELT $ 11) (SPADCALL |c| (QREFELT $ 21))))) +\end{verbatim} +Again we need to get the domain vector, this time from the CHAR domain. +The domain vector has all of the information about a domain including +what functions are referenced and what data values are used. The +QREFELT is a "quick elt" function which resolved to a highly +type optimized function call. The SPADCALL function funcalls the +second argument to SPADCALL with the first argument to SPADCALL +effectively giving: +\begin{verbatim} +(funcall (qrefelt $ 21) |c|) +\end{verbatim} +So we modify the CHAR;coerce;\$Of;12 function to capture the domain +vector thus: +\begin{verbatim} +(DEFUN |CHAR;coerce;$Of;12| (|c| $) + (format t "|CHAR;coerce;$Of;12| called") + (setq tpd1 $) + (ELT (QREFELT $ 10) + (+ (QREFELT $ 11) (SPADCALL |c| (QREFELT $ 21))))) +\end{verbatim} +Again we rerun the failing function and now tpd1 contains the +domain vector for the domain CHAR: + + + + + + +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol4 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} + diff --git a/books/bookvol5.pamphlet b/books/bookvol5.pamphlet new file mode 100644 index 0000000..56ffef1 --- /dev/null +++ b/books/bookvol5.pamphlet @@ -0,0 +1,6134 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 5: Axiom Interpreter}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{The Interpreter} +The Axiom interpreter is a large common lisp program. +It has several forms of interaction and run from +terminal in a standalone fashion, run under the control of a +session handler program, run as a web server, or run in a unix +pipe. +\chapter{The Fundamental Data Structures} +\chapter{Starting Axiom} +Axiom starts by invoking a function value of the lisp symbol +[[*top-level-hook*]]. The function invocation path to from this +point until the prompt is approximates (skipping initializations): +\begin{verbatim} + lisp -> restart + -> |spad| + -> |runspad| + -> |ncTopLevel| + -> |ncIntLoop| + -> |intloop| + -> |SpadInterpretStream| + -> |intloopReadConsole| +\end{verbatim} +The [[|intloopReadConsole|]] function does tail-recursive calls to +itself (don't break this) and never exits. +\section{Variables Used} +\section{Data Structures} +\section{Functions} +\subsection{defun set-restart-hook} +When a lisp image containing code is reloaded there is a hook to +allow a function to be called. In our case it is the [[restart]] +function which is the entry to the Axiom interpreter. +<>= +(defun set-restart-hook () + #+KCL (setq system::*top-level-hook* 'restart) + #+Lucid (setq boot::restart-hook 'restart) + 'restart + ) + +@ +\subsection{defun restart} +The restart function is the real root of the world. It sets up memory +if we are working in a GCL/akcl version of the system. It sets the +current package to be the ``BOOT'' package which is the standard +package in which the interpreter runs. It calls initroot \cite{1} +to set the \$spadroot variable (usually the \$AXIOM variable). + +The [[compiler::*compile-verbose*]] flag has been set to nil globally. +We do not want to know about the microsteps of GCL's compile facility. + +The [[compiler::*suppress-compiler-warnings*]] flag has been set to t. +We do not care that certain generated variables are not used. + +The [[compiler::*suppress-compiler-notes*]] flag has been set to t. +We do not care that tail recursion occurs. +<>= +(defun restart () +#+:akcl + (init-memory-config :cons 500 :fixnum 200 :symbol 500 :package 8 + :array 400 :string 500 :cfun 100 :cpages 3000 :rpages 1000 :hole 2000) +#+:akcl (setq compiler::*compile-verbose* nil) +#+:akcl (setq compiler::*suppress-compiler-warnings* t) +#+:akcl (setq compiler::*suppress-compiler-notes* t) +#-:CCL + (in-package "BOOT") +#+:CCL + (setq *package* (find-package "BOOT")) +#+:CCL (setpchar "") ;; Turn off CCL read prompts +#+(OR :akcl :CCL) (initroot) +#+:akcl (system:gbc-time 0) +#+:akcl + (when (and $openServerIfTrue (fboundp '|openServer|)) + (prog (os) + (setq os (|openServer| $SpadServerName)) + (if (zerop os) + (progn + (setq $openServerIfTrue nil) + (setq |$SpadServer| t))))) +;; We do the following test at runtime to allow us to use the same images +;; with Saturn and Sman. MCD 30-11-95 +#+:CCL + (when + (and (memq :unix *features*) $openServerIfTrue (fboundp '|openServer|)) + (prog (os) + (setq os (|openServer| $SpadServerName)) + (if (zerop os) + (progn + (setq $openServerIfTrue nil) + (setq |$SpadServer| t))))) + (setq |$IOindex| 1) + (setq |$InteractiveFrame| (|makeInitialModemapFrame|)) + (setq |$printLoadMsgs| t) +#+(and :lucid :ibm/370) + (setq *default-pathname-defaults* "") +#+:CCL + (setq *default-pathname-defaults* (get-current-directory)) +#-(or :CCL (and :lucid :ibm/370)) + (setq *default-pathname-defaults* (probe-file "./")) +#+(and :lucid :ibm/370) + (setq vmlisp::$current-directory "") +#-(and :lucid :ibm/370) + (setq vmlisp::$current-directory + (make-directory *default-pathname-defaults*)) + (|loadExposureGroupData|) + (|statisticsInitialization|) + (|initHist|) + (|initializeInterpreterFrameRing|) + + (when |$displayStartMsgs| + (|spadStartUpMsgs|)) + (setq |$currentLine| nil) + (restart0) + (|readSpadProfileIfThere|) + (|spad|)) + +@ +\subsection{defun spad} +\begin{verbatim} +spad() == + -- starts the interpreter but does not read in profiles, etc. + $PrintCompilerMessageIfTrue: local + $inLispVM : local := nil + setOutputAlgebra "%initialize%" + runspad() + 'EndOfSpad +\end{verbatim} +<>= +(defun |spad| () + (prog (|$PrintCompilerMessageIfTrue| |$inLispVM|) + (declare (special |$PrintCompilerMessageIfTrue| |$inLispVM|)) + (return + (progn + (spadlet |$PrintCompilerMessageIfTrue| nil) + (spadlet |$inLispVM| nil) + (|setOutputAlgebra| '|%initialize%|) + (|runspad|) + '|EndOfSpad|)))) + +@ +\subsection{defun runspad} +\begin{verbatim} +runspad() == + mode:='restart + while mode='restart repeat + resetStackLimits() + CATCH($quitTag, CATCH('coerceFailure, + mode:=CATCH('top__level, ncTopLevel()))) +\end{verbatim} +<>= +(defun |runspad| () + (prog (mode) + (return + (seq + (progn + (spadlet mode '|restart|) + (do () + ((null (boot-equal mode '|restart|)) NIL) + (seq + (exit + (progn + (|resetStackLimits|) + (catch |$quitTag| + (catch '|coerceFailure| + (spadlet mode (catch '|top_level| (|ncTopLevel|)))))))))))))) + +@ +\subsection{defun ncTopLevel} +\begin{verbatim} +ncTopLevel() == +-- Top-level read-parse-eval-print loop for the interpreter. Uses +-- the Bill Burge's parser. + IN_-STREAM: fluid := CURINSTREAM + _*EOF_*: fluid := NIL + $InteractiveMode :fluid := true + $BOOT: fluid := NIL + $NEWSPAD: fluid := true + $SPAD: fluid := true + $e:fluid := $InteractiveFrame + ncIntLoop() +\end{verbatim} +<>= +(defun |ncTopLevel| () + (prog (|$e| $spad $newspad $boot |$InteractiveMode| *eof* in-stream) + (declare (special |$e| $spad $newspad $boot |$InteractiveMode| *eof* + in-stream |$InteractiveFrame|)) + (return + (progn + (setq in-stream curinstream) + (setq *eof* nil) + (setq |$InteractiveMode| t) + (setq $boot nil) + (setq $newspad t) + (setq $spad t) + (setq |$e| |$InteractiveFrame|) + (|ncIntLoop|))))) + +@ +\subsection{defun ncIntLoop} +<>= +(defun |ncIntLoop| () + (let ((curinstream *standard-output*) + (curoutstream *standard-input*)) + (declare (special curinstream curoutstream)) + (|intloop|))) + +@ +\subsection{defun intloop} +Note that the [[SpadInterpretStream]] function uses a list of +three strings as an argument. The values in the list seem to have +no use and can eventually be removed. +\begin{verbatim} +intloop () == + mode := $intRestart + while mode = $intRestart repeat + resetStackLimits() + mode := CATCH($intTopLevel, + SpadInterpretStream(1, ["TIM", "DALY", "?"], true)) + +\end{verbatim} +<>= +(defun |intloop| () + (prog (mode) + (declare (special |$intTopLevel| |$intRestart|)) + (return + (progn + (setq mode |$intRestart|) + ((lambda () + (loop + (cond + ((not (equal mode |$intRestart|)) + (return nil)) + (t + (progn + (|resetStackLimits|) + (setq mode + (catch |$intTopLevel| + (|SpadInterpretStream| 1 + (list 'tim 'daly '?) t))))))))))))) + +@ +\subsection{defun SpadInterpretStream} +The [[SpadInterpretStream]] function takes three arguments +\begin{list}{} +\item [[str]] This is passed as an argument to [[intloopReadConsole]] +\item [[source]] This is the name of a source file but appears not +to be used. It is set to the list [[(tim daly ?)]]. +\item [[interactive?]] If this is false then various messages are +suppressed and input does not use piles. If this is true then the +library loading routines might output messages and piles are expected +on input (as from a file). +\end{list} +\begin{verbatim} +SpadInterpretStream(str, source, interactive?) == + $fn : local := source + pile? := not interactive? + $libQuiet : local := not interactive? + $newcompMode : local := false +-- following seems useless and causes ccl package problems +-- $InteractiveMode : local := false + + $newcompErrorCount: local := 0 -- SMW Feb 2/90. + -- Used in highComplete, ncHardError etc. + + $okToExecuteMachineCode: local := true -- set false on error + $inclAssertions: local := ["AIX", "CommonLisp"] -- Jan 28/90 + + + $lastPos : local := $nopos ------------>!!! + $erMsgToss : local := false --------------->!!! + $ncMsgList : local := nil + + $systemCommandFunction : local := function InterpExecuteSpadSystemCommand + $shoeReadLineFunction : local := function serverReadLine + $promptMsg : local := 'S2CTP023 + + interactive? => + PRINC(MKPROMPT()) + intloopReadConsole('"", str) + [] + intloopInclude (source,0) + [] + + ----------------------------------------------------------------- +\end{verbatim} +<>= +(defun |SpadInterpretStream| (str source interactive?) + (prog (|$promptMsg| |$shoeReadLineFunction| |$systemCommandFunction| + |$ncMsgList| |$erMsgToss| |$lastPos| |$inclAssertions| + |$okToExecuteMachineCode| |$newcompErrorCount| |$newcompMode| + |$libQuiet| |$fn|) + (declare (special |$promptMsg| |$shoeReadLineFunction| + |$systemCommandFunction| |$ncMsgList| |$erMsgToss| |$lastPos| + |$inclAssertions| |$okToExecuteMachineCode| |$newcompErrorCount| + |$newcompMode| |$libQuiet| |$fn| |$nopos|)) + (return + (progn + (setq |$fn| source) + (setq |$libQuiet| (null interactive?)) + (setq |$newcompMode| nil) + (setq |$newcompErrorCount| 0) + (setq |$okToExecuteMachineCode| t) + (setq |$inclAssertions| (list 'aix '|CommonLisp|)) + (setq |$lastPos| |$nopos|) + (setq |$erMsgToss| nil) + (setq |$ncMsgList| nil) + (setq |$systemCommandFunction| #'|InterpExecuteSpadSystemCommand|) + (setq |$shoeReadLineFunction| #'|serverReadLine|) + (setq |$promptMsg| 'S2CTP023) + (cond + (interactive? + (progn + (princ (mkprompt)) + (|intloopReadConsole| "" str) + nil)) + (t + (progn + (|intloopInclude| source 0) + nil))))))) + +@ +\section{The Read-Eval-Print Loop} +\subsection{defun intloopReadConsole} +Note that this function relies on the fact that lisp can do tail-recursion. +The function recursively invokes itself. + +The serverReadLine function is a special readline function that handles +communication with the session manager code, which is a separate process +running in parallel. + +We read a line from standard input. +\begin{itemize} +\item If it is a null line then we exit Axiom. +\item If it is a zero length line we prompt and recurse +\item If \$dalymode and open-paren we execute lisp code, prompt and recurse +The \$dalymode will interpret any input that begins with an open-paren +as a lisp expression rather than Axiom input. This is useful for debugging +purposes when most of the input lines will be lisp. Setting \$dalymode +non-nil will certainly break user expectations and is to be used with +caution. +\item If it is ``)fi'' or ``)fin'' we drop into lisp. Use the (restart) + function to return to the interpreter loop. +\item If it starts with ``)'' we process the command, prompt, and recurse +\item If it is a command then we remember the current line, process the + command, prompt, and recurse. +\item If the input has a trailing underscore (Axiom line-continuation) + then we cut off the continuation character and pass the truncated + string to ourselves, prompt, and recurse +\item otherwise we process the input, prompt, and recurse. +\end{itemize} +Notice that all but two paths (a null input or a ``)fi'' or a ``)fin'') +will end up as a recursive call to ourselves. +<>= +(defun |intloopReadConsole| (b n) + (declare (special $dalymode)) + (let (c d pfx input) + (setq input (|serverReadLine| *standard-input*)) + (when (null (stringp input)) (|leaveScratchpad|)) + (when (eql (length input) 0) + (princ (mkprompt)) + (|intloopReadConsole| "" n)) + (when (and $dalymode (|intloopPrefix?| "(" input)) + (|intnplisp| input) + (princ (mkprompt)) + (|intloopReadConsole| "" n)) + (setq pfx (|intloopPrefix?| ")fi" input)) + (when (and pfx (or (string= pfx ")fi") (string= pfx ")fin"))) + (throw '|top_level| nil)) + (when (and (equal b "") (setq d (|intloopPrefix?| ")" input))) + (|setCurrentLine| d) + (setq c (|ncloopCommand| d n)) + (princ (mkprompt)) + (|intloopReadConsole| "" c)) + (setq input (concat b input)) + (when (|ncloopEscaped| input) + (|intloopReadConsole| (subseq input 0 (- (length input) 1)) n)) + (setq c (|intloopProcessString| input n)) + (princ (mkprompt)) + (|intloopReadConsole| "" c))) + +@ +\section{Helper Functions} +\subsection{defun getenviron} +<>= +(defun getenviron (shellvar) + #+allegro (sys::getenv (string var)) + #+clisp (ext:getenv (string var)) + #+(or cmu scl) + (cdr + (assoc (string var) ext:*environment-list* :test #'equalp :key #'string)) + #+(or kcl akcl gcl) (si::getenv (string var)) + #+lispworks (lw:environment-variable (string var)) + #+lucid (lcl:environment-variable (string var)) + #+mcl (ccl::getenv var) + #+sbcl (sb-ext:posix-getenv var) + ) +@ + +\subsection{defun init-memory-config} +Austin-Kyoto Common Lisp (AKCL), now known as Gnu Common Lisp (GCL) +requires some changes to the default memory setup to run Axiom efficently. +This function performs those setup commands. +<>= +(defun init-memory-config (&key + (cons 500) + (fixnum 200) + (symbol 500) + (package 8) + (array 400) + (string 500) + (cfun 100) + (cpages 3000) + (rpages 1000) + (hole 2000) ) + ;; initialize AKCL memory allocation parameters + #+:AKCL + (progn + (system:allocate 'cons cons) + (system:allocate 'fixnum fixnum) + (system:allocate 'symbol symbol) + (system:allocate 'package package) + (system:allocate 'array array) + (system:allocate 'string string) + (system:allocate 'cfun cfun) + (system:allocate-contiguous-pages cpages) + (system:allocate-relocatable-pages rpages) + (system:set-hole-size hole)) + #-:AKCL + nil) + +@ + +\subsection{defun initroot} +Sets up the system to use the {\bf AXIOM} shell variable if we can +and default to the {\bf \$spadroot} variable (which was the value +of the {\bf AXIOM} shell variable at build time) if we can't. +<>= +(defun initroot (&optional (newroot (BOOT::|getEnv| "AXIOM"))) + (reroot (or newroot $spadroot (error "setenv AXIOM or (setq $spadroot)")))) + +@ + +\subsection{defun intloopPrefix?} +If the prefix string is the same as the whole string initial characters +(ignoring spaces in the whole string) then we return the whole string +minus any leading spaces. +<>= +(defun |intloopPrefix?| (prefix whole) + (let ((newprefix (string-left-trim '(#\space) prefix)) + (newwhole (string-left-trim '(#\space) whole))) + (when (<= (length newprefix) (length newwhole)) + (when (string= newprefix newwhole :end2 (length prefix)) + newwhole)))) + +@ +\subsection{defun loadExposureGroupData} +<>= +#+:AKCL +(defun |loadExposureGroupData| () + (cond + ((load "./exposed" :verbose nil :if-does-not-exist nil) + '|done|) + ((load (concat (system:getenv "AXIOM") "/algebra/exposed") + :verbose nil :if-does-not-exist nil) + '|done|) + (t '|failed|) )) + +#+:CCL +(defun |loadExposureGroupData| () + (cond + ((load "./exposed.lsp" :verbose NIL :if-does-not-exist NIL) '|done|) + ((load (concat (BOOT::|getEnv| "AXIOM") "/../../src/algebra/exposed.lsp") + :verbose nil :if-does-not-exist nil) '|done|) + (t nil) )) + +@ + +\subsection{make-absolute-filename} +Prefix a filename with the {\bf AXIOM} shell variable. +<>= +(defun make-absolute-filename (name) + (concatenate 'string $spadroot name)) + +@ + +\subsection{defun makeInitialModemapFrame} +\begin{verbatim} +makeInitialModemapFrame() == COPY $InitialModemapFrame +\end{verbatim} +<>= +(defun |makeInitialModemapFrame| () + (copy |$InitialModemapFrame|)) + +@ + +\subsection{defun ncloopEscaped} +The ncloopEscaped function will return true if the last non-blank +character of a line is an underscore, the Axiom line-continuation +character. Otherwise, it returns nil. +<>= +(defun |ncloopEscaped| (x) + (let ((l (length x))) + (dotimes (i l) + (when (char= (char x (- l i 1)) #\_) (return t)) + (unless (char= (char x (- l i 1)) #\space) (return nil))))) + +@ + +\subsection{defun reclaim} +Call the garbage collector on various platforms. +<>= +#+abcl +(defun reclaim () (ext::gc)) +#+:allegro +(defun reclaim () (excl::gc t)) +#+:CCL +(defun reclaim () (gc)) +#+clisp +(defun reclaim () (#+lisp=cl ext::gc #-lisp=cl lisp::gc)) +#+(or :cmulisp :cmu) +(defun reclaim () (ext:gc)) +#+cormanlisp +(defun reclaim () (cl::gc)) +#+(OR IBCL KCL GCL) +(defun reclaim () (si::gbc t)) +#+lispworks +(defun reclaim () (hcl::normal-gc)) +#+Lucid +(defun reclaim () (lcl::gc)) +#+sbcl +(defun reclaim () (sb-ext::gc)) +@ + +\subsection{defun reroot} +The reroot function is used to reset the important variables used by +the system. In particular, these variables are sensitive to the +{\bf AXIOM} shell variable. That variable is renamed internally to +be {\bf \$spadroot}. The {\bf reroot} function will change the +system to use a new root directory and will have the same effect +as changing the {\bf AXIOM} shell variable and rerunning the system +from scratch. Note that we have changed from the +NAG distribution back to the original form. If you need the NAG +version you can push {\bf :tpd} on the {\bf *features*} variable +before compiling this file. A correct call looks like: +\begin{verbatim} +(in-package "BOOT") +(reroot "/spad/mnt/${SYS}") +\end{verbatim} +where the [[${SYS}]] variable is the same one set at build time. +<>= +(defun reroot (dir) + (setq $spadroot dir) + (setq $directory-list + (mapcar #'make-absolute-filename $relative-directory-list)) + (setq $library-directory-list + (mapcar #'make-absolute-filename $relative-library-directory-list)) + (setq |$defaultMsgDatabaseName| + (pathname (make-absolute-filename "/doc/msgs/s2-us.msgs"))) + (setq |$msgDatabaseName| ()) + (setq *default-pathname-defaults* + (pathname (make-absolute-filename ""))) + (setq $current-directory $spadroot)) + +@ + +\subsection{defun setCurrentLine} +Remember the current line. The cases are: +\begin{itemize} +\item If there is no \$currentLine set it to the input +\item Is the current line a string and the input a string? + Make them into a list +\item Is \$currentLine not a cons cell? Make it one. +\item Is the input a string? Cons it on the end of the list. +\item Otherwise stick it on the end of the list +\end{itemize} +Note I suspect the last two cases do not occur in practice since +they result in a dotted pair if the input is not a cons. However, +this is what the current code does so I won't change it. +<>= +(defun |setCurrentLine| (s) + (cond + ((null |$currentLine|) + (setq |$currentLine| s)) + ((and (stringp |$currentLine|) (stringp s)) + (setq |$currentLine| (list |$currentLine| s))) + ((not (consp |$currentLine|)) + (setq |$currentLine| (cons |$currentLine| s))) + ((stringp s) + (rplacd (last |$currentLine|) (cons s nil))) + (t + (rplacd (last |$currentLine|) s))) + |$currentLine|) + +@ + +\subsection{defun statisticsInitialization} +<>= +(defun |statisticsInitialization| () + "initialize the garbage collection timer" + #+:akcl (system:gbc-time 0) + nil) + +@ +\chapter{System Command Handling} +\subsection{defvar \$systemCommands} +The system commands are the top-level commands available in Axiom +that can all be invoked by prefixing the symbol with a closed-paren. +Thus, to see they copyright you type: +\begin{verbatim} + )copyright +\end{verbatim} +New commands need to be added to this table. The command invoked will +be the first entry of the pair and the ``user level'' of the command +will be the second entry. +<>= +(defvar |$systemCommands| nil) + +(eval-when (eval load) + (setq |$systemCommands| + '( + (|abbreviations| . |compiler| ) + (|boot| . |development|) + (|browse| . |development|) + (|cd| . |interpreter|) + (|clear| . |interpreter|) + (|close| . |interpreter|) + (|compiler| . |compiler| ) + (|copyright| . |interpreter|) + (|credits| . |interpreter|) + (|display| . |interpreter|) + (|edit| . |interpreter|) + (|fin| . |development|) + (|frame| . |interpreter|) + (|help| . |interpreter|) + (|history| . |interpreter|) +;; (|input| . |interpreter|) + (|lisp| . |development|) + (|library| . |interpreter|) + (|load| . |interpreter|) + (|ltrace| . |interpreter|) + (|pquit| . |interpreter|) + (|quit| . |interpreter|) + (|read| . |interpreter|) + (|savesystem| . |interpreter|) + (|set| . |interpreter|) + (|show| . |interpreter|) + (|spool| . |interpreter|) + (|summary| . |interpreter|) + (|synonym| . |interpreter|) + (|system| . |interpreter|) + (|trace| . |interpreter|) + (|undo| . |interpreter|) + (|what| . |interpreter|) + (|with| . |interpreter|) + (|workfiles| . |development|) + (|zsystemdevelopment| . |interpreter|) + ))) + +@ + +\subsection{defvar \$SYSCOMMANDS} +This table is used to look up a symbol to see if it might be a command. +<>= +(defvar $SYSCOMMANDS nil) +(eval-when (eval load) + (setq $SYSCOMMANDS (mapcar #'car |$systemCommands|))) + +@ +\subsection{defvar \$noParseCommands} +This is a list of the commands which have their arguments passed verbatim. +Certain functions, such as the lisp function need to be able to handle +all kinds of input that will not be acceptable to the interpreter. +<>= +(defvar |$noParseCommands| nil) +(eval-when (eval load) + (setq |$noParseCommands| + '( |boot| + |copyright| + |credits| + |fin| + |lisp| + |pquit| + |quit| + |suspend| + |synonym| + |system| + ))) + +@ +\subsection{defvar \$tokenCommands} +This is a list of the commands that expect the interpreter to parse +their arguments. Thus the history command expects that Axiom will have +tokenized and validated the input before calling the history function. +<>= +(defvar |$tokenCommands| nil) +(eval-when (eval load) + (setq |$tokenCommands| + '( |abbreviations| + |cd| + |clear| + |close| + |compiler| + |depends| + |display| + |edit| + |frame| + |frame| + |help| + |history| + |input| + |library| + |load| + |ltrace| + |read| + |savesystem| + |set| + |spool| + |undo| + |what| + |with| + |workfiles| + |zsystemdevelopment| + ))) + +@ + +\subsection{defvar \$InitialCommandSynonymAlist} +Axiom can create ``synonyms'' for commands. We create an initial table +of synonyms which are in common use. +<>= +(defvar |$InitialCommandSynonymAlist| nil) +(eval-when (eval load) + (setq |$InitialCommandSynonymAlist| + '( + (|?| . "what commands") + (|ap| . "what things") + (|apr| . "what things") + (|apropos| . "what things") + (|cache| . "set functions cache") + (|cl| . "clear") + (|cls| . "zsystemdevelopment )cls") + (|cms| . "system") + (|co| . "compiler") + (|d| . "display") + (|dep| . "display dependents") + (|dependents| . "display dependents") + (|e| . "edit") + (|expose| . "set expose add constructor") + (|fc| . "zsystemdevelopment )c") + (|fd| . "zsystemdevelopment )d") + (|fdt| . "zsystemdevelopment )dt") + (|fct| . "zsystemdevelopment )ct") + (|fctl| . "zsystemdevelopment )ctl") + (|fe| . "zsystemdevelopment )e") + (|fec| . "zsystemdevelopment )ec") + (|fect| . "zsystemdevelopment )ect") + (|fns| . "exec spadfn") + (|fortran| . "set output fortran") + (|h| . "help") + (|hd| . "system hypertex &") + (|kclam| . "boot clearClams ( )") + (|killcaches| . "boot clearConstructorAndLisplibCaches ( )") + (|patch| . "zsystemdevelopment )patch") + (|pause| . "zsystemdevelopment )pause") + (|prompt| . "set message prompt") + (|recurrence| . "set functions recurrence") + (|restore| . "history )restore") + (|save| . "history )save") + (|startGraphics| . "system $AXIOM/lib/viewman &") + (|startNAGLink| . "system $AXIOM/lib/nagman &") + (|stopGraphics| . "lisp (|sockSendSignal| 2 15)") + (|stopNAGLink| . "lisp (|sockSendSignal| 8 15)") + (|time| . "set message time") + (|type| . "set message type") + (|unexpose| . "set expose drop constructor") + (|up| . "zsystemdevelopment )update") + (|version| . "lisp *yearweek*") + (|w| . "what") + (|wc| . "what categories") + (|wd| . "what domains") + (|who| . "lisp (pprint credits)") + (|wp| . "what packages") + (|ws| . "what synonyms") +))) + +@ +\subsection{defvar \$CommandSynonymAlist} +The actual list of synonyms is initialized to be the same as the +above initial list of synonyms. The user synonyms that are added +during a session are pushed onto this list for later lookup. +<>= +(defvar |$CommandSynonymAlist| nil) +(eval-when (eval load) + (setq |$CommandSynonymAlist| (copy-alist |$InitialCommandSynonymAlist|))) + +@ + +\subsection{defun ncloopCommand} +The \$systemCommandFunction is set in SpadInterpretStream +to point to the function InterpExecuteSpadSystemCommand. +<>= +(defun |ncloopCommand| (line n) + (declare (special |$systemCommandFunction|)) + (let (a) + (cond + ((setq a (|ncloopPrefix?| ")include" line)) + (|ncloopInclude1| a n)) + (t + (funcall |$systemCommandFunction| line) + n)))) + +@ +\subsection{defun ncloopPrefix?} +If we find the prefix string in the whole string starting at position zero +we return the remainder of the string without the leading prefix. +<>= +(defun |ncloopPrefix?| (prefix whole) + (when (eql (search prefix whole) 0) + (subseq whole (length prefix)))) + +@ +\subsection{defun ncloopInclude1} +<>= +(defun |ncloopInclude1| (name n) + (let (a) + (if (setq a (|ncloopIncFileName| name)) + (|ncloopInclude| a n) + n))) + +@ +\subsection{defun ncloopIncFileName} +Returns the first non-blank substring of the given string. +<>= +(defun |ncloopIncFileName| (string) + (let (fn) + (unless (setq fn (|incFileName| string)) + (write-line (concat string " not found"))) + fn)) + +@ + +\subsection{defun ncloopInclude} +Open the file and read it in. The ncloopInclude0 function is part +of the parser and lives in int-top.boot. +<>= +(defun |ncloopInclude| (name n) + (with-open-file (st name) (|ncloopInclude0| st name n))) + +@ + +\subsection{defun incFileName} +Given a string we return the first token from the string which is +the first non-blank substring. +<>= +(defun |incFileName| (x) + (car (|incBiteOff| x))) + +@ + +\subsection{defun incBiteOff} +Takes a sequence and returns the a list of the first token and the +remaining string characters. If there are no remaining string characters +the second string is of length 0. Effectively it "bites off" the first +token in the string. If the string only 0 or more blanks it returns nil. +<>= +(defun |incBiteOff| (x) + (let (blank nonblank) + (setq x (string x)) + (when (setq nonblank (position #\space x :test-not #'char=)) + (setq blank (position #\space x :start nonblank)) + (if blank + (list (subseq x nonblank blank) (subseq x blank)) + (list (subseq x nonblank) ""))))) + +@ +\chapter{The History Mechanism} +\section{)history} +\index{ugSysCmdhistory} + +\index{history} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )history )on} +\item{\tt )history )off} +\item{\tt )history )write} {\it historyInputFileName} +\item{\tt )history )show [{\it n}] [both]} +\item{\tt )history )save} {\it savedHistoryName} +\item{\tt )history )restore} [{\it savedHistoryName}] +\item{\tt )history )reset} +\item{\tt )history )change} {\it n} +\item{\tt )history )memory} +\item{\tt )history )file} +\item{\tt \%} +\item{\tt \%\%({\it n})} +\item{\tt )set history on | off} +\end{list} + +\par\noindent{\bf Command Description:} + +The {\it history} facility within Axiom allows you to restore your +environment to that of another session and recall previous +computational results. +Additional commands allow you to review previous +input lines and to create an {\bf .input} file of the lines typed to +\index{file!input} +Axiom. + +Axiom saves your input and output if the history facility is +turned on (which is the default). +This information is saved if either of +\begin{verbatim} +)set history on +)history )on +\end{verbatim} +has been issued. +Issuing either +\begin{verbatim} +)set history off +)history )off +\end{verbatim} +will discontinue the recording of information. +\index{history )on} +\index{set history on} +\index{set history off} +\index{history )off} + +Whether the facility is disabled or not, +the value of {\tt \%} in Axiom always +refers to the result of the last computation. +If you have not yet entered anything, +{\tt \%} evaluates to an object of type +{\tt Variable('\%)}. +The function {\tt \%\%} may be used to refer +to other previous results if the history facility is enabled. +In that case, +{\tt \%\%(n)} is the output from step {\tt n} if {\tt n > 0}. +If {\tt n < 0}, the step is computed relative to the current step. +Thus {\tt \%\%(-1)} is also the previous step, +{\tt \%\%(-2)}, is the step before that, and so on. +If an invalid step number is given, Axiom will signal an error. + +The {\it environment} information can either be saved in a file or entirely in +memory (the default). +Each frame +(\ref{ugSysCmdframe} on page~\pageref{ugSysCmdframe}) +has its own history database. +When it is kept in a file, some of it may also be kept in memory for +efficiency. +When the information is saved in a file, the name of the file is +of the form {\bf FRAME.axh} where ``{\bf FRAME}'' is the name of the +current frame. +The history file is placed in the current working directory +(see \ref{ugSysCmdcd} on page~\pageref{ugSysCmdcd}). +Note that these history database files are not text files (in fact, +they are directories themselves), and so are not in human-readable +format. + +The options to the {\tt )history} command are as follows: + +\begin{description} +\item[{\tt )change} {\it n}] +will set the number of steps that are saved in memory to {\it n}. +This option only has effect when the history data is maintained in a +file. +If you have issued {\tt )history )memory} (or not changed the default) +there is no need to use {\tt )history )change}. +\index{history )change} + +\item[{\tt )on}] +will start the recording of information. +If the workspace is not empty, you will be asked to confirm this +request. +If you do so, the workspace will be cleared and history data will begin +being saved. +You can also turn the facility on by issuing {\tt )set history on}. + +\item[{\tt )off}] +will stop the recording of information. +The {\tt )history )show} command will not work after issuing this +command. +Note that this command may be issued to save time, as there is some +performance penalty paid for saving the environment data. +You can also turn the facility off by issuing {\tt )set history off}. + +\item[{\tt )file}] +indicates that history data should be saved in an external file on disk. + +\item[{\tt )memory}] +indicates that all history data should be kept in memory rather than +saved in a file. +Note that if you are computing with very large objects it may not be +practical to kept this data in memory. + +\item[{\tt )reset}] +will flush the internal list of the most recent workspace calculations +so that the data structures may be garbage collected by the underlying +Common Lisp system. +Like {\tt )history )change}, this option only has real effect when +history data is being saved in a file. + +\item[{\tt )restore} [{\it savedHistoryName}]] +completely clears the environment and restores it to a saved session, if +possible. +The {\tt )save} option below allows you to save a session to a file +with a given name. If you had issued +{\tt )history )save jacobi} +the command +{\tt )history )restore jacobi} +would clear the current workspace and load the contents of the named +saved session. If no saved session name is specified, the system looks +for a file called {\bf last.axh}. + +\item[{\tt )save} {\it savedHistoryName}] +is used to save a snapshot of the environment in a file. +This file is placed in the current working directory +(see \ref{ugSysCmdcd} on page~\pageref{ugSysCmdcd}). +Use {\tt )history )restore} to restore the environment to the state +preserved in the file. +This option also creates an input file containing all the lines of input +since you created the workspace frame (for example, by starting your +Axiom session) or last did a {\tt )clear all} or +{\tt )clear completely}. + +\item[{\tt )show} [{\it n}] [{\tt both}]] +can show previous input lines and output results. +{\tt )show} will display up to twenty of the last input lines +(fewer if you haven't typed in twenty lines). +{\tt )show} {\it n} will display up to {\it n} of the last input lines. +{\tt )show both} will display up to five of the last input lines and +output results. +{\tt )show} {\it n} {\tt both} will display up to {\it n} of the last +input lines and output results. + +\item[{\tt )write} {\it historyInputFile}] +creates an {\bf .input} file with the input lines typed since the start +of the session/frame or the last {\tt )clear all} or {\tt )clear +completely}. +If {\it historyInputFileName} does not contain a period (``.'') in the filename, +{\bf .input} is appended to it. +For example, +{\tt )history )write chaos} +and +{\tt )history )write chaos.input} +both write the input lines to a file called {\bf chaos.input} in your +current working directory. +If you issued one or more {\tt )undo} commands, +{\tt )history )write} +eliminates all +input lines backtracked over as a result of {\tt )undo}. +You can edit this file and then use {\tt )read} to have Axiom process +the contents. +\end{description} + +\par\noindent{\bf Also See:} +{\tt )frame} \index{ugSysCmdframe}, +{\tt )read} \index{ugSysCmdread}, +{\tt )set} \index{ugSysCmdset}, and +{\tt )undo} \index{ugSysCmdundo}. + + +History recording is done in two different ways: +\begin{itemize} +\item all changes in variable bindings (i.e. previous values) are + written to [[$HistList]], which is a circular list +\item all new bindings (including the binding to [[%]]) are written to a + file called [[histFileName()]] + one older session is accessible via the file [[$oldHistFileName()]] +\end{itemize} + +\section{Variables Used} +The following global variables are used: +\begin{list}{} +\item [[$HistList]], [[$HistListLen]] and [[$HistListAct]] which is the + actual number of ``undoable'' steps) +\item [[$HistRecord]] collects the input line, all variable bindings + and the output of a step, before it is written to the file + [[histFileName()]]. +\item [[$HiFiAccess]] is a flag, which is reset by [[)history )off]] +\end{list} +The result of step n can be accessed by [[%n]], which is translated +into a call of [[fetchOutput(n)]]. The +[[updateHist]] is called after every interpreter step. The +[[putHist]] function records all changes in the environment to [[$HistList]] + and [[$HistRecord]] + +\subsection{Initialized history variables} +<>= +(defvar |$oldHistoryFileName| '|last| "vm/370 filename name component") +(defvar |$historyFileType| '|axh| "vm/370 filename type component") +(defvar |$historyDirectory| 'A "vm/370 filename disk component") +(defvar |$HiFiAccess| t "t means turn on history mechanism") +(defvar |$useInternalHistoryTable| t "t means keep history in core") + +@ +\section{Data Structures} +\section{Functions} +\subsection{defun makeHistFileName} +\begin{verbatim} +makeHistFileName(fname) == + makePathname(fname,$historyFileType,$historyDirectory) +\end{verbatim} +<>= +(defun |makeHistFileName| (fname) + (|makePathname| fname |$historyFileType| |$historyDirectory|)) + +@ +\subsection{defun oldHistFileName} +\begin{verbatim} +oldHistFileName() == + makeHistFileName($oldHistoryFileName) +\end{verbatim} +<>= +(defun |oldHistFileName| () + (|makeHistFileName| |$oldHistoryFileName|)) + +@ +\subsection{defun histFileName} +\begin{verbatim} +histFileName() == + makeHistFileName($interpreterFrameName) +\end{verbatim} +<>= +(defun |histFileName| () + (|makeHistFileName| |$interpreterFrameName|)) + +@ +\subsection{defun histInputFileName} +\begin{verbatim} +histInputFileName(fn) == + null fn => + makePathname($interpreterFrameName,'INPUT,$historyDirectory) + makePathname(fn,'INPUT,$historyDirectory) +\end{verbatim} +<>= +(defun |histInputFileName| (fn) + (if (null fn) + (|makePathname| |$interpreterFrameName| 'input |$historyDirectory|) + (|makePathname| fn 'input |$historyDirectory|))) +@ + +\subsection{defun initHist} +\begin{verbatim} +initHist() == + $useInternalHistoryTable => initHistList() + oldFile := oldHistFileName() + newFile := histFileName() + -- see if history directory is writable + histFileErase oldFile + if MAKE_-INPUT_-FILENAME newFile then $REPLACE(oldFile,newFile) + $HiFiAccess:= 'T + initHistList() +\end{verbatim} +<>= +(defun |initHist| () + (prog (oldFile newFile) + (return + (cond + (|$useInternalHistoryTable| + (|initHistList|)) + (t + (spadlet oldFile (|oldHistFileName|)) + (spadlet newFile (|histFileName|)) + (|histFileErase| oldFile) + (when (make-input-filename newFile) + ($replace oldFile newFile)) + (spadlet |$HiFiAccess| t) + (|initHistList|)))))) + +@ +\subsection{defun initHistList} +\begin{verbatim} +initHistList() == + -- creates $HistList as a circular list of length $HistListLen + -- and $HistRecord + $HistListLen:= 20 + $HistList:= LIST NIL + li:= $HistList + for i in 1..$HistListLen repeat li:= CONS(NIL,li) + RPLACD($HistList,li) + $HistListAct:= 0 + $HistRecord:= NIL +\end{verbatim} +<>= +(defun |initHistList| () + (prog (li) + (return + (seq + (progn + (spadlet |$HistListLen| 20) + (spadlet |$HistList| (list nil)) + (spadlet li |$HistList|) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| |$HistListLen|) nil) + (seq + (exit + (spadlet li (cons nil li))))) + (rplacd |$HistList| li) + (spadlet |$HistListAct| 0) + (spadlet |$HistRecord| NIL)))))) + +@ +\subsection{defun history} +\begin{verbatim} +history l == + l or null $options => sayKeyedMsg("S2IH0006",NIL) + historySpad2Cmd() +\end{verbatim} +<>= +(defun |history| (l) + (cond + ((or l (null |$options|)) + (|sayKeyedMsg| 'S2IH0006 nil)) ; syntax error + (t + (|historySpad2Cmd|)))) + +@ +\subsection{defun historySpad2Cmd} +\begin{verbatim} +historySpad2Cmd() == + -- history is a system command which can call resetInCoreHist + -- and changeHistListLen, and restore last session + histOptions:= + '(on off yes no change reset restore write save show file memory) + opts:= [ [selectOptionLC(opt,histOptions,'optionError),:optargs] + for [opt,:optargs] in $options] + for [opt,:optargs] in opts repeat + opt in '(on yes) => + $HiFiAccess => sayKeyedMsg("S2IH0007",NIL) + $IOindex = 1 => -- haven't done anything yet + $HiFiAccess:= 'T + initHistList() + sayKeyedMsg("S2IH0008",NIL) + x := UPCASE queryUserKeyedMsg("S2IH0009",NIL) + MEMQ(STRING2ID_-N(x,1),'(Y YES)) => + histFileErase histFileName() + $HiFiAccess:= 'T + $options := nil + clearSpad2Cmd '(all) + sayKeyedMsg("S2IH0008",NIL) + initHistList() + sayKeyedMsg("S2IH0010",NIL) + opt in '(off no) => + null $HiFiAccess => sayKeyedMsg("S2IH0011",NIL) + $HiFiAccess:= NIL + disableHist() + sayKeyedMsg("S2IH0012",NIL) + opt = 'file => setHistoryCore NIL + opt = 'memory => setHistoryCore true + opt = 'reset => resetInCoreHist() + opt = 'save => saveHistory optargs + opt = 'show => showHistory optargs + opt = 'change => changeHistListLen first optargs + opt = 'restore => restoreHistory optargs + opt = 'write => writeInputLines(optargs,1) + 'done +\end{verbatim} +<>= +(defun |historySpad2Cmd| () + (prog (histOptions opts opt optargs x) + (return + (seq + (progn + (spadlet histOptions + '(|on| |off| |yes| |no| |change| |reset| |restore| |write| + |save| |show| |file| |memory|)) + (spadlet opts + (prog (tmp1) + (spadlet tmp1 nil) + (return + (do ((tmp2 |$options| (cdr tmp2)) (tmp3 NIL)) + ((or (atom tmp2) + (progn + (setq tmp3 (car tmp2)) + nil) + (progn + (progn + (spadlet opt (car tmp3)) + (spadlet optargs (cdr tmp3)) + tmp3) + nil)) + (nreverse0 tmp1)) + (seq + (exit + (setq tmp1 + (cons + (cons + (|selectOptionLC| opt histOptions '|optionError|) + optargs) + tmp1)))))))) + (do ((tmp4 opts (cdr tmp4)) (tmp5 nil)) + ((or (atom tmp4) + (progn + (setq tmp5 (car tmp4)) + nil) + (progn + (progn + (spadlet opt (car tmp5)) + (spadlet optargs (cdr tmp5)) + tmp5) + nil)) + nil) + (seq + (exit + (cond + ((|member| opt '(|on| |yes|)) + (cond + (|$HiFiAccess| + (|sayKeyedMsg| 'S2IH0007 nil)) ; history already on + ((eql |$IOindex| 1) + (spadlet |$HiFiAccess| t) + (|initHistList|) + (|sayKeyedMsg| 'S2IH0008 nil)) ; history now on + (t + (spadlet x ; really want to turn history on? + (upcase (|queryUserKeyedMsg| 'S2IH0009 nil))) + (cond + ((memq (string2id-n x 1) '(Y YES)) + (|histFileErase| (|histFileName|)) + (spadlet |$HiFiAccess| t) + (spadlet |$options| nil) + (|clearSpad2Cmd| '(|all|)) + (|sayKeyedMsg| 'S2IH0008 nil) ; history now on + (|initHistList|)) + (t + (|sayKeyedMsg| 'S2IH0010 nil)))))) ; history still off + ((|member| opt '(|off| |no|)) + (cond + ((null |$HiFiAccess|) + (|sayKeyedMsg| 'S2IH0011 nil)) ; history already off + (t + (spadlet |$HiFiAccess| nil) + (|disableHist|) + (|sayKeyedMsg| 'S2IH0012 nil)))) ; history now off + ((boot-equal opt '|file|) + (|setHistoryCore| nil)) + ((boot-equal opt '|memory|) + (|setHistoryCore| t)) + ((boot-equal opt '|reset|) + (|resetInCoreHist|)) + ((boot-equal opt '|save|) + (|saveHistory| optargs)) + ((boot-equal opt '|show|) + (|showHistory| optargs)) + ((boot-equal opt '|change|) + (|changeHistListLen| (CAR optargs))) + ((boot-equal opt '|restore|) + (|restoreHistory| optargs)) + ((boot-equal opt '|write|) + (|writeInputLines| optargs 1)))))) + '|done|))))) + +@ +\subsection{defun setHistoryCore} +We [[case]] on the [[inCore]] argument value +\begin{list}{} +\item If history is already on and is kept in the same location as requested +(file or memory) then complain. +\item If history is not in use then start using the file or memory as +requested. This is done by simply setting the [[$useInternalHistoryTable]] +to the requested value, where [[T]] means use memory and [[NIL]] means +use a file. We tell the user. +\item If history should be in memory, that is [[inCore]] is not [[NIL]], +and the history file already contains information we read the information +from the file, store it in memory, and erase the history file. We modify +[[$useInternalHistoryTable]] to [[T]] to indicate that we're maintining +the history in memory and tell the user. +\item Otherwise history must be on and in memory. We erase any old history +file and then write the in-memory history to a new file +\end{list} +\begin{verbatim} +setHistoryCore inCore == + inCore = $useInternalHistoryTable => + sayKeyedMsg((inCore => "S2IH0030"; "S2IH0029"),NIL) + not $HiFiAccess => + $useInternalHistoryTable := inCore + inCore => sayKeyedMsg("S2IH0032",NIL) + sayKeyedMsg("S2IH0031",NIL) + inCore => + $internalHistoryTable := NIL + if $IOindex ^= 0 then + -- actually put something in there + l := LENGTH RKEYIDS histFileName() + for i in 1..l repeat + vec:= UNWIND_-PROTECT(readHiFi(i),disableHist()) + $internalHistoryTable := CONS([i,:vec],$internalHistoryTable) + histFileErase histFileName() + $useInternalHistoryTable := true + sayKeyedMsg("S2IH0032",NIL) + $HiFiAccess:= 'NIL + histFileErase histFileName() + str := RDEFIOSTREAM ['(MODE . OUTPUT),['FILE,:histFileName()]] + for [n,:rec] in reverse $internalHistoryTable repeat + SPADRWRITE(object2Identifier n,rec,str) + RSHUT str + $HiFiAccess:= 'T + $internalHistoryTable := NIL + $useInternalHistoryTable := NIL + sayKeyedMsg("S2IH0031",NIL) +\end{verbatim} +<>= +(defun |setHistoryCore| (inCore) + (prog (l vec str n rec) + (cond + ((boot-equal inCore |$useInternalHistoryTable|) + (if inCore + (|sayKeyedMsg| 'S2IH0030 NIL) ; memory history already in use + (|sayKeyedMsg| 'S2IH0029 NIL))) ; file history already in use + ((null |$HiFiAccess|) + (spadlet |$useInternalHistoryTable| inCore) + (if inCore + (|sayKeyedMsg| 'S2IH0032 NIL) ; use memory history + (|sayKeyedMsg| 'S2IH0031 NIL))) ; use file history + (inCore + (spadlet |$internalHistoryTable| nil) + (cond + ((nequal |$IOindex| 0) + (spadlet l (length (rkeyids (|histFileName|)))) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| l) NIL) + (seq + (exit + (progn + (spadlet vec (unwind-protect (|readHiFi| |i|) (|disableHist|))) + (spadlet |$internalHistoryTable| + (cons (cons |i| vec) |$internalHistoryTable|)))))) + (|histFileErase| (|histFileName|)))) + (spadlet |$useInternalHistoryTable| t) + (|sayKeyedMsg| 'S2IH0032 nil)) ; use memory history + (t + (spadlet |$HiFiAccess| nil) + (|histFileErase| (|histFileName|)) + (spadlet str + (rdefiostream + (cons + '(mode . output) + (cons + (cons 'file (|histFileName|)) + nil)))) + (do ((tmp0 (reverse |$internalHistoryTable|) (cdr tmp0)) + (tmp1 NIL)) + ((or (atom tmp0) + (progn + (setq tmp1 (car tmp0)) + nil) + (progn + (progn + (spadlet n (car tmp1)) + (spadlet rec (cdr tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (spadrwrite (|object2Identifier| n) rec str)))) + (rshut str) + (spadlet |$HiFiAccess| t) + (spadlet |$internalHistoryTable| nil) + (spadlet |$useInternalHistoryTable| nil) + (|sayKeyedMsg| 'S2IH0031 NIL))))) ; use file history + +@ +\subsection{defun writeInputLines} +\begin{verbatim} +writeInputLines(fn,initial) == + -- writes all input lines into file histInputFileName() + not $HiFiAccess => sayKeyedMsg("S2IH0013",NIL) ; history not on + null fn => + throwKeyedMsg("S2IH0038", nil) ; missing file name + maxn := 72 + breakChars := [" ","+"] + for i in initial..$IOindex - 1 repeat + vecl := CAR readHiFi i + if STRINGP vecl then vecl := [vecl] + for vec in vecl repeat + n := SIZE vec + while n > maxn repeat + -- search backwards for a blank + done := nil + for j in 1..maxn while ^done repeat + k := 1 + maxn - j + MEMQ(vec.k,breakChars) => + svec := STRCONC(SUBSTRING(vec,0,k+1),UNDERBAR) + lineList := [svec,:lineList] + done := true + vec := SUBSTRING(vec,k+1,NIL) + n := SIZE vec + -- in case we can't find a breaking point + if ^done then n := 0 + lineList := [vec,:lineList] + file := histInputFileName(fn) + histFileErase file + inp:= DEFIOSTREAM(['(MODE . OUTPUT),['FILE,:file]],255,0) + for x in removeUndoLines NREVERSE lineList repeat WRITE_-LINE(x,inp) + -- see file "undo" for definition of removeUndoLines + if fn ^= 'redo then sayKeyedMsg("S2IH0014",[namestring file]) + SHUT inp + NIL +\end{verbatim} +<>= +(defun |writeInputLines| (fn initial) + (prog (maxn breakChars vecl k svec done vec n lineList file inp) + (return + (seq + (cond + ((null |$HiFiAccess|) + (|sayKeyedMsg| 'S2IH0013 nil)) ; history is not on + ((null fn) + (|throwKeyedMsg| 'S2IH0038 nil)) ; missing file name + (t + (spadlet maxn 72) + (spadlet breakChars (cons '| | (cons '+ nil))) + (do ((tmp0 (spaddifference |$IOindex| 1)) + (|i| initial (+ |i| 1))) + ((> |i| tmp0) nil) + (seq + (exit + (progn + (spadlet vecl (car (|readHiFi| |i|))) + (cond + ((stringp vecl) (spadlet vecl (cons vecl nil)))) + (do ((tmp1 vecl (cdr tmp1)) (vec nil)) + ((or (atom tmp1) (progn (setq vec (car tmp1)) nil)) nil) + (seq + (exit + (progn + (spadlet n (size vec)) + (do () + ((null (> n maxn)) nil) + (seq + (exit + (progn + (spadlet done nil) + (do ((|j| 1 (qsadd1 |j|))) + ((or (qsgreaterp |j| maxn) (null (null done))) nil) + (seq + (exit + (progn + (spadlet k (spaddifference (plus 1 maxn) |j|)) + (cond + ((memq (ELT vec k) breakChars) + (progn + (spadlet svec (strconc + (substring vec 0 (plus k 1)) underbar)) + (spadlet lineList (cons svec lineList)) + (spadlet done t) + (spadlet vec (substring vec (plus k 1) nil)) + (spadlet n (size vec))))))))) + (cond + ((null done) (spadlet n 0)) + (t nil)))))) + (spadlet lineList (cons vec lineList)))))))))) + (spadlet file (|histInputFileName| fn)) + (|histFileErase| file) + (spadlet inp + (defiostream + (cons + '(mode . output) + (cons (cons 'file file) nil)) 255 0)) + (do ((tmp2 (|removeUndoLines| (nreverse lineList)) (cdr tmp2)) + (x nil)) + ((or (atom tmp2) + (progn + (setq x (car tmp2)) + nil)) + nil) + (seq + (exit + (write-line x inp)))) + (cond + ((nequal fn '|redo|) + (|sayKeyedMsg| 'S2IH0014 ; edit this file to see input lines + (cons (|namestring| file) nil)))) + (shut inp) + nil)))))) + +@ +\subsection{defun resetInCoreHist} +\begin{verbatim} +resetInCoreHist() == + -- removes all pointers from $HistList + $HistListAct:= 0 + for i in 1..$HistListLen repeat + $HistList:= CDR $HistList + RPLACA($HistList,NIL) +\end{verbatim} +<>= +(defun |resetInCoreHist| () + (seq + (progn + (spadlet |$HistListAct| 0) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| |$HistListLen|) nil) + (seq + (exit + (progn + (spadlet |$HistList| (cdr |$HistList|)) + (rplaca |$HistList| nil)))))))) + +@ +\subsection{defun changeHistListLen} +\begin{verbatim} +changeHistListLen(n) == + -- changes the length of $HistList. n must be nonnegative + NULL INTEGERP n => sayKeyedMsg("S2IH0015",[n]) + dif:= n-$HistListLen + $HistListLen:= n + l:= CDR $HistList + if dif > 0 then + for i in 1..dif repeat l:= CONS(NIL,l) + if dif < 0 then + for i in 1..-dif repeat l:= CDR l + if $HistListAct > n then $HistListAct:= n + RPLACD($HistList,l) + 'done +\end{verbatim} +<>= +(defun |changeHistListLen| (n) + (prog (dif l) + (return + (seq + (cond + ((null (integerp n)) + (|sayKeyedMsg| 'S2IH0015 (cons n nil))) ; only positive integers + (t + (spadlet dif (spaddifference n |$HistListLen|)) + (spadlet |$HistListLen| n) + (spadlet l (cdr |$HistList|)) + (cond + ((> dif 0) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| dif) nil) + (seq + (exit + (spadlet l (cons nil l))))))) + (cond + ((minusp dif) + (do ((tmp0 (spaddifference dif)) + (|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| tmp0) nil) + (seq + (exit + (spadlet l (cdr l))))) + (cond + ((> |$HistListAct| n) (spadlet |$HistListAct| n)) + (t nil)))) + (rplacd |$HistList| l) + '|done|)))))) + +@ +\subsection{defun updateHist} +\begin{verbatim} +updateHist() == + -- updates the history file and calls updateInCoreHist + null $IOindex => nil + startTimingProcess 'history + updateInCoreHist() + if $HiFiAccess then + UNWIND_-PROTECT(writeHiFi(),disableHist()) + $HistRecord:= NIL + $IOindex:= $IOindex+1 + updateCurrentInterpreterFrame() + $mkTestInputStack := nil + $currentLine := nil + stopTimingProcess 'history +\end{verbatim} +<>= +(defun |updateHist| () + (cond + ((null |$IOindex|) nil) + (t + (|startTimingProcess| '|history|) + (|updateInCoreHist|) + (when |$HiFiAccess| + (unwind-protect (|writeHiFi|) (|disableHist|)) + (spadlet |$HistRecord| nil)) + (spadlet |$IOindex| (plus |$IOindex| 1)) + (|updateCurrentInterpreterFrame|) + (spadlet |$mkTestInputStack| nil) + (spadlet |$currentLine| nil) + (|stopTimingProcess| '|history|)))) + +@ +\subsection{defun updateInCoreHist} +\begin{verbatim} +updateInCoreHist() == + -- updates $HistList and $IOindex + $HistList:= CDR($HistList) + RPLACA($HistList,NIL) + if $HistListAct < $HistListLen then $HistListAct:= $HistListAct+1 +\end{verbatim} +<>= +(defun |updateInCoreHist| () + (progn + (spadlet |$HistList| (cdr |$HistList|)) + (rplaca |$HistList| nil) + (COND + ((> |$HistListLen| |$HistListAct|) + (spadlet |$HistListAct| (plus |$HistListAct| 1))) + (t nil)))) + +@ +\subsection{defun putHist} +\begin{verbatim} +putHist(x,prop,val,e) == + -- records new value to $HistRecord and old value to $HistList + -- then put is called with e + if not (x='%) then recordOldValue(x,prop,get(x,prop,e)) + if $HiFiAccess then recordNewValue(x,prop,val) + putIntSymTab(x,prop,val,e) +\end{verbatim} +<>= +(defun |putHist| (x prop val e) + (progn + (when (null (boot-equal x '%)) + (|recordOldValue| x prop (|get| x prop e))) + (when |$HiFiAccess| + (|recordNewValue| x prop val)) + (|putIntSymTab| x prop val e))) + +@ +\subsection{defun recordNewValue} +\begin{verbatim} +recordNewValue(x,prop,val) == + startTimingProcess 'history + recordNewValue0(x,prop,val) + stopTimingProcess 'history +\end{verbatim} +<>= +(defun |recordNewValue| (x prop val) + (progn + (|startTimingProcess| '|history|) + (|recordNewValue0| x prop val) + (|stopTimingProcess| '|history|))) + +@ +\subsection{defun recordNewValue0} +\begin{verbatim} +recordNewValue0(x,prop,val) == + -- writes (prop . val) into $HistRecord + -- updateHist writes this stuff out into the history file + p1:= ASSQ(x,$HistRecord) => + p2:= ASSQ(prop,CDR p1) => + RPLACD(p2,val) + RPLACD(p1,CONS(CONS(prop,val),CDR p1)) + p:= CONS(x,list CONS(prop,val)) + $HistRecord:= CONS(p,$HistRecord) +\end{verbatim} +<>= +(defun |recordNewValue0| (x prop val) + (prog (p1 p2 p) + (return + (cond + ((spadlet p1 (ASSQ x |$HistRecord|)) + (cond + ((spadlet p2 (assq prop (cdr p1))) (rplacd p2 val)) + (t (rplacd p1 (cons (cons prop val) (cdr p1)))))) + (t + (spadlet p (cons x (list (cons prop val)))) + (spadlet |$HistRecord| (cons p |$HistRecord|))))))) + +@ +\subsection{defun recordOldValue} +\begin{verbatim} +recordOldValue(x,prop,val) == + startTimingProcess 'history + recordOldValue0(x,prop,val) + stopTimingProcess 'history +\end{verbatim} +<>= +(defun |recordOldValue| (x prop val) + (progn + (|startTimingProcess| '|history|) + (|recordOldValue0| x prop val) + (|stopTimingProcess| '|history|))) + +@ +\subsection{defun recordOldValue0} +\begin{verbatim} +recordOldValue0(x,prop,val) == + -- writes (prop . val) into $HistList + p1:= ASSQ(x,CAR $HistList) => + not ASSQ(prop,CDR p1) => + RPLACD(p1,CONS(CONS(prop,val),CDR p1)) + p:= CONS(x,list CONS(prop,val)) + RPLACA($HistList,CONS(p,CAR $HistList)) +\end{verbatim} +<>= +(defun |recordOldValue0| (x prop val) + (prog (p1 p) + (return + (seq + (when (spadlet p1 (assq x (car |$HistList|))) + (exit + (when (null (assq prop (cdr p1))) + (exit + (rplacd p1 (cons (cons prop val) (cdr p1))))))) + (spadlet p (cons x (list (cons prop val)))) + (rplaca |$HistList| (cons p (car |$HistList|))))))) + +@ +\subsection{defun undoInCore} +\begin{verbatim} +undoInCore(n) == + -- undoes the last n>0 steps using $HistList + -- resets $InteractiveFrame + li:= $HistList + for i in n..$HistListLen repeat li:= CDR li + undoChanges(li) + n:= $IOindex-n-1 + n>0 and + $HiFiAccess => + vec:= CDR UNWIND_-PROTECT(readHiFi(n),disableHist()) + val:= ( p:= ASSQ('%,vec) ) and ( p1:= ASSQ('value,CDR p) ) and + CDR p1 + sayKeyedMsg("S2IH0019",[n]) + $InteractiveFrame:= putHist('%,'value,val,$InteractiveFrame) + updateHist() +\end{verbatim} +<>= +(defun |undoInCore| (n) + (prog (li vec p p1 val) + (return + (seq + (progn + (spadlet li |$HistList|) + (do ((i n (+ i 1))) + ((> i |$HistListLen|) nil) + (seq + (exit + (spadlet li (cdr li))))) + (|undoChanges| li) + (spadlet n (spaddifference (spaddifference |$IOindex| n) 1)) + (and + (> n 0) + (cond + (|$HiFiAccess| + (spadlet vec + (cdr (unwind-protect (|readHiFi| n) (|disableHist|)))) + (spadlet val + (and + (spadlet p (assq '% vec)) + (spadlet p1 (assq '|value| (cdr p))) + (cdr p1)))) + (t + (|sayKeyedMsg| 'S2IH0019 (cons n nil))))) ; no history file + (spadlet |$InteractiveFrame| + (|putHist| '% '|value| val |$InteractiveFrame|)) + (|updateHist|)))))) + +@ +\subsection{defun undoChanges} +\begin{verbatim} +undoChanges(li) == + -- undoes all changes of list 'li' + if not CDR li = $HistList then undoChanges CDR li + for p1 in CAR li repeat + x:= CAR p1 + for p2 in CDR p1 repeat + putHist(x,CAR p2,CDR p2,$InteractiveFrame) +\end{verbatim} +<>= +(defun |undoChanges| (li) + (prog (x) + (return + (seq + (progn + (when (null (boot-equal (cdr li) |$HistList|)) + (|undoChanges| (cdr li))) + (do ((tmp0 (car li) (cdr tmp0)) (p1 NIL)) + ((or (atom tmp0) (progn (setq p1 (car tmp0)) nil)) nil) + (seq + (exit + (progn + (spadlet x (car p1)) + (do ((tmp1 (cdr p1) (cdr tmp1)) (p2 nil)) + ((or (atom tmp1) (progn (setq p2 (car tmp1)) nil)) nil) + (seq + (exit + (|putHist| x (car p2) (cdr p2) |$InteractiveFrame|) + )))))))))))) + +@ +\subsection{defun undoFromFile} +\begin{verbatim} +undoFromFile(n) == + -- makes a clear and redoes all the assignments until step n + for [x,:varl] in CAAR $InteractiveFrame repeat + for p in varl repeat + [prop,:val]:= p + val => + if not (x='%) then recordOldValue(x,prop,val) + if $HiFiAccess then recordNewValue(x,prop,val) + RPLACD(p,NIL) + for i in 1..n repeat + vec:= UNWIND_-PROTECT(CDR readHiFi(i),disableHist()) + for p1 in vec repeat + x:= CAR p1 + for p2 in CDR p1 repeat + $InteractiveFrame:= putHist(x,CAR p2,CDR p2,$InteractiveFrame) + val:= ( p:= ASSQ('%,vec) ) and ( p1:= ASSQ('value,CDR p) ) and CDR p1 + $InteractiveFrame:= putHist('%,'value,val,$InteractiveFrame) + updateHist() +\end{verbatim} +<>= +(defun |undoFromFile| (n) + (prog (varl prop vec x p p1 val) + (return + (seq + (progn + (do ((tmp0 (caar |$InteractiveFrame|) (cdr tmp0)) (tmp1 nil)) + ((or (atom tmp0) + (progn (setq tmp1 (car tmp0)) nil) + (progn + (progn + (spadlet x (car tmp1)) + (spadlet varl (cdr tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (do ((tmp2 varl (cdr tmp2)) (p nil)) + ((or (atom tmp2) (progn (setq p (car tmp2)) nil)) nil) + (seq + (exit + (progn + (spadlet prop (car p)) + (spadlet val (cdr p)) + (when val + (progn + (when (null (boot-equal x '%)) + (|recordOldValue| x prop val)) + (when |$HiFiAccess| + (|recordNewValue| x prop val)) + (rplacd p nil)))))))))) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| n) nil) + (seq + (exit + (progn + (spadlet vec + (unwind-protect (cdr (|readHiFi| |i|)) (|disableHist|))) + (do ((tmp3 vec (cdr tmp3)) (p1 nil)) + ((or (atom tmp3) (progn (setq p1 (car tmp3)) nil)) nil) + (seq + (exit + (progn + (spadlet x (car p1)) + (do ((tmp4 (cdr p1) (cdr tmp4)) (p2 nil)) + ((or (atom tmp4) (progn (setq p2 (car tmp4)) nil)) nil) + (seq + (exit + (spadlet |$InteractiveFrame| + (|putHist| x (car p2) (CDR p2) |$InteractiveFrame|) + )))))))))))) + (spadlet val + (and + (spadlet p (assq '% vec)) + (spadlet p1 (assq '|value| (cdr p))) + (cdr p1))) + (spadlet |$InteractiveFrame| + (|putHist| '% '|value| val |$InteractiveFrame|)) + (|updateHist|)))))) + +@ +\subsection{defun saveHistory} +\begin{verbatim} +saveHistory(fn) == + $seen: local := MAKE_-HASHTABLE 'EQ + not $HiFiAccess => sayKeyedMsg("S2IH0016",NIL) + not $useInternalHistoryTable and + null MAKE_-INPUT_-FILENAME histFileName() => sayKeyedMsg("S2IH0022",NIL) + null fn => + throwKeyedMsg("S2IH0037", nil) + savefile := makeHistFileName(fn) + inputfile := histInputFileName(fn) + writeInputLines(fn,1) + histFileErase savefile + + if $useInternalHistoryTable + then + saveStr := RDEFIOSTREAM ['(MODE . OUTPUT),['FILE,:savefile]] + for [n,:rec] in reverse $internalHistoryTable repeat + val := SPADRWRITE0(object2Identifier n,rec,saveStr) + val = 'writifyFailed => + sayKeyedMsg("S2IH0035", [n, inputfile]) ; unable to save step + RSHUT saveStr + sayKeyedMsg("S2IH0018",[namestring(savefile)]) ; saved hist file named + nil +\end{verbatim} +<>= +(defun |saveHistory| (fn) + (prog (|$seen| savefile inputfile saveStr n rec val) + (declare (special |$seen|)) + (return + (seq + (progn + (spadlet |$seen| (make-hashtable 'eq)) + (cond + ((null |$HiFiAccess|) + (|sayKeyedMsg| 'S2IH0016 nil)) ; the history file is not on + ((and (null |$useInternalHistoryTable|) + (null (make-input-filename (|histFileName|)))) + (|sayKeyedMsg| 'S2IH0022 nil)) ; no history saved yet + ((null fn) + (|throwKeyedMsg| 'S2IH0037 nil)) ; need to specify a history filename + (t + (spadlet savefile (|makeHistFileName| fn)) + (spadlet inputfile (|histInputFileName| fn)) + (|writeInputLines| fn 1) + (|histFileErase| savefile) + (when |$useInternalHistoryTable| + (spadlet saveStr + (rdefiostream + (cons '(mode . output) + (cons (cons 'file savefile) nil)))) + (do ((tmp0 (reverse |$internalHistoryTable|) (cdr tmp0)) + (tmp1 nil)) + ((or (atom tmp0) + (progn (setq tmp1 (car tmp0)) nil) + (progn + (progn + (spadlet n (car tmp1)) + (spadlet rec (cdr tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (progn + (spadlet val + (spadrwrite0 (|object2Identifier| n) rec saveStr)) + (when (boot-equal val '|writifyFailed|) + (|sayKeyedMsg| 'S2IH0035 ; can't save the value of step + (cons n (cons inputfile nil)))))))) + (rshut saveStr)) + (|sayKeyedMsg| 'S2IH0018 ; saved history file is + (cons (|namestring| savefile) nil)) + nil))))))) + +@ +\subsection{defun restoreHistory} +\begin{verbatim} +restoreHistory(fn) == + -- uses fn $historyFileType to recover an old session + -- if fn = NIL, then use $oldHistoryFileName + if null fn then fn' := $oldHistoryFileName + else if fn is [fn'] and IDENTP(fn') then fn' := fn' + else throwKeyedMsg("S2IH0023",[fn']) + restfile := makeHistFileName(fn') + null MAKE_-INPUT_-FILENAME restfile => + sayKeyedMsg("S2IH0024",[namestring(restfile)]) ; no history file + + -- if clear is changed to be undoable, this should be a reset-clear + $options: local := nil + clearSpad2Cmd '(all) + + curfile := histFileName() + histFileErase curfile + _$FCOPY(restfile,curfile) + + l:= LENGTH RKEYIDS curfile + $HiFiAccess:= 'T + oldInternal := $useInternalHistoryTable + $useInternalHistoryTable := NIL + if oldInternal then $internalHistoryTable := NIL + for i in 1..l repeat + vec:= UNWIND_-PROTECT(readHiFi(i),disableHist()) + if oldInternal then $internalHistoryTable := + CONS([i,:vec],$internalHistoryTable) + LINE:= CAR vec + for p1 in CDR vec repeat + x:= CAR p1 + for p2 in CDR p1 repeat + $InteractiveFrame:= putHist(x,CAR p2,CDR p2,$InteractiveFrame) + updateInCoreHist() + $e := $InteractiveFrame + for [a,:.] in CAAR $InteractiveFrame repeat + get(a,'localModemap,$InteractiveFrame) => + rempropI(a,'localModemap) + rempropI(a,'localVars) + rempropI(a,'mapBody) + $IOindex:= l+1 + $useInternalHistoryTable := oldInternal + sayKeyedMsg("S2IH0025",[namestring(restfile)]) + clearCmdSortedCaches() + nil +\end{verbatim} +<>= +(defun |restoreHistory| (fn) + (prog (|$options| fnq restfile curfile l oldInternal vec line x a) + (declare (special |$options|)) + (return + (seq + (progn + (cond + ((null fn) + (spadlet fnq |$oldHistoryFileName|)) + ((and (pairp fn) + (eq (qcdr fn) nil) + (progn + (spadlet fnq (qcar fn)) + t) + (identp fnq)) + (spadlet fnq fnq)) + (t (|throwKeyedMsg| 'S2IH0023 (cons fnq nil)))) ; invalid filename + (spadlet restfile (|makeHistFileName| fnq)) + (cond + ((null (make-input-filename restfile)) + (|sayKeyedMsg| 'S2IH0024 ; file does not exist + (cons (|namestring| restfile) nil))) + (t + (spadlet |$options| NIL) + (|clearSpad2Cmd| '(|all|)) + (spadlet curfile (|histFileName|)) + (|histFileErase| curfile) + ($fcopy restfile curfile) + (spadlet l (length (rkeyids curfile))) + (spadlet |$HiFiAccess| t) + (spadlet oldInternal |$useInternalHistoryTable|) + (spadlet |$useInternalHistoryTable| nil) + (when oldInternal + (spadlet |$internalHistoryTable| nil)) + (do ((|i| 1 (qsadd1 |i|))) + ((qsgreaterp |i| l) nil) + (seq + (exit + (progn + (spadlet vec (unwind-protect (|readHiFi| |i|) (|disableHist|))) + (when oldInternal + (spadlet |$internalHistoryTable| + (cons (cons |i| vec) |$internalHistoryTable|))) + (spadlet line (car vec)) + (do ((tmp0 (cdr vec) (cdr tmp0)) (p1 nil)) + ((or (atom tmp0) (progn (setq p1 (car tmp0)) nil)) nil) + (seq + (exit + (progn + (spadlet x (car p1)) + (do ((tmp1 (cdr p1) (cdr tmp1)) (p2 nil)) + ((or (atom tmp1) (progn (setq p2 (car tmp1)) nil)) nil) + (seq + (exit + (spadlet |$InteractiveFrame| + (|putHist| x + (car p2) (cdr p2) |$InteractiveFrame|))))))))) + (|updateInCoreHist|))))) + (spadlet |$e| |$InteractiveFrame|) + (seq + (do ((tmp2 (caar |$InteractiveFrame|) (cdr tmp2)) (tmp3 nil)) + ((or (atom tmp2) + (progn + (setq tmp3 (car tmp2)) + nil) + (progn + (progn + (spadlet a (car tmp3)) + tmp3) + nil)) + nil) + (seq + (exit + (when (|get| a '|localModemap| |$InteractiveFrame|) + (exit + (progn + (|rempropI| a '|localModemap|) + (|rempropI| a '|localVars|) + (|rempropI| a '|mapBody|))))))) + (spadlet |$IOindex| (plus l 1)) + (spadlet |$useInternalHistoryTable| oldInternal) + (|sayKeyedMsg| 'S2IH0025 ; workspace restored + (cons (|namestring| restfile) nil)) + (|clearCmdSortedCaches|) + nil)))))))) + +@ +\subsection{defun showHistory} +\begin{verbatim} +-- the following used to be the show command when that was used to +-- show history. +showHistory(arg) == + -- arg can be of form + -- NIL show at most last 20 input lines + -- (n) show at most last n input lines + -- (lit) where lit is an abbreviation for 'input or 'both + -- if 'input, same as NIL + -- if 'both, show last 5 input and outputs + -- (n lit) show last n input lines + last n output lines + -- if lit expands to 'both + $evalTimePrint: local:= 0 + $printTimeSum: local:= 0 + -- ugh!!! these are needed for timedEvaluateStream + -- displays the last n steps, default n=20 + not $HiFiAccess => sayKeyedMsg("S2IH0026",['show]) + showInputOrBoth := 'input + n := 20 + nset := nil + if arg then + arg1 := CAR arg + if INTEGERP arg1 then + n := arg1 + nset := true + KDR arg => arg1 := CADR arg + arg1 := NIL + arg1 => + arg2 := selectOptionLC(arg1,'(input both),nil) + if arg2 + then ((showInputOrBoth := arg2) = 'both) and (null nset) => n:= 5 + else sayMSG + concat('" ",bright arg1,'"is an invalid argument.") + if n >= $IOindex then n:= $IOindex-1 + mini:= $IOindex-n + maxi:= $IOindex-1 + showInputOrBoth = 'both => + UNWIND_-PROTECT(showInOut(mini,maxi),setIOindex(maxi+1)) + showInput(mini,maxi) +\end{verbatim} +<>= +(defun |showHistory| (arg) + (prog (|$evalTimePrint| |$printTimeSum| nset arg1 arg2 + showInputOrBoth n mini maxi) + (declare (special |$evalTimePrint| |$printTimeSum|)) + (return + (seq + (progn + (spadlet |$evalTimePrint| 0) + (spadlet |$printTimeSum| 0) + (cond + ((null |$HiFiAccess|) + (|sayKeyedMsg| 'S2IH0026 (cons '|show| nil))) ; history not on + (t + (spadlet showInputOrBoth '|input|) + (spadlet n 20) + (spadlet nset nil) + (when arg + (spadlet arg1 (car arg)) + (when (integerp arg1) + (spadlet n arg1) + (spadlet nset t) + (cond + ((kdr arg) (spadlet arg1 (cadr arg))) + (t (spadlet arg1 nil)))) + (when arg1 + (progn + (spadlet arg2 (|selectOptionLC| arg1 '(|input| |both|) nil)) + (seq + (cond + (arg2 + (when (and (boot-equal + (spadlet showInputOrBoth arg2) '|both|) + (null nset)) + (exit (spadlet n 5)))) + (t + (|sayMSG| + (|concat| + (makestring " ") + (|bright| arg1) + (makestring "is an invalid argument."))))))))) + (when (>= n |$IOindex|) + (spadlet n (spaddifference |$IOindex| 1))) + (spadlet mini (spaddifference |$IOindex| n)) + (spadlet maxi (spaddifference |$IOindex| 1)) + (cond + ((boot-equal showInputOrBoth '|both|) + (unwind-protect + (|showInOut| mini maxi) + (|setIOindex| (plus maxi 1)))) + (t (|showInput| mini maxi)))))))))) + +@ +\subsection{defun setIOindex} +\begin{verbatim} +setIOindex(n) == + -- set $IOindex to n + $IOindex:= n +\end{verbatim} +<>= +(defun |setIOindex| (n) + (spadlet |$IOindex| n)) + +@ +\subsection{defun showInput} +\begin{verbatim} +showInput(mini,maxi) == + -- displays all input lines from mini to maxi + for ind in mini..maxi repeat + vec:= UNWIND_-PROTECT(readHiFi(ind),disableHist()) + if ind<10 then TAB 2 else if ind<100 then TAB 1 + l := CAR vec + STRINGP l => + sayMSG ['" [",ind,'"] ",CAR vec] + sayMSG ['" [",ind,'"] " ] + for ln in l repeat + sayMSG ['" ", ln] +\end{verbatim} +<>= +(defun |showInput| (mini maxi) + (prog (vec l) + (return + (seq + (do ((|ind| mini (+ |ind| 1))) + ((> |ind| maxi) NIL) + (seq + (exit + (progn + (spadlet vec (unwind-protect (|readHiFi| |ind|) (|disableHist|))) + (cond + ((> 10 |ind|) (tab 2)) + ((> 100 |ind|) (tab 1)) + (t nil)) + (spadlet l (car vec)) + (cond + ((stringp l) + (|sayMSG| + (cons + (makestring " [") + (cons |ind| + (cons (makestring "] ") + (cons (car vec) nil)))))) + (t + (|sayMSG| + (cons (makestring " [") + (cons |ind| + (cons (makestring "] ") nil)))) + (do ((tmp0 l (cdr tmp0)) (|ln| nil)) + ((or (atom tmp0) (progn (setq |ln| (car tmp0)) nil)) nil) + (seq + (exit + (|sayMSG| + (cons (makestring " ") (cons |ln| nil)))))))))))))))) + +@ +\subsection{defun showInOut} +\begin{verbatim} +showInOut(mini,maxi) == + -- displays all steps from mini to maxi + for ind in mini..maxi repeat + vec:= UNWIND_-PROTECT(readHiFi(ind),disableHist()) + sayMSG [CAR vec] + Alist:= ASSQ('%,CDR vec) => + triple:= CDR ASSQ('value,CDR Alist) + $IOindex:= ind + spadPrint(objValUnwrap triple,objMode triple) +\end{verbatim} +<>= +(defun |showInOut| (mini maxi) + (prog (vec Alist triple) + (return + (seq + (do ((ind mini (+ ind 1))) + ((> ind maxi) nil) + (seq + (exit + (progn + (spadlet vec (unwind-protect (|readHiFi| ind) (|disableHist|))) + (|sayMSG| (cons (car vec) nil)) + (cond + ((spadlet Alist (assq '% (cdr vec))) + (progn + (spadlet triple (cdr (assq '|value| (cdr Alist)))) + (spadlet |$IOindex| ind) + (|spadPrint| + (|objValUnwrap| triple) (|objMode| triple))))))))))))) + +@ +\subsection{defun fetchOutput} +\begin{verbatim} +fetchOutput(n) == + -- result is the output of step n + (n = -1) and (val := getI("%",'value)) => val + $HiFiAccess => + n:= + n < 0 => $IOindex+n + n + n >= $IOindex => throwKeyedMsg("S2IH0001",[n]) + n < 1 => throwKeyedMsg("S2IH0002",[n]) + vec:= UNWIND_-PROTECT(readHiFi(n),disableHist()) + Alist:= ASSQ('%,CDR vec) => + val:= CDR ASSQ('value,CDR Alist) => val + throwKeyedMsg("S2IH0003",[n]) + throwKeyedMsg("S2IH0003",[n]) + throwKeyedMsg("S2IH0004",NIL) +\end{verbatim} +<>= +(defun |fetchOutput| (n) + (prog (vec Alist val) + (return + (cond + ((and (boot-equal n (spaddifference 1)) + (spadlet val (|getI| '% '|value|))) + val) + (|$HiFiAccess| + (spadlet n + (cond + ((minusp n) (plus |$IOindex| n)) + (t n))) + (cond + ((>= n |$IOindex|) + (|throwKeyedMsg| 'S2IH0001 (cons n nil))) ; no step n yet + ((> 1 n) + (|throwKeyedMsg| 'S2IH0002 (cons n nil))) ; only nonzero steps + (t + (spadlet vec (unwind-protect (|readHiFi| n) (|disableHist|))) + (cond + ((spadlet Alist (assq '% (cdr vec))) + (cond + ((spadlet val (cdr (assq '|value| (cdr Alist)))) + val) + (t + (|throwKeyedMsg| 'S2IH0003 (cons n nil))))) ; no step value + (t (|throwKeyedMsg| 'S2IH0003 (cons n nil))))))) ; no step value + (t (|throwKeyedMsg| 'S2IH0004 nil)))))) ; history not on + +@ +\subsection{defun readHiFi} +\begin{verbatim} +readHiFi(n) == + -- reads the file using index n + if $useInternalHistoryTable + then + pair := assoc(n,$internalHistoryTable) + ATOM pair => keyedSystemError("S2IH0034",NIL) + vec := QCDR pair + else + HiFi:= RDEFIOSTREAM ['(MODE . INPUT),['FILE,:histFileName()]] + vec:= SPADRREAD(object2Identifier n,HiFi) + RSHUT HiFi + vec +\end{verbatim} +<>= +(defun |readHiFi| (n) + (prog (pair HiFi vec) + (return + (progn + (cond + (|$useInternalHistoryTable| + (spadlet pair (|assoc| n |$internalHistoryTable|)) + (cond + ((atom pair) + (|keyedSystemError| 'S2IH0034 nil)) ; missing element + (t + (spadlet vec (qcdr pair))))) + (t + (spadlet HiFi + (rdefiostream + (cons + '(mode . input) + (cons + (cons 'file (|histFileName|)) nil)))) + (spadlet vec (spadrread (|object2Identifier| n) HiFi)) + (rshut HiFi))) + vec)))) + +@ +\subsection{defun writeHiFi} +\begin{verbatim} +writeHiFi() == + -- writes the information of the current step out to history file + if $useInternalHistoryTable + then + $internalHistoryTable := CONS([$IOindex,$currentLine,:$HistRecord], + $internalHistoryTable) + else + HiFi:= RDEFIOSTREAM ['(MODE . OUTPUT),['FILE,:histFileName()]] + SPADRWRITE(object2Identifier $IOindex, CONS($currentLine,$HistRecord),HiFi) + RSHUT HiFi +\end{verbatim} +<>= +(defun |writeHiFi| () + (prog (HiFi) + (return + (cond + (|$useInternalHistoryTable| + (spadlet |$internalHistoryTable| + (cons + (cons |$IOindex| + (cons |$currentLine| |$HistRecord|)) + |$internalHistoryTable|))) + (t + (spadlet HiFi + (rdefiostream + (cons + '(mode . output) + (cons (cons 'file (|histFileName|)) nil)))) + (spadrwrite (|object2Identifier| |$IOindex|) + (cons |$currentLine| |$HistRecord|) HiFi) + (rshut HiFi)))))) + +@ +\subsection{defun disableHist} +\begin{verbatim} +disableHist() == + -- disables the history mechanism if an error occurred in the protected + -- piece of code + not $HiFiAccess => histFileErase histFileName() + NIL +\end{verbatim} +<>= +(defun |disableHist| () + (cond + ((null |$HiFiAccess|) + (|histFileErase| (|histFileName|))) + (t nil))) + +@ +\subsection{defun writeHistModesAndValues} +\begin{verbatim} +writeHistModesAndValues() == + for [a,:.] in CAAR $InteractiveFrame repeat + x := get(a,'value,$InteractiveFrame) => + putHist(a,'value,x,$InteractiveFrame) + x := get(a,'mode,$InteractiveFrame) => + putHist(a,'mode,x,$InteractiveFrame) + NIL +\end{verbatim} +<>= +(defun |writeHistModesAndValues| () + (prog (a x) + (return + (seq + (progn + (do ((tmp0 (caar |$InteractiveFrame|) (cdr tmp0)) (tmp1 nil)) + ((or (atom tmp0) + (progn + (setq tmp1 (car tmp0)) + nil) + (progn + (progn + (spadlet a (car tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (cond + ((spadlet x (|get| a '|value| |$InteractiveFrame|)) + (|putHist| a '|value| x |$InteractiveFrame|)) + ((spadlet x (|get| a '|mode| |$InteractiveFrame|)) + (|putHist| a '|mode| x |$InteractiveFrame|)))))) + nil))))) + +@ +\section{Lisplib output transformations} +--% Lisplib output transformations +-- Some types of objects cannot be saved by LISP/VM in lisplibs. +-- These functions transform an object to a writable form and back. +-- SMW +\subsection{defun SPADRWRITE0} +\begin{verbatim} +SPADRWRITE0(vec, item, stream) == + val := safeWritify item + val = 'writifyFailed => val + rwrite(vec, val, stream) + item +\end{verbatim} +<>= +(defun spadrwrite0 (vec item stream) + (prog (val) + (return + (progn + (spadlet val (|safeWritify| item)) + (cond + ((boot-equal val '|writifyFailed|) val) + (t (|rwrite| vec val stream) item)))))) + +@ +\subsection{defun SPADRWRITE} +\begin{verbatim} +SPADRWRITE(vec, item, stream) == + val := SPADRWRITE0(vec, item, stream) + val = 'writifyFailed => + throwKeyedMsg("S2IH0036", nil) ; cannot save value to file + item +\end{verbatim} +<>= +(defun spadrwrite (vec item stream) + (prog (val) + (return + (progn + (spadlet val (spadrwrite0 vec item stream)) + (cond + ((boot-equal val '|writifyFailed|) + (|throwKeyedMsg| 'S2IH0036 nil)) ; cannot save value to file + (t item)))))) + +@ +\subsection{defun SPADRREAD} +\begin{verbatim} +SPADRREAD(vec, stream) == + dewritify rread(vec, stream, nil) +\end{verbatim} +<>= +(defun spadrread (vec stream) + (|dewritify| (|rread| vec stream nil))) + +@ +\subsection{defun unwritable?} +\begin{verbatim} +unwritable? ob == + PAIRP ob or VECP ob => false -- first for speed + COMPILED_-FUNCTION_-P ob or HASHTABLEP ob => true + PLACEP ob or READTABLEP ob => true + FLOATP ob => true + false +\end{verbatim} +<>= +(defun |unwritable?| (ob) + (cond + ((or (pairp ob) (vecp ob)) nil) + ((or (compiled-function-p ob) (hashtablep ob)) t) + ((or (placep ob) (readtablep ob)) t) + ((floatp ob) t) + (t nil))) + +@ +\subsection{defun writifyComplain} +\begin{verbatim} +-- Create a full isomorphic object which can be saved in a lisplib. +-- Note that dewritify(writify(x)) preserves UEQUALity of hashtables. +-- HASHTABLEs go both ways. +-- READTABLEs cannot presently be transformed back. + +writifyComplain s == + $writifyComplained = true => nil + $writifyComplained := true + sayKeyedMsg("S2IH0027",[s]) +\end{verbatim} +<>= +(defun |writifyComplain| (s) + (cond + ((boot-equal |$writifyComplained| t) NIL) + (t + (spadlet |$writifyComplained| t) + (|sayKeyedMsg| 'S2IH0027 (cons s nil))))) ; cannot save value + +@ +\subsection{defun safeWritify} +\begin{verbatim} +safeWritify ob == + CATCH('writifyTag, writify ob) +\end{verbatim} +<>= +(defun |safeWritify| (ob) + (catch '|writifyTag| (|writify| ob))) + +@ +\subsection{defun writify} +\begin{verbatim} +writify ob == + not ScanOrPairVec(function(unwritable?), ob) => ob + $seen: local := MAKE_-HASHTABLE 'EQ + $writifyComplained: local := false + + writifyInner ob where + writifyInner ob == + null ob => nil + (e := HGET($seen, ob)) => e + + PAIRP ob => + qcar := QCAR ob + qcdr := QCDR ob + (name := spadClosure? ob) => + d := writifyInner QCDR ob + nob := ['WRITIFIED_!_!, 'SPADCLOSURE, d, name] + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + (ob is ['LAMBDA_-CLOSURE, ., ., x, :.]) and x => + THROW('writifyTag, 'writifyFailed) + nob := CONS(qcar, qcdr) + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + qcar := writifyInner qcar + qcdr := writifyInner qcdr + QRPLACA(nob, qcar) + QRPLACD(nob, qcdr) + nob + VECP ob => + isDomainOrPackage ob => + d := mkEvalable devaluate ob + nob := ['WRITIFIED_!_!, 'DEVALUATED, writifyInner d] + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + n := QVMAXINDEX ob + nob := MAKE_-VEC(n+1) + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + for i in 0..n repeat + QSETVELT(nob, i, writifyInner QVELT(ob,i)) + nob + ob = 'WRITIFIED_!_! => + ['WRITIFIED_!_!, 'SELF] + -- In CCL constructors are also compiled functions, so we + -- need this line: + constructor? ob => ob + COMPILED_-FUNCTION_-P ob => + THROW('writifyTag, 'writifyFailed) + HASHTABLEP ob => + nob := ['WRITIFIED_!_!] + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + keys := HKEYS ob + QRPLACD(nob, + ['HASHTABLE, + HASHTABLE_-CLASS ob, + writifyInner keys, + [writifyInner HGET(ob,k) for k in keys]]) + nob + PLACEP ob => + nob := ['WRITIFIED_!_!, 'PLACE] + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + -- The next three types cause an error on de-writifying. + -- Create an object of the right shape, nonetheless. + READTABLEP ob => + THROW('writifyTag, 'writifyFailed) + -- Default case: return the object itself. + STRINGP ob => + EQ(ob, $NullStream) => ['WRITIFIED_!_!, 'NULLSTREAM] + EQ(ob, $NonNullStream) => ['WRITIFIED_!_!, 'NONNULLSTREAM] + ob + FLOATP ob => + ob = READ_-FROM_-STRING STRINGIMAGE ob => ob + ['WRITIFIED_!_!, 'FLOAT, ob,: + MULTIPLE_-VALUE_-LIST INTEGER_-DECODE_-FLOAT ob] + ob +\end{verbatim} +<>= +(defun |writify,writifyInner| (ob) + (prog (e name tmp1 tmp2 tmp3 x qcar qcdr d n keys nob) + (return + (seq + (when (null ob) + (exit nil)) + (when (spadlet e (hget |$seen| ob)) + (exit e)) + (when (pairp ob) + (exit + (seq + (spadlet qcar (qcar ob)) + (spadlet qcdr (qcdr ob)) + (when (spadlet name (|spadClosure?| ob)) + (exit + (seq + (spadlet d (|writify,writifyInner| (qcdr ob))) + (spadlet nob + (cons 'writified!! + (cons 'spadclosure + (cons d (cons name nil))))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (when + (and + (and (pairp ob) + (eq (qcar ob) 'lambda-closure) + (progn + (spadlet tmp1 (qcdr ob)) + (and (pairp tmp1) + (progn + (spadlet tmp2 (qcdr tmp1)) + (and + (pairp tmp2) + (progn + (spadlet tmp3 (qcdr tmp2)) + (and (pairp tmp3) + (progn + (spadlet x (qcar tmp3)) + t)))))))) x) + (exit + (throw '|writifyTag| '|writifyFailed|))) + (spadlet nob (cons qcar qcdr)) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (spadlet qcar (|writify,writifyInner| qcar)) + (spadlet qcdr (|writify,writifyInner| qcdr)) + (qrplaca nob qcar) + (qrplacd nob qcdr) + (exit nob)))) + (when (vecp ob) + (exit + (seq + (when (|isDomainOrPackage| ob) + (exit + (seq + (spadlet d (|mkEvalable| (|devaluate| ob))) + (spadlet nob + (cons 'writified!! + (cons 'devaluated + (cons (|writify,writifyInner| d) nil)))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (spadlet n (qvmaxindex ob)) + (spadlet nob (make-vec (plus n 1))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (do ((|i| 0 (qsadd1 |i|))) + ((qsgreaterp |i| n) NIL) + (seq + (exit + (qsetvelt nob |i| (|writify,writifyInner| (QVELT ob |i|)))))) + (exit nob)))) + (when (boot-equal ob 'writified!!) + (exit + (cons 'writified!! (cons 'self nil)))) + (when (|constructor?| ob) + (exit ob)) + (when (compiled-function-p ob) + (exit + (throw '|writifyTag| '|writifyFailed|))) + (when (hashtablep ob) + (exit + (seq + (spadlet nob (cons 'writified!! nil)) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (spadlet keys (hkeys ob)) + (qrplacd nob + (cons + 'hashtable + (cons + (hashtable-class ob) + (cons + (|writify,writifyInner| keys) + (cons + (prog (tmp0) + (spadlet tmp0 nil) + (return + (do ((tmp1 keys (cdr tmp1)) (k nil)) + ((or (atom tmp1) + (progn + (setq k (car tmp1)) + nil)) + (nreverse0 tmp0)) + (seq + (exit + (setq tmp0 + (cons + (|writify,writifyInner| (HGET ob k)) + tmp0))))))) + nil))))) + (exit nob)))) + (when (placep ob) + (exit + (seq + (spadlet nob (cons 'writified!! (cons 'place nil))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (when (readtablep ob) + (exit + (throw '|writifyTag| '|writifyFailed|))) + (when (stringp ob) + (exit + (seq + (when (eq ob |$NullStream|) + (exit + (cons 'writified!! (cons 'nullstream nil)))) + (when (eq ob |$NonNullStream|) + (exit + (cons 'writified!! (cons 'nonnullstream nil)))) + (exit ob)))) + (when (floatp ob) + (exit + (seq + (when (boot-equal ob (read-from-string (stringimage ob))) + (exit ob)) + (exit + (cons 'writified!! + (cons 'float + (cons ob + (multiple-value-list (integer-decode-float ob))))))))) + (exit ob))))) + +@ +<>= +(defun |writify| (ob) + (prog (|$seen| |$writifyComplained|) + (declare (special |$seen| |$writifyComplained|)) + (return + (cond + ((null (|ScanOrPairVec| (|function| |unwritable?|) ob)) + ob) + (t + (spadlet |$seen| (make-hashtable 'eq)) + (spadlet |$writifyComplained| nil) + (|writify,writifyInner| ob)))))) + +@ +\subsection{defun spadClosure?} +\begin{verbatim} +spadClosure? ob == + fun := QCAR ob + not (name := BPINAME fun) => nil + vec := QCDR ob + not VECP vec => nil + name +\end{verbatim} +<>= +(defun |spadClosure?| (ob) + (prog (fun name vec) + (return + (progn + (spadlet fun (qcar ob)) + (cond + ((null (spadlet name (bpiname fun))) nil) + (t + (spadlet vec (qcdr ob)) + (cond + ((null (vecp vec)) nil) + (t name)))))))) + +@ +\subsection{defun dewritify} +\begin{verbatim} +dewritify ob == + (not ScanOrPairVec(function is?, ob) + where is? a == a = 'WRITIFIED_!_!) => ob + + $seen: local := MAKE_-HASHTABLE 'EQ + + dewritifyInner ob where + dewritifyInner ob == + null ob => nil + e := HGET($seen, ob) => e + + PAIRP ob and CAR ob = 'WRITIFIED_!_! => + type := ob.1 + type = 'SELF => + 'WRITIFIED_!_! + type = 'BPI => + oname := ob.2 + f := + INTP oname => EVAL GENSYMMER oname + SYMBOL_-FUNCTION oname + not COMPILED_-FUNCTION_-P f => + error '"A required BPI does not exist." + #ob > 3 and HASHEQ f ^= ob.3 => + error '"A required BPI has been redefined." + HPUT($seen, ob, f) + f + type = 'HASHTABLE => + nob := MAKE_-HASHTABLE ob.2 + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + for k in ob.3 for e in ob.4 repeat + HPUT(nob, dewritifyInner k, dewritifyInner e) + nob + type = 'DEVALUATED => + nob := EVAL dewritifyInner ob.2 + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + type = 'SPADCLOSURE => + vec := dewritifyInner ob.2 + name := ob.3 + not FBOUNDP name => + error STRCONC('"undefined function: ", SYMBOL_-NAME name) + nob := CONS(SYMBOL_-FUNCTION name, vec) + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + type = 'PLACE => + nob := READ MAKE_-INSTREAM NIL + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + nob + type = 'READTABLE => + error '"Cannot de-writify a read table." + type = 'NULLSTREAM => $NullStream + type = 'NONNULLSTREAM => $NonNullStream + type = 'FLOAT => + [fval, signif, expon, sign] := CDDR ob + fval := SCALE_-FLOAT( FLOAT(signif, fval), expon) + sign<0 => -fval + fval + error '"Unknown type to de-writify." + + PAIRP ob => + qcar := QCAR ob + qcdr := QCDR ob + nob := CONS(qcar, qcdr) + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + QRPLACA(nob, dewritifyInner qcar) + QRPLACD(nob, dewritifyInner qcdr) + nob + VECP ob => + n := QVMAXINDEX ob + nob := MAKE_-VEC(n+1) + HPUT($seen, ob, nob) + HPUT($seen, nob, nob) + for i in 0..n repeat + QSETVELT(nob, i, dewritifyInner QVELT(ob,i)) + nob + -- Default case: return the object itself. + ob +\end{verbatim} +<>= +(defun |dewritify,is?| (a) + (boot-equal a 'writified!!)) + +@ +<>= +(defun |dewritify,dewritifyInner| (ob) + (prog (e type oname f vec name tmp1 signif expon sign fval qcar qcdr n nob) + (return + (seq + (when (null ob) + (exit nil)) + (when (spadlet e (hget |$seen| ob)) + (exit e)) + (when (and (pairp ob) (boot-equal (car ob) 'writified!!)) + (exit + (seq + (spadlet type (elt ob 1)) + (when (boot-equal type 'self) + (exit 'writified!!)) + (when (boot-equal type 'bpi) + (exit + (seq + (spadlet oname (elt ob 2)) + (spadlet f + (seq + (when (intp oname) (exit (eval (gensymmer oname)))) + (exit (symbol-function oname)))) + (when (null (compiled-function-p f)) + (exit (|error| (makestring "A required BPI does not exist.")))) + (when (and (> (|#| ob) 3) (nequal (hasheq f) (elt ob 3))) + (exit (|error| (makestring "A required BPI has been redefined.")))) + (hput |$seen| ob f) + (exit f)))) + (when (boot-equal type 'hashtable) + (exit + (seq + (spadlet nob (make-hashtable (elt ob 2))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (do ((tmp0 (elt ob 3) (cdr tmp0)) + (k nil) + (tmp1 (elt ob 4) (cdr tmp1)) + (e nil)) + ((or (atom tmp0) + (progn + (setq k (car tmp0)) + nil) + (atom tmp1) + (progn + (setq e (car tmp1)) + nil)) + nil) + (seq + (exit + (hput nob (|dewritify,dewritifyInner| k) + (|dewritify,dewritifyInner| e))))) + (exit nob)))) + (when (boot-equal type 'devaluated) + (exit + (seq + (spadlet nob (eval (|dewritify,dewritifyInner| (elt ob 2)))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (when (boot-equal type 'spadclosure) + (exit + (seq + (spadlet vec (|dewritify,dewritifyInner| (elt ob 2))) + (spadlet name (ELT ob 3)) + (when (null (fboundp name)) + (exit + (|error| + (strconc (makestring "undefined function: ") + (symbol-name name))))) + (spadlet nob (cons (symbol-function name) vec)) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (when (boot-equal type 'place) + (exit + (seq + (spadlet nob (vmread (make-instream nil))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (exit nob)))) + (when (boot-equal type 'readtable) + (exit (|error| (makestring "Cannot de-writify a read table.")))) + (when (boot-equal type 'nullstream) + (exit |$NullStream|)) + (when (boot-equal type 'nonnullstream) + (exit |$NonNullStream|)) + (when (boot-equal type 'float) + (exit + (seq + (progn + (spadlet tmp1 (cddr ob)) + (spadlet fval (car tmp1)) + (spadlet signif (cadr tmp1)) + (spadlet expon (caddr tmp1)) + (spadlet sign (cadddr tmp1)) + tmp1) + (spadlet fval (scale-float (float signif fval) expon)) + (when (minusp sign) + (exit (spaddifference fval))) + (exit fval)))) + (exit (|error| (makestring "Unknown type to de-writify.")))))) + (when (pairp ob) + (exit + (seq + (spadlet qcar (qcar ob)) + (spadlet qcdr (qcdr ob)) + (spadlet nob (cons qcar qcdr)) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (qrplaca nob (|dewritify,dewritifyInner| qcar)) + (qrplacd nob (|dewritify,dewritifyInner| qcdr)) + (exit nob)))) + (when (vecp ob) + (exit + (seq + (spadlet n (qvmaxindex ob)) + (spadlet nob (make-vec (plus n 1))) + (hput |$seen| ob nob) + (hput |$seen| nob nob) + (do ((|i| 0 (qsadd1 |i|))) + ((qsgreaterp |i| n) nil) + (seq + (exit + (qsetvelt nob |i| + (|dewritify,dewritifyInner| (qvelt ob |i|)))))) + (exit nob)))) + (exit ob))))) + +@ +<>= +(defun |dewritify| (ob) + (prog (|$seen|) + (declare (special |$seen|)) + (return + (cond + ((null (|ScanOrPairVec| (|function| |dewritify,is?|) ob)) + ob) + (t + (spadlet |$seen| (make-hashtable 'EQ)) + (|dewritify,dewritifyInner| ob)))))) + +@ +\subsection{defun ScanOrPairVec} +\begin{verbatim} +ScanOrPairVec(f, ob) == + $seen: local := MAKE_-HASHTABLE 'EQ + + CATCH('ScanOrPairVecAnswer, ScanOrInner(f, ob)) where + ScanOrInner(f, ob) == + HGET($seen, ob) => nil + PAIRP ob => + HPUT($seen, ob, true) + ScanOrInner(f, QCAR ob) + ScanOrInner(f, QCDR ob) + nil + VECP ob => + HPUT($seen, ob, true) + for i in 0..#ob-1 repeat ScanOrInner(f, ob.i) + nil + FUNCALL(f, ob) => + THROW('ScanOrPairVecAnswer, true) + nil +\end{verbatim} +<>= +(defun |ScanOrPairVec,ScanOrInner| (f ob) + (seq + (when (hget |$seen| ob) + (exit nil)) + (when (pairp ob) + (exit + (seq + (hput |$seen| ob t) + (|ScanOrPairVec,ScanOrInner| f (qcar ob)) + (|ScanOrPairVec,ScanOrInner| f (qcdr ob)) + (exit nil)))) + (when (vecp ob) + (exit + (seq + (hput |$seen| ob t) + (do ((tmp0 (spaddifference (|#| ob) 1)) (|i| 0 (qsadd1 |i|))) + ((qsgreaterp |i| tmp0) nil) + (seq + (exit (|ScanOrPairVec,ScanOrInner| f (elt ob |i|))))) + (exit nil)))) + (when (funcall f ob) + (exit + (throw '|ScanOrPairVecAnswer| t))) + (exit nil))) + +(defun |ScanOrPairVec| (f ob) + (prog (|$seen|) + (declare (special |$seen|)) + (return + (progn + (spadlet |$seen| (make-hashtable 'eq)) + (catch '|ScanOrPairVecAnswer| (|ScanOrPairVec,ScanOrInner| f ob)))))) + +@ +\subsection{defun gensymInt} +\begin{verbatim} +gensymInt g == + not GENSYMP g => error '"Need a GENSYM" + p := PNAME g + n := 0 + for i in 2..#p-1 repeat n := 10 * n + charDigitVal p.i + n +\end{verbatim} +<>= +(defun |gensymInt| (g) + (prog (p n) + (return + (seq + (cond + ((null (gensymp g)) + (|error| (makestring "Need a GENSYM"))) + (t + (spadlet p (pname g)) + (spadlet n 0) + (do ((tmp0 (spaddifference (|#| p) 1)) (|i| 2 (qsadd1 |i|))) + ((qsgreaterp |i| tmp0) nil) + (seq + (exit + (spadlet n (plus (times 10 n) (|charDigitVal| (elt p |i|))))))) + n)))))) + +@ +\subsection{defun charDigitVal} +\begin{verbatim} +charDigitVal c == + digits := '"0123456789" + n := -1 + for i in 0..#digits-1 while n < 0 repeat + if c = digits.i then n := i + n < 0 => error '"Character is not a digit" + n +\end{verbatim} +<>= +(defun |charDigitVal| (c) + (prog (digits n) + (return + (seq + (progn + (spadlet digits (makestring "0123456789")) + (spadlet n (spaddifference 1)) + (do ((tmp0 (spaddifference (|#| digits) 1)) (|i| 0 (qsadd1 |i|))) + ((or (qsgreaterp |i| tmp0) (null (minusp n))) nil) + (seq + (exit + (cond + ((boot-equal c (elt digits |i|)) (spadlet n |i|)) + (t nil))))) + (cond + ((minusp n) (|error| (makestring "Character is not a digit"))) + (t n))))))) + +@ +\subsection{defun histFileErase} +\begin{verbatim} +histFileErase file == + --OBEY STRCONC('"rm -rf ", file) + PROBE_-FILE(file) and DELETE_-FILE(file) +\end{verbatim} +<>= +(defun |histFileErase| (file) + (when (probe-file file) + (delete-file file))) + +@ +\section{History File Messages} +<>= +S2IH0001 + You have not reached step %1b yet, and so its value cannot be + supplied. +S2IH0002 + Cannot supply value for step %1b because 1 is the first step. +S2IH0003 + Step %1b has no value. +S2IH0004 + The history facility is not on, so you cannot use %b %% %d . +S2IH0006 + You have not used the correct syntax for the %b history %d command. + Issue %b )help history %d for more information. +S2IH0007 + The history facility is already on. +S2IH0008 + The history facility is now on. +S2IH0009 + Turning on the history facility will clear the contents of the + workspace. + Please enter %b y %d or %b yes %d if you really want to do this: +S2IH0010 + The history facility is still off. +S2IH0011 + The history facility is already off. +S2IH0012 + The history facility is now off. +S2IH0013 + The history facility is not on, so the .input file containing your user input + cannot be created. +S2IH0014 + Edit %b %1 %d to see the saved input lines. +S2IH0015 + The argument %b n %d for %b )history )change n must be a nonnegative + integer and your argument, %1b , is not one. +S2IH0016 + The history facility is not on, so no information can be saved. +S2IH0018 + The saved history file is %1b . +S2IH0019 + There is no history file, so value of step %1b is + undefined. +S2IH0022 + No history information had been saved yet. +S2IH0023 + %1b is not a valid filename for the history file. +S2IH0024 + History information cannot be restored from %1b because the file does + not exist. +S2IH0025 + The workspace has been successfully restored from the history file + %1b . +S2IH0026 + The history facility command %1b cannot be performed because the + history facility is not on. +S2IH0027 + A value containing a %1b is being saved in a history file or a + compiled input file INLIB. This type + is not yet usable in other history operations. You might want to issue + %b )history )off %d +S2IH0029 + History information is already being maintained in an external file + (and not in memory). +S2IH0030 + History information is already being maintained in memory (and not + in an external file). +S2IH0031 + When the history facility is active, history information will be + maintained in a file (and not in an internal table). +S2IH0032 + When the history facility is active, history information will be + maintained in memory (and not in an external file). +S2IH0034 + Missing element in internal history table. +S2IH0035 + Can't save the value of step number %1b. You can re-generate this value + by running the input file %2b. +S2IH0036 + The value specified cannot be saved to a file. +S2IH0037 + You must specify a file name to the history save command +S2IH0038 + You must specify a file name to the history write command +@ + +\chapter{The Frame Mechanism} +\section{)frame} +%\label{ugSysCmdframe} +%\index{frame} +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )frame new {\it frameName}} +\item{\tt )frame drop {\it [frameName]}} +\item{\tt )frame next} +\item{\tt )frame last} +\item{\tt )frame names} +\item{\tt )frame import {\it frameName} {\it [objectName1 [objectName2 ...]]}} +\item{\tt )set message frame on | off} +\item{\tt )set message prompt frame} +\end{list} + +\par\noindent{\bf Command Description:} + +A {\it frame} can be thought of as a logical session within the +physical session that you get when you start the system. You can +have as many frames as you want, within the limits of your computer's +storage, paging space, and so on. +Each frame has its own {\it step number}, {\it environment} and {\it history.} +You can have a variable named {\tt a} in one frame and it will +have nothing to do with anything that might be called {\tt a} in +any other frame. + +Some frames are created by the HyperDoc program and these can +have pretty strange names, since they are generated automatically. +\index{frame names} +To find out the names +of all frames, issue +\begin{verbatim} +)frame names +\end{verbatim} +It will indicate the name of the current frame. + +You create a new frame +\index{frame new} +``{\bf quark}'' by issuing +\begin{verbatim} +)frame new quark +\end{verbatim} +The history facility can be turned on by issuing either +{\tt )set history on} or {\tt )history )on}. +If the history facility is on and you are saving history information +in a file rather than in the Axiom environment +then a history file with filename {\bf quark.axh} will +be created as you enter commands. +If you wish to go back to what +you were doing in the +\index{frame next} +``{\bf initial}'' frame, use +\index{frame last} +\begin{verbatim} +)frame next +\end{verbatim} +or +\begin{verbatim} +)frame last +\end{verbatim} +to cycle through the ring of available frames to get back to +``{\bf initial}''. + +If you want to throw +away a frame (say ``{\bf quark}''), issue +\begin{verbatim} +)frame drop quark +\end{verbatim} +If you omit the name, the current frame is dropped. +\index{frame drop} + +If you do use frames with the history facility on and writing to a file, +you may want to delete some of the older history files. +\index{file!history} +These are directories, so you may want to issue a command like +{\tt rm -r quark.axh} to the operating system. + +You can bring things from another frame by using +\index{frame import} +{\tt )frame import}. +For example, to bring the {\tt f} and {\tt g} from the frame ``{\bf quark}'' +to the current frame, issue +\begin{verbatim} +)frame import quark f g +\end{verbatim} +If you want everything from the frame ``{\bf quark}'', issue +\begin{verbatim} +)frame import quark +\end{verbatim} +You will be asked to verify that you really want everything. + +There are two {\tt )set} flags +\index{set message frame} +to make it easier to tell where you are. +\begin{verbatim} +)set message frame on | off +\end{verbatim} +will print more messages about frames when it is set on. +By default, it is off. +\begin{verbatim} +)set message prompt frame +\end{verbatim} +will give a prompt +\index{set message prompt frame} +that looks like +\begin{verbatim} +initial (1) -> +\end{verbatim} +\index{prompt!with frame name} +when you start up. In this case, the frame name and step make up the +prompt. + +\par\noindent{\bf Also See:} +{\tt )history} \index{ugSysCmdhistory} and +{\tt )set} \index{ugSysCmdset}. + +@ +\section{Variables Used} +The frame mechanism uses several dollar variables. +\subsection{Primary variables} +Primary variables are those which exist solely to make the frame +mechanism work. + +The \$interpreterFrameName contains a symbol which is the name +of the current frame in use. + +The \$interpreterFrameRing contains a list of all of the existing +frames. The first frame on the list is the ``current'' frame. When +AXIOMsys is started directly there is only one frame named ``initial''. + +If the system is started under sman (using the axiom shell script, +for example), there are two frames, ``initial'' and ``frame0''. In +this case, ``frame0'' is the current frame. This can cause subtle +problems because functions defined in the axiom initialization file +(.axiom.input) will be defined in frame ``initial'' but the current +frame will be ``frame0''. They will appear to be undefined. However, +if the user does ``)frame next'' they can switch to the ``initial'' +frame and see the functions correctly defined. + +The \$frameMessages variable controls when frame messages will be +displayed. The variable is initially NIL. It can be set on (T) or off (NIL) +using the system command: +\begin{verbatim} + )set message frame on | off +\end{verbatim} +Setting frame messages on will output a line detailing the +current frame after every output is complete. + +\subsection{Used variables} + +The frame collects and uses a few top level variables. These are: +\$InteractiveFrame, \$IOindex, \$HiFiAccess, \$HistList, \$HistListLen, +\$HistListAct, \$HistRecord, \$internalHistoryTable, and \$localExposureData. + +These variables can also be changed by the frame mechanism when the user +requests changing to a different frame. + +\section{Data Structures} +\subsection{Frames and the Interpreter Frame Ring} + +Axiom has the notion of ``frames''. A frame is a data structure which +holds all the vital data from an Axiom session. There can be multiple +frames and these live in a top-level variable called +\$interpreterFrameRing. This variable holds a circular list of frames. +The parts of a frame and their initial, default values are: + +\begin{verbatim} + $interpreterFrameName a string, named on creation + $InteractiveFrame (list (list nil)) + $IOindex an integer, 1 + $HiFiAccess $HiFiAccess, see the variable description + $HistList $HistList, see the variable description + $HistListLen $HistListLen, see the variable description + $HistListAct $HistListAct, see the variable description + $HistRecord $HistRecord, see the variable description + $internalHistoryTable nil + $localExposureData a copy of $localExposureData +\end{verbatim} + +\section{Accessor Functions} +These could be macros but we wish to export them to the API code +in the algebra so we keep them as functions. +\subsection{0th Frame Component -- frameName} +<>= +(defun frameName (frame) + (car frame)) + +@ +\subsection{1st Frame Component -- frameInteractive} +<>= +(defun frameInteractive (frame) + (nth 1 frame)) + +@ +\subsection{2nd Frame Component -- frameIOIndex} +<>= +(defun frameIOIndex (frame) + (nth 2 frame)) + +@ +\subsection{3rd Frame Component -- frameHiFiAccess} +<>= +(defun frameHiFiAccess (frame) + (nth 3 frame)) + +@ +\subsection{4th Frame Component -- frameHistList} +<>= +(defun frameHistList (frame) + (nth 4 frame)) + +@ +\subsection{5th Frame Component -- frameHistListLen} +<>= +(defun frameHistListLen (frame) + (nth 5 frame)) + +@ +\subsection{6th Frame Component -- frameHistListAct} +<>= +(defun frameHistListAct (frame) + (nth 6 frame)) + +@ +\subsection{7th Frame Component -- frameHistRecord} +<>= +(defun frameHistRecord (frame) + (nth 7 frame)) + +@ +\subsection{8th Frame Component -- frameHistoryTable} +<>= +(defun frameHistoryTable (frame) + (nth 8 frame)) + +@ +\subsection{9th Frame Component -- frameExposureData} +<>= +(defun frameExposureData (frame) + (nth 9 frame)) + +@ + +\section{Functions} +\subsection{Initializing the Interpreter Frame Ring} + +Now that we know what a frame looks like we need a function to +initialize the list of frames. This function sets the initial frame +name to ``initial'' and creates a list of frames containing an empty +frame. This list is the interpreter frame ring and is not actually +circular but is managed as a circular list. + +As a final step we update the world from this frame. This has the +side-effect of resetting all the important global variables to their +initial values. + +<>= +(defun |initializeInterpreterFrameRing| () + (setq |$interpreterFrameName| '|initial|) + (setq |$interpreterFrameRing| + (list (|emptyInterpreterFrame| |$interpreterFrameName|))) + (|updateFromCurrentInterpreterFrame|) nil) + +@ + +The initial values of an empty frame are created here. This function +returns a single frame that will be placed in the frame ring. + +\subsection{Creating an Empty Frame} +<>= +(defun |emptyInterpreterFrame| (name) + (list name + (list (list nil)) + 1 + |$HiFiAccess| + |$HistList| + |$HistListLen| + |$HistListAct| + |$HistRecord| + nil + (copy-seq |$localExposureDataDefault|))) + +@ +\subsection{Collecting up the Environment into a Frame} + +We can collect up all the current environment information into +one frame element with this call. It creates a list of the current +values of the global variables and returns this as a frame element. + +<>= +(defun |createCurrentInterpreterFrame| () + (list + |$interpreterFrameName| + |$InteractiveFrame| + |$IOindex| + |$HiFiAccess| + |$HistList| + |$HistListLen| + |$HistListAct| + |$HistRecord| + |$internalHistoryTable| + |$localExposureData|)) + +@ +\subsection{Updating from the Current Frame} + +The frames are kept on a circular list. The first element on that +list is known as ``the current frame''. This will initialize all +of the interesting interpreter data structures from that frame. + +<>= +(defun |updateFromCurrentInterpreterFrame| () + (let (tmp1) + (setq tmp1 (first |$interpreterFrameRing|)) + (setq |$interpreterFrameName| (nth 0 tmp1)) + (setq |$InteractiveFrame| (nth 1 tmp1)) + (setq |$IOindex| (nth 2 tmp1)) + (setq |$HiFiAccess| (nth 3 tmp1)) + (setq |$HistList| (nth 4 tmp1)) + (setq |$HistListLen| (nth 5 tmp1)) + (setq |$HistListAct| (nth 6 tmp1)) + (setq |$HistRecord| (nth 7 tmp1)) + (setq |$internalHistoryTable| (nth 8 tmp1)) + (setq |$localExposureData| (nth 9 tmp1)) + (when |$frameMessages| + (|sayMessage| + (cons " Current interpreter frame is called" |$interpreterFrameName|))))) + +@ + +\subsection{Creating a List of all of the Frame Names} + +This function simply walks across the frame in the frame ring and +returns a list of the name of each frame. + +<>= +(defun |frameNames| () + (let (result) + (dolist (frame |$interpreterFrameRing|) + (push (frameName frame) result)) + (nreverse result))) + +@ +\subsection{Find a Frame in the Frame Ring by Name} + +Each frame contains its name as the 0th element. We simply walk all +the frames and if we find one we return it. + +<>= +(defun |findFrameInRing| (name) + (block () + (dolist (frame |$interpreterFrameRing|) + (when (eq (frameName frame) name) (return frame))))) + +@ +\subsection{Get Named Frame Environment (aka Interactive)} + +If the frame is found we return the environment portion of the frame +otherwise we construct an empty environment and return it. + +<>= +(defun |frameEnvironment| (fname) + (let ((frame (|findFrameInRing| fname))) + (if frame + (frameInteractive frame) + (list (list nil))))) + +@ +\subsection{Update the Current Interpreter Frame} + +This function collects the normal contents of the world into a +frame object, places it first on the frame list, and then sets +the current values of the world from the frame object. + +<>= +(defun |updateCurrentInterpreterFrame| () + (rplaca |$interpreterFrameRing| (|createCurrentInterpreterFrame|)) + (|updateFromCurrentInterpreterFrame|) + nil) + +@ +\subsection{defun nextInterpreterFrame} + +This function updates the current frame to make sure all of the +current information is recorded. If there are more frame elements +in the list then this will destructively move the current frame +to the end of the list, that is, assume the frame list reads (1 2 3) +this function will destructively change it to (2 3 1). + +Note: the nconc2 function destructively inserts the second list at the +end of the first. + +\begin{verbatim} +nextInterpreterFrame() == + updateCurrentInterpreterFrame() + null rest $interpreterFrameRing => NIL -- nothing to do + $interpreterFrameRing := + NCONC2(rest $interpreterFrameRing,[first $interpreterFrameRing]) + updateFromCurrentInterpreterFrame() +\end{verbatim} +<>= +(defun |nextInterpreterFrame| () + (|updateCurrentInterpreterFrame|) + (when (cdr |$interpreterFrameRing|) + (setq |$interpreterFrameRing| + (nconc2 (cdr |$interpreterFrameRing|) + (list (car |$interpreterFrameRing|)))) + (|updateFromCurrentInterpreterFrame|))) + +@ +\subsection{defun changeToNamedInterpreterFrame} +\begin{verbatim} +changeToNamedInterpreterFrame(name) == + updateCurrentInterpreterFrame() + frame := findFrameInRing(name) + null frame => NIL + $interpreterFrameRing := [frame,:NREMOVE($interpreterFrameRing, frame)] + updateFromCurrentInterpreterFrame() +\end{verbatim} +<>= +(defun |changeToNamedInterpreterFrame| (name) + (prog (frame) + (return + (progn + (|updateCurrentInterpreterFrame|) + (spadlet frame (|findFrameInRing| name)) + (cond + ((null frame) + nil) + (t + (spadlet |$interpreterFrameRing| + (cons frame (nremove |$interpreterFrameRing| frame))) + (|updateFromCurrentInterpreterFrame|))))))) + +@ +\subsection{defun previousInterpreterFrame} +\begin{verbatim} +previousInterpreterFrame() == + updateCurrentInterpreterFrame() + null rest $interpreterFrameRing => NIL -- nothing to do + [:b,l] := $interpreterFrameRing + $interpreterFrameRing := NCONC2([l],b) + updateFromCurrentInterpreterFrame() +\end{verbatim} +<>= +(defun |previousInterpreterFrame| () + (prog (tmp1 l b) + (return + (progn + (|updateCurrentInterpreterFrame|) + (cond + ((null (cdr |$interpreterFrameRing|)) + nil) + (t + (spadlet tmp1 (reverse |$interpreterFrameRing|)) + (spadlet l (car tmp1)) + (spadlet b (nreverse (cdr tmp1))) + (spadlet |$interpreterFrameRing| (nconc2 (cons l nil) b)) + (|updateFromCurrentInterpreterFrame|))))))) + +@ +\subsection{defun addNewInterpreterFrame} +\begin{verbatim} +addNewInterpreterFrame(name) == + null name => throwKeyedMsg("S2IZ0018",NIL) + updateCurrentInterpreterFrame() + -- see if we already have one by that name + for f in $interpreterFrameRing repeat + name = frameName(f) => throwKeyedMsg("S2IZ0019",[name]) + initHistList() + $interpreterFrameRing := CONS(emptyInterpreterFrame(name), + $interpreterFrameRing) + updateFromCurrentInterpreterFrame() + _$ERASE histFileName() +\end{verbatim} +<>= +(defun |addNewInterpreterFrame| (name) + (seq + (cond + ((null name) + (|throwKeyedMsg| 'S2IZ0018 nil)) ; you must provide a name for new frame + (t + (|updateCurrentInterpreterFrame|) + (seq + (do ((tmp0 |$interpreterFrameRing| (cdr tmp0)) (f nil)) + ((or (atom tmp0) + (progn (setq f (car tmp0)) nil)) + nil) + (seq + (exit + (when (boot-equal name (frameName f)) + (exit + (|throwKeyedMsg| 'S2IZ0019 ; existing frame with same name + (cons name nil))))))) + (|initHistList|) + (spadlet |$interpreterFrameRing| + (cons (|emptyInterpreterFrame| name) |$interpreterFrameRing|)) + (|updateFromCurrentInterpreterFrame|) + ($erase (|histFileName|))))))) + +@ +\subsection{defun closeInterpreterFrame} +\begin{verbatim} +closeInterpreterFrame(name) == + -- if name = NIL then it means the current frame + null rest $interpreterFrameRing => + name and (name ^= $interpreterFrameName) => + throwKeyedMsg("S2IZ0020",[$interpreterFrameName]) + throwKeyedMsg("S2IZ0021",NIL) + if null name then $interpreterFrameRing := rest $interpreterFrameRing + else -- find the frame + found := nil + ifr := NIL + for f in $interpreterFrameRing repeat + found or (name ^= frameName(f)) => ifr := CONS(f,ifr) + found := true + not found => throwKeyedMsg("S2IZ0022",[name]) + _$ERASE makeHistFileName(name) + $interpreterFrameRing := nreverse ifr + updateFromCurrentInterpreterFrame() +\end{verbatim} +<>= +(defun |closeInterpreterFrame| (name) + (prog (ifr found) + (return + (seq + (cond + ((null (cdr |$interpreterFrameRing|)) + (cond + ((and name (nequal name |$interpreterFrameName|)) + (|throwKeyedMsg| 'S2IZ0020 ; 1 frame left. not the correct name. + (cons |$interpreterFrameName| nil))) + (t (|throwKeyedMsg| 'S2IZ0021 nil)))) ; only 1 frame left, not closed + (t + (cond + ((null name) + (spadlet |$interpreterFrameRing| (cdr |$interpreterFrameRing|))) + (t + (spadlet found nil) + (spadlet ifr nil) + (do ((tmp0 |$interpreterFrameRing| (cdr tmp0)) (f nil)) + ((or (atom tmp0) (progn (setq f (car tmp0)) nil)) nil) + (seq + (exit + (cond + ((or found (nequal name (frameName f))) + (spadlet ifr (cons f ifr))) + (t + (spadlet found t)))))) + (cond + ((null found) + (|throwKeyedMsg| 'S2IZ0022 (cons name nil))) + (t + ($erase (|makeHistFileName| name)) + (spadlet |$interpreterFrameRing| (nreverse ifr)))))) + (|updateFromCurrentInterpreterFrame|))))))) + +@ +\subsection{defun displayFrameNames} +\begin{verbatim} +displayFrameNames() == + fs := "append"/[ ['%l,'" ",:bright frameName f] for f in + $interpreterFrameRing] + sayKeyedMsg("S2IZ0024",[fs]) +\end{verbatim} +<>= +(defun |displayFrameNames| () + (prog (fs) + (return + (seq + (progn + (spadlet fs + (prog (tmp0) + (spadlet tmp0 NIL) + (return + (do ((tmp1 |$interpreterFrameRing| (cdr tmp1)) (f nil)) + ((or (atom tmp1) + (progn (setq f (car tmp1)) nil)) + tmp0) + (seq + (exit + (setq tmp0 + (append tmp0 (cons '|%l| + (cons (makestring " ") (|bright| (frameName f)))))))))))) + (|sayKeyedMsg| 'S2IZ0024 (cons fs nil))))))) ; frame names are ... + +@ +\subsection{defun importFromFrame} +\begin{verbatim} +importFromFrame args == + -- args should have the form [frameName,:varNames] + if args and atom args then args := [args] + null args => throwKeyedMsg("S2IZ0073",NIL) + [fname,:args] := args + not member(fname,frameNames()) => + throwKeyedMsg("S2IZ0074",[fname]) + fname = frameName first $interpreterFrameRing => + throwKeyedMsg("S2IZ0075",NIL) + fenv := frameEnvironment fname + null args => + x := UPCASE queryUserKeyedMsg("S2IZ0076",[fname]) + MEMQ(STRING2ID_-N(x,1),'(Y YES)) => + vars := NIL + for [v,:props] in CAAR fenv repeat + v = "--macros" => + for [m,:.] in props repeat vars := cons(m,vars) + vars := cons(v,vars) + importFromFrame [fname,:vars] + sayKeyedMsg("S2IZ0077",[fname]) + for v in args repeat + plist := GETALIST(CAAR fenv,v) + plist => + -- remove anything with the same name in the current frame + clearCmdParts ['propert,v] + for [prop,:val] in plist repeat + putHist(v,prop,val,$InteractiveFrame) + (m := get("--macros--",v,fenv)) => + putHist("--macros--",v,m,$InteractiveFrame) + sayKeyedMsg("S2IZ0079",[v,fname]) + sayKeyedMsg("S2IZ0078",[fname]) +\end{verbatim} +<>= +(defun |importFromFrame| (args) + (prog (temp1 fname fenv x v props vars plist prop val m) + (return + (seq + (progn + (when (and args (atom args)) + (spadlet args (cons args nil))) + (cond + ((null args) + (|throwKeyedMsg| 'S2IZ0073 nil)) ; missing frame name + (t + (spadlet temp1 args) + (spadlet fname (car temp1)) + (spadlet args (cdr temp1)) + (cond + ((null (|member| fname (|frameNames|))) + (|throwKeyedMsg| 'S2IZ0074 (cons fname nil))) ; not frame name + ((boot-equal fname (frameName (car |$interpreterFrameRing|))) + (|throwKeyedMsg| 'S2IZ0075 NIL)) ; cannot import from curr frame + (t + (spadlet fenv (|frameEnvironment| fname)) + (cond + ((null args) + (spadlet x + (upcase (|queryUserKeyedMsg| 'S2IZ0076 (cons fname nil)))) + ; import everything? + (cond + ((memq (string2id-n x 1) '(y yes)) + (spadlet vars nil) + (do ((tmp0 (caar fenv) (cdr tmp0)) (tmp1 nil)) + ((or (atom tmp0) + (progn (setq tmp1 (car tmp0)) nil) + (progn + (progn + (spadlet v (car tmp1)) + (spadlet props (cdr tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (cond + ((boot-equal v '|--macros|) + (do ((tmp2 props (cdr tmp2)) + (tmp3 nil)) + ((or (atom tmp2) + (progn (setq tmp3 (car tmp2)) nil) + (progn + (progn (spadlet m (car tmp3)) tmp3) + nil)) + nil) + (seq + (exit + (spadlet vars (cons m vars)))))) + (t (spadlet vars (cons v vars))))))) + (|importFromFrame| (cons fname vars))) + (t + (|sayKeyedMsg| 'S2IZ0077 (cons fname nil))))) + (t + (do ((tmp4 args (cdr tmp4)) (v nil)) + ((or (atom tmp4) (progn (setq v (car tmp4)) nil)) nil) + (seq + (exit + (progn + (spadlet plist (getalist (caar fenv) v)) + (cond + (plist + (|clearCmdParts| (cons '|propert| (cons v nil))) + (do ((tmp5 plist (cdr tmp5)) (tmp6 nil)) + ((or (atom tmp5) + (progn (setq tmp6 (car tmp5)) nil) + (progn + (progn + (spadlet prop (car tmp6)) + (spadlet val (cdr tmp6)) + tmp6) + nil)) + nil) + (seq + (exit (|putHist| v prop val |$InteractiveFrame|))))) + ((spadlet m (|get| '|--macros--| v fenv)) + (|putHist| '|--macros--| v m |$InteractiveFrame|)) + (t + (|sayKeyedMsg| 'S2IZ0079 ; frame not found + (cons v (cons fname nil))))))))) + (|sayKeyedMsg| 'S2IZ0078 ; import complete + (cons fname nil))))))))))))) + +@ +\subsection{defun frame} +\begin{verbatim} +-- the system command + +frame l == frameSpad2Cmd l +\end{verbatim} +<>= +(defun |frame| (l) + (|frameSpad2Cmd| l)) + +@ +\subsection{defun frameSpad2Cmd} +\begin{verbatim} +frameSpad2Cmd args == + frameArgs := '(drop import last names new next) + $options => throwKeyedMsg("S2IZ0016",['")frame"]) + null(args) => helpSpad2Cmd ['frame] + arg := selectOptionLC(first args,frameArgs,'optionError) + args := rest args + if args is [a] then args := a + if ATOM args then args := object2Identifier args + arg = 'drop => + args and PAIRP(args) => throwKeyedMsg("S2IZ0017",[args]) + closeInterpreterFrame(args) + arg = 'import => importFromFrame args + arg = 'last => previousInterpreterFrame() + arg = 'names => displayFrameNames() + arg = 'new => + args and PAIRP(args) => throwKeyedMsg("S2IZ0017",[args]) + addNewInterpreterFrame(args) + arg = 'next => nextInterpreterFrame() + + NIL +\end{verbatim} +<>= +(defun |frameSpad2Cmd| (args) + (prog (frameArgs arg a) + (return + (progn + (spadlet frameArgs '(|drop| |import| |last| |names| |new| |next|)) + (cond + (|$options| + (|throwKeyedMsg| 'S2IZ0016 ; frame command does not take options + (cons (makestring ")frame") nil))) + ((null args) + (|helpSpad2Cmd| (cons '|frame| nil))) + (t + (spadlet arg + (|selectOptionLC| (car args) frameArgs '|optionError|)) + (spadlet args (cdr args)) + (cond + ((and (pairp args) + (eq (qcdr args) nil) + (progn (spadlet a (qcar args)) t)) + (spadlet args a))) + (when (atom args) + (spadlet args (|object2Identifier| args))) + (cond + ((boot-equal arg '|drop|) + (cond + ((and args (pairp args)) + (|throwKeyedMsg| 'S2IZ0017 ; not a valid frame name + (cons args nil))) + (t (|closeInterpreterFrame| args)))) + ((boot-equal arg '|import|) + (|importFromFrame| args)) + ((boot-equal arg '|last|) + (|previousInterpreterFrame|)) + ((boot-equal arg '|names|) + (|displayFrameNames|)) + ((boot-equal arg '|new|) + (cond + ((and args (pairp args)) + (|throwKeyedMsg| 'S2IZ0017 ; not a valid frame name + (cons args nil))) + (t + (|addNewInterpreterFrame| args)))) + ((boot-equal arg '|next|) + (|nextInterpreterFrame|)) + (t nil)))))))) + +@ +\section{Frame File Messages} +<>= +S2IZ0016 + The %1b system command takes arguments but no options. +S2IZ0017 + %1b is not a valid frame name +S2IZ0018 + You must provide a name for the new frame. +S2IZ0019 + You cannot use the name %1b for a new frame because an existing + frame already has that name. +S2IZ0020 + There is only one frame active and therefore that cannot be closed. + Furthermore, the frame name you gave is not the name of the current frame. + The current frame is called %1b . +S2IZ0021 + The current frame is the only active one. Issue %b )clear all %d to + clear its contents. +S2IZ0022 + There is no frame called %1b and so your command cannot be + processed. +S2IZ0024 + The names of the existing frames are: %1 %l + The current frame is the first one listed. +S2IZ0073 + %b )frame import %d must be followed by the frame name. The names + of objects in that frame can then optionally follow the frame name. + For example, + %ceon %b )frame import calculus %d %ceoff + imports all objects in the %b calculus %d frame, and + %ceon %b )frame import calculus epsilon delta %d %ceoff + imports the objects named %b epsilon %d and %b delta %d from the + frame %b calculus %d . + Please note that if the current frame contained any information + about objects with these names, then that information would be + cleared before the import took place. +S2IZ0074 + You cannot import anything from the frame %1b because that is not + the name of an existing frame. +S2IZ0075 + You cannot import from the current frame (nor is there a need!). +S2IZ0076 + User verification required: + do you really want to import everything from the frame %1b ? + If so, please enter %b y %d or %b yes %d : +S2IZ0077 + On your request, AXIOM will not import everything from frame %1b. +S2IZ0078 + Import from frame %1b is complete. Please issue %b )display all %d + if you wish to see the contents of the current frame. +S2IZ0079 + AXIOM cannot import %1b from frame %2b because it cannot be found. +@ +\chapter{The Undo Mechanism} +\section{)undo} +\index{ugSysCmdundo} + +\index{undo} + + +\par\noindent{\bf User Level Required:} interpreter + +\par\noindent{\bf Command Syntax:} +\begin{list}{} +\item{\tt )undo} +\item{\tt )undo} {\it integer} +\item{\tt )undo} {\it integer [option]} +\item{\tt )undo} {\tt )redo} +\end{list} +% +where {\it option} is one of +% +\begin{list}{} +\item{\tt )after} +\item{\tt )before} +\end{list} + +\par\noindent{\bf Command Description:} + +This command is used to +restore the state of the user environment to an earlier +point in the interactive session. +The argument of an {\tt )undo} is an integer which must designate some +step number in the interactive session. + +\begin{verbatim} +)undo n +)undo n )after +\end{verbatim} +These commands return the state of the interactive +environment to that immediately after step {\tt n}. +If {\tt n} is a positive number, then {\tt n} refers to step nummber +{\tt n}. If {\tt n} is a negative number, it refers to the \tt n-th +previous command (that is, undoes the effects of the last $-n$ +commands). + +A {\tt )clear all} resets the {\tt )undo} facility. +Otherwise, an {\tt )undo} undoes the effect of {\tt )clear} with +options {\tt properties}, {\tt value}, and {\tt mode}, and +that of a previous {\tt undo}. +If any such system commands are given between steps $n$ and +$n + 1$ ($n > 0$), their effect is undone +for {\tt )undo m} for any $0 < m \leq n$.. + +The command {\tt )undo} is equivalent to {\tt )undo -1} (it undoes +the effect of the previous user expression). +The command {\tt )undo 0} undoes any of the above system commands +issued since the last user expression. + +\begin{verbatim} +)undo n )before +\end{verbatim} +This command returns the state of the interactive +environment to that immediately before step {\tt n}. +Any {\tt )undo} or {\tt )clear} system commands +given before step {\tt n} will not be undone. + +\begin{verbatim} +)undo )redo +\end{verbatim} +This command reads the file {\tt redo.input}. +created by the last {\tt )undo} command. +This file consists of all user input lines, excluding those +backtracked over due to a previous {\tt )undo}. + +\par\noindent{\bf Also See:} +{\tt )history} \index{ugSysCmdhistory}. +The command {\tt )history )write} will eliminate the ``undone'' command +lines of your program. +\section{Variables Used} +\section{Data Structures} +[[$frameRecord = [delta1, delta2,... ] ]] where +[[delta(i)]] contains changes in the ``backwards'' direction. +Each [[delta(i)]] has the form [[((var . proplist)...)]] where +proplist denotes an ordinary proplist. For example, an entry +of the form [[((x (value) (mode (Integer)))...)]] indicates that +to undo 1 step, [[x]]'s value is cleared and its mode should be set +to [[(Integer)]]. + +A [[delta(i)]] of the form [[(systemCommand . delta)]] is a special +delta indicating changes due to system commands executed between +the last command and the current command. By recording these deltas +separately, it is possible to undo to either BEFORE or AFTER +the command. These special [[delta(i)]]s are given ONLY when a +a system command is given which alters the environment. + +Note: [[recordFrame('system)]] is called before a command is executed, and +[[recordFrame('normal)]] is called after (see processInteractive1). +If no changes are found for former, no special entry is given. + +The [[$previousBindings]] is a copy of the +[[CAAR $InteractiveFrame]]. This is used to +compute the [[delta(i)]]s stored in [[$frameRecord]]. +\section{Functions} +\subsection{Initial Undo Variables} +\begin{verbatim} +$undoFlag := true --Default setting for undo is "on" +$frameRecord := nil --Initial setting for frame record +$previousBindings := nil +\end{verbatim} +<>= +(defvar |$undoFlag| t "t means we record undo information") +(defvar |$frameRecord| nil "a list of value changes") +(defvar |$previousBindings| nil "a copy of Interactive Frame info for undo") +(defvar |$reportUndo| nil "t means we report the steps undo takes") +@ +\subsection{defun undo} +\begin{verbatim} +undo(l) == +--undo takes one option ")redo" which simply reads "redo.input", +-- a file created by every normal )undo command (see below) + undoWhen := 'after + if $options is [[key]] then + stringPrefix?(s := PNAME key,'"redo") => + $options := nil --clear $options so that "read" won't see them + read '(redo_.input) + not stringPrefix?(s,'"before") => + userError '"only option to undo is _")redo_"" + undoWhen := 'before + n := + null l => -1 + first l + if IDENTP n then + n := PARSE_-INTEGER PNAME n + if not FIXP n then userError '"undo argument must be an integer" + $InteractiveFrame := undoSteps(undoCount n,undoWhen) + nil +\end{verbatim} +<>= +(defun |undo| (l) + (prog (tmp1 key s undoWhen n) + (return + (progn + (spadlet undoWhen '|after|) + (when + (and (pairp |$options|) + (eq (qcdr |$options|) nil) + (progn + (spadlet tmp1 (qcar |$options|)) + (and (pairp tmp1) + (eq (qcdr tmp1) nil) + (progn (spadlet key (qcar tmp1)) t))) + (cond + ((|stringPrefix?| (spadlet s (pname key)) (makestring "redo")) + (spadlet |$options| nil) + (|read| '(|redo.input|))) + ((null (|stringPrefix?| s (makestring "before"))) + (|userError| (makestring "only option to undo is \")redo\""))) + (t + (spadlet undoWhen '|before|))))) + (if (null l) + (spadlet n (spaddifference 1)) + (spadlet n (car l))) + (when (identp n) + (spadlet n (parse-integer (pname n))) + (cond + ((null (fixp n)) + (|userError| (makestring "undo argument must be an integer"))) + (t + nil))) + (spadlet |$InteractiveFrame| (|undoSteps| (|undoCount| n) undoWhen)) + nil)))) + +@ +\subsection{defun recordFrame} +\begin{verbatim} +recordFrame(systemNormal) == + null $undoFlag => nil --do nothing if facility is turned off + currentAlist := KAR $frameRecord + delta := diffAlist(CAAR $InteractiveFrame,$previousBindings) + if systemNormal = 'system then + null delta => return nil --do not record + delta := ['systemCommand,:delta] + $frameRecord := [delta,:$frameRecord] + $previousBindings := --copy all but the individual properties + [CONS(CAR x,[CONS(CAR y,CDR y) for y in CDR x]) for x in CAAR $InteractiveFrame] + first $frameRecord +\end{verbatim} +<>= +(defun |recordFrame| (systemNormal) + (prog (currentAlist delta) + (return + (seq + (cond + ((null |$undoFlag|) nil) + (t + (spadlet currentAlist (kar |$frameRecord|)) + (spadlet delta + (|diffAlist| (caar |$InteractiveFrame|) |$previousBindings|)) + (cond + ((boot-equal systemNormal '|system|) + (cond + ((null delta) + (return nil)) + (t + (spadlet delta (cons '|systemCommand| delta)))))) + (spadlet |$frameRecord| (cons delta |$frameRecord|)) + (spadlet |$previousBindings| + (prog (tmp0) + (spadlet tmp0 nil) + (return + (do ((tmp1 (caar |$InteractiveFrame|) (cdr tmp1)) (x nil)) + ((or (atom tmp1) + (progn (setq x (car tmp1)) nil)) + (nreverse0 tmp0)) + (seq + (exit + (setq tmp0 + (cons + (cons + (car x) + (prog (tmp2) + (spadlet tmp2 nil) + (return + (do ((tmp3 (cdr x) (cdr tmp3)) (|y| nil)) + ((or (atom tmp3) + (progn (setq |y| (car tmp3)) nil)) + (nreverse0 tmp2)) + (seq + (exit + (setq tmp2 (cons (cons (car |y|) (cdr |y|)) tmp2)))))))) + tmp0)))))))) + (car |$frameRecord|))))))) + +@ +\subsection{defun diffAlist} +\begin{verbatim} +diffAlist(new,old) == +--record only those properties which are different + for (pair := [name,:proplist]) in new repeat + -- name has an entry both in new and old world + -- (1) if the old world had no proplist for that variable, then + -- record NIL as the value of each new property + -- (2) if the old world does have a proplist for that variable, then + -- a) for each property with a value: give the old value + -- b) for each property missing: give NIL as the old value + oldPair := ASSQ(name,old) => + null (oldProplist := CDR oldPair) => + --record old values of new properties as NIL + acc := [[name,:[[prop] for [prop,:.] in proplist]],:acc] + deltas := nil + for (propval := [prop,:val]) in proplist repeat + null (oldPropval := ASSOC(prop,oldProplist)) => --missing property + deltas := [[prop],:deltas] + EQ(CDR oldPropval,val) => 'skip + deltas := [oldPropval,:deltas] + deltas => acc := [[name,:NREVERSE deltas],:acc] + acc := [[name,:[[prop] for [prop,:.] in proplist]],:acc] +--record properties absent on new list (say, from a )cl all) + for (oldPair := [name,:r]) in old repeat + r and null LASSQ(name,new) => + acc := [oldPair,:acc] + -- name has an entry both in new and old world + -- (1) if the new world has no proplist for that variable + -- (a) if the old world does, record the old proplist + -- (b) if the old world does not, record nothing + -- (2) if the new world has a proplist for that variable, it has + -- been handled by the first loop. + res := NREVERSE acc + if BOUNDP '$reportUndo and $reportUndo then reportUndo res + res +\end{verbatim} +<>= +(defun |diffAlist| (new old) + (prog (proplist oldPair oldProplist val oldPropval deltas prop name r acc res) + (return + (seq + (progn + (do ((tmp0 new (cdr tmp0)) (pair nil)) + ((or (atom tmp0) + (progn (setq pair (car tmp0)) nil) + (progn + (progn + (spadlet name (car pair)) + (spadlet proplist (cdr pair)) + pair) + nil)) + nil) + (seq + (exit + (cond + ((spadlet oldPair (assq name old)) + (cond + ((null (spadlet oldProplist (cdr oldPair))) + (spadlet acc + (cons + (cons + name + (prog (tmp1) + (spadlet tmp1 nil) + (return + (do ((tmp2 proplist (cdr tmp2)) (tmp3 nil)) + ((or (atom tmp2) + (progn (setq tmp3 (car tmp2)) nil) + (progn + (progn (spadlet prop (car tmp3)) tmp3) + nil)) + (nreverse0 tmp1)) + (seq + (exit + (setq tmp1 (cons (cons prop nil) tmp1)))))))) + acc))) + (t + (spadlet deltas nil) + (do ((tmp4 proplist (cdr tmp4)) (|propval| nil)) + ((or (atom tmp4) + (progn (setq |propval| (car tmp4)) nil) + (progn + (progn + (spadlet prop (car |propval|)) + (spadlet val (cdr |propval|)) + |propval|) + nil)) + nil) + (seq + (exit + (cond + ((null (spadlet oldPropval (|assoc| prop oldProplist))) + (spadlet deltas (cons (cons prop nil) deltas))) + ((eq (cdr oldPropval) val) '|skip|) + (t (spadlet deltas (cons oldPropval deltas))))))) + (when deltas + (spadlet acc + (cons (cons name (nreverse deltas)) acc)))))) + (t + (spadlet acc + (cons + (cons + name + (prog (tmp5) + (spadlet tmp5 nil) + (return + (do ((tmp6 proplist (cdr tmp6)) (tmp7 nil)) + ((or (atom tmp6) + (progn (setq tmp7 (CAR tmp6)) nil) + (progn + (progn (spadlet prop (CAR tmp7)) tmp7) + nil)) + (nreverse0 tmp5)) + (seq + (exit + (setq tmp5 (cons (cons prop nil) tmp5)))))))) + acc))))))) + (seq + (do ((tmp8 old (cdr tmp8)) (oldPair nil)) + ((or (atom tmp8) + (progn (setq oldPair (car tmp8)) nil) + (progn + (progn + (spadlet name (car oldPair)) + (spadlet r (cdr oldPair)) + oldPair) + nil)) + nil) + (seq + (exit + (cond + ((and r (null (lassq name new))) + (exit + (spadlet acc (cons oldPair acc)))))))) + (spadlet res (nreverse acc)) + (cond + ((and (boundp '|$reportUndo|) |$reportUndo|) + (|reportUndo| res))) + (exit res))))))) + +@ +\subsection{defun reportUndo} +This function is enabled by setting [[|$reportUndo]] to a non-nil value. +An example of the output generated is: +\begin{verbatim} +r := binary(22/7) + + + ___ + (1) 11.001 + Type: BinaryExpansion +Properties of % :: + value was: NIL + value is: ((|BinaryExpansion|) WRAPPED . #(1 (1 1) NIL (0 0 1))) +Properties of r :: + value was: NIL + value is: ((|BinaryExpansion|) WRAPPED . #(1 (1 1) NIL (0 0 1))) + +\end{verbatim} + +\begin{verbatim} +reportUndo acc == + for [name,:proplist] in acc repeat + sayBrightly STRCONC("Properties of ",PNAME name,'" ::") + curproplist := LASSOC(name,CAAR $InteractiveFrame) + for [prop,:value] in proplist repeat + sayBrightlyNT ['" ",prop,'" was: "] + pp value + sayBrightlyNT ['" ",prop,'" is: "] + pp LASSOC(prop,curproplist) +\end{verbatim} +<>= +(defun |reportUndo| (acc) + (prog (name proplist curproplist prop value) + (return + (seq + (do ((tmp0 acc (cdr tmp0)) (tmp1 nil)) + ((or (atom tmp0) + (progn (setq tmp1 (car tmp0)) nil) + (progn + (progn + (spadlet name (car tmp1)) + (spadlet proplist (cdr tmp1)) + tmp1) + nil)) + nil) + (seq + (exit + (progn + (|sayBrightly| + (strconc '|Properties of | (pname name) (makestring " ::"))) + (spadlet curproplist (lassoc name (caar |$InteractiveFrame|))) + (do ((tmp2 proplist (cdr tmp2)) (tmp3 nil)) + ((or (atom tmp2) + (progn (setq tmp3 (car tmp2)) nil) + (progn + (progn + (spadlet prop (car tmp3)) + (spadlet value (cdr tmp3)) + tmp3) + nil)) + nil) + (seq + (exit + (progn + (|sayBrightlyNT| + (cons + (makestring " ") + (cons prop (cons (makestring " was: ") nil)))) + (|pp| value) + (|sayBrightlyNT| + (cons + (makestring " ") + (cons prop (cons (makestring " is: ") nil)))) + (|pp| (lassoc prop curproplist)))))))))))))) + +@ +\subsection{defun clearFrame} +\begin{verbatim} +clearFrame() == + clearCmdAll() + $frameRecord := nil + $previousBindings := nil +\end{verbatim} +<>= +(defun |clearFrame| () + (progn + (|clearCmdAll|) + (spadlet |$frameRecord| nil) + (spadlet |$previousBindings| nil))) + +@ +\subsection{defun undoCount} +\begin{verbatim} +--======================================================================= +-- Undoing previous m commands +--======================================================================= +undoCount(n) == --computes the number of undo's, given $IOindex +--pp ["IOindex = ",$IOindex] + m := + n >= 0 => $IOindex - n - 1 + -n + m >= $IOindex => userError STRCONC('"Magnitude of undo argument must be less than step number (",STRINGIMAGE $IOindex,'").") + m +\end{verbatim} +<>= +(defun |undoCount| (n) + (prog (m) + (return + (progn + (spadlet m + (cond + ((>= n 0) (spaddifference (spaddifference |$IOindex| n) 1)) + (t (spaddifference n)))) + (cond + ((>= m |$IOindex|) + (|userError| + (strconc + (makestring + "Magnitude of undo argument must be less than step number (") + (stringimage |$IOindex|) (makestring ").")))) + (t m)))))) + +@ +\subsection{defun undoSteps} +\begin{verbatim} +undoSteps(m,beforeOrAfter) == +-- undoes m previous commands; if )before option, then undo one extra at end +--Example: if $IOindex now is 6 and m = 2 then general layout of $frameRecord, +-- after the call to recordFrame below will be: +-- ( +-- ( +-- ( +-- ( +-- +-- ) where system +-- command entries are optional and identified by (systemCommand . change). +-- For a ")undo 3 )after", m = 2 and undoStep swill restore the environment +-- up to, but not including . +-- An "undo 3 )before" will additionally restore . +-- Thus, the later requires one extra undo at the end. + writeInputLines('redo,$IOindex - m) + recordFrame('normal) --do NOT mark this as a system command change + --do this undo FIRST (i=0 case) + env := COPY CAAR $InteractiveFrame + for i in 0..m for framelist in tails $frameRecord repeat + env := undoSingleStep(first framelist,env) + framelist is [.,['systemCommand,:systemDelta],:.] => +-- pp '"===============> AHA <=============" + framelist := rest framelist --undoing system commands given + env := undoSingleStep(systemDelta,env) -- before command line + lastTailSeen := framelist + if beforeOrAfter = 'before then --do one additional undo for )before + env := undoSingleStep(first rest lastTailSeen,env) + $frameRecord := rest $frameRecord --flush the effect of extra recordFrame + $InteractiveFrame := LIST LIST env +\end{verbatim} +<>= +(defun |undoSteps| (m beforeOrAfter) + (prog (tmp1 tmp2 systemDelta framelist lastTailSeen env) + (return + (seq + (progn + (|writeInputLines| '|redo| (spaddifference |$IOindex| m)) + (|recordFrame| '|normal|) + (spadlet env (copy (caar |$InteractiveFrame|))) + (do ((|i| 0 (qsadd1 |i|)) (framelist |$frameRecord| (cdr framelist))) + ((or (qsgreaterp |i| m) (atom framelist)) nil) + (seq + (exit + (progn + (spadlet env (|undoSingleStep| (CAR framelist) env)) + (cond + ((and (pairp framelist) + (progn + (spadlet tmp1 (qcdr framelist)) + (and (pairp tmp1) + (progn + (spadlet tmp2 (qcar tmp1)) + (and (pairp tmp2) + (eq (qcar tmp2) '|systemCommand|) + (progn + (spadlet systemDelta (qcdr tmp2)) + t)))))) + (spadlet framelist (cdr framelist)) + (spadlet env (|undoSingleStep| systemDelta env))) + (t (spadlet lastTailSeen framelist))))))) + (cond + ((boot-equal beforeOrAfter '|before|) + (spadlet env (|undoSingleStep| (car (cdr lastTailSeen)) env)))) + (spadlet |$frameRecord| (cdr |$frameRecord|)) + (spadlet |$InteractiveFrame| (list (list env)))))))) + +@ +\subsection{defun undoSingleStep} +\begin{verbatim} +undoSingleStep(changes,env) == +--Each change is a name-proplist pair. For each change: +-- (1) if there exists a proplist in env, then for each prop-value change: +-- (a) if the prop exists in env, RPLAC in the change value +-- (b) otherwise, CONS it onto the front of prop-values for that name +-- (2) add change to the front of env +-- pp '"----Undoing 1 step--------" +-- pp changes + for (change := [name,:changeList]) in changes repeat + if LASSOC('localModemap,changeList) then + changeList := undoLocalModemapHack changeList + pairlist := ASSQ(name,env) => + proplist := CDR pairlist => + for (pair := [prop,:value]) in changeList repeat + node := ASSQ(prop,proplist) => RPLACD(node,value) + RPLACD(proplist,[CAR proplist,:CDR proplist]) + RPLACA(proplist,pair) + RPLACD(pairlist,changeList) + env := [change,:env] + env +\end{verbatim} +<>= +(defun |undoSingleStep| (changes env) + (prog (name changeList pairlist proplist prop value node) + (return + (seq + (progn + (do ((tmp0 changes (cdr tmp0)) (|change| nil)) + ((or (atom tmp0) + (progn (setq |change| (car tmp0)) nil) + (progn + (progn + (spadlet name (car |change|)) + (spadlet changeList (cdr |change|)) + |change|) + nil)) + nil) + (seq + (exit + (progn + (when (lassoc '|localModemap| changeList) + (spadlet changeList (|undoLocalModemapHack| changeList))) + (cond + ((spadlet pairlist (assq name env)) + (cond + ((spadlet proplist (cdr pairlist)) + (do ((tmp1 changeList (cdr tmp1)) (pair nil)) + ((or (atom tmp1) + (progn (setq pair (car tmp1)) nil) + (progn + (progn + (spadlet prop (car pair)) + (spadlet value (cdr pair)) + pair) + nil)) + nil) + (seq + (exit + (cond + ((spadlet node (assq prop proplist)) + (rplacd node value)) + (t + (rplacd proplist + (cons (car proplist) (cdr proplist))) + (rplaca proplist pair))))))) + (t (rplacd pairlist changeList)))) + (t + (spadlet env (cons |change| env)))))))) + env))))) + +@ +\subsection{defun undoLocalModemapHack} +\begin{verbatim} +undoLocalModemapHack changeList == + [newPair for (pair := [name,:value]) in changeList | newPair] where newPair == + name = 'localModemap => [name] + pair +\end{verbatim} +<>= +(defun |undoLocalModemapHack| (changeList) + (prog (name value) + (return + (seq + (prog (tmp0) + (spadlet tmp0 NIL) + (return + (do ((tmp1 changeList (cdr tmp1)) (pair nil)) + ((or (atom tmp1) + (progn (setq pair (car tmp1)) nil) + (progn + (progn + (spadlet name (car pair)) + (spadlet value (cdr pair)) + pair) + nil)) + (nreverse0 tmp0)) + (seq + (exit + (cond + ((cond + ((boot-equal name '|localModemap|) (cons name nil)) + (t pair)) + (setq tmp0 + (cons + (cond + ((boot-equal name '|localModemap|) (cons name nil)) + (t pair)) tmp0))))))))))))) + +@ +\subsection{defun removeUndoLines} +Removing undo lines from [[)hist )write linelist]] +\begin{verbatim} +removeUndoLines u == --called by writeInputLines + xtra := + STRINGP $currentLine => [$currentLine] + REVERSE $currentLine + xtra := [x for x in xtra | not stringPrefix?('")history",x)] + u := [:u, :xtra] + not (or/[stringPrefix?('")undo",x) for x in u]) => u + --(1) reverse the list + --(2) walk down the (reversed) list: when >n appears remove: + -- (a) system commands + -- (b) if n > 0: (replace n by n-1; remove a command; repeat (a-b)) + savedIOindex := $IOindex --save value + $IOindex := 1 + for y in tails u repeat + (x := first y).0 = char '_) => + stringPrefix?('")undo",s := trimString x) => --parse "undo )option" + s1 := trimString SUBSTRING(s,5,nil) + if s1 ^= '")redo" then + m := charPosition(char '_),s1,0) + code := + m < MAXINDEX s1 => s1.(m + 1) + char 'a + s2 := trimString SUBSTRING(s1,0,m) + n := + s1 = '")redo" => 0 + s2 ^= '"" => undoCount PARSE_-INTEGER s2 + -1 + RPLACA(y,CONCAT('">",code,STRINGIMAGE n)) + nil + $IOindex := $IOindex + 1 --referenced by undoCount + acc := nil + for y in tails NREVERSE u repeat + (x := first y).0 = char '_> => + code := x . 1 --code = a,b, or r + n := PARSE_-INTEGER SUBSTRING(x,2,nil) --n = number of undo steps + y := rest y --kill >n line + while y repeat + c := first y + c.0 = char '_) or c.0 = char '_> => y := rest y --kill system commands + n = 0 => return nil --including undos + n := n - 1 + y := rest y --kill command + y and code^= char 'b => acc := [c,:acc] --add last unless )before + acc := [x,:acc] + $IOindex := savedIOindex + acc +\end{verbatim} +<>= +(defun |removeUndoLines| (u) + (prog (xtra savedIOindex s s1 m s2 x code c n y acc) + (return + (seq + (progn + (spadlet xtra + (cond + ((stringp |$currentLine|) (cons |$currentLine| nil)) + (t (reverse |$currentLine|)))) + (spadlet xtra + (prog (tmp0) + (spadlet tmp0 nil) + (return + (do ((tmp1 xtra (cdr tmp1)) (x nil)) + ((or (atom tmp1) + (progn (setq x (car tmp1)) nil)) + (nreverse0 tmp0)) + (seq + (exit + (cond + ((null (|stringPrefix?| (makestring ")history") x)) + (setq tmp0 (cons x tmp0)))))))))) + (spadlet u (append u xtra)) + (cond + ((null + (prog (tmp2) + (spadlet tmp2 nil) + (return + (do ((tmp3 nil tmp2) (tmp4 u (cdr tmp4)) (x nil)) + ((or tmp3 (atom tmp4) (progn (setq x (car tmp4)) nil)) tmp2) + (seq + (exit + (setq tmp2 + (or tmp2 (|stringPrefix?| (makestring ")undo") x))))))))) u) + (t + (spadlet savedIOindex |$IOindex|) + (spadlet |$IOindex| 1) + (do ((y u (cdr y))) + ((atom y) nil) + (seq + (exit + (cond + ((boot-equal (elt (spadlet x (car y)) 0) (|char| '|)|)) + (cond + ((|stringPrefix?| (makestring ")undo") + (spadlet s (|trimString| x))) + (spadlet s1 (|trimString| (substring s 5 nil))) + (cond + ((nequal s1 (makestring ")redo")) + (spadlet m (|charPosition| (|char| '|)|) s1 0)) + (spadlet code + (cond + ((> (maxindex s1) m) (elt s1 (plus m 1))) + (t (|char| '|a|)))) + (spadlet s2 (|trimString| (substring s1 0 m))))) + (spadlet n + (cond + ((boot-equal s1 (makestring ")redo")) + 0) + ((nequal s2 (makestring "")) + (|undoCount| (parse-integer s2))) + (t (spaddifference 1)))) + (rplaca y + (concat (makestring ">") code (stringimage n)))) + (t nil))) + (t (spadlet |$IOindex| (plus |$IOindex| 1))))))) + (spadlet acc nil) + (do ((y (nreverse u) (cdr y))) + ((atom y) nil) + (seq + (exit + (cond + ((boot-equal (elt (spadlet x (car y)) 0) (|char| '>)) + (spadlet code (elt x 1)) + (spadlet n (parse-integer (substring x 2 nil))) + (spadlet y (cdr y)) + (do () + ((null y) nil) + (seq + (exit + (progn + (spadlet c (car y)) + (cond + ((or (boot-equal (elt c 0) (|char| '|)|)) + (boot-equal (elt c 0) (|char| '>))) + (spadlet y (cdr y))) + ((eql n 0) + (return nil)) + (t + (spadlet n (spaddifference n 1)) + (spadlet y (cdr y)))))))) + (cond + ((and y (nequal code (|char| '|b|))) + (spadlet acc (cons c acc))))) + (t (spadlet acc (cons x acc))))))) + (spadlet |$IOindex| savedIOindex) + acc))))))) + +@ + +\chapter{The Spad Server Mechanism} +<>= +(defvar $openServerIfTrue t "t means try starting an open server") +(defconstant $SpadServerName "/tmp/.d" "the name of the spad server socket") +(defvar |$SpadServer| nil "t means Scratchpad acts as a remote server") + +@ + +\chapter{The Help Browser Mechanism} +The Axiom book on the help browser is a complete rewrite of the +hyperdoc mechanism. There are several components that were needed +to make this function. Most of the web browser components are +described in bookvol11.pamphlet. This portion describes some of +the design issues needed to support the interface. + +The axServer command takes a port (defaulting to 8085) and a +program to handle the browser interaction (defaulting to multiServ). +The axServer function opens the port, constructs the stream, and +passes the stream to multiServ. The multiServ loop processes one +interaction at a time. + +So the basic process is that the Axiom ``)browse'' command opens a +socket and listens for http requests. Based on the type of request +(either 'GET' or 'POST') and the content of the request, which is +one of: +\begin{itemize} +\item command - algebra request/response +\item lispcall - a lisp s-expression to be evaluated +\item showcall - an Axiom )show command +\end{itemize} +the multiServ function will call a handler function to evaluate +the command line and construct a response. GET requests result +in a new browser page. POST requests result in an inline result. + +Most responses contain the fields: +\begin{itemize} +\item stepnum - this is the Axiom step number +\item command - this is the original command from the browser +\item algebra - this is the Axiom 2D algebra output +\item mathml - this is the MathML version of the Axiom algebra +\item type - this is the type of the Axiom result +\end{itemize} + +\section{Browsers, MathML, and Fonts} +This work has the Firefox browser as its target. Firefox has built-in +support for MathML, javascript, and XMLHttpRequest handling. More details +are available in bookvol11.pamphlet but the very basic machinery for +communication with the browser involves a dance between the browser +and the multiServ function (see the axserver.spad.pamphlet). + +In particular, a simple request is embedded in a web page as: +\begin{verbatim} +
      +
    • + +
      +
    • +
    +\end{verbatim} +which says that this is an html ``input'' field of type ``submit''. +The CSS display class is ``subbut'' which is of a different color +than the surrounding text to make it obvious that you can click on +this field. Clickable fields that have no response text are of class +``noresult''. + +The javascript call to ``makeRequest'' gives the ``id'' of this input +field, which must be unique in the page, as an argument. In this case, +the argument is 'p3'. The ``value'' field holds the display text which +will be passed back to Axiom as a command. + +When the result arrives the ``showanswer'' function will select out +the mathml field of the response, construct the ``id'' of the html +div to hold the response by concatenating the string ``ans'' (answer) +to the ``id'' of the request resulting, in this case, as ``ansp3''. +The ``showanswer'' function will find this div and replace it with a +div containing the mathml result. + +The ``makeRequest'' function is: +\begin{verbatim} + function makeRequest(arg) { + http_request = new XMLHttpRequest(); + var command = commandline(arg); + //alert(command); + http_request.open('POST', '127.0.0.1:8085', true); + http_request.onreadystatechange = handleResponse; + http_request.setRequestHeader('Content-Type', 'text/plain'); + http_request.send("command="+command); + return(false); +\end{verbatim} +It contains a request to open a local server connection to Axiom, +sets ``handleResponse'' as the function to call on reply, sets up +the type of request, fills in the command field, and sends off the +http request. + +When a response is received, the ``handleResponse'' function checks +for the correct reply state, strips out the important text, and +calls ``showanswer''. +\begin{verbatim} + function handleResponse() { + if (http_request.readyState == 4) { + if (http_request.status == 200) { + showanswer(http_request.responseText,'mathAns'); + } else + { + alert('There was a problem with the request.'+ http_request.statusText); + } + } + } +\end{verbatim} +See bookvol11.pamphlet for further details. + +\section{The axServer/multiServ loop} +The basic call to start an Axiom browser listener is: +\begin{verbatim} + )set message autoload off + )set output mathml on + axServer(8085,multiServ)$AXSERV +\end{verbatim} + +This call sets the port, opens a socket, attaches it to a stream, +and then calls ``multiServ'' with that stream. The ``multiServ'' +function loops serving web responses to that port. + +\section{The )browse command} +In order to make the whole process cleaner the function ``)browse'' +handles the details. This code creates the command-line function for )browse + +The browse function does the internal equivalent of the following 3 command +line statments: +\begin{verbatim} + )set message autoload off + )set output mathml on + axServer(8085,multiServ)$AXSERV +\end{verbatim} +which causes Axiom to start serving web pages on port 8085 + +For those unfamiliar with calling algebra from lisp there are a +few points to mention. + +The loadLib needs to be called to load the algebra code into the image. +Normally this is automatic but we are not using the interpreter so +we need to do this ``by hand''. + +Each algebra file contains a "constructor function" which builds the +domain, which is a vector, and then caches the vector so that every +call to the contructor returns an EQ vector, that is, the same vector. +In this case, we call the constructor $\vert$AxiomServer$\vert$ + +The axServer function was mangled internally to +$\vert$AXSERV;axServer;IMV;2$\vert$. +The multiServ function was mangled to $\vert$AXSERV;multiServ;SeV;3$\vert$ +Note well that if you change axserver.spad these names might change +which will generate the error message along the lines of: +\begin{verbatim} + System error: + The function $\vert$AXSERV;axServer;IMV;2$\vert$ is undefined. +\end{verbatim} + +To fix this you need to look at int/algebra/AXSERV.nrlib/code.lsp +and find the new mangled function name. A better solution would +be to dynamically look up the surface names in the domain vector. + +Each Axiom function expects the domain vector as the last argument. +This is not obvious from the call as the interpreter supplies it. +We must do that ``by hand''. + +We don't call the multiServ function. We pass it as a parameter to +the axServer function. When it does get called by the SPADCALL +macro it needs to be a lisp pair whose car is the function and +whose cdr is the domain vector. We construct that pair here as +the second argument to axServer. The third, hidden, argument to +axServer is the domain vector which we supply ``by hand''. + +The socket can be supplied on the command line but defaults to 8085. +Axiom supplies the arguments as a list. +<>= +(defun |browse| (socket) + (let (axserv browser) + (if socket + (setq socket (car socket)) + (setq socket 8085)) + (|set| '(|mes| |auto| |off|)) + (|set| '(|out| |mathml| |on|)) + (|loadLib| '|AxiomServer|) + (setq axserv (|AxiomServer|)) + (setq browser + (|AXSERV;axServer;IMV;2| socket + (cons #'|AXSERV;multiServ;SeV;3| axserv) axserv)))) + +@ +Now we have to bolt it into Axiom. This involves two lookups. + +We create the lisp pair +\begin{verbatim} +(|browse| . |development|) +\end{verbatim} +and cons it into the \$systemCommands command table. This allows the +command to be executed in development mode. This lookup decides if +this command is allowed. It also has the side-effect of putting the +command into the \$SYSCOMMANDS variable which is used to determine +if the token is a command. + +\section{The server support code} + +\chapter{Axiom Build-time Functions} +\subsection{defun spad-save} +The {\bf spad-save} function is just a cover function for more +lisp system specific save functions. There is no standard name +for saving a lisp image so we make one and conditionalize it +at compile time. + +This function is passed the name of an image that will be saved. +The saved image contains all of the loaded functions. + +This is used in the [[src/interp/Makefile.pamphlet]] in three places: +\begin{list}{} +\item creating depsys, an image for compiling axiom. + +Some of the Common Lisp code we compile uses macros which +are assumed to be available at compile time. The {\bf DEPSYS} +image is created to contain the compile time environment +and saved. We pipe compile commands into this environment +to compile from Common Lisp to machine dependent code. +\begin{verbatim} +DEPSYS= ${OBJ}/${SYS}/bin/depsys +\end{verbatim} + +\item creating savesys, an image for running axiom. + +Once we've compile all of the Common Lisp files we fire up +a clean lisp image called {\bf LOADSYS}, load all of the +final executable code and save it out as {\bf SAVESYS}. +The {\bf SAVESYS} image is copied to the [[${MNT}/${SYS}/bin]] +subdirectory and becomes the axiom executable image. +\begin{verbatim} +LOADSYS= ${OBJ}/${SYS}/bin/lisp +SAVESYS= ${OBJ}/${SYS}/bin/interpsys +AXIOMSYS= ${MNT}/${SYS}/bin/AXIOMsys +\end{verbatim} + + +\item creating debugsys, an image with all interpreted functions loaded. + +Occasionally we need to really get into the system internals. +The best way to do this is to run almost all of the lisp code +interpreted rather than compiled (note that cfuns.lisp and sockio.lisp +still need to be loaded in compiled form as they depend on the +loader to link with lisp internals). This image is nothing more +than a load of the file src/interp/debugsys.lisp.pamphlet. If +you need to make test modifications you can add code to that +file and it will show up here. +\begin{verbatim} +DEBUGSYS=${OBJ}/${SYS}/bin/debugsys +\end{verbatim} +\end{list} +<>= +(defun user::spad-save (save-file) + (setq |$SpadServer| nil) + (setq $openServerIfTrue t) +#+:AKCL + (system::save-system save-file) +#+:allegro + (if (fboundp 'boot::restart) + (excl::dumplisp :name save-file :restart-function #'boot::restart) + (excl::dumplisp :name save-file)) +#+Lucid + (if (fboundp 'boot::restart) + (sys::disksave save-file :restart-function #'boot::restart) + (sys::disksave save-file)) +#+:CCL + (preserve) +) + +@ + +\chapter{The Interpreter} +<>= +(in-package "BOOT") +<> + + +<> + +<> + +<> +<> +<> +<> +<> +<> + +<> +<> +<> +<> +<> +<> + +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> +<> + +<> +<> +<> +<> +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> + +<> +<> +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> + +<> +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +<> +<> +<> +<> +<> +<> + +@ +\chapter{The Global Variables} +\section{Star Global Variables} +\begin{tabular}{lll} +NAME & SET & USE \\ +*default-pathname-defaults* & reroot & restart \\ +*eof* & ncTopLevel & \\ +*features* & & restart \\ +*package* & & restart \\ +*standard-input* & & ncIntLoop \\ +*standard-output* & & ncIntLoop \\ +*top-level-hook* & set-restart-hook & \\ +\end{tabular} +\subsection{*default-pathname-defaults*} +The [[*default-pathname-defaults*]] variable is set by +[[make-absolute-filename]] called on the empty string. This has +the effect of setting the value to the [[AXIOM]] variable as this +function just concatenates the [[AXIOM]] variable onto the given string. +We pass this string to the common lisp [[pathname]] function to set it +to a real pathname. + +The [[*default-pathname-defaults*]] defaults common lisp variable is +set in [[restart]] to the current directory in most cases. If we are +working in Lucid Common Lisp ([[:lucid]]) on an IBM/370 mainframe +([[:ibm/370]]) then it is set to the empty string. Using Lucid on a +mainframe seems to use the variable [[vmlisp::$current-directory]]. + +\subsection{*eof*} +The [[*eof*]] variable is set to [[NIL]] in [[ncTopLevel]]. +\subsection{*features*} +The [[*features*]] variable from common lisp is tested for the presence +of the [[:unix]] keyword. Apparently this controls the use of Saturn, +a previous Axiom frontend. The Saturn frontend was never released as +open source and so this test and the associated variables are probably +not used. + +\subsection{*package*} +The [[*package*]] variable, from common lisp, is set in [[restart]] +to the [[BOOT]] package where the intepreter lives. +\subsection{*standard-input*} +The [[*standard-input*]] common lisp variable is used to set the +[[curinstream]] variable in [[ncIntLoop]]. + +This variable is an argument to [[serverReadLine]] in +the [[intloopReadConsole]] function. + +\subsection{*standard-output*} +The [[*standard-output*]] common lisp variable is used to set the +[[curoutstream]] variable in [[ncIntLoop]]. + +\subsection{*top-level-hook*} +The [[*top-level-hook*]] common lisp variable contains the name of +a function to invoke when an image is started. In our case it is +called [[restart]]. This is the entry point to the Axiom interpreter. + +\section{Dollar Global Variables} +\begin{tabular}{lll} +NAME & SET & USE \\ +\$boot & ncTopLevel & \\ +coerceFailure & & runspad \\ +curinstream & ncIntLoop & \\ +curoutstream & ncIntLoop & \\ +vmlisp::\$current-directory & restart & \\ + & reroot & \\ +\$currentLine & restart & removeUndoLines \\ +\$dalymode & & intloopReadConsole \\ +\$defaultMsgDatabaseName & reroot & \\ +\$directory-list & reroot & \\ +\$displayStartMsgs & & restart \\ +\$e & ncTopLevel & \\ +\$erMsgToss & SpadInterpretStream & \\ +\$fn & SpadInterpretStream & \\ +\$frameRecord & initvars & \\ + & clearFrame & \\ + & undoSteps & undoSteps \\ + & recordFrame & recordFrame \\ +\$HiFiAccess & initHist & historySpad2Cmd \\ + & historySpad2Cmd & \\ + & & setHistoryCore \\ +\$HistList & initHist & \\ +\$HistListAct & initHist & \\ +\$HistListLen & initHistList & \\ +\$HistRecord & initHistList & \\ +\$historyDirectory & & makeHistFileName \\ + & & makeHistFileName \\ +\$historyFileType & initvars & histInputFileName \\ +\$inclAssertions & SpadInterpretStream & \\ +\$inLispVM & spad & \\ +\$InteractiveFrame & restart & ncTopLevel \\ + & undo & recordFrame \\ + & undoSteps & undoSteps \\ + & & reportUndo \\ +\$InteractiveMode & ncTopLevel & \\ +\$internalHistoryTable & initvars & \\ +\$interpreterFrameName & initializeInterpreterFrameRing & \\ +\$interpreterFrameRing & initializeInterpreterFrameRing & \\ +\$InitialModemapFrame & & makeInitialModemapFrame \\ +\$intRestart & & intloop \\ +\$intTopLevel & intloop & \\ +\$IOindex & restart & historySpad2Cmd \\ + & removeUndoLines & undoCount \\ +\$genValue & bookvol5 & i-toplev \\ + & & i-analy \\ + & & i-syscmd \\ + & & i-spec1 \\ + & & i-spec2 \\ + & & i-map \\ +\$lastPos & SpadInterpretStream & \\ +\$libQuiet & SpadInterpretStream & \\ +\$library-directory-list & reroot & \\ +\$msgDatabaseName & reroot * \\ +\$ncMsgList & SpadInterpretStream & \\ +\$newcompErrorCount & SpadInterpretStream & \\ +\$newcompMode & SpadInterpretStream & \\ +\$newspad & ncTopLevel & \\ +\$nopos & & SpadInterpretStream \\ +\$okToExecuteMachineCode & SpadInterpretStream & \\ +\$oldHistoryFileName & initvars & oldHistFileName \\ +\$openServerIfTrue & restart & restart \\ + & spad-save & \\ + & initvars & \\ +\$options & & history \\ + & historySpad2Cmd & historySpad2Cmd \\ + & & undo \\ +\$previousBindings & initvars & \\ + & clearFrame & \\ + & recordFrame & recordFrame \\ +\$printLoadMsgs & restart & \\ +\$PrintCompilerMessageIfTrue & spad & \\ +\$promptMsg & SpadInterpretStream & \\ +\$relative-directory-list & & reroot \\ +\$relative-library-directory-list & & reroot \\ +\$reportUndo & initvars & diffAlist \\ +\$shoeReadLineFunction & SpadInterpretStream & \\ +\$spad & ncTopLevel & \\ +\$spadroot & reroot & initroot \\ + & & make-absolute-filename \\ + & & reroot \\ +\$SpadServer & restart & \\ +\$SpadServerName & initvars & restart \\ +\$systemCommandFunction & SpadInterpretStream & \\ +top\_level & & runspad \\ +\$quitTag & & runspad \\ +\$useInternalHistoryTable & initvars & initHist \\ + & setHistoryCore & setHistoryCore \\ +\$undoFlag & initvars & recordFrame \\ +\end{tabular} + +\subsection{\$boot} +The [[$boot]] variable is set to [[NIL]] in [[ncTopLevel]]. + +\subsection{coerceFailure} +The [[coerceFailure]] symbol is a catch tag used in [[runspad]] +to catch an exit from [[ncTopLevel]]. + +\subsection{curinstream} +The [[curinstream]] variable is set to the value of the +[[*standard-input*]] common lisp +variable in [[ncIntLoop]]. While not using the +``dollar'' convention this variable is still ``global''. + +\subsection{curinstream} +The [[curoutstream]] variable is set to the value of the +[[*standard-output*]] common lisp variable in [[ncIntLoop]]. +While not using the ``dollar'' convention this variable is still ``global''. + +\subsection{vmlisp::\$current-directory} +When running in Lucid Common Lisp ([[:lucid]]) on an IBM/370 mainframe +([[:ibm/370]]) this variable is used in place of the +[[*default-pathname-defaults*]] common lisp variable. +Otherwise this variable is +set to the empty string in [[restart]]. + +The [[reroot]] function sets this variable to the value of +[[$spadroot]] which itself has the value of the argument to the +[[reroot]] function. Since the argument to the [[reroot]] function is +an string which represents an absolute pathname pointing to AXIOM the +net result is that the [[$current-directory]] is set to point to the +shell [[AXIOM]] variable. + +So during execute both [[$current-directory]] and [[$spadroot]] reflect +the value of the [[AXIOM]] shell variable. + +\subsection{\$currentLine} +The [[$currentLine]] line is set to [[NIL]] in [[restart]]. +It is used in [[removeUndoLines]] in the undo mechanism. + +\subsection{\$dalymode} +The [[$dalymode]] variable is used in a case statement in +[[intloopReadConsole]]. This variable can be set to any non-nil +value. When not nil the interpreter will send any line that begins +with an ``[[(]]'' to be sent to the underlying lisp. This is useful +for debugging Axiom. The normal value of this variable is [[NIL]]. + +This variable was created as an alternative to prefixing every lisp +command with [[)lisp]]. When doing a lot of debugging this is tedious +and error prone. This variable was created to shortcut that process. +Clearly it breaks some semantics of the language accepted by the +interpreter as parens are used for grouping expressions. + +\subsection{\$defaultMsgDatabaseName} +The [[$defaultMsgDatabaseName]] is the absolute path to the +[[s2-us.msgs]] file which contains all of the english language +messages output by the system. + +\subsection{\$directory-list} +The [[$directory-list]] is a list of absolute directory names. +These names are made absolute by mapping the [[make-absolute-filename]] +over the variable [[$relative-directory-list]]. + +\subsection{\$displayStartMsgs} +The [[$displayStartMsgs]] variable is used in [[restart]] but is not +set so this is likely a bug. + +\subsection{\$e} +The [[$e]] variable is set to the value of +[[$InteractiveFrame]] which is set in [[restart]] to the value of the +call to the [[makeInitialModemapFrame]] function. This function simply +returns a copy of the variable [[$InitialModemapFrame]]. + +Thus [[$e]] is a copy of the variable [[$InitialModemapFrame]]. + +This variable is used in the undo mechanism. + +\subsection{\$erMsgToss} +The [[$erMsgToss]] variable is set to [[NIL]] in [[SpadInterpretStream]]. + +\subsection{\$fn} +The [[$fn]] variable is set in [[SpadInterpretStream]]. It is set to +the second argument which is a list. It appears that this list has the +same structure as an argument to the LispVM [[rdefiostream]] function. + +\subsection{\$frameRecord} +[[$frameRecord = [delta1, delta2,... ] ]] where +[[delta(i)]] contains changes in the ``backwards'' direction. +Each [[delta(i)]] has the form [[((var . proplist)...)]] where +proplist denotes an ordinary proplist. For example, an entry +of the form [[((x (value) (mode (Integer)))...)]] indicates that +to undo 1 step, [[x]]'s value is cleared and its mode should be set +to [[(Integer)]]. + +A [[delta(i)]] of the form [[(systemCommand . delta)]] is a special +delta indicating changes due to system commands executed between +the last command and the current command. By recording these deltas +separately, it is possible to undo to either BEFORE or AFTER +the command. These special [[delta(i)]]s are given ONLY when a +a system command is given which alters the environment. + +Note: [[recordFrame('system)]] is called before a command is executed, and +[[recordFrame('normal)]] is called after (see processInteractive1). +If no changes are found for former, no special entry is given. + +This is part of the undo mechanism. + +\subsection{\$genValue} +If the [[$genValue]] variable is true then evaluate generated code, +otherwise leave code unevaluated. If [[$genValue]] is false then we +are compiling. This variable is only defined and used locally. +<>= +(defvar |$genValue| nil "evaluate generated code if true") + +@ + +\subsection{\$HiFiAccess} +The [[$HiFiAccess]] is set by [[initHist]] to [[T]]. It is a flag +used by the history mechanism to record whether the history function +is currently on. It can be reset by using the axiom +command +\begin{verbatim} + )history off +\end{verbatim} +It appears that the name means ``History File Access''. + +The [[$HiFiAccess]] variable is used by [[historySpad2Cmd]] to check +whether history is turned on. [[T]] means it is, [[NIL]] means it is not. + +\subsection{\$HistList} +Thie [[$HistList]] variable is set by [[initHistList]] to an initial +value of [[NIL]] elements. The last element of the list is smashed to +point to the first element to make the list circular. +This is a circular list of length [[$HistListLen]]. + +\subsection{\$HistListAct} +The [[$HistListAct]] variable is set by [[initHistList]] to [[0]]. +This variable holds the actual number of elements in the history list. +This is the number of ``undoable'' steps. + +\subsection{\$HistListLen} +The [[$HistListLen]] variable is set by [[initHistList]] to [[20]]. +This is the length of a circular list maintained in the variable +[[$HistList]]. + +\subsection{\$HistRecord} +The [[$HistRecord]] variable is set by [[initHistList]] to [[NIL]]. +[[$HistRecord]] collects the input line, all variable bindings +and the output of a step, before it is written to the file named by +the function [[histFileName]]. + +\subsection{\$historyFileType} +The [[$historyFileType]] is set at load time by a call to +[[initvars]] to a value of ``[[axh]]''. It appears that this +is intended to be used as a filetype extension. +It is part of the history mechanism. It is used in [[makeHistFileName]] +as part of the history file name. + +\subsection{\$inclAssertions} +The [[$inclAssertions]] is set +in the function [[SpadInterpretStream]] to the list [[(aix |CommonLisp|)]] + +\subsection{\$internalHistoryTable} +The [[$internalHistoryTable]] variable is set at load time by a call to +[[initvars]] to a value of [[NIL]]. +It is part of the history mechanism. + +\subsection{\$interpreterFrameName} +The [[$interpreterFrameName]] variable, set in +[[initializeInterpreterFrameRing]] to the constant +[[initial]] to indicate that this is the initial (default) frame. + +Frames are structures that capture all of the variables defined in a +session. There can be multiple frames and the user can freely switch +between them. Frames are kept in a ring data structure so you can +move around the ring. + +\subsection{\$interpreterFrameRing} +The [[$interpreterFrameRing]] is set to a pair whose car is set to +the result of [[emptyInterpreterFrame]] + +\subsection{\$InitialModemapFrame} +This variable is copied and returned by the function +[[makeInitialModemapFrame]]. There is no initial value so this +is probably a bug. + +\subsection{\$inLispVM} +The [[$inLispVM]] is set to [[NIL]] in [[spad]]. LispVM is a +non-common lisp that runs on IBM/370 mainframes. This is probably dead +code. It appears that this list has the same structure as an argument +to the LispVM [[rdefiostream]] function. + +\subsection{\$InteractiveFrame} +The [[$InteractiveFrame]] is set in [[restart]] to the value of the +call to the [[makeInitialModemapFrame]] function. This function simply +returns a copy of the variable [[$InitialModemapFrame]] + +\subsection{\$InteractiveMode} +The [[$InteractiveMode]] is set to [[T]] in [[ncTopLevel]]. + +\subsection{\$intRestart} +The [[$intRestart]] variable is used in [[intloop]] but has no value. +This is probably a bug. While the variable's value is unchanged the +system will continually reenter the [[SpadInterpretStream]] function. + +\subsection{\$intTopLevel} +The [[$intTopLevel]] is a catch tag. Throwing to this tags which is +caught in the [[intloop]] will +restart the [[SpadInterpretStream]] function. + +\subsection{\$IOindex} +The [[$IOindex]] index variable is set to [[1]] in [[restart]]. +This variable is used in the [[historySpad2Cmd]] function in the +history mechanism. It is set in the [[removeUndoLines]] function +in the undo mechanism. + +This is used in the undo mechanism in function [[undoCount]] +to compute the number of undos. You can't undo more actions then +have already happened. + +\subsection{\$lastPos} +The [[$lastPos]] variable is set in [[SpadInterpretStream]] +to the value of the [[$nopos]] variable. +Since [[$nopos]] appears to have no value +this is likely a bug. + +\subsection{\$libQuiet} +The [[$libQuiet]] variable is set to the third argument of the +[[SpadInterpretStream]] function. This is passed from [[intloop]] +with the value of [[T]]. This variable appears to be intended to +control the printing of library loading messages which would need +to be suppressed if input was coming from a file. + +\subsection{\$library-directory-list} +The [[$library-directory-list]] variable is set by [[reroot]] by +mapping the function [[make-absolute-filename]] across the +[[$relative-library-directory-list]] variable which is not yet set so this +is probably a bug. + +\subsection{\$msgDatabaseName} +The [[$msgDatabaseName]] is set to [[NIL]] in [[reroot]]. + +\subsection{\$ncMsgList} +The [[$ncMsgList]] is set to [[NIL]] in [[SpadInterpretStream]]. + +\subsection{\$newcompErrorCount} +The [[$newcompErrorCount]] is set to [[0]] in [[SpadInterpretStream]]. + +\subsection{\$newcompMode} +The [[$newcompMode]] is set to [[NIL]] in [[SpadInterpretStream]]. + +\subsection{\$newspad} +The [[$newspad]] is set to [[T]] in [[ncTopLevel]]. + +\subsection{\$nopos} +The [[$nopos]] variable is used in [[SpadInterpretStream]] but does +not appear to have a value and is likely a bug. + +\subsection{\$oldHistoryFileName} +The [[$oldHistoryFileName]] is set at load time by a call to +[[initvars]] to a value of ``[[last]]''. +It is part of the history mechanism. It is used in the function +[[oldHistFileName]] and [[restoreHistory]]. + +\subsection{\$okToExecuteMachineCode} +The [[$okToExecuteMachineCode]] is set to [[T]] in [[SpadInterpretStream]]. + +\subsection{\$options} +The [[$options]] variable is tested by the [[history]] function. +If it is [[NIL]] then output the message +\begin{verbatim} + You have not used the correct syntax for the history command. + Issue )help history for more information. +\end{verbatim} + +The [[$options]] variable is tested in the [[historySpad2Cmd]] function. +It appears to record the options that were given to a spad command on +the input line. The function [[selectOptionLC]] appears to take a list +off options to scan. + +This variable is not yet set and is probably a bug. + +\subsection{\$previousBindings} +The [[$previousBindings]] is a copy of the +[[CAAR $InteractiveFrame]]. This is used to +compute the [[delta(i)]]s stored in [[$frameRecord]]. +This is part of the undo mechanism. + +\subsection{\$printLoadMsgs} +The [[$printLoadMsgs]] variable is set to [[T]] in [[restart]]. + +\subsection{\$PrintCompilerMessageIfTrue} +The [[$PrintCompilerMessageIfTrue]] variable is set to [[NIL]] in [[spad]]. + +\subsection{\$openServerIfTrue} +The [[$openServerIfTrue]] is tested in [[restart]] before it has been +set (and is thus a bug). It appears to control whether the interpreter +will be used as an open server, probably for OpenMath use. + +If an open server is not requested then this variable to [[NIL]] + +\subsection{\$promptMsg} +The [[$promptMsg]] variable is set to the constant [[S2CTP023]]. This +constant points to a message in [[src/doc/msgs/s2-us.msgs]]. This message +does nothing but print the argument value. + +\subsection{\$relative-directory-list} +The [[$relative-directory-list]] is used in [[reroot]] to create +[[$directory-list]] which is a list of absolute directory names. +It is not yet set and is probably a bug. + +\subsection{\$relative-library-directory-list} +The [[$relative-library-directory-list]] is used in [[reroot]] to create +a list of absolute directory names from [[$library-directory-list]] (which is +It is not yet set and is probably a bug). + +\subsection{\$reportUndo} +The [[$reportUndo]] variable is used in [[diffAlist]]. It was not normally +bound but has been set to [[T]] in [[initvars]]. If the variable is set +to [[T]] then we call [[reportUndo]]. + +It is part of the undo mechanism. + +\subsection{\$shoeReadLineFunction} +The [[$shoeReadLineFunction]] is set in [[SpadInterpretStream]] +to point to the +[[serverReadLine]] + +\subsection{\$spadroot} +The [[$spadroot]] variable is the internal name for the [[AXIOM]] +shell variable. + +The [[$spadroot]] variable is set in [[reroot]] to the value of the +argument. The argument is expected to be a directory name. + +The [[$spadroot]] variable is tested in [[initroot]]. + +The [[$spadroot]] variable is used by the function +[[make-absolute-filename]]. It concatenates this variable to the +front of a relative pathname to make it absolute. +\subsection{\$spad} +The [[$spad]] variable is set to [[T]] in [[ncTopLevel]]. + +\subsection{\$SpadServer} +If an open server is not requested then this variable to [[T]]. +It has no value before this time (and is thus a bug). + +\subsection{\$SpadServerName} +The [[$SpadServerName]] is passed to the [[openServer]] function, if the +function exists. + +\subsection{\$systemCommandFunction} +The [[$systemCommandFunction]] is set in [[SpadInterpretStream]] +to point to the function +[[InterpExecuteSpadSystemCommand]]. + +\subsection{top\_level} +The [[top\_level]] symbol is a catch tag used in [[runspad]] +to catch an exit from [[ncTopLevel]]. + +\subsection{\$quitTag} +The [[$quitTag]] is used as a variable in a [[catch]] block. +It appears that it can be thrown somewhere below [[ncTopLevel]]. + +\subsection{\$useInternalHistoryTable} +The [[$useInternalHistoryTable]] variable is set at load time by a call to +[[initvars]] to a value of [[NIL]]. It is part of the history mechanism. + +\subsection{\$undoFlag} +The [[$undoFlag]] is used in [[recordFrame]] to decide whether to do +undo recording. It is initially set to [[T]] in [[initvars]]. +This is part of the undo mechanism. + +\chapter{Makefile.bookvol5} +<<*>>= +LATEX=/usr/bin/latex +LISP=${AXIOM}/obj/linux/bin/lisp +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE -delay + +all: bookvol5 + @echo 0 done + +bookvol5: bookvol5.pamphlet + @echo 1 extracting the bookvol5reter + ${WEAVE} bookvol5.pamphlet >bookvol5.tex + ${LATEX} bookvol5.tex + ${LATEX} bookvol5.tex + ${TANGLE} -R"Interpreter" bookvol5.pamphlet >bookvol5.lisp + +remake: + @echo 2 rebuilding the makefile + @${TANGLE} bookvol5.pamphlet >Makefile.bookvol5 + +@ +\eject +\begin{thebibliography}{99} +\bibitem nothing +\end{thebibliography} +\end{document} diff --git a/books/bookvol6.pamphlet b/books/bookvol6.pamphlet new file mode 100644 index 0000000..4c44735 --- /dev/null +++ b/books/bookvol6.pamphlet @@ -0,0 +1,7170 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 6: Axiom Command}} +\end{titlepage} +\pagenumbering{roman} + +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group +Ltd. All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or +without modification, are permitted provided that the following +conditions are met: +\begin{itemize} +\item Redistributions of source code must retain the above + copyright notice, this list of conditions and the following + disclaimer. + +\item Redistributions in binary form must reproduce the above + copyright notice, this list of conditions and the following + disclaimer in the documentation and/or other materials + provided with the distribution. + +\item Neither the name of The Numerical ALgorithms Group Ltd. nor + the names of its contributors may be used to endorse or promote + products derived from this software without specific prior + written permission. +\end{itemize} + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND +CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, +INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF +MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR +CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, +INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, +BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; +LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER +CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT +LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN +ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE +POSSIBILITY OF SUCH DAMAGE. + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Overview} +The axiom system consists of a set of processes managed by the +superman process. The superman process, called sman, is normally +invoked from the axiom shell script in order to start a tree of +subprocesses. + +The {\tt axiom} command is a shell script that collects the +command line options for the {\tt sman} process, sets some shell +variables, and then invokes {\tt sman}. + +The {\tt sman} process starts the following tree of processes: +\begin{verbatim} + --xterm---bash---sman-|-AXIOMsys + |-clef---spadclient + |-hypertex + |-session + |-sman + |-viewman +\end{verbatim} + +\chapter{The axiom Command} +The {\tt axiom} command starts everything for Axiom. The options +for the {\tt axiom} command are: +\begin{verbatim} +axiom + [-ht |-noht] whether to use HyperDoc + [-gr |-nogr] whether to use Graphics + [-clef |-noclef] whether to use Clef + [-nonag |-nag] whether to use NAG + [-noiw |-iw] start in interpreter in a separate window + [-ihere |-noihere] start an interpreter in this window + [-nox] don't use X Windows + [-go |-nogo] whether to start system + [-ws wsname] use named workspace + [-list] list workspaces only + [-grprog fname] use named program for Graphics + [-nagprog fname] use named program for Nag + [-htprog fname] use named program for HyperDoc + [-clefprog fname] use named program for Clef + [-sessionprog fname] use named program for session + [-clientprog fname] use named program for spadclient + [-h] show usage +\end{verbatim} + +In detail, the command options are: +\subsection{[-ht $\vert$ -noht]} +\begin{verbatim} + [-ht |-noht] whether to use HyperDoc +\end{verbatim} +{\tt Hyperdoc}\cite{7} is the documentation tool for Axiom. The +{\tt -ht} option, enabled by default, will start this tool. +See Jenks\cite{1} Chapter 3 for further information on the +{\tt hyperdoc} subsystem. + +\subsection{[-gr $\vert$ -nogr]} +\begin{verbatim} + [-gr |-nogr] whether to use Graphics +\end{verbatim} +The {\tt graphics}\cite{8} subsystem is enabled using the +{\tt -gr} option, enabled by default. Graphics will appear as a +result of a draw command, such as +\begin{verbatim} + draw(sin(x),x=0..1) +\end{verbatim} +Note that attempting to use draw commands when the graphics is +disabled will simply hang the interpreter waiting for a response. +See Jenks\cite{1} Chapter 7 for further information on the +{\tt graphics} subsystem. + +\subsection{[-clef $\vert$ -noclef]} +\begin{verbatim} + [-clef |-noclef] whether to use Clef +\end{verbatim} +The {\tt clef} (Command Line Edit Facility) allows for command completion. +The list of command completion strings is in the last chapter of this +document. If {\tt clef}, enabled by default, is running then you can type: +\begin{verbatim} + x:Dena +\end{verbatim} +and this will automatically be expanded to: +\begin{verbatim} + x:DenavitHartenbergMatrix +\end{verbatim} + +The {\tt clef} program also allows command line editing. The commands are +special keyboard keys. +\begin{itemize} +\item HOME move to beginning of the line +\item END move to the end of the line +\item CTRL-END delete to end of the line +\item TAB command completion (multiple tabs give new choices) +\item UPARROW move back thru commands +\item DOWNARROW move forward thru commands +\item LEFTARROW move left on the line +\item RIGHTARROW move right on the line +\item INSERT toggle insert/overstrike +\end{itemize} +See Jenks\cite{1} page 21 for further information on the {\tt clef} command. + +\subsection{[-nonag $\vert$ -nag]} +\begin{verbatim} + [-nonag |-nag] whether to use NAG +\end{verbatim} +The {\tt nag} option, disabled by default, will attempt to start +the {\tt nagman} program in the \$AXIOM/lib subdirectory. Since +the open source version does not include the NAG numeric libraries +this option does not work. + +\subsection{[-noiw $\vert$ -iw]} +\begin{verbatim} + [-noiw |-iw] start in interpreter in a separate window +\end{verbatim} +The {\tt iw} option, disabled by default, will start a second +interpreter in its own window with its own frame. The fact that +the second interpreter is in its own frame can be seen using the +{\tt )frame} command. For instance, if you type +\begin{verbatim} + axiom -iw +\end{verbatim} +there will be two interpreter windows available, one in the current +window and one in a new window. In the current window if you type: +\begin{verbatim} + )frame names +\end{verbatim} +you will see: +\begin{verbatim} + The names of the existing frames are: + frame0 + frame1 + initial + The current frame is the first one listed. +\end{verbatim} + +In the second window, if you type +\begin{verbatim} + )frame names +\end{verbatim} +you will see: +\begin{verbatim} + The names of the existing frames are: + frame1 + frame0 + initial + The current frame is the first one listed. +\end{verbatim} +Setting +\begin{verbatim} + x:=3 +\end{verbatim} +in the second window will set the variable $x$ in the frame {\tt frame1}. +Switching to the first window and typing: +\begin{verbatim} + x +\end{verbatim} +gives: +\begin{verbatim} + + (1) x + Type: Variable x +\end{verbatim} +since the first window is in {\tt frame0} and the variable $x$ is +defined in {\tt frame1}. +But we can switch frames in the first window using +\begin{verbatim} + )frame next +\end{verbatim} +and then +\begin{verbatim} + x +\end{verbatim} +gives: +\begin{verbatim} + + (2) 3 + Type: PositiveInteger +\end{verbatim} +and now the two windows share the same frame space. +See Jenks\cite{1} page 579 for further information on the {\tt frame} command. + +\subsection{[-ihere $\vert$ -noihere]} +\begin{verbatim} + [-ihere |-noihere] start an interpreter in this window +\end{verbatim} +This option determines whether Axiom will start in the current window. +Using this option alone is not particularly useful and it is generally +used in combination with the {\tt -iw} option: +\begin{verbatim} + axiom -noihere -iw & +\end{verbatim} + +However, used alone, as in: +\begin{verbatim} + axiom -noihere & +\end{verbatim} +it will start Axiom and show the Hyperdoc window. Graphics will also +work from the Hyperdoc pages. + +\subsection{[-nox]} +\begin{verbatim} + [-nox] don't use X Windows +\end{verbatim} +allows Axiom to start the interpreter without Hyperdoc or the graphics +subsystem. This is useful for starting Axiom in an emacs buffer. + +\subsection{[-go $\vert$ -nogo]} +\begin{verbatim} + [-go |-nogo] whether to start system +\end{verbatim} +uses the {\tt -go} option, enabled by default, controls whether +the system starts from the command line. If the {\tt -nogo} +option is chosen the system prints the command line that would +have been issued. This is useful for finding out what the command +line options to {\tt sman} will be. For instance: +\begin{verbatim} + axiom -nogo -iw +\end{verbatim} +does not start Axiom but types out: +\begin{verbatim} +Would now start the processes. +exec ~/mnt/linux/bin/sman -iw -ws ~/mnt/linux/bin/AXIOMsys +\end{verbatim} + +\subsection{[-ws wsname]} +\begin{verbatim} + [-ws wsname] use named workspace +\end{verbatim} +In the {\tt -nogo} command above you can see that the default +workspace name is +\begin{verbatim} +-ws ~/mnt/linux/bin/AXIOMsys +\end{verbatim} +This option allows you to change that. This is useful for +debugging new system builds. During build a debugging version +of Axiom is created in the {\tt obj/linux/bin} directory. The +{\tt debugsys} image uses interpreted lisp code rather than +compiled code. This makes it possible to do deep debugging. To +use this workspace you would incant: +\begin{verbatim} + cd youraxiombuild + export AXIOM=`pwd`/mnt/linux + export PATH=$AXIOM/bin:$PATH + axiom -ws obj/linux/bin/debugsys +\end{verbatim} + + +\subsection{[-list]} +\begin{verbatim} + [-list] list workspaces only +\end{verbatim} +shows you the executable workspaces. Generally in a built system +there is only one, called\\ +{\tt \$AXIOM/bin/AXIOMsys}. + +\subsection{[-grprog fname]} +\begin{verbatim} + [-grprog fname] use named program for Graphics +\end{verbatim} +allows you to specify which program to use for the graphics. +By default this is\\ +{\tt \$AXIOM/lib/viewman}. + +\subsection{[-nagprog fname]} +\begin{verbatim} + [-nagprog fname] use named program for Nag +\end{verbatim} +allows you to specify which program to use for the NAG library +connection. By default this is\\ +{\tt \$AXIOM/lib/nagman} but it is disabled by default. + +\subsection{[-htprog fname]} +\begin{verbatim} + [-htprog fname] use named program for Hyperdoc +\end{verbatim} +allows you tp specify which program to use for Hyperdoc. By +default it is\\ +{\tt \$AXIOM/bin/hypertex -s}. + +\subsection{[-clefprog fname]} +\begin{verbatim} + [-clefprog fname] use named program for Clef +\end{verbatim} +allows you to specify which program to use for clef. By default +it is\\ +{\tt \$AXIOM/bin/clef -f \$AXIOM/lib/command.list -e}. + +\subsection{[-sessionprog fname]} +\begin{verbatim} + [-sessionprog fname] use named program for session +\end{verbatim} +allows you to specify the session manager program. By default it is\\ +{\tt \$AXIOM/lib/session}. + +\subsection{[-clientprog fname]} +\begin{verbatim} + [-clientprog fname] use named program for spadclient +\end{verbatim} +allows you to specify the spadclient program. By default it is\\ +{\tt \$AXIOM/lib/spadclient}. + +\subsection{[-h]} +\begin{verbatim} + [-h] show usage +\end{verbatim} + +<>= +#!/bin/sh + +@ +The {\tt MALLOCTYPE} shell variable is an {\tt IBM} {\tt AIX} +shell variable that controls buckets based extensions in the default +memory allocator which may enhance performance. AIX uses a new +memory management routine that does not zero {\tt malloc} memory +and does not round up to the nearest power of 2, unlike most non-AIX +systems. This can cause failures so we protect against that here. +See the AIX Performance Tuning Guide\cite{9} for details. +<>= + +MALLOCTYPE=3.1 +export MALLOCTYPE + +@ +The {\tt nagman} process needs to know the hostname +<>= +HOST=`hostname` +export HOST + +@ +There are 4 basic utilities used by this script. +The {\tt ciao} script for immediate exit: +<>= +ciao() { + echo "Goodbye." + exit 1 +} + +@ +The {\tt needsubopt} script which is used to issue an error message +when one of the command line options requires an option: +<>= +needsubopt () { + echo "The $1 option requires an argument." + ciao +} +@ +The {\tt showuse} script which gives basic command line help: +<>= +showuse() { +echo "axiom" +echo " [-ht |-noht] whether to use HyperDoc" +echo " [-gr |-nogr] whether to use Graphics" +echo " [-clef |-noclef] whether to use Clef" +echo " [-nonag |-nag] whether to use NAG" +echo " [-noiw |-iw] start in interpreter in a separate window" +echo " [-ihere |-noihere] start an interpreter in this window" +echo " [-nox] don't use X Windows" +echo " [-go |-nogo] whether to start system" +echo " [-ws wsname] use named workspace" +echo " [-list] list workspaces only" +echo " [-grprog fname] use named program for Graphics" +echo " [-nagprog fname] use named program for Nag" +echo " [-htprog fname] use named program for HyperDoc" +echo " [-clefprog fname] use named program for Clef" +echo " [-sessionprog fname] use named program for session" +echo " [-clientprog fname] use named program for spadclient" +echo " [-h] show usage" +} + +@ +List the various workspaces if asked. +<>= +listwspaces() +{ + echo "$1" + ls -l $2 | grep "sys$" + echo "" +} + +@ +Step 1. Ensure the environment is set. + +Just process ``-h''. If it exists in the command line then we +print out the simple command line help menu. +<>= + +if [ "$*" = "-h" ] ; then + showuse +fi +@ + +We assume that Axiom is installed in the standard place on +a linux system. We will modify this assumption as we process +the environment and command line. The term {\tt spad} is an +historical shortened version of the name {\tt scratchpad}, +the original name of the {\tt Axiom} system. +<>= +SPADDEFAULT=/usr/local/axiom/mnt/linux + +@ +If the {\tt \$AXIOM} shell variable is set then we use it.\\ +If not, then if the {\tt \$SPAD} shell variable is set then we use it.\\ +If not, then we try to use the default value above.\\ +If not, we simply fail. +<>= +if [ "$SPAD" = "" ] ; then + if [ "$AXIOM" = "" ] ; then + SPAD=$SPADDEFAULT + echo "AXIOM variable is not set" + echo "assuming AXIOM = $SPAD" + AXIOM=$SPAD + export AXIOM + else + SPAD=$AXIOM + fi + export SPAD +else + if [ "$AXIOM" = "" ] ; then + echo "AXIOM variable is not set" + echo "but SPAD = $SPAD" + echo "Using AXIOM = $SPAD" + AXIOM=$SPAD + export AXIOM + else + if [ ! "$SPAD" = "$AXIOM" ] ; then + echo "ignoring SPAD variable" + SPAD=$AXIOM + fi + fi +fi + +@ +If we get here then all attempts to find axiom have failed +so we complain and exit. +<>= +if [ ! -d "$SPAD" ] ; then + echo "The directory for Axiom, $SPAD, does not exist." + ciao +fi + +@ +If we get here we now +know where axiom lives. We need to see if we can find the +Aldor compiler. Aldor is optional and is not part of the open source +version of Axiom due to license issues. However you can install it +under the location specified by the {\tt \$AXIOMXLROOT} shell variable. +If the compiler is found we add it to the {\tt PATH}. +This shell variable defaults to the location:\\ +{\tt \$AXIOM/compiler} +<>= +if [ "$AXIOMXLROOT" = "" ] ; then +AXIOMXLROOT=${AXIOM}/compiler +fi +export AXIOMXLROOT +PATH=$AXIOM/bin:$AXIOMXLROOT/bin:${PATH} +export PATH + +@ +Name the workspace directories. +<>= +rootwsdir=$SPAD/bin + +@ +Step 2. Process command line arguments. + +First we set up the defaults for command-line arguments. +We don't want just a list by default +<>= +list=no + +@ +We default to actually executing the workspace. +<>= +go=yes + +@ +We default to the {\tt AXIOMsys} workspace. +<>= +wsname=AXIOMsys + +@ +And all other options are unset. +<>= +otheropts="" + +@ +For each option on the command line do +<>= +while [ "$*" != "" ] ; do +@ +<>= + + case $1 in +@ +If the user specified list anywhere then we give the workspace list +and exit. +<>= + + -list) list=yes + go=no;; +@ +If the user specified {\tt go} or {\tt nogo} we handle that case +<>= + -go) go=yes ;; + -nogo) go=no ;; + +@ +The workspace option requires an argument which follows immediately. +If the argument is missing we complain and exit. +<>= + -ws) + if [ "$2" = "" ] ; then needsubopt "$1" ; fi + shift + wsname="$1" + ;; + +@ +We can specify the various subprograms to use. +<>= + -nagprog|-grprog|-htprog|-clefprog|-sessionprog|-clientprog) + if [ "$2" = "" ] ; then needsubopt "$1" ; fi + otheropts="$otheropts $1 $2" + shift + ;; +@ +These options were not explained earlier and are only for developer use. +<>= + -paste|-rm|-rv) + if [ "$2" = "" ] ; then needsubopt "$1" ; fi + otheropts="$otheropts $1 $2" + shift + ;; +@ +We handle the various [{\tt -option} $\vert$ {\tt -nooption}] cases +<>= + -clef|-noclef|-gr|-nogr|-ht|-noht|-iw|-noiw) + otheropts="$otheropts $1" + ;; + -ihere|-noihere|-nox|-nag|-nonag) + otheropts="$otheropts $1" + ;; +@ +The user wanted help so we will not execute. +<>= + -h) + go=no + ;; +@ +The user is confused. Complain and exit. +<>= + *) echo "Unknown option: $1" + echo "To use a specific workspace use, e.g.: spad -ws $1" + ciao + ;; + esac +@ +Move to the next option and loop. +<>= + + shift +done + +@ +Step 3. Handle options that require special case handling. + +The user just wanted to know what workspaces are available. +<>= +if [ $list = yes ] ; then + listwspaces "AXIOM workspaces in \$AXIOM/bin = $rootwsdir: " $rootwsdir +fi + +@ +Try to ensure a suitable workspace on this host. +<>= +if [ `expr $wsname : '.*/.*'` = 0 ] ; then + serverws=$rootwsdir/$wsname +else + serverws=$wsname +fi +@ +If we can't find the executable then we complain and exit. +<>= + +if [ ! -x $serverws ] ; then + echo "Cannot find the executable $serverws" + showuse + ciao +fi +@ +The user just wanted to see what would happen so we output the +command line and exit. +<>= +if [ $go = no ] ; then + echo "Would now start the processes." + echo exec $SPAD/bin/sman $otheropts -ws $serverws + exit 0 +fi +@ +All of the options have been processed so we start {\tt sman} +<>= +exec $SPAD/bin/sman $otheropts -ws $serverws + +@ +\chapter{The {\tt sman} program} +\section{sman.h} +The spad\_proc structure holds information about the process id +of a child process, what to do when it dies, and the shell command +line necessary to restart the process. There is a linked list of +these structures which maintains the process list for axiom. +<>= +/* Process control definitions. Used by fork_you and spawn_of_hell */ + +/* When a process dies it kills off everything else */ +#define Die 1 +/* When a process dies, do nothing */ +#define NadaDelShitsky 2 +/* When a process dies start it up again */ +#define DoItAgain 3 +/* When hypertex dies, clean its socket */ +#define CleanHypertexSocket 4 + +typedef struct spad_proc { + int proc_id; /* process id of child */ + int death_action; /* one of the above constants */ + char *command; /* sh command line to restart the process */ + struct spad_proc *next; +} SpadProcess; + +@ +\section{sman} +\subsection{includes} +<>= +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#if defined(SUN4OS5platform) || defined(HP10platform) +#include +#endif + +#include "com.h" +#include "bsdsignal.h" +#include "sman.h" + +#include "bsdsignal.h1" +#include "sockio-c.h1" +#include "openpty.h1" +#include "sman.h1" + +@ +\subsection{variables} +<>= +char *ws_path; /* location of the AXIOM executable */ +int start_clef; /* start clef under spad */ +int start_graphics; /* start the viewman */ +int start_nagman; /* start the nagman */ +int start_ht; /* start hypertex */ +int start_spadclient; /* Start the client spad buffer */ +int start_local_spadclient; /* Start the client spad buffer */ +int use_X; /* Use the X windows environment */ +int server_num; /* AXIOM server number */ +@ +We add a debug flag so we can print information about what [[sman]] +is trying to do. This change is pervasive as it touches nearly every +routine. +<>= +int tpd=0; /* to-print-debug information */ + +/************************************************/ +/* definitions of programs which sman can start */ +/************************************************/ + +<> +<> +<> +<> +<> +<> +char *PasteFile = NULL; +char *MakeRecordFile = NULL; +char *VerifyRecordFile = NULL; + +SpadProcess *spad_process_list = NULL; +/***************************/ +/* sman defaults file name */ +/***************************/ + +#define SpadDefaultFile "spadprof.input" + +char ClefCommandLine[256]; + +#define BufSize 4096 /* size of communication buffer */ +char big_bad_buf[BufSize]; /* big I/O buffer */ + +Sock *session_io = NULL; /* socket connecting to session manager */ + +/***********************************************************/ +/* Some characters used and externally defined in edible.h */ +/***********************************************************/ + +unsigned char _INTR, _QUIT, _ERASE, _KILL, _EOF, _EOL, _RES1, _RES2; + +/*************************************/ +/* Stuff for opening pseudo-terminal */ +/*************************************/ + +int ptsNum, ptcNum; +char ptsPath[20], ptcPath[20]; + +char **new_envp; /* new environment for AXIOM */ +int child_pid; /* child's process id */ +struct termios oldbuf; /* the original settings */ +struct termios childbuf; /* terminal structure for user i/o */ + +int nagman_signal=0; +int death_signal = 0; + +@ +\subsection{process\_arguments} +<>= +static void +process_arguments(int argc,char ** argv) +{ + int arg; + if (tpd == 1) fprintf(stderr,"sman:process_arguments entered\n"); + for (arg = 1; arg < argc; arg++) { + if (strcmp(argv[arg], "-debug") == 0) + tpd = 1; + else if (strcmp(argv[arg], "-noclef") == 0) + start_clef = 0; + else if (strcmp(argv[arg], "-clef") == 0) + start_clef = 1; + else if (strcmp(argv[arg], "-gr") == 0) + start_graphics = 1; + else if (strcmp(argv[arg], "-nogr") == 0) + start_graphics = 0; + else if (strcmp(argv[arg], "-nag") == 0) + start_nagman = 1; + else if (strcmp(argv[arg], "-nonag") == 0) + start_nagman = 0; + else if (strcmp(argv[arg], "-ht") == 0) + start_ht = 1; + else if (strcmp(argv[arg], "-noht") == 0) + start_ht = 0; + else if (strcmp(argv[arg], "-iw") == 0) + start_spadclient = 1; + else if (strcmp(argv[arg], "-ihere") == 0) + start_local_spadclient = 1; + else if (strcmp(argv[arg], "-noihere") == 0) + start_local_spadclient = 0; + else if (strcmp(argv[arg], "-noiw") == 0) + start_spadclient = 0; + else if (strcmp(argv[arg], "-ws") == 0) + ws_path = argv[++arg]; + else if (strcmp(argv[arg], "-comp") == 0) + ws_path = "$AXIOM/etc/images/comp"; + else if (strcmp(argv[arg], "-nox") == 0) + { + use_X = 0; + start_local_spadclient = 1; + start_spadclient = 0; + start_ht = 0; + start_graphics = 0; + } + else if (strcmp(argv[arg], "-grprog") == 0) + GraphicsProgram = argv[++arg]; + else if (strcmp(argv[arg], "-nagprog") == 0) + NagManagerProgram = argv[++arg]; + else if (strcmp(argv[arg], "-htprog") == 0) + HypertexProgram = argv[++arg]; + else if (strcmp(argv[arg], "-clefprog") == 0) { + strcpy(ClefCommandLine,argv[++arg]); + ClefProgram = + strcat(ClefCommandLine, " -f $AXIOM/lib/command.list -e "); + } + else if (strcmp(argv[arg], "-sessionprog") == 0) + SessionManagerProgram = argv[++arg]; + else if (strcmp(argv[arg], "-clientprog") == 0) + SpadClientProgram = argv[++arg]; + else if (strcmp(argv[arg], "-rm") == 0) + MakeRecordFile = argv[++arg]; + else if (strcmp(argv[arg], "-rv") == 0) + VerifyRecordFile = argv[++arg]; + else if (strcmp(argv[arg], "-paste") == 0) + PasteFile = argv[++arg]; + else { + fprintf(stderr, "Usage: sman <-clef|-noclef> <-gr|-nogr> <-ht|-noht>"); + fprintf(stderr, " <-iw|-noiw> <-nag|-nonag> <-nox> <-comp>"); + fprintf(stderr, " <-ws spad_workspace> <-grprog path> <-htprog path>"); + fprintf(stderr, " <-clefprog path> <-sessionprog path> <-nagprog path>"); + fprintf(stderr, " <-clientprog path>\n"); + exit(-1); + } + } + if (tpd == 1) + { fprintf(stderr," sman "); + if (start_clef == 0) + fprintf(stderr,"-noclef "); + else + fprintf(stderr,"-clef "); + if (start_graphics == 0) + fprintf(stderr,"-nogr "); + else + fprintf(stderr,"-gr "); + if (start_nagman == 0) + fprintf(stderr,"-nonag "); + else + fprintf(stderr,"-nag "); + if (start_ht == 0) + fprintf(stderr,"-noht "); + else + fprintf(stderr,"-ht "); + if (start_spadclient == 0) + fprintf(stderr,"-noiw "); + else + fprintf(stderr,"-iw "); + if (start_local_spadclient == 0) + fprintf(stderr,"-noihere "); + else + fprintf(stderr,"-ihere "); + if (start_local_spadclient == 0) + fprintf(stderr,"-noihere "); + else + fprintf(stderr,"-ihere "); + if (use_X == 0) + fprintf(stderr,"-nox "); + fprintf(stderr,"-ws "); + fprintf(stderr,"'%s' ",ws_path); + fprintf(stderr,"-grprog "); + fprintf(stderr,"'%s' ",GraphicsProgram); + fprintf(stderr,"-nagprog "); + fprintf(stderr,"'%s' ",NagManagerProgram); + fprintf(stderr,"-htprog "); + fprintf(stderr,"'%s' ",HypertexProgram); + fprintf(stderr,"-clefprog "); + fprintf(stderr,"'%s' ",ClefCommandLine); + fprintf(stderr,"-sessionprog "); + fprintf(stderr,"'%s' ",SessionManagerProgram); + fprintf(stderr,"-clientprog "); + fprintf(stderr,"'%s' ",SpadClientProgram); + fprintf(stderr,"-rm "); + fprintf(stderr,"'%s' ",MakeRecordFile); + fprintf(stderr,"-rv "); + fprintf(stderr,"'%s' ",VerifyRecordFile); + fprintf(stderr,"-paste "); + fprintf(stderr,"'%s' ",PasteFile); + fprintf(stderr,"\n"); + } + if (tpd == 1) fprintf(stderr,"sman:process_arguments exit\n"); +} + +@ +\subsection{should\_I\_clef} +<>= +static int +should_I_clef(void) +{ + return(1); +} + +@ +\subsection{in\_X} +<>= +static int +in_X(void) +{ + if (getenv("DISPLAY")) return 1; + return 0; +} + +@ +\subsection{set\_up\_defaults} +These are the default values for sman. A '1' value means that +sman will try to start the given process, a '0' value means not +starting the process. + +We do not have replacement code for the [[nagman]] process nor +do we have a copy of the [[nag fortran library]] to test the process. +Until this changes we set [[start_nagman = 0]] in order to disable +starting this process by default. +<>= +static void +set_up_defaults(void) +{ + if (tpd == 1) fprintf(stderr,"sman:set_up_defaults entered\n"); + start_clef = should_I_clef(); + start_graphics = 1; + start_nagman = 0; + start_ht = 1; + start_spadclient = 0; + start_local_spadclient = 1; + use_X = isatty(0) && in_X(); + ws_path = "$AXIOM/bin/AXIOMsys"; + if (tpd == 1) fprintf(stderr,"sman:set_up_defaults exit\n"); +} + +@ +\subsection{process\_options} +<>= +static void +process_options(int argc, char **argv) +{ + if (tpd == 1) fprintf(stderr,"sman:process_options entered\n"); + set_up_defaults(); + process_arguments(argc, argv); + if (tpd == 1) fprintf(stderr,"sman:process_options exit\n"); +} + +@ +\subsection{death\_handler} +<>= +static void +death_handler(int sig) +{ + death_signal = 1; +} + +@ +\subsection{nagman\_handler} +<>= +static void +nagman_handler(int sig) +{ + nagman_signal=1; +} + +@ +\subsection{sman\_catch\_signals} +<>= +static void +sman_catch_signals(void) +{ + + /* Set up the signal handlers for sman */ + bsdSignal(SIGINT, SIG_IGN,RestartSystemCalls); + bsdSignal(SIGTERM, death_handler,RestartSystemCalls); + bsdSignal(SIGQUIT, death_handler,RestartSystemCalls); + bsdSignal(SIGHUP, death_handler,RestartSystemCalls); + bsdSignal(SIGILL, death_handler,RestartSystemCalls); + bsdSignal(SIGTRAP, death_handler,RestartSystemCalls); + bsdSignal(SIGIOT, death_handler,RestartSystemCalls); + bsdSignal(SIGBUS, death_handler,RestartSystemCalls); + bsdSignal(SIGSEGV, death_handler,RestartSystemCalls); + /* don't restart wait call on SIGUSR1 */ + bsdSignal(SIGUSR1, nagman_handler,DontRestartSystemCalls); + /* ONLY nagman should send this. + If an error (such as C-c) interrupts a NAGLINK call, nagman + gets a signal to clean up. We need to start another nagman + almost immediately to process the next NAGLINK request. + Since nagman takes a while to clean up, we treat it specially. + nagman should send a signal (USR1) to sman. + sman should respond by spawning a new nagman. + + so nagman is NOT a DoItAgain but a NadaDelShitsky. + + The USR1 mechanism does not work for HPUX 9 - use DoItAgain + */ + +} + +@ +\subsection{fix\_env} +insert SPADSERVER and SPADNUM variables into the environemnt +<>= +static void +fix_env(char **envp, int spadnum) +{ + int len, i; + char *sn; + for(len = 0; envp[len] != NULL; len++); + new_envp = (char **) malloc((len + 3) * sizeof(char *)); + new_envp[0] = "SPADSERVER=TRUE"; + sn = (char *) malloc(20 * sizeof(char)); + sprintf(sn, "SPADNUM=%d", spadnum); + new_envp[1] = sn; + for(i=0; i<=len; i++) + new_envp[i+2] = envp[i]; +} + +@ +\subsection{init\_term\_io} +<>= +static void +init_term_io(void) +{ + if(!isatty(0)) return; + if( tcgetattr(0, &oldbuf) == -1) { + perror("getting termios"); + return ; /* exit(-1); */ + } + if( tcgetattr(0, &childbuf) == -1) { + perror("getting termios"); + return ; /* exit(-1); */ + } + _INTR = oldbuf.c_cc[VINTR]; + _QUIT = oldbuf.c_cc[VQUIT]; + _ERASE = oldbuf.c_cc[VERASE]; + _KILL = oldbuf.c_cc[VKILL]; + _EOF = oldbuf.c_cc[VEOF]; + _EOL = oldbuf.c_cc[VEOL]; +} + +@ +\subsection{strPrefix} +<>= +static char * +strPrefix(char *prefix,char * s) +{ + while (*prefix != '\0' && *prefix == *s) { + prefix++; + s++; + } + if (*prefix == '\0') return s; + return NULL; +} + +@ +\subsection{check\_spad\_proc} +<>= +static void +check_spad_proc(char *file, char *prefix) +{ + char *num; + int pid; + if ((num = strPrefix(prefix, file))) { + pid = atoi(num); + if (pid > 2) { + kill(pid, 0); + if (kill(pid, 0) == -1 && errno == ESRCH) { + unlink(file); + } + } + } +} + +@ +\subsection{clean\_up\_old\_sockets} +<>= +static void +clean_up_old_sockets(void) +{ + char com[512], tmp_file[128]; + FILE *file; + int len; + sprintf(tmp_file, "/tmp/socks.%d", server_num); + sprintf(com, "ls /tmp/.d* /tmp/.s* /tmp/.i* /tmp/.h* 2> %s > %s", + tmp_file, tmp_file); + system(com); + file = fopen(tmp_file, "r"); + if (file == NULL) { + fprintf(stderr, "Can't open socket listing file\n"); + return; + } + while(fgets(com, 512, file) != NULL) { + len = strlen(com); + if (len) com[len-1] = '\0'; + else break; + check_spad_proc(com, "/tmp/.d"); + check_spad_proc(com, "/tmp/.s"); + check_spad_proc(com, "/tmp/.i"); + check_spad_proc(com, "/tmp/.h"); + } + fclose(file); + unlink(tmp_file); +} + +@ +\subsection{fork\_you} +<>= +static SpadProcess * +fork_you(int death_action) +{ + /* fork a new process, giving it a default death action */ + /* return NULL in child, SpadProcess in parent */ + int child_pid = fork(); + SpadProcess *proc; + if (!child_pid) return NULL; + proc = (SpadProcess *) malloc(sizeof(SpadProcess)); + proc->proc_id = child_pid; + proc->death_action = death_action; + proc->command = NULL; + proc->next = spad_process_list; + spad_process_list = proc; + return proc; +} + +@ +\subsection{exec\_command\_env} +Note that the next-to-last argument of {\tt execle} must be an +explicit NULL pointer. The previous naked 0 value was not correct. +<>= +static void +exec_command_env(char *command,char ** env) +{ + char new_command[512]; + sprintf(new_command, "exec %s", command); + execle("/bin/sh","/bin/sh", "-c", new_command, (char *)0, env); +} + +@ +\subsection{spawn\_of\_hell} +<>= +static SpadProcess * +spawn_of_hell(char *command, int death_action) +{ + SpadProcess *proc = fork_you(death_action); + if (proc != NULL) { + proc->command = command; + return proc; + } + exec_command_env(command, new_envp); + return NULL; +} + +@ +\subsection{start\_the\_spadclient} +run a AXIOM client in the main process +<>= +static void +start_the_spadclient(void) +{ + char command[256]; + if (start_clef) +#ifdef RIOSplatform + sprintf(command, + "aixterm -sb -sl 500 -name axiomclient -n AXIOM -T AXIOM -e %s %s", + ClefProgram, SpadClientProgram); +#else + sprintf(command, + "xterm -sb -sl 500 -name axiomclient -n AXIOM -T AXIOM -e %s %s", + ClefProgram, SpadClientProgram); +#endif + else +#ifdef RIOSplatform + sprintf(command, + "aixterm -sb -sl 500 -name axiomclient -n AXIOM -T AXIOM -e %s", + SpadClientProgram); +#else + sprintf(command, + "xterm -sb -sl 500 -name axiomclient -n AXIOM -T AXIOM -e %s", + SpadClientProgram); +#endif + if (tpd == 1) + fprintf(stderr,"sman:start_the_spadclient: %s\n",command); + spawn_of_hell(command, NadaDelShitsky); +} + +@ +\subsection{start\_the\_local\_spadclient} +<>= +static void +start_the_local_spadclient(void) +{ + char command[256]; + if (start_clef) + sprintf(command, "%s %s", ClefProgram, SpadClientProgram); + else + sprintf(command, "%s", SpadClientProgram); + if (tpd == 1) + fprintf(stderr,"sman:start_the_local_spadclient: %s\n",command); + spawn_of_hell(command, NadaDelShitsky); +} + +@ +\subsection{start\_the\_nagman} +<>= +static void +start_the_nagman(void) +{ +#if defined(HP9platform) + spawn_of_hell(NagManagerProgram,DoItAgain); +#else + spawn_of_hell(NagManagerProgram,NadaDelShitsky ); +#endif +} + +@ +\subsection{start\_the\_session\_manager} +<>= +static void +start_the_session_manager(void) +{ + spawn_of_hell(SessionManagerProgram, Die); +} + +@ +\subsection{start\_the\_hypertex} +<>= +static void +start_the_hypertex(void) +{ + char prog[512]; + + if (PasteFile){ + sprintf(prog, "%s -k -ip %s", HypertexProgram, PasteFile); + spawn_of_hell(prog, NadaDelShitsky); + } + else if (MakeRecordFile){ + sprintf(prog, "%s -k -rm %s", HypertexProgram,MakeRecordFile ); + spawn_of_hell(prog, NadaDelShitsky); + } + else if (VerifyRecordFile){ + sprintf(prog, "%s -k -rv %s", HypertexProgram, VerifyRecordFile); + spawn_of_hell(prog, NadaDelShitsky); + } + /* If we restart hyperdoc from the axiom command prompt */ + else spawn_of_hell(HypertexProgram, CleanHypertexSocket); +} + +@ +\subsection{start\_the\_graphics} +<>= +static void +start_the_graphics(void) +{ + spawn_of_hell(GraphicsProgram, DoItAgain); +} + +@ +\subsection{fork\_Axiom} +<>= +/* Start the AXIOM session in a separate process, */ +/* using a pseudo-terminal to catch all input and output */ +static void +fork_Axiom(void) +{ + char augmented_ws_path[256]; /* will append directory path */ + char *tmp_pointer; + SpadProcess *proc; + + proc = fork_you(Die); + child_pid = (proc == NULL ? 0 : proc->proc_id); + switch(child_pid) { + case -1 : + fprintf(stderr, "Can't create a new process \n"); + exit(0); + case 0: + /* Dissasociate from my parents group so all my child processes */ + /* look at my terminal as the controlling terminal for the */ + /* group */ + + if(setsid() < 0) { + perror("Dissassociating from parents group"); + exit(-1); + } + + close(ptsNum); + /* Now reopen the server side, so that pg, su, etc. work properly */ + + if ((ptsNum = open(ptsPath, O_RDWR)) < 0 ) { + perror("fork_Axiom: Failed to reopen server"); + exit(-1); + } +#if defined(SUN4OS5platform) || defined(HP10platform) + ioctl(ptsNum,I_PUSH,"ptem"); + ioctl(ptsNum,I_PUSH,"ldterm"); +#endif + + /* since I am the child, I can close ptc, and dup pts for all its */ + /* standard descriptors */ + + if( (dup2(ptsNum, 0) == -1) || + (dup2(ptsNum, 1) == -1) || + (dup2(ptsNum, 2) == -1) ) { + perror("trying to dupe the child"); + exit(-1); + } + close(ptcNum); + close(ptsNum); + + + /* I also have to turn off echoing, since I am echoing all the */ + /* input myself */ + + childbuf.c_lflag &= ~ECHO; + if( tcsetattr(0, TCSAFLUSH, &childbuf) == -1) { + perror("setting the term buffer"); + exit(-1); + } + strcpy(augmented_ws_path,ws_path); /* write the name */ + strcat(augmented_ws_path," "); /* space */ + strcat(augmented_ws_path,ws_path); /* name again */ + tmp_pointer = (char *) + strrchr(augmented_ws_path,'/'); /*pointer to last / */ + *(++tmp_pointer) = '\0'; + exec_command_env(augmented_ws_path, new_envp); + + /* fprintf(stderr, "Cannot execute the %s system.\n", ws_path); */ + + exit(0); + } +} + +@ +\subsection{start\_the\_Axiom} +<>= +static void +start_the_Axiom(char **envp) +{ + server_num = make_server_number(); + clean_up_old_sockets(); + if (server_num == -1) { + fprintf(stderr, "could not get an AXIOM server number\n"); + exit(-1); + } + if (ptyopen(&ptcNum, &ptsNum, ptcPath, ptsPath) == -1) { + perror("start_the_Axiom: ptyopen failed"); + exit(-1); + } + fix_env(envp, server_num); + fork_Axiom(); + close(ptsNum); +} + +@ +\subsection{clean\_up\_sockets} +In order to be able to restart hyperdoc from the axiom command prompt +we need to remove the socket for this server. +<>= +static void +clean_hypertex_socket(void) +{ + char name[256]; + sprintf(name, "%s%d", MenuServerName, server_num); + unlink(name); +} + +static void +clean_up_sockets(void) +{ + char name[256]; + sprintf(name, "%s%d", SpadServer, server_num); + unlink(name); + sprintf(name, "%s%d", SessionServer, server_num); + unlink(name); + sprintf(name, "%s%d", SessionIOName, server_num); + unlink(name); + clean_hypertex_socket(); +} + +@ +\subsection{read\_from\_spad\_io} +<>= +static void +read_from_spad_io(int ptcNum) +{ + int ret_code = 0, i=0; + static int mes_len =0; + ret_code = read(ptcNum, big_bad_buf, BufSize); + if (ret_code == -1) { + clean_up_sockets(); + exit(-1); + } + if (session_io == NULL) { + if (ret_code < mes_len) + mes_len -= ret_code; + else { + if (mes_len > 0) { + i = mes_len; + mes_len = 0; + } + else + i = 0; + ret_code = write(1, big_bad_buf+i, ret_code-i); + } + } + else + ret_code = swrite(session_io, big_bad_buf, ret_code, + "writing to session man"); + if (ret_code == -1) { + perror("writing output to session manager"); + clean_up_sockets(); + exit(-1); + } +} + +@ +\subsection{read\_from\_manager} +<>= +static void +read_from_manager(int ptcNum) +{ + int ret_code; + ret_code = sread(session_io, big_bad_buf, BufSize, "reading session io"); + if (ret_code == -1) { + return; + } + ret_code = write(ptcNum, big_bad_buf, ret_code); + if (ret_code == -1) { + return; + } +} + +@ +\subsection{manage\_spad\_io} +<>= +static void +manage_spad_io(int ptcNum) +{ + int ret_code, i, p; + fd_set rd; + while (1) { + rd = socket_mask; + FD_SET(ptcNum, &rd); + if (session_io != NULL) + FD_SET(session_io->socket, &rd); + ret_code = sselect(FD_SETSIZE, &rd, 0, 0, NULL); + if (ret_code == -1) { + perror("Session manager select"); + clean_up_sockets(); + exit(-1); + } + if (FD_ISSET(ptcNum, &rd)) { + read_from_spad_io(ptcNum); + } + for(i=0; i<2; i++) { + if (server[i].socket > 0 && FD_ISSET(server[i].socket, &rd)) { + p = accept_connection(server+i); + switch(p) { + case SessionIO: + session_io = purpose_table[SessionIO]; + /* printf("connected session manager\n\r");*/ + printf("\n"); + break; + default: + printf("sman: Unkown connection request type: %d\n", p); + break; + } + } + } + if (session_io != NULL && FD_ISSET(session_io->socket, &rd)) { + read_from_manager(ptcNum); + } + } +} + +@ +\subsection{init\_spad\_process\_list} +<>= +static void +init_spad_process_list(void) +{ + spad_process_list = NULL; +} + +@ +\subsection{print\_spad\_process\_list} +<>= +#if 0 +static void +print_spad_process_list() +{ + SpadProcess *proc; + for(proc = spad_process_list; proc != NULL; proc = proc->next) + fprintf(stderr, "proc_id = %d, death_action = %d\n", proc->proc_id, + proc->death_action); +} +#endif + +@ +\subsection{find\_child} +<>= +static SpadProcess * +find_child(int proc_id) +{ + SpadProcess *proc; + for(proc = spad_process_list; proc != NULL; proc = proc->next) + if (proc->proc_id == proc_id) return proc; + return NULL; +} + +@ +\subsection{kill\_all\_children} +<>= +static void +kill_all_children(void) +{ + char name[256]; + SpadProcess *proc; + + + for(proc = spad_process_list; proc != NULL; proc = proc->next) { + kill(proc->proc_id, SIGTERM); + } + sprintf(name, "/tmp/hyper%d.input",server_num); + unlink(name); + +} + +@ +\subsection{clean\_up\_terminal} +<>= +static void +clean_up_terminal(void) +{ + tcsetattr(0, TCSAFLUSH, &oldbuf); +} + +@ +\subsection{monitor\_children} +<>= +static void +monitor_children(void) +{ + int dead_baby, stat; + SpadProcess *proc; + while (1) { + stat = 0; + dead_baby = wait(&stat); + /* Check the value of dead_baby, since wait may have returned + a pid but subsequently we have received a signal. Yeuch! + In order to restart hyperdoc from the axiom command prompt + we no longer call clean_up_terminal */ + if (dead_baby == -1 && death_signal) { + kill_all_children(); + clean_up_sockets(); + sleep(2); + exit(0); + } + /* Check the value of dead_baby, since wait may have returned + a pid but subsequently we have received a signal. Yeuch! */ + if(dead_baby == -1 && nagman_signal) { + nagman_signal=0; + spawn_of_hell(NagManagerProgram,NadaDelShitsky); + continue; + } + + if (dead_baby == -1) { + fprintf(stderr, "sman: wait returned -1\n"); + continue; + } + proc = find_child(dead_baby); + if (proc == NULL) { + /* fprintf(stderr, "sman: %d is not known to be a child process\n", + dead_baby); + */ + continue; + } + switch(proc->death_action) { + /* In order to restart hyperdoc from the axiom command prompt + we no longer call clean_up_terminal. Instead we've added a + case to just clean up the socket. */ + case Die: + kill_all_children(); + clean_up_sockets(); + sleep(2); + exit(0); + case NadaDelShitsky: + break; + case DoItAgain: + spawn_of_hell(proc->command, DoItAgain); + break; + case CleanHypertexSocket: + clean_hypertex_socket(); + break; + } + } +} + +@ +\subsection{main sman} +The main procedure should return an [[int]]. We change the return value +here and in [[src/include/sman.h1]]. +<>= + return(0); +@ +<>= +int +main(int argc, char *argv[],char *envp[]) +{ + if (tpd == 1) fprintf(stderr,"sman:main entered\n"); + bsdSignal(SIGINT, SIG_IGN,RestartSystemCalls); + process_options(argc, argv); + + init_term_io(); + init_spad_process_list(); + start_the_Axiom(envp); + if (open_server(SessionIOName) == -2) { + fprintf(stderr, "Fatal error opening I/O socket\n"); + clean_up_sockets(); + exit(-1); + } + start_the_session_manager(); + if (start_spadclient) start_the_spadclient(); + if (start_local_spadclient) start_the_local_spadclient(); + if (start_nagman) start_the_nagman(); + if (start_ht) start_the_hypertex(); + if (start_graphics) start_the_graphics(); + sleep(1); + + if (fork_you(Die) != NULL) { + sman_catch_signals(); + monitor_children(); + exit(0); + } + manage_spad_io(ptcNum); + if (tpd == 1) fprintf(stderr,"sman:main exit\n"); +<> +} + +@ +\subsection{sman} +<>= +#define _SMAN_C + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +@ +\chapter{Support Routines} +\section{Command Completion} +Hyperdoc has the ability to do command completion. The known commands +are listed, one entry per line, in a file called command.list. +\chapter{The {\tt viewman} program} +<>= +char *GraphicsProgram = "$AXIOM/lib/viewman"; +@ +\chapter{The {\tt nagman} program} +<>= +char *NagManagerProgram = "$AXIOM/lib/nagman"; +@ +\section{nag.x} +<>= + +/* + * msg.x: Remote message printing protocol + */ +const MAXASP = 10; + +/* + * the nago structure is essentially a variable length string + */ + +struct nago { + opaque z <>; + }; +struct nagerr { +nago p; +nago q; +}; + +struct host{ +nago h <>; +}; + +struct nagst { + +/* Okay, if you understand this bit you know the essentials of how the link + * works. h <> is an array of nago, which is an array of fortran source + * code, the length of the array being the no. of asps (0 for most routines). + * y is the actual (XDR) input data for the routine. nm is the name of the + * routine. id is a tag identifying the host/axiom session. Finally per is a + * number telling whether or not to erase old fortran files on the remote + * machine (persistence - the number per distinct fortran files will be + * stored, any more than this and earlier ones will be deleted. + */ + + nago h <>; + nago y; + nago nm; + nago id; + int per; + }; +program NAGPROG { + version NAGVERS { + nagerr CALLNAG(nagst) = 1; + nago NAGMON(int)=2; + void AXEND(nago)=3; + } = 1; +/* + * the following number is very important. It tells the + * portmapper what number to register the nag daemon under. + * There are rules about which number to pick - check SUN + * technical info for more details + */ +} = 100088; + + +@ +\section{nagman} +\subsection{includes} +<>= +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include /* always needed */ +#include +#include +#include +#include +#include "nag.h" /* generated by rpcgen */ +#include "com.h" +#include "bsdsignal.h" +#include "sockio-c.h1" +#include "bsdsignal.h1" +#include "nagman.h1" + +@ +\subsection{variables} +<>= +#ifdef ALPHAplatform +extern int getdomainname( char *, int ); +#endif +#ifdef SUN4OS5platform +extern int getdomainname( char *, int ); +extern int gethostname( char *, int ); +#endif + +nagerr * callnag_1(nagst *,CLIENT *); +nago * nagmon_1(int *,CLIENT *); +void * axend_1(nago *,CLIENT *); + +#define DO 1 +#define DONT 0 + +int hnum, vmax; +char *datafile, *resultsfile; + +struct hostnode { + char * name; + struct hostnode *next; +} *hlist=NULL; + +nagst nag4; +Sock *sock1; + +@ +\subsection{term} +this code runs when the user quits axiom. before nagman dies, it does +an rpc call to nagd to tell it to get rid of files etc. The rpc call in +question is [[axend_1]] +we also send a USR1 to sman to beget another nagman +<>= +static void +term(int sig) +{ + CLIENT *cld; + void *res; + struct hostnode *pnode; + +#ifndef HP9platform /* can't figure out a way to do this on HP/UX 9 */ + kill(atoi(getenv("SPADNUM")) , SIGUSR1); +#endif + + + if(hnum!=0) + { + unlink(datafile); + unlink(resultsfile); + } + + for(pnode=hlist;pnode!=NULL;pnode=pnode->next) + { + cld=clnt_create(pnode->name,NAGPROG, NAGVERS, "tcp"); + if (cld == NULL) + goto NOHOST; + + res=axend_1(&(nag4.id),cld); + NOHOST: + clnt_destroy(cld); + } + exit(0); +} + +@ +\subsection{size\_of\_file} +<>= +static long +size_of_file(char *filename) +{ + struct stat buf_stat; + + stat(filename,&buf_stat); + return (buf_stat.st_size); + +} + +@ +\subsection{rpcloop} +<>= +static void +rpcloop(void) +{ + CLIENT *cl; + int res,j,v=0,u,showMessage; + long i; + register struct hostent *alias1, *alias2; + struct in_addr *addrnum; + u_long junk; + struct timeval tv; + nagerr *result; + char *Buf , *buf1; + char *ffile[MAXASP]; + char routine[12], naghost[256]; + FILE *nfp1, *nfp2, *nfp3; + struct hostnode *phost; + int fd; + + for (;;) + { + + if((Buf=get_string(sock1))==NULL) term(1); /* one string carries all */ + + if(hnum!=0) + { + /* call parameters */ + free(nag4.nm.z.z_val); /* the routine name */ + free(nag4.y.z.z_val); /* the XDR data */ + for(i=0;ivmax)? v : vmax; + } + + + + + buf1=strtok(Buf," "); + if (buf1) strcpy(naghost,buf1); + else printf("can't parse the naghost\n"); + /* INFO printf("%s\n",naghost);*/ + + buf1=strtok(NULL," "); + if (buf1) strcpy(routine,buf1); + else printf("can't parse the routine\n"); + /* INFO printf("%s\n",routine);*/ + + /* make copy of filenames because we will reuse Buf before deleting the files*/ + buf1=strtok(NULL," "); + if (buf1) resultsfile=strdup(buf1); + else printf("can't parse the resultsfile file\n"); + /* INFO printf("%s\n",resultsfile);*/ + + buf1=strtok(NULL," "); + if (buf1) datafile=strdup(buf1); + else printf("can't parse the datafile file\n"); + /* INFO printf("%s\n",datafile);*/ + + buf1=strtok(NULL," "); + if (buf1) nag4.per=atoi(buf1); + else printf("can't parse the persistence\n"); + /* INFO printf("%d\n",nag4.per);*/ + + buf1=strtok(NULL," "); + if (buf1) { + if (!strcmp(buf1,"on")) showMessage=DO; + else showMessage=DONT; + } + else printf("can't parse the messages flag\n"); + /* INFO printf("%s\n",buf1);*/ + + v=0; /* asp counter */ + while( (buf1=strtok(NULL," ")) ) + { + ffile[v++]=strdup(buf1); + /* INFO printf("%s\n",ffile[v-1]);*/ + } + + /* INFO printf("number of asps seen %d\n",v);*/ + + if(showMessage==DO) printf("nagman:acknowledging request for %s\n",routine); + + res=0; /* prepare result integer to be sent to Lisp */ + + if((nfp3=fopen(resultsfile,"w"))==NULL) + { + printf("can't open output file\n"); + goto END; + } + + /* nag4.h is the XDR array of asp text */ + nag4.h.h_len=v; + nag4.h.h_val=(nago *)malloc((v)*sizeof(nago)); + + + /* get asp text in call argument */ + for(u=0;uh_name); + else + if((alias1=gethostbyname(naghost))!=NULL) + strcpy(naghost,alias1->h_name); + } + else + if((alias1=gethostbyname(naghost))!=NULL) + strcpy(naghost,alias1->h_name); + + + + + cl = clnt_create(naghost, NAGPROG, NAGVERS, "tcp"); + if (cl == NULL) + { + /* + * Couldn't establish connection with server. + * Print error message and die. + */ + clnt_pcreateerror(naghost); + goto END; + } + else + if (showMessage==DO) + printf("nagman:connection successful to %s\n",naghost); + + /* + * this number here sets the "timeout" for the rpc call. after this number + * of seconds, the call will quit if no response is received + * + */ + + tv.tv_sec=1000000; + tv.tv_usec=0; + clnt_control(cl,CLSET_TIMEOUT,(char *)&tv); + + + result = callnag_1(&nag4, cl); + + for(phost=hlist;phost!=NULL;phost=phost->next) + { + /* + * hlist is the "hostlist" of sites that have been contacted by nagman. + * here we check if this call is contacting a new site, and if so add it + * to the hostlist + * + */ + + if(!strcmp(phost->name,naghost)) + goto SKIP; + } + + if(hnum==0) { + hlist=(struct hostnode *)malloc(sizeof(struct hostnode)); + hlist->name=strdup(naghost); + hlist->next=NULL; + } + + else { + phost=(struct hostnode *)malloc(sizeof(struct hostnode)); + phost->name=strdup(naghost); + phost->next=hlist; + hlist=phost; + } + hnum++; + + + SKIP: + if (result == NULL) + { + /* + * An error occurred while calling the server. + * Print error message and die. + */ + if (showMessage==DO) + printf("nagman:no results (error) from %s\n",naghost); + clnt_perror(cl,naghost); + clnt_destroy(cl); + goto END; + } + + /* + * (*result).p is the part of the result with the XDRed results in it + * (numbers). (*result).q is the part with (text) error messages that + * have come from the NAG library. If there is neither an XDR result, + * nor a text error message from the library, then something is wrong + * so we just print out the "no result or error returned" message. + * + */ + + else if ((*result).p.z.z_len==0) + { + if((*result).q.z.z_len==0) + { + if (showMessage==DO) + printf("nagman:empty result (error) from %s\n",naghost); + clnt_destroy(cl); + goto END; + } + else + { + if (showMessage==DO) + printf("nagman:receiving results from %s\n\n",naghost); + for(j=0;j<(*result).q.z.z_len;j++) + printf("%c",(*result).q.z.z_val[j]); + clnt_destroy(cl); + goto END; + } + } + else + if (showMessage==DO) + printf("nagman:receiving results from %s\n\n",naghost); + + if (showMessage==DO) + fwrite(result->q.z.z_val,sizeof(char),result->q.z.z_len,stdout); + + /*INFO printf("\nRESULTS of length %d\n",(*result).p.z.z_len);*/ + + fwrite(result->p.z.z_val,sizeof(char),result->p.z.z_len, nfp3); + res=1; + clnt_destroy(cl); + + /* + * in case of any type of error, a goto END in the above code causes + * nagman to skip here and return to AXIOM + * + */ + + + END: + fclose(nfp3); + /* + * if everything has gone alright, send_int returns the integer res=1. If + * not it returns res=0. This is detected by the boot code which acts + * accordingly. + */ + send_int(sock1,res); + free(Buf); + } + + +} + +@ +\subsection{catchSignals} +catchSignals sets up signal handling. If nagman gets a sigterm it does not +die but goes back to rpcloop +<>= +static void +catchSignals(void) +{ + bsdSignal(SIGTERM,term,RestartSystemCalls); + bsdSignal(SIGSEGV,term,RestartSystemCalls); +} + +@ +\subsection{main nagman} +<>= +void +main(int argc,char **argv) +{ + char this[256],*hname,*dname,*spadnum; + int stat; + + catchSignals(); + stat=gethostname(this,256); + if (stat!=0) perror("gethostname"); + hname=strdup(this); + + stat=getdomainname(this,256); + if (stat!=0) perror("getdomainname"); + dname=strdup(this); + spadnum=getenv("SPADNUM"); + if (spadnum==0) { + fprintf(stderr,"nagman error: SPADNUM is not in the environment\n"); + exit(0); + } + + /* some machines return a full name from hostname + need to check hname has a . in it */ + + if (strchr(hname,'.')) + /* '.' found */ + sprintf(this,"%s_%i",hname,atoi(spadnum)); + else + /* substring not found */ + sprintf(this,"%s.%s_%i",hname,dname,atoi(spadnum)); + + /* this must contain the Internet address of the current host */ + nag4.id.z.z_val=strdup(this); + nag4.id.z.z_len=strlen(nag4.id.z.z_val); + hnum=0; + vmax=0; + /* + * this line sets up a socket for communication with the lisp + */ + + sock1 = connect_to_local_server(SpadServer, DebugWindow, 120 /*seconds*/); + if (!sock1) exit(0); + + rpcloop(); +} + +@ +\subsection{nagman} +<>= +#define _NAGMAN_C +<> +<> +<> +<> +<> +<> +<> +@ + +\chapter{The {\tt hypertex} program} +<>= +char *HypertexProgram = "$AXIOM/bin/hypertex -s"; +@ +\chapter{The {\tt clef} program} +<>= +char *ClefProgram = "$AXIOM/bin/clef -f $AXIOM/lib/command.list -e "; +@ +\chapter{The {\tt session} program} +<>= +char *SessionManagerProgram = "$AXIOM/lib/session"; +@ +\section{session} +\subsection{includes} +<>= +#include +#include +#include +#include +#include +#ifdef SGIplatform +#include +#endif +#include "com.h" +#include "bsdsignal.h" +#include "sockio-c.h1" +#include "bsdsignal.h1" +#include "session.h1" + +@ +\subsection{variables} +<>= +#define BufSize 4096 /* size of communication buffer */ + +typedef struct sock_list { /* linked list of Sock */ + Sock Socket; + struct sock_list *next; +} Sock_List; + +Sock *spad_io = (Sock *) 0; /* to_server socket for SessionIO */ +Sock *spad_server = (Sock *) 0; /* to_server socket for SpadServer */ +Sock *menu_client = (Sock *) 0; /* to_client socket for MenuServerName */ +Sock *active_session = (Sock *) 0; /* pointer to currently active session */ + +Sock_List *plSock = (Sock_List *) 0; + +char big_bad_buf[BufSize]; /* big I/O buffer */ +int num_active_clients = 0; /* number of InterpWindows attached */ +int reading_output = 0; +fd_set session_socket_mask; + +@ +\subsection{usr1\_handler} +<>= +static void +usr1_handler(int sig) +{ + return; +} + +@ +\subsection{usr2\_handler} +SIGUSR2 is generated by spadclients. +We interpret it as an interrupt for the Lisp. +<>= +static void +usr2_handler(int sig) +{ + send_signal(spad_server, SIGINT); + return; +} + +@ +\subsection{term\_handler} +<>= +static void +term_handler(int sig) +{ + exit(1); +} + +@ +\subsection{pr} +<>= +static void +pr() +{ + Sock_List *pSock; + + fprintf(stderr,"The socket list:\n"); + for(pSock=plSock;pSock!=(Sock_List *)0;pSock=pSock->next){ + fprintf(stderr,"(%d,%d,%d)\t", + pSock->Socket.pid, 2<<(pSock->Socket.socket), pSock->Socket.frame); + } + fprintf(stderr,"\n"); +} + +@ +\subsection{close\_client} +<>= +static void +close_client(int frame) +{ + Sock_List *pSock,*locSock; + int socket_fd; + + /* we will check for frame equality, + kill with send_signal, + notify HyperTex so that it updates its list (if it's a spadbuf), + repair the list, + unset the active_session, + update num_active_clients + */ + + + /* first check head */ +#ifdef DEBUG +fprintf(stderr,"close_client(%d)\n",frame); +#endif + + if ( (plSock) && (plSock->Socket.frame == frame) ){ + socket_fd = plSock->Socket.socket; + send_signal((Sock *)plSock, SIGTERM); + if ( menu_client != (Sock *) 0){ + send_int(menu_client,CloseClient); + send_int(menu_client,(*plSock).Socket.pid); + } +#ifdef DEBUG +fprintf(stderr,"trying to clear %u\n",socket_fd); +#endif + FD_CLR(socket_fd,&session_socket_mask); + locSock = plSock; + if ((*plSock).next == (Sock_List *) 0) + {plSock = (Sock_List *) 0;} + else + {plSock = plSock->next;} + active_session = (Sock *) 0; + num_active_clients--; + free(locSock); + } + + /* now check the rest */ + + else { + for (pSock=plSock; pSock->next != (Sock_List *) 0 ; pSock=pSock->next) + if (pSock->next->Socket.frame == frame){ + socket_fd = pSock->next->Socket.socket; + send_signal((Sock *)pSock->next, SIGTERM); + if ( menu_client != (Sock *) 0){ + send_int(menu_client,CloseClient); + send_int(menu_client,(*plSock).Socket.pid); + } +#ifdef DEBUG +fprintf(stderr,"trying to clear %u\n",socket_fd); +#endif + FD_CLR(socket_fd,&session_socket_mask); + locSock = pSock->next; + if ( pSock->next->next == (Sock_List *) 0 ) + { pSock->next= (Sock_List *) 0;} + else + { pSock->next = pSock->next->next;} + num_active_clients--; + active_session = (Sock *) 0; + free(locSock); + break; + } + } +#ifdef DEBUG +pr(); +#endif +} + +@ +\subsection{read\_SpadServer\_command} +<>= +static void +read_SpadServer_command(void) +{ + int cmd, frame, num; + cmd = get_int(spad_server); + switch (cmd) { + case EndOfOutput: + if (menu_client != (Sock *) 0) send_signal(menu_client, SIGUSR2); + if (reading_output != 0) reading_output = 0; + break; + case QueryClients: + /* don't count MenuServer */ + num = num_active_clients ; + send_int(spad_server, num); + break; + case CloseClient: + frame = get_int(spad_server); + if (frame != -1) close_client(frame); + break; + case SendXEventToHyperTeX: + break; + default: + fprintf(stderr, "session : unknown command from SpadServer %d\n", cmd); + break; + } +} + +@ +\subsection{test\_sock\_for\_process} +<>= +static int +test_sock_for_process(Sock *sock) +{ + if (sock == (Sock *)0 ) return -1; + return kill(sock->pid, 0); +} + +@ +\subsection{read\_menu\_client\_command} +<>= +static void +read_menu_client_command(void) +{ + int cmd,frame, i,socket_fd; + Sock_List *pSock; + + /* save it for possible clearing */ + socket_fd = menu_client->socket; + + if (test_sock_for_process(menu_client) == -1) { + FD_CLR(socket_fd,&session_socket_mask); + menu_client = (Sock *) 0; + reading_output = 0; + return; + } + cmd = get_int(menu_client); + switch(cmd) { + case -1: /* socket closed */ + FD_CLR(socket_fd,&session_socket_mask); + menu_client = (Sock *) 0; + reading_output = 0; + break; + case SwitchFrames: +#ifdef DEBUG +fprintf(stderr,"menu:SwitchFrames\n"); +#endif + frame = get_int(menu_client); + send_int(spad_server, SwitchFrames); + send_int(spad_server, frame); + for(i=0,pSock=plSock; pSock != (Sock_List *) 0 ; i++,pSock=pSock->next) + if ((pSock->Socket.frame == frame)) { + active_session = (Sock *)pSock; + reading_output = 1; + break; + } + if (i == num_active_clients) { + /* fprintf(stderr, "Couldn't find socket for frame %d\n", frame); */ + } + break; + case QuerySpad: +#ifdef DEBUG +fprintf(stderr,"menu:QuerySpad\n"); +#endif + send_int(menu_client, reading_output); + break; + default: + fprintf(stderr, "session : unknown command from MenuServer: %d\n", cmd); + menu_client = (Sock *) 0; + break; + } +} + +@ +\subsection{read\_from\_spad\_io} +<>= +static void +read_from_spad_io(void) +{ + int ret_code; + ret_code = sread(spad_io, big_bad_buf, BufSize, "session: stdout socket"); + if (ret_code == -1) return; + if(active_session != (Sock *) 0) { + ret_code = swrite(active_session, big_bad_buf, ret_code, + NULL); + } +} + +@ +\subsection{kill\_spad} +<>= +static void +kill_spad(void) +{ + int i; + Sock_List *pSock; + + send_signal(spad_server, SIGTERM); + for (pSock=plSock,i=0; + (inext) { + if ((pSock->Socket).socket != 0) + send_signal((Sock *)pSock, SIGTERM); + } + if (menu_client != (Sock *) 0) send_signal(menu_client, SIGTERM); + exit(0); +} + +@ +\subsection{accept\_session\_connection} +<>= +static int +accept_session_connection(Sock *server_sock) +{ + int sock_fd, ret_code; + Sock_List *pls; + + /* Could be three things : KillSpad MenuServer InterpWindow */ + + pls = (Sock_List *) malloc(sizeof (Sock_List)); + sock_fd = accept(server_sock->socket, 0, 0); + if (sock_fd == -1) { + perror("session : accepting connection"); + return -1; + } + (pls->Socket).socket = sock_fd; + get_socket_type((Sock *)pls); + + switch((pls->Socket).purpose) { + case KillSpad: + kill_spad(); + return KillSpad; + free(pls); + case MenuServer: +#ifdef DEBUG + fprintf(stderr,"session: accepted MenuServer , fd = %d\n",sock_fd); +#endif + menu_client = &(pls->Socket); + FD_SET(menu_client->socket, &session_socket_mask); + return MenuServer; + case InterpWindow: +#ifdef DEBUG + fprintf(stderr,"session: accepted InterpWindow , fd = %d\n",sock_fd); +#endif + + /* new Sock is put at the head of the list */ + if (plSock == (Sock_List *)0 ) { + plSock = pls; + plSock->next = (Sock_List *)0 ; + } + else{ + pls->next = plSock; + plSock = pls; + } + + /* we need to maintain session_socket_mask here + since we roll our own accept */ + + FD_SET(plSock->Socket.socket, &session_socket_mask); + send_int(spad_server, CreateFrame); + { + int command = get_int(spad_server); + /* XXX hack -- the whole protocol looks broken, we just + try to detect losage */ + if (command != CreateFrameAnswer) { + fprintf(stderr, "session: non-fatal, got out of sync " + "with Spad server\n (lost race)\n"); + /* exit(1); */ + } + } + plSock->Socket.frame = get_int(spad_server); + active_session = (Sock *)plSock; + get_string_buf(spad_server, big_bad_buf, BufSize); + ret_code = swrite((Sock *)plSock, big_bad_buf, strlen(big_bad_buf)+1, + "session: writing to InterpWindow"); + if (ret_code == -1) + return -1; + num_active_clients++; +#ifdef DEBUG +pr(); +#endif + return plSock->Socket.purpose; + } + return (-1); +} + +@ +\subsection{read\_from\_session} +<>= +static void +read_from_session(Sock *sock) +{ + int ret_code; + if (sock != active_session) { + send_int(spad_server, SwitchFrames); + send_int(spad_server, sock->frame); + } + active_session = sock; + ret_code = sread(sock, big_bad_buf, BufSize, + "session: reading InterpWindow"); + if (ret_code == -1) { + active_session = (Sock *) 0; + reading_output = 0; + return; + } + ret_code = swrite(spad_io, big_bad_buf, ret_code, + "session: writing SessionIO"); + if (ret_code == -1) { + active_session = (Sock *)0 ; + reading_output = 0; + return; + } + reading_output = 1; +} + +@ +\subsection{manage\_sessions} +<>= +static void +manage_sessions(void) +{ + int ret_code; + fd_set rd, wr, ex; + Sock_List *pSock; + + reading_output = 0; + while (1) { + FD_ZERO(&rd); + FD_ZERO(&wr); + FD_ZERO(&ex); + + /* Allow server socket and all connections if not waiting for output + socket_mask is maintained by libspad.a */ +#ifdef DEBUG +fprintf(stderr,"session_socket_mask=%u ",*((long *)session_socket_mask.fds_bits)); +#endif + rd = session_socket_mask; + if (!reading_output) { + rd = session_socket_mask; + } + + /* Allow the active_session if set */ + if (active_session) FD_SET(active_session->socket, &rd); +#ifdef DEBUG +fprintf(stderr,"[rd=%u ",*((long *)rd.fds_bits)); +#endif + + ret_code = sselect(FD_SETSIZE, &rd, &wr, &ex, NULL); + if (ret_code == -1) { + break; + } +#ifdef DEBUG +fprintf(stderr,"rd=%u]\n",*((long *)rd.fds_bits)); +#endif + + if ((menu_client != (Sock *) 0) && FD_ISSET(menu_client->socket, &rd)) { + /* MenuServer wants to talk */ + read_menu_client_command(); } + + + if (FD_ISSET(spad_io->socket, &rd)) { + /* Lisp has output */ + read_from_spad_io(); } + + + if (FD_ISSET(server[1].socket, &rd)) { + /* Someone wants to connect to our server socket */ + accept_session_connection(server+1); } + + + for(pSock=plSock; pSock != (Sock_List *) 0 ; pSock=pSock->next) { + if ((active_session == (Sock *)pSock || !reading_output) && + (pSock->Socket).socket>0 && FD_ISSET(pSock->Socket.socket, &rd)) { + /* An InterpWindow */ + read_from_session((Sock *)pSock); } + } + + + if (FD_ISSET(spad_server->socket, &rd)) { + /* The Lisp socket */ + read_SpadServer_command(); } + } +} + +@ +\subsection{main sessionmanager} +<>= +int +main(void) +{ + +#ifdef DEBUG2 + /* delay for attaching with debugger before interesting things happen */ + sleep(30); +#endif + + /* spad_server connects to Lisp server socket + read_SpadServer_command handles requests */ + spad_server = connect_to_local_server(SpadServer, SessionManager, Forever); + if (spad_server == (Sock *) 0) { + fprintf(stderr, "session: Cannot connect to AXIOM server!\n"); + exit(0); + } + else { +#ifdef DEBUG + fprintf(stderr, "session: connected SpadServer , fd = %d\n", + spad_server->socket); +#endif + FD_SET(spad_server->socket, &session_socket_mask); + } + + + /* spad_io connects to SessionIOName server socket + this is Lisp std IO read_from_spad_io handles requests */ + spad_io = connect_to_local_server(SessionIOName, SessionIO, Forever); + if (spad_io == (Sock *) 0) { + fprintf(stderr, "session: Cannot connect to AXIOM IO!\n"); + exit(0); + } + else { +#ifdef DEBUG + fprintf(stderr,"session: connected SessionIOName , fd = %d\n", + spad_io->socket); +#endif + FD_SET(spad_io->socket, &session_socket_mask); + } + bsdSignal(SIGUSR2, usr2_handler,DontRestartSystemCalls); + bsdSignal(SIGUSR1, usr1_handler,RestartSystemCalls); + bsdSignal(SIGINT, SIG_IGN,RestartSystemCalls); + bsdSignal(SIGTERM, term_handler,RestartSystemCalls); + + /* open_server opens the server socket so that we can accept connections + we expect connections from spadbuf/spadclient(purpose:InterpWindow) + and hypertex (MenuServer) */ + + if (open_server(SessionServer) == -2) { + fprintf(stderr, "session: Cannot make server socket!\n"); + exit(-1); + } + else { +#ifdef DEBUG + fprintf(stderr, "session: opened SessionServer , fd = %d\n", + server[1].socket); +#endif + FD_SET(server[1].socket,&session_socket_mask); + } + manage_sessions(); + return(0); +} + +@ +\subsection{session} +<>= +/* #define DEBUG */ +#define _SESSION_C + +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> +<> + +@ +\chapter{The {\tt spadclient} program} +<>= +char *SpadClientProgram = "$AXIOM/lib/spadclient"; +@ +\section{spadclient} +<>= +#define _SPADCLIENT_C + +#include +#include +#include "com.h" +#include "bsdsignal.h" + +#include "bsdsignal.h1" +#include "sockio-c.h1" +#include "spadclient.h1" + +Sock *sock; + +static void +inter_handler(int sig) +{ + send_signal(sock, SIGUSR2); + fflush(stderr); +} + + +int +main(void) +{ + sock = connect_to_local_server(SessionServer, InterpWindow, Forever); + bsdSignal(SIGINT, inter_handler,RestartSystemCalls); + remote_stdio(sock); + return(0); +} + +@ +\chapter{The Command Completion List} +<>= +- +/ +/\ +^ +^= +~ +~= +* +** +\/ +# ++ +< +<= += +> +>= +0 +1 +abelianGroup +abs +absolutelyIrreducible? +accuracyIF +acos +acosh +acoshIfCan +acosIfCan +acot +acoth +acothIfCan +acotIfCan +acsc +acsch +acschIfCan +acscIfCan +aCubic +adaptive +adaptive? +adaptive3D? +addBadValue +addChild! +addData! +addField! +addiag +addMatch +addMatchRestricted +addmod +addPoint +addPoint2 +addPointLast +adjoint +airyAi +airyBi +Aleph +algDsolve +algebraic? +algebraicCoefficients? +algebraicDecompose +algebraicOf +algebraicSort +algebraicVariables +algint +algintegrate +algSplitSimple +aLinear +allRootsOf +alphabetic +alphabetic? +alphanumeric +alphanumeric? +alternating +alternatingGroup +alternative? +An +AND +And +and +anfactor +antiAssociative? +antiCommutative? +antiCommutator +anticoord +antisymmetric? +antisymmetricTensors +any +any? +append +appendPoint +apply +applyQuote +applyRules +approximants +approximate +approxNthRoot +approxSqrt +aQuadratic +aQuartic +areEquivalent? +arg1 +arg2 +argscript +argument +argumentList! +argumentListOf +arity +aromberg +arrayStack +asec +asech +asechIfCan +asecIfCan +asimpson +asin +asinh +asinhIfCan +asinIfCan +aspFilename +assert +assign +assoc +associatedEquations +associatedSystem +associates? +associative? +associator +associatorDependence +atan +atanh +atanhIfCan +atanIfCan +atom? +atoms +atrapezoidal +att2Result +augment +autoReduced? +axes +axesColorDefault +B1solve +back +backOldPos +badNum +badValues +bag +balancedBinaryTree +balancedFactorisation +bandedHessian +bandedJacobian +base +baseRDE +baseRDEsys +BasicMethod +basicSet +basis +basisOfCenter +basisOfCentroid +basisOfCommutingElements +basisOfLeftAnnihilator +basisOfLeftNucleus +basisOfLeftNucloid +basisOfMiddleNucleus +basisOfNucleus +basisOfRightAnnihilator +basisOfRightNucleus +basisOfRightNucloid +bat +bat1 +beauzamyBound +belong? +bernoulli +bernoulliB +besselI +besselJ +besselK +besselY +Beta +bezoutDiscriminant +bezoutMatrix +bezoutResultant +bfEntry +bfKeys +binary +binaryFunction +binarySearchTree +binaryTournament +binaryTree +binomial +binomThmExpt +bipolar +bipolarCylindrical +biRank +birth +bit? +bitCoef +bitLength +bits +bitTruth +bivariate? +bivariatePolynomials +bivariateSLPEBR +blankSeparate +block +blue +bombieriNorm +bool +bool? +bottom! +boundOfCauchy +box +brace +bracket +branchIfCan +branchPoint? +branchPointAtInfinity? +bright +brillhartIrreducible? +brillhartTrials +bringDown +bsolve +btwFact +bubbleSort! +build +BumInSepFFE +bumprow +bumptab +bumptab1 +BY +c02aff +c02agf +c05adf +c05nbf +c05pbf +c06eaf +c06ebf +c06ecf +c06ekf +c06fpf +c06fqf +c06frf +c06fuf +c06gbf +c06gcf +c06gqf +c06gsf +cache +cAcos +cAcosh +cAcot +cAcoth +cAcsc +cAcsch +calcRanges +call +cap +car +cardinality +cartesian +cAsec +cAsech +cAsin +cAsinh +cAtan +cAtanh +cCos +cCosh +cCot +cCoth +cCsc +cCsch +cdr +ceiling +center +central? +certainlySubVariety? +cExp +cfirst +chainSubResultants +changeBase +changeMeasure +changeName +changeNameToObjf +changeThreshhold +changeVar +changeWeightLevel +char +character? +characteristic +characteristicPolynomial +characteristicSerie +characteristicSet +charClass +charpol +charthRoot +chebyshevT +chebyshevU +check +checkCxResult +checkForZero +checkMxCDF +checkMxDF +checkPrecision +checkResult +checkRur +child +child? +children +chineseRemainder +chiSquare +chiSquare1 +choosemon +chvar +Ci +className +clearCache +clearDenominator +clearFortranOutputStack +clearTable! +clearTheFTable +clearTheIFTable +clearTheSymbolTable +clikeUniv +clip +clipBoolean +clipParametric +clipPointsDefault +clipSurface +clipWithRanges +cLog +close +close! +closeComponent +closed? +closedCurve +closedCurve? +cn +code +coef +coefChoose +coefficient +coefficients +coerce +coerceImages +coerceListOfPairs +coerceP +coercePreimagesImages +coHeight +coleman +collect +collectQuasiMonic +collectUnder +collectUpper +color +colorDef +colorFunction +column +combineFeatureCompatibility +commaSeparate +comment +common +commonDenominator +commutative? +commutativeEquality +commutator +comp +compactFraction +companionBlocks +comparison +compBound +compdegd +compile +compiledFunction +complement +complementaryBasis +complete +completeEchelonBasis +completeEval +completeHensel +completeHermite +completeSmith +complex +complex? +complexEigenvalues +complexEigenvectors +complexElementary +complexExpand +complexForm +complexIntegrate +complexLimit +complexNormalize +complexNumeric +complexNumericIfCan +complexRoots +complexSolve +complexZeros +component +components +compose +composite +composites +computeBasis +computeCycleEntry +computeCycleLength +computeInt +computePowers +concat +concat! +cond +condition +conditionP +conditions +conditionsForIdempotents +conical +conjHerm +conjug +conjugate +conjugates +connect +connect? +cons +consnewpol +const +constant +constant? +constantCoefficientRicDE +constantIfCan +constantKernel +constantLeft +constantOperator +constantOpIfCan +constantRight +constantToUnaryFunction +constDsolve +construct +contains? +content +continue +continuedFraction +contract +contractSolve +controlPanel +convergents +convert +coord +coordinate +coordinates +copies +copy +copy! +copyInto! +corrPoly +cos +cos2sec +cosh +cosh2sech +coshIfCan +cosIfCan +cosSinInfo +cot +cot2tan +cot2trig +coth +coth2tanh +coth2trigh +cothIfCan +cotIfCan +count +countable? +countRealRoots +countRealRootsMultiple +cPower +cRationalPower +create +create3Space +createGenericMatrix +createIrreduciblePoly +createLowComplexityNormalBasis +createLowComplexityTable +createMultiplicationMatrix +createMultiplicationTable +createNormalElement +createNormalPoly +createNormalPrimitivePoly +createPrimitiveElement +createPrimitiveNormalPoly +createPrimitivePoly +createRandomElement +createThreeSpace +createZechTable +credPol +crest +critB +critBonD +critM +critMonD1 +critMTonD1 +critpOrder +critT +cross +crushedSet +csc +csc2sin +csch +csch2sinh +cschIfCan +cscIfCan +cSec +cSech +cSin +cSinh +csubst +cTan +cTanh +cubic +cup +currentSubProgram +curry +curryLeft +curryRight +curve +curve? +curveColor +curveColorPalette +cycle +cycleElt +cycleEntry +cycleLength +cyclePartition +cycleRagits +cycles +cycleSplit! +cycleTail +cyclic +cyclic? +cyclicCopy +cyclicEntries +cyclicEqual? +cyclicGroup +cyclicParents +cyclicSubmodule +cyclotomic +cyclotomicDecomposition +cyclotomicFactorization +cylindrical +D +d01ajf +d01akf +d01alf +d01amf +d01anf +d01apf +d01aqf +d01asf +d01bbf +d01fcf +d01gaf +d01gbf +d02bbf +d02bhf +d02cjf +d02ejf +d02gaf +d02gbf +d02kef +d02raf +d03edf +d03eef +d03faf +dAndcExp +dark +datalist +ddFact +debug +debug3D +dec +decimal +declare +declare! +decompose +decomposeFunc +decrease +decreasePrecision +deepCopy +deepestInitial +deepestTail +deepExpand +defineProperty +definingEquations +definingInequation +definingPolynomial +degree +degreePartition +degreeSubResultant +degreeSubResultantEuclidean +delay +delete +delete! +deleteProperty! +deleteRoutine! +delta +denom +denominator +denominators +denomLODE +denomRicDE +depth +dequeue +dequeue! +deref +deriv +derivationCoordinates +derivative +destruct +determinant +df2ef +df2fi +df2mf +df2st +dflist +dfRange +diag +diagonal +diagonal? +diagonalMatrix +diagonalProduct +diagonals +dictionary +diff +difference +differentialVariables +differentiate +digamma +digit +digit? +digits +dihedral +dihedralGroup +dilog +dim +dimension +dimensionOfIrreducibleRepresentation +dimensions +dimensionsOf +diophantineSystem +dioSolve +direction +directory +directProduct +directSum +discreteLog +discriminant +discriminantEuclidean +display +dispose! +distance +distdfact +distFact +distribute +div +divergence +divide +divideExponents +divideIfCan +divideIfCan! +divisor +divisorCascade +divisors +dmp2rfi +dmpToHdmp +dmpToP +dn +dom +domainOf +dominantTerm +dot +double +double? +doubleComplex? +doubleDisc +doubleRank +doubleResultant +doublyTransitive? +draw +drawComplex +drawComplexVectorField +drawCurves +drawStyle +drawToScale +droot +duplicates +duplicates? +e +e01baf +e01bef +e01bff +e01bgf +e01bhf +e01daf +e01saf +e01sbf +e01sef +e01sff +e02adf +e02aef +e02agf +e02ahf +e02ajf +e02akf +e02baf +e02bbf +e02bcf +e02bdf +e02bef +e02daf +e02dcf +e02ddf +e02def +e02dff +e02gaf +e02zaf +e04dgf +e04fdf +e04gcf +e04jaf +e04mbf +e04naf +e04ucf +e04ycf +edf2df +edf2ef +edf2efi +edf2fi +ef2edf +Ei +eigenMatrix +eigenvalues +eigenvector +eigenvectors +eisensteinIrreducible? +elColumn2! +elem? +element? +elementary +elements +elliptic +elliptic? +ellipticCylindrical +elRow1! +elRow2! +elt +empty +empty? +endOfFile? +endSubProgram +enqueue! +enterInCache +enterPointData +entries +entry +entry? +enumerate +epilogue +EQ +eq +eq? +equality +equation +erf +error +errorInfo +errorKind +escape +euclideanGroebner +euclideanNormalForm +euclideanSize +euler +eulerE +eulerPhi +eval +evaluate +evaluateInverse +even? +evenInfiniteProduct +evenlambert +every? +exactQuotient +exactQuotient! +exists? +exp +exp1 +expand +expandLog +expandPower +expandTrigProducts +expenseOfEvaluation +expenseOfEvaluationIF +expextendedint +expIfCan +expint +expintegrate +expintfldpoly +explicitEntries? +explicitlyEmpty? +explicitlyFinite? +explimitedint +explogs2trigs +exponent +exponential +exponential1 +exponentialOrder +exponents +expPot +expr +expressIdealMember +exprHasAlgebraicWeight +exprHasLogarithmicWeights +exprHasWeightCosWXorSinWX +exprToGenUPS +exprToUPS +exprToXXP +expt +exptMod +exQuo +exquo +extend +extendedEuclidean +extendedint +extendedIntegrate +extendedResultant +extendedSubResultantGcd +extendIfCan +extension +extensionDegree +exteriorDifferential +external? +externalList +extract! +extractBottom! +extractClosed +extractIfCan +extractIndex +extractPoint +extractProperty +extractSplittingLeaf +extractTop! +eyeDistance +F +f01brf +f01bsf +f01maf +f01mcf +f01qcf +f01qdf +f01qef +f01rcf +f01rdf +f01ref +f02aaf +f02abf +f02adf +f02aef +f02aff +f02agf +f02ajf +f02akf +f02awf +f02axf +f02bbf +f02bjf +f02fjf +f02wef +f02xef +f04adf +f04arf +f04asf +f04atf +f04axf +f04faf +f04jgf +f04maf +f04mbf +f04mcf +f04qaf +f07adf +f07aef +f07fdf +f07fef +f2df +F2FG +f2st +factor +factor1 +factorAndSplit +factorByRecursion +factorFraction +factorGroebnerBasis +factorial +factorials +factorList +factorOfDegree +factorPolynomial +factors +factorset +factorSFBRlcUnit +factorsOfCyclicGroupSize +factorsOfDegree +factorSquareFree +factorSquareFreeByRecursion +factorSquareFreePolynomial +failed +failed? +false +ffactor +FG2F +fglmIfCan +fi2df +fibonacci +field +fields +figureUnits +filename +fill! +fillPascalTriangle +filterUntil +filterWhile +find +findCycle +finite? +finiteBasis +finiteBound +fintegrate +first +firstDenom +firstNumer +firstSubsetGray +firstUncouplingMatrix +fixedDivisor +fixedPoint +fixedPointExquo +fixedPoints +fixPredicate +flagFactor +flatten +flexible? +flexibleArray +float +float? +floatlist +floatlist? +floor +fmecg +forLoop +FormatArabic +FormatRoman +formula +fortran +fortranCarriageReturn +fortranCharacter +fortranCompilerName +fortranComplex +fortranDouble +fortranDoubleComplex +fortranInteger +fortranLinkerArgs +fortranLiteral +fortranLiteralLine +fortranLogical +fortranReal +fortranTypeOf +fprindINFO +fracPart +fractionFreeGauss! +fractionPart +fractRadix +fractRagits +freeOf? +Frobenius +frobenius +front +froot +frst +fTable +fullDisplay +fullPartialFraction +function +functionIsContinuousAtEndPoints +functionIsFracPolynomial? +functionIsOscillatory +Gamma +gbasis +gcd +gcdcofact +gcdcofactprim +gcdPolynomial +gcdprim +gcdPrimitive +gderiv +GE +generalInfiniteProduct +generalizedContinuumHypothesisAssumed +generalizedContinuumHypothesisAssumed? +generalizedEigenvector +generalizedEigenvectors +generalizedInverse +generalLambert +generalPosition +generalSqFr +generalTwoFactor +generate +generateIrredPoly +generator +generators +generic +generic? +genericLeftDiscriminant +genericLeftMinimalPolynomial +genericLeftNorm +genericLeftTrace +genericLeftTraceForm +genericPosition +genericRightDiscriminant +genericRightMinimalPolynomial +genericRightNorm +genericRightTrace +genericRightTraceForm +genus +geometric +getBadValues +getButtonValue +getCode +getCurve +getDatabase +getExplanations +getGoodPrime +getGraph +gethi +getlo +getMatch +getMeasure +getMultiplicationMatrix +getMultiplicationTable +getOrder +getPickedPoints +getRef +getStream +getVariableOrder +getZechTable +GF2FG +goodnessOfFit +goodPoint +GospersMethod +goto +gradient +graeffe +gramschmidt +graphCurves +graphImage +graphs +graphState +graphStates +green +groebgen +groebner +groebner? +groebnerFactorize +groebnerIdeal +groebSolve +ground +ground? +GT +halfExtendedResultant1 +halfExtendedResultant2 +halfExtendedSubResultantGcd1 +halfExtendedSubResultantGcd2 +harmonic +has? +hash +hasHi +hasoln +hasPredicate? +hasSolution? +hasTopPredicate? +Hausdorff +hclf +hconcat +hcrf +hdmpToDmp +hdmpToP +head +headReduce +headReduced? +headRemainder +heap +heapSort +height +henselFact +HenselLift +hermite +hermiteH +HermiteIntegrate +hessian +hex +hexDigit +hexDigit? +hi +high +highCommonTerms +hitherPlane +hMonic +HMS +homogeneous? +horizConcat +hspace +htrigs +hue +hyperelliptic +hypergeometric0F1 +iCompose +id +ideal +idealiser +idealiserMatrix +idealSimplify +identification +identity +identityMatrix +identitySquareMatrix +iExquo +iflist2Result +iFTable +ignore? +iiabs +iiacos +iiacosh +iiacot +iiacoth +iiacsc +iiacsch +iiasec +iiasech +iiasin +iiasinh +iiatan +iiatanh +iibinom +iicos +iicosh +iicot +iicoth +iicsc +iicsch +iidprod +iidsum +iiexp +iifact +iiGamma +iilog +iiperm +iipow +iisec +iisech +iisin +iisinh +iisqrt2 +iisqrt3 +iitan +iitanh +imag +imagE +imagI +imagi +imaginary +imagJ +imagj +imagK +imagk +implies +in? +inc +incr +increase +increasePrecision +increment +incrementBy +incrementKthElement +index +index? +indices +indiceSubResultant +indiceSubResultantEuclidean +indicialEquation +indicialEquationAtInfinity +indicialEquations +inf +infieldint +infieldIntegrate +infinite? +infiniteProduct +infinity +infinityNorm +infix +infix? +infLex? +infRittWu? +inGroundField? +inHallBasis? +init +initial +initializeGroupForWordProblem +initiallyReduce +initiallyReduced? +initials +initTable! +innerEigenvectors +innerint +innerSolve +innerSolve1 +input +inR? +inRadical? +inrootof +insert +insert! +insertBottom! +insertionSort! +insertMatch +insertRoot! +insertTop! +inspect +int +int? +intChoose +intcompBasis +integer +integer? +integerBound +integerIfCan +integers +integral +integral? +integralAtInfinity? +integralBasis +integralBasisAtInfinity +integralCoordinates +integralDerivationMatrix +integralLastSubResultant +integralMatrix +integralMatrixAtInfinity +integralRepresents +integrate +intensity +intermediateResultsIF +internal? +internalAugment +internalDecompose +internalInfRittWu? +internalIntegrate +internalIntegrate0 +internalLastSubResultant +internalSubPolSet? +internalSubQuasiComponent? +internalZeroSetSplit +interpolate +interpret +interpretString +interReduce +intersect +interval +intlist +intlist? +intPatternMatch +inv +inverse +inverseColeman +inverseIntegralMatrix +inverseIntegralMatrixAtInfinity +inverseLaplace +invertible? +invertibleElseSplit? +invertibleSet +invertIfCan +invmod +invmultisect +invWrite +iomode +ipow +iprint +iroot +irreducible? +irreducibleFactor +irreducibleFactors +irreducibleRepresentation +Is +is? +isAbsolutelyIrreducible? +isExpt +isList +isMult +isobaric? +isOp +isPlus +isPower +isQuotient +isTimes +iter +iteratedInitials +jacobi +jacobian +jacobiIdentity? +janko2 +jordanAdmissible? +jordanAlgebra? +karatsuba +karatsubaDivide +karatsubaOnce +kernel +kernels +key +key? +keys +kmax +knownInfBasis +kovacic +kroneckerDelta +KrullNumber +ksec +label +lagrange +LagrangeInterpolation +laguerre +laguerreL +lambda +lambert +laplace +laplacian +largest +last +lastSubResultant +lastSubResultantElseSplit +lastSubResultantEuclidean +latex +laurent +laurentIfCan +laurentRep +Lazard +Lazard2 +LazardQuotient +LazardQuotient2 +lazy? +lazyEvaluate +lazyGintegrate +lazyIntegrate +lazyIrreducibleFactors +lazyPquo +lazyPrem 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+VectorFunctions2 +ViewDefaultsPackage +ViewportPackage +WeierstrassPreparation +WildFunctionFieldIntegralBasis +XExponentialPackage +ZeroDimensionalSolvePackage +@ +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. \\ +``Axiom -- The Scientific Computation System''\\ +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming''\\ +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie,\\ +``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman\\ +``Noweb -- A Simple, Extensible Tool for Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\bibitem{7} Axiom Book Volume 7 -- Hyperdoc\\ +{\bf file://usr/local/axiom/src/hyper/bookvol7.pamphlet} +\bibitem{8} Axiom Book Volume 8 -- Graphics\\ +{\bf file://usr/local/axiom/src/graph/bookvol8.pamphlet} +\bibitem{9} AIX Version 3.2 and 4 Performance Tuning Guide\\ +{\bf +http://www.rs6000.ibm.com/doc\_link/en\_US/\\ +{\hbox {\hskip 1.0cm}}a\_doc\_lib/aixbman/prftungd/toc.htm} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol7.pamphlet b/books/bookvol7.pamphlet new file mode 100644 index 0000000..9dcb878 --- /dev/null +++ b/books/bookvol7.pamphlet @@ -0,0 +1,1277 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 7: Axiom Hyperdoc}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Overview} +This book contains the hyperdoc routines for Axiom. +\chapter{Support Routines} +\chapter{Hyperdoc} +\section{hyper.h} +The [[hypertex]] function, of which this is the top level, is a browser +for Axiom information. It works off a database of pages. The pages are +stored in the [[$AXIOM/doc/hypertex/pages]] subdirectory and there is +a key file called [[ht.db]] in that subdirectory which contains +critical information about each page. If you add or delete pages you +must rerun the [[htadd]] command. +(See the [[htadd]] command in [[src/hyper/htadd.pamphlet]].) + +Generally, if you add or delete pages you can recreate a proper +[[pages/ht.db]] file by doing: +\begin{verbatim} +cd $AXIOM/doc/hypertex +htadd -f pages -n pages/* +\end{verbatim} + + +The [[hypertex]] function looks in [[$AXIOM/doc/hypertex/pages]] by +default. This can be over-ridden by setting the [[HTPATH]] shell +variable to point to the desired directory containing the pages and +the ht.db file. +<>= +<> +#ifndef _HYPER_H_ +#define _HYPER_H_ 1 + +#include +#include +#include +#include + +#include +#include +#include + +#include "com.h" +#include "token.h" +#include "hash.h" + +#define boolean unsigned short int + +#ifndef TRUE +#define TRUE ((boolean) 0x1) +#endif +#ifndef FALSE +#define FALSE ((boolean) 0x0) +#endif + +/* Struct forward declarations */ + +struct text_node; +struct input_box; +struct input_window; +struct paste_node; +struct radio_boxes; +struct group_item; + +#define Scrollupbutton 1 +#define Scrolldownbutton 2 +#define Noopbutton 6 + +#define Scrolling 1 +#define Header 2 +#define Footer 3 +#define Title 4 + +#ifndef HTADD +extern int MenuServerOpened; + +/* These are all the colors one can use in HyperDoc. */ + +extern int gActiveColor, + gAxiomColor, + gBackgroundColor, + gBfColor, + gControlBackgroundColor, + gControlForegroundColor, + gEmColor, + gInputBackgroundColor, + gInputForegroundColor, + gItColor, + gRmColor, + gSlColor, + gTtColor; + +/* These are all the different fonts one can use in HyperDoc. */ + +extern XFontStruct *gActiveFont, + *gAxiomFont, + *gBfFont, + *gEmFont, + *gInputFont, + *gItFont, + *gRmFont, + *gSlFont, + *gTitleFont, + *gTtFont; + + +#endif + + +/** I am implementing a state node stack, this is the structure I store **/ + +typedef struct state_node { + int last_ch, last_token, input_type; + long fpos, keyword_fpos; + long page_start_fpos; + Token token; + char *input_string; + FILE *cfile; + int keyword; + struct state_node *next; +} StateNode; + +#ifndef HTADD +/** pointer to the top of the state node graph **/ +extern StateNode *top_state_node; +#endif + + +/* structure for a hyper text link */ +typedef struct hyper_link { + int type; /* Memolink, Spadlink, Downlink etc. */ + Window win; /* X11 window containing active area */ + union { + struct text_node *node; /* ID of link to which link refers */ + struct input_box *box; + struct input_window *string; + struct paste_node *paste; /* the paste node area */ + } reference; + int x,y; /* relative position inside page */ +} HyperLink; + + +typedef struct if_node { + struct text_node *cond; /* the condition nodes*/ + struct text_node *thennode; + struct text_node *elsenode; +} IfNode; + +typedef struct item_stack { + int indent; + int item_indent; + int in_item; + struct item_stack *next; +} ItemStack; + +typedef struct paste_node { + char *name; + int where; /* where should I be parsing from? */ + short int hasbutton; + short int haspaste; + struct group_item *group; + ItemStack *item_stack; + struct text_node *arg_node; + struct text_node *end_node; + struct text_node *begin_node; + struct input_window *paste_item; +} PasteNode; + +/* Structure for formatted hypertext */ + +typedef struct text_node { + short type; /* type of node (text, link, etc.) */ + int x,y, width, height; /* relative location on screen */ + int space; /* was there space in front of me ? */ + union { + char *text; /* piece of text to display */ + struct text_node *node; /* argument text */ + struct if_node *ifnode; + } data; + HyperLink *link; /* link for active text */ + union { + Pixmap pm; /* pixmap for bit images */ + XImage *xi; /* pixmap image */ + } image; + struct text_node *next; /* next node in formatted text */ +} TextNode; + +/** Structure used to store pixmaps and bitmaps **/ + +typedef struct image_struct { + int width,height; /** It's width and height **/ + union { + Pixmap pm; + XImage *xi; + } image; + char *filename; /** The filename used to reference it **/ +} ImageStruct; + +/* Structure for locating HyperDoc pages in a source file */ + +typedef struct { + char *name; /* file name */ + long pos; /* position in file */ + int ln; /* the line number */ +} FilePosition; + +/*** The structure needed for storing a macro **/ + +typedef struct macro_store { + short int loaded; + FilePosition fpos; + char *name; + char *macro_string; + short number_parameters; +} MacroStore; + + +/** Structure needed for storing a patch **/ +typedef struct patch_store { + short int loaded; + FilePosition fpos; + char *name; + char *string; +} PatchStore; + +/* Here are the structures needed for doing input to HyperDoc windows. */ + +typedef struct line_struct { + char *buffer; + int changed; /* Has the line changed */ + int line_number; + int buff_pntr; + int len; + struct line_struct *prev, *next; +} LineStruct; + +typedef struct input_window { + char *name; /* symbol name **/ + int size; /* the length of the window **/ + int cursor_x; /* x-coordinate for the cursor **/ + int entered; /* tells me whether I have typed here + before */ + int num_lines; /* number of lines needed to store + buffer */ + LineStruct *lines; + LineStruct *curr_line; /* the current line on which the cursor */ + Window win; + struct input_window *next; +} InputItem; + + +/* structure for storing input boxes **/ +typedef struct input_box { + char *name; + ImageStruct *selected, *unselected; + short int picked; + struct input_box *next; + struct radio_boxes *rbs; + Window win; +} InputBox; + +typedef struct radio_boxes { + char *name; + InputBox *boxes; + ImageStruct *selected, *unselected; + int width, height; + struct radio_boxes *next; +} RadioBoxes; + +/* Structure for spadcommand dependencies hash table entries */ +typedef struct spadcom_depend { + char *label; /* dependency label */ + TextNode *spadcom; /* spadcommand defining the label */ + short executed; /* true iff spadcommand has benn executed */ +} SpadcomDepend; + +typedef struct button_list { + int x0,y0,x1,y1; + HyperLink *link; + Window win; + struct button_list *next; +} ButtonList; + +/* Stucture for unformatted hyper text page */ + +typedef struct hyperdoc_page { + short type; /* Normal, Quitbutton, Upbutton etc. */ + char *name; /* ID of page */ + char *filename; /* The name of the file in which the page + occurs, Null if not */ + int scroll_off; /* The offset in the scrolling region */ + int bot_scroll_margin; /* bottom of the scrolling region */ + int top_scroll_margin; /* top of the scrolling region */ + TextNode *title; /* the title of the page */ + TextNode *header; /* formatted version of page */ + TextNode *scrolling; /* Top of scrolling region */ + TextNode *footer; /* top of non-scrolling region at bottom */ + Sock *sock; /* socket connection for spad buffer */ + HashTable *fLinkHashTable; /* active link hash table */ + ButtonList *s_button_list; /* active buttons on page */ + ButtonList *button_list; /* active buttons on page */ + HashTable *depend_hash; /* Hash tables of spadcommand dependencies */ + InputItem *input_list; /* List of input structures */ + InputItem *current_item; /* a pntr to the currently active item */ + HashTable *box_hash; /* place where all the boxes are stored */ + RadioBoxes *radio_boxes; /* a linked list of radio boxes */ + short page_flags; /* A list of flags for the page */ + char *helppage; /* the name of the helppage */ +} HyperDocPage; + +/* Structure for an unloaded page */ + +typedef struct unloaded_page { + short type; /* indicator of unloaded page */ + char *name; /* name of page */ + FilePosition fpos; /* where to find the page */ +} UnloadedPage; + +/* Structure for a HyperDoc Window */ + +typedef struct { + Window fMainWindow; /* The main text field window. */ + Window fScrollWindow; /* The scrolling area of the window */ + Window fDisplayedWindow; /* The current window of the above two, */ + /* being filled by display */ + + Window fScrollUpWindow; /* Window for scrolling up a line */ + Window fScrollDownWindow; /* Window for scrolling down a line */ + + Window scrollbar; /* the window for scrolling */ + Window scroller; /* the scroller window */ + + Window fTitleBarButton1; /* 1st titlebar bitmap button */ + Window fTitleBarButton2; /* 2nd titlebar bitmap button */ + Window fTitleBarButton3; /* 3rd titlebar bitmap button */ + Window fTitleBarButton4; /* 4th titlebar bitmap button */ + + int fScrollerTopPos; /* where the top of the scroller is */ + int fScrollerHeight; /* the height of the scroller */ + int fScrollBarHeight; /* the height for the scrollbar */ + + int scrollwidth; /* the width of the scrolling area */ + int scrollheight; /* the height of the scrolling area */ + int scrollupy; /* Current y position of the scroll up */ + /* button */ + int scrolldowny; /* Current y position of the scroll */ + /* downbutton */ + int scrollbary; /* Current y position of teh scrollbar */ + int scrollx; /* X coordinates for all of the above */ + int border_width; /* Width of the border */ + HyperDocPage *page; /* currently displayed page */ + int width, height; /* in pixels */ + int columns; /* Width in characters, only setable */ + /* for form pages */ + HyperDocPage **fMemoStack; /* stack of memo links */ + HyperDocPage **fDownLinkStack;/* stack of down links */ + + int *fDownLinkStackTop; /* stack of down links */ + int fMemoStackIndex; /* memo stack pointer */ + int fDownLinkStackIndex; /* downlink stack pointer */ + + HashTable *fWindowHashTable; /* hash table of active subwindows */ + HashTable *fPageHashTable; /* hash table of HyperDoc pages */ + HashTable *fPasteHashTable; /* hash table for paste in areas */ + HashTable *fMacroHashTable; /* hash table of HyperDoc macros */ + HashTable *fCondHashTable; /* hash table for values */ + HashTable *fPatchHashTable; /* hash table for patch locations */ + + int fAxiomFrame; /* Axiom frame number initializing window */ + GC fStandardGC; /* Graphics context for window */ + GC fInputGC; /* Graphics context for the input windows */ + GC fCursorGC; /* Graphics context for the cursors */ + GC fControlGC; /* Graphics context for the buttons */ + Cursor fDisplayedCursor; /* The currently displayed cursor */ +} HDWindow; + +/* Structure for identifying appropriate link hash tables */ + +typedef struct { + int code; /* code of active area */ + HyperDocPage *page; /* page for which hash table applies */ +} LinkHashID; + +/*** Flags for the page ***/ + +#define NOLINES 0000001 /* Ibid, for the bottom of the page ***/ + + +/* external variables and functions. See the source file for a description + of their purposes */ + +extern HashTable gSessionHashTable; /* hash table of HD windows */ + +extern HDWindow *gParentWindow; /* the parent window. The one that + * appears when you first start HD */ + +extern HyperLink *quitLink; /** a special link to the protected quit page **/ + + +#ifndef HTADD +/* From hyper.c */ +extern int gXScreenNumber; +extern Display *gXDisplay; +extern int gSwitch_to_mono; +extern unsigned long * spadColors; +extern int gIsEndOfOutput; +extern HDWindow *gWindow; +extern Sock *session_server; +extern Sock *spad_socket; +extern HashTable gFileHashTable; +extern HashTable gImageHashTable; /* A global hash table for images */ +extern Cursor gNormalCursor; /* The normal mouse cursor */ +extern Cursor gActiveCursor; /* The cursor in active regions */ +extern Cursor gBusyCursor; /* The clock cursor for when I am busy */ +extern int gIsAxiomServer; /* true iff HyperDoc is acting as an Axiom server */ +extern int gArgc; /* original argc from main */ +extern char **gArgv; /* original argv from main */ +/* from lex.c */ +extern long fpos, keyword_fpos; +extern Token token; +extern int last_token, input_type, last_ch; +extern char *input_string; +extern FILE *cfile; +/* from input.c */ +extern XImage *picked; +extern int picked_height; +extern int picked_width; +extern XImage *unpicked; +extern int unpicked_height; +extern int unpicked_width; +/* from display.c */ +extern int line_height; +extern int need_scroll_up_button; +extern int scrolling; +extern int need_scroll_down_button; +extern int space_width; +#endif + +/* Here are some of the functions and constants declared and needed in + htadd.c ******/ + +#define NoChar -9999 +#define temp_dir "/tmp/" +#define db_file_name "ht.db" +#define def_spad "/usr/local/axiom" + + +/* Types of HyperDoc pages */ + +#define UlUnknownPage 9993 /*I hate this hack, but I have to know whether*/ +#define UnknownPage 9994 /*this page has been loaded or not. */ +#define ErrorPage 9995 +#define Unixfd 9996 +#define SpadGen 9997 +#define Normal 9998 +#define UnloadedPageType 9999 + +/* Commands from Axiom */ + +#define EndOfPage 99 +#define SendLine 98 +#define StartPage 97 /* A normal HyperDoc page */ +#define LinkToPage 96 +#define PopUpPage 95 /* A pop-up page */ +#define PopUpNamedPage 94 +#define KillPage 93 +#define ReplacePage 92 +#define ReplaceNamedPage 91 +#define SpadError 90 + +/* Constants declaring size of page stacks */ + +#define MaxMemoDepth 25 /* max nesting level for memolinks */ +#define MaxDownlinkDepth 50 /* max downlink nesting depth */ + +/* Constants defining the size of various hash tables */ + +#define PageHashSize 1000 +#define FileHashSize 30 +#define SessionHashSize 10 +#define MacroHashSize 100 +#define ImageHashSize 100 +#define CondHashSize 100 +#define BoxHashSize 20 +#define PasteHashSize 100 +#define PatchHashSize 100 + +/* A couple of macros for memo and down links */ + +#define need_up_button \ + (gWindow->fMemoStackIndex ? gWindow->fDownLinkStackIndex >= \ + gWindow->fDownLinkStackTop[gWindow->fMemoStackIndex-1] \ + : gWindow->fDownLinkStackIndex) + +#define need_return_button (gWindow->fMemoStackIndex) + +#define need_help_button (gWindow->page->helppage != NULL) + +#define max(x,y) ((x) > (y) ? (x) : (y)) + + +#define pick_box(box) fill_box(box->win, box->selected) +#define unpick_box(box) fill_box(box->win, box->unselected) + +#define TopLevelHelpPage "ugHyperPage" +#define NoMoreHelpPage "NoMoreHelpPage" +#define KeyDefsHelpPage "ugHyperKeysPage" +#define InputAreaHelpPage "ugHyperInputPage" + +/* definitions for connecting to the Axiom server */ + +#define Connected 0 +#define NotConnected 1 +#define SpadBusy 2 + +/* some GUI-dependent stuff */ + +#define BeepAtTheUser() /* (XBell(gXDisplay, 5)) */ +#define LoudBeepAtTheUser() /* (XBell(gXDisplay, 50)) */ + + +/*** default fonts ***/ + +#if defined(RTplatform) || defined(PS2platform) || defined(RIOSplatform) || defined(AIX370platform) +#define RmFontDefault "Rom14" +#define TtFontDefault "Erg14" +#define ActiveFontDefault "Bld14" +#define AxiomFontDefault "Erg14" +#define EmphasizeFontDefault "Itl14" +#define BoldFontDefault "Bld14" +#endif + +#if defined(SUNplatform) || defined (SUN4OS5platform) || defined(SGIplatform) || defined (HP9platform) || defined(HP10platform) || defined (ALPHAplatform) || defined(LINUXplatform) || defined(MACOSXplatform) || defined(BSDplatform) +#define RmFontDefault "-adobe-courier-medium-r-normal--18-*-*-*-m-*-iso8859-1" +#define TtFontDefault "-adobe-courier-medium-r-normal--18-*-*-*-m-*-iso8859-1" +#define ActiveFontDefault "-adobe-courier-bold-r-normal--18-*-*-*-m-*-iso8859-1" +#define AxiomFontDefault "-adobe-courier-bold-o-normal--18-*-*-*-m-*-iso8859-1" +#define EmphasizeFontDefault "-adobe-courier-medium-o-normal--18-*-*-*-m-*-iso8859-1" +#define BoldFontDefault "-adobe-courier-bold-r-normal--18-*-*-*-m-*-iso8859-1" +#endif + + + + + + +typedef struct group_item { + int cur_color; + XFontStruct *cur_font; + int center; + struct group_item *next; +} GroupItem; + + +extern GroupItem *gTopOfGroupStack; + + +typedef struct cond_node { + char *label; + char *cond; +} CondNode; + +typedef struct parameter_list_type { + char **list; /** The parameters in string form **/ + short number; /** How many parameters are there **/ + struct parameter_list_type *next; +} *ParameterList; + +#endif +@ +\section{hyper.c} +<>= +/* + * This is the main module of the HyperDoc program. It contains the main + * routine which initializes all the X stuff, and the tables. Then it passes + * control over to the main event loop. + */ + +/* #define DEBUG 1 */ + +/* Include all the needed include files */ +#define _HYPER_C +#include "debug.h" + + +#include "hyper.h" + +#include +#include +#include +#include +#include + +#include "keyin.h" +#include "initx.h" +#include "event.h" +#include "parse-aux.h" +#include "bsdsignal.h" + +#include "all-hyper-proto.h1" +#include "sockio-c.h1" +#include "bsdsignal.h1" +/* + * Here is a flag used to tell me whether I made a good connection to the + * menu server. Needed so I don't send spad commands when I should not + */ + +int MenuServerOpened = 1; + +/* include icon bitmap data */ + +#define BITMAPDEPTH 1 + +/* X11 display and screen variables */ + +Display *gXDisplay; +int gXScreenNumber; + +/* + * Information about the top level HyperDoc window + */ + +HDWindow *gWindow = NULL; /* the current window */ +HDWindow *gParentWindow =NULL; /* the parent window. The one that appears + * when you first start HyperDoc */ + +HashTable gSessionHashTable; /* hash table of HD windows */ +HashTable init_page_hash; /* initial hash table of HD pages */ +HashTable init_macro_hash; /* initial hash table of HD macros */ +HashTable init_patch_hash; /* initial hash table of HD patches */ + +/* The various Cursors we use */ + +Cursor gNormalCursor; /* The normal mouse cursor */ +Cursor gActiveCursor; /* The cursor in active regions */ +Cursor gBusyCursor; /* The clock cursor for when I am busy */ + + +HashTable gFileHashTable; /* hash table of HyperDoc files */ +HashTable gImageHashTable; /* hash table for images */ + + +/* Some things needed for Handling interupts properly */ + +int gIsEndOfOutput; /* set to true when spad has finished output */ +int received_window_request = 0;/* true iff Spad wants a pop-up */ +int in_next_event = 0; /* true when in XNextEvent */ +int make_input_file = 0; /* true when making input files from ht */ +int make_patch_files = 0; /* true when making patch files from ht */ +int gmake_record_file= 0; /* true when making record files from ht */ +int gverify_record_file = 0; /* true when verifying record files from ht */ +int gverify_dates = 0; /* true when we want hypertex to verify ht.db dates */ + +Sock *session_server; /* socket connecting to session manager */ + +int gIsAxiomServer = 0; /* true iff HyperDoc is acting as a */ + /* an Axiom server */ + +int kill_spad = 0; /* kill spad when finished with paste file */ + +int gSwitch_to_mono=0; /* will be set to 1 if at any time we don't have + enough colours for the images. We will know this + when read_pixmap_file returns -1. We will use this + when deciding what to do in case of \inputimage */ + +int gTtFontIs850=0; /* a flag that tells us if the Tt font is a IBM pagecode 850 + font and hence supports the graphics chars + set when the TtFont is opened*/ + +/* + * Global copies of the command line arguments, so they never have to be + * passed as parameters. This is also so any child process starting up also + * has the same values. + */ + +int gArgc; +char **gArgv; + +char **input_file_list; +int input_file_count; + +/* + * SIGUSR2 is raised by the spadbuf program when it is done with the current + * command + */ + +void +sigusr2_handler(int sig) +{ + gIsEndOfOutput = 1; + return ; +} + +void +sigcld_handler(int sig) +{ + + /* why were we waiting after the child had already died ?? + because we don't want zombies */ + + int x; + wait(&x); + +} + +extern jmp_buf env; + + +/* Clean up spad sockets on exit */ +void +clean_socket(void ) +{ + char name[256]; + + make_server_name(name, MenuServerName); + unlink(name); +} + +/* + * initialize hash tables, signal handlers and windows, then call the main + * event handling loop + */ + +int +main(int argc, char **argv) +{ + int ret_status; + + /* Initialize some global values */ +/* fprintf(stderr,"hyper:main:entered\n");*/ + gArgc = argc; + gArgv = argv; + gIsEndOfOutput = 1; + +/* fprintf(stderr,"hyper:main:calling check_arguments\n");*/ + check_arguments(); +/* fprintf(stderr,"hyper:main:returned check_arguments\n");*/ + + /* + * initialize the hash tables for the files and the windows and images + */ +/* fprintf(stderr,"hyper:main:calling init_hash\n");*/ + init_hash(); +/* fprintf(stderr,"hyper:main:returned init_hash\n");*/ + + /* + * initialize the parser keyword hash table + */ +/* fprintf(stderr,"hyper:main:calling parser_init\n");*/ + parser_init(); +/* fprintf(stderr,"hyper:main:returned parser_init\n");*/ + +/* fprintf(stderr,"hyper:main:calling read_ht_db\n");*/ + read_ht_db(&init_page_hash, &init_macro_hash, &init_patch_hash); +/* fprintf(stderr,"hyper:main:returned read_ht_db\n");*/ + + /* + * Now initialize x. This includes opening the display, setting the + * screen and display global values, and also gets all the fonts and + * colors we will need. + */ + + if (!make_input_file && !gmake_record_file && !gverify_record_file) { +/* fprintf(stderr,"hyper:main:calling initializeWindowSystem\n");*/ + initializeWindowSystem(); +/* fprintf(stderr,"hyper:main:returned initializeWindowSystem\n");*/ + + /* + * Initialize some of the global values used by the input string + * routines + */ +/* fprintf(stderr,"hyper:main:calling init_keyin\n");*/ + init_keyin(); +/* fprintf(stderr,"hyper:main:returned init_keyin\n");*/ + + /* + * regardless of what else happened, we should always pop up an + * initial window. + */ + +/* fprintf(stderr,"hyper:main:calling init_top_window\n");*/ + ret_status = init_top_window("RootPage"); +/* fprintf(stderr,"hyper:main:returned init_top_window\n");*/ + gParentWindow = gWindow; + if (ret_status == -1) { + fprintf(stderr, + "(HyperDoc) Could not find RootPage for top-level window.\n"); + exit(-1); + } + + /* + * Tell it how to handle the user defined signals I may get + */ + bsdSignal(SIGUSR2, sigusr2_handler,RestartSystemCalls); + bsdSignal(SIGUSR1, SIG_IGN,RestartSystemCalls); +#if defined(BSDplatform) || defined(MACOSXplatform) + bsdSignal(SIGCHLD, sigcld_handler,RestartSystemCalls); +#else + bsdSignal(SIGCLD, sigcld_handler,RestartSystemCalls); +#endif + bsdSignal(SIGINT, SIG_IGN,RestartSystemCalls); + + /* + * Now go to the main event loop. I will never return, so just end + * the main routine after that + */ + + /* + * make an input file if requested + */ + } + else { + + /* + * Try to establish all the socket connections I need. If I am an + * gIsAxiomServer and the routine fails, it will exit for me + */ +/* fprintf(stderr,"hyper:main:in else case\n");*/ +/* fprintf(stderr,"hyper:main:calling make_server_connections\n");*/ + make_server_connections(); +/* fprintf(stderr,"hyper:main:returned make_server_connections\n");*/ + + + if (make_input_file) ht2_input(); + if (gmake_record_file) make_record(); + if (gverify_record_file) verify_record(); + exit(0); + } + + /* + * Try to establish all the socket connections I need. If I am an + * gIsAxiomServer and the routine fails, it will exit for me + */ +/* fprintf(stderr,"hyper:main:calling make_server_connections\n");*/ + make_server_connections(); +/* fprintf(stderr,"hyper:main:returned make_server_connections\n");*/ + + +/* fprintf(stderr,"hyper:main:calling mainEventLoop\n");*/ + mainEventLoop(); +/* fprintf(stderr,"hyper:main:returned mainEventLoop\n");*/ + + return 0; +} + +/* + * Initializes the hash table for Files, and Windows + */ + +static void +init_hash(void) +{ + hash_init(&gFileHashTable, + FileHashSize, + (EqualFunction)string_equal, + (HashcodeFunction) string_hash); + hash_init(&gSessionHashTable, + SessionHashSize, + (EqualFunction) window_equal, + (HashcodeFunction) window_code); + hash_init(&gImageHashTable, + ImageHashSize, + (EqualFunction) string_equal, + (HashcodeFunction) string_hash); +} + +/* initialize the HyperDoc page hierarchy data structures */ + +void +init_page_structs(HDWindow *w) +{ + int i; + + w->fMemoStackIndex = 0; + for (i = 0; i < MaxMemoDepth; i++) { + w->fMemoStack[i] = NULL; + w->fDownLinkStackTop[i] = 0; + } + w->fDownLinkStackIndex = 0; + for (i = 0; i < MaxDownlinkDepth; i++) + w->fDownLinkStack[i] = NULL; +} + +static void +check_arguments(void) +{ + int i; + + /* + * Now check the command line arguments, to see if I am supposed to be a + * server or not + */ + for (i = 1; i < gArgc; i++) { + if (gArgv[i][0] == '-') + switch (gArgv[i][1]) { + case 'p': + gverify_dates=1; + break; + case 's': + if (!MenuServerOpened) { + fprintf(stderr, "(HyperDoc) Server already in use.\n"); + exit(-1); + } + gIsAxiomServer = 1; + break; + case 'i': + if (gArgv[i][2] == 'p') + make_patch_files = 1; + make_input_file = 1; + input_file_list = gArgv + i + 1; + input_file_count = gArgc - i - 1; + break; + case 'k': + kill_spad = 1; + break; + case 'r': + if (gArgv[i][2] == 'm') + gmake_record_file=1; + else if (gArgv[i][2] == 'v') + gverify_record_file=1; + else + fprintf(stderr, "(HyperDoc) v or m must follow -r\n"); + input_file_list = gArgv + i + 1; + input_file_count = gArgc - i - 1; + break; + default: + fprintf(stderr, "(HyperDoc) Unexpected Command Line Argument %s\n", gArgv[i]); + fprintf(stderr, " Usage: hypertex [-s]\n"); + break; + } + } +} + +static void +make_server_connections(void) +{ + int i, wait_time; + + /* + * Try to open the menuserver socket, if I can not, then set a flag + */ + + if (open_server(MenuServerName) == -2) { + fprintf(stderr, "(HyperDoc) Warning: Not connected to AXIOM Server!\n"); + MenuServerOpened = 0; + } + else + MenuServerOpened = 1; + + + /* + * If I have opened the MenuServer socket, then I should also try to open + * the SpadServer socket, so I can send stuff right to SPAD. + */ + + if (MenuServerOpened) { + + /* + * If I am a ht server, then I should not continue on unless I + * establish some sort of connection + */ + + /* + * Modified on 11/20 so that it prints an error message every ten for + * ten tries at opeing the socket. If it fails all ten times, it + * gives up and exits. + */ + + if (!gIsAxiomServer) + wait_time = 2; + else + wait_time = 1000; + + for (i = 0, spad_socket = NULL; i < 2 && spad_socket == NULL; i++) { + spad_socket = connect_to_local_server(SpadServer, + MenuServer, wait_time); + if (gIsAxiomServer && spad_socket == NULL) + fprintf(stderr, "(HyperDoc) Error opening AXIOM server. Retrying ...\n"); + else + i = 11; + } + if (! spad_socket) { + fprintf(stderr, "(HyperDoc) Couldn't connect to AXIOM server!\n"); + if (!gIsAxiomServer) + MenuServerOpened = 0; + else { + fprintf(stderr, "(HyperDoc) Couldn't connect to AXIOM server!\n"); + exit(-1); + } + } + else { + + /* + * Do the same thing for the SessionServer + */ + + for (i = 0, session_server = NULL; i < 2 && session_server == NULL + ; i++) { + session_server = + connect_to_local_server(SessionServer, MenuServer, + wait_time); + if (gIsAxiomServer && session_server == NULL) { + fprintf(stderr, + "(HyperDoc) Error opening SessionServer, Retrying ...\n"); + } + else + i = 11; + } + if (session_server == NULL) { + fprintf(stderr, "(HyperDoc) Connection attempt to session manager timed out.\n"); + if (gIsAxiomServer) { + fprintf(stderr, + "(HyperDoc) Server unable to connect to session server\n"); + exit(-1); + } + else { + MenuServerOpened = 0; + } + } + } + } +} +@ +\section{License} +<>= +/* +Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +*/ + +@ +<<*>>= +<> +<> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol8.pamphlet b/books/bookvol8.pamphlet new file mode 100644 index 0000000..eed4adb --- /dev/null +++ b/books/bookvol8.pamphlet @@ -0,0 +1,219 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 8: Axiom Graphics}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Overview} +This book contains the hyperdoc routines for Axiom. +\chapter{Support Routines} +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/bookvol9.pamphlet b/books/bookvol9.pamphlet new file mode 100644 index 0000000..4acb8b2 --- /dev/null +++ b/books/bookvol9.pamphlet @@ -0,0 +1,333 @@ +\documentclass{book} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +% struggle with latex figure-floating behavior +\renewcommand\floatpagefraction{.9} +\renewcommand\topfraction{.9} +\renewcommand\bottomfraction{.9} +\renewcommand\textfraction{.1} +\setcounter{totalnumber}{50} +\setcounter{topnumber}{50} +\setcounter{bottomnumber}{50} + + +%% spadgraph are the actual text that you type at the axiom prompt for draw +\providecommand{\spadgraph}[1]% +{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } + +% spadfunFrom records the function name and domain in the index +\providecommand{\spadfunFrom}[2]% +{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} + +%% spadsig gives the standard -> notation for signatures +\providecommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} + +% special meanings for math characters +\providecommand{\N}{\mbox{\bbold N}} +\providecommand{\Natural}{\mbox{\bbold N}} +\providecommand{\Z}{\mbox{\bbold Z}} +\providecommand{\Integer}{\mbox{\bbold Z}} +\providecommand{\Rational}{\mbox{\bbold Q}} +\providecommand{\Q}{\mbox{\bbold Q}} +\providecommand{\Complex}{\mbox{\bbold C}} +\providecommand{\C}{{\mathcal C}} +\providecommand{\Real}{\mbox{\bbold R}} +\providecommand{\F}{{\mathcal F}} +\providecommand{\R}{{\mathcal R}} + +% draw a box around a text block +\providecommand\boxed[2]{% +\begin{center} +\begin{tabular}{|c|} +\hline +\begin{minipage}{#1} +\normalsize +{#2} +\end{minipage}\\ +\hline +\end{tabular} +\end{center}} + +\providecommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} +\providecommand{\argDef}[1]{{\tt ({#1})}} +\providecommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} +\providecommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} +\providecommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} + +\def\glossaryTerm#1{{\bf #1}\index{#1}} +\def\glossaryTermNoIndex#1{{\bf #1}} +\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} +\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} +\def\csch{\mathop{\rm csch}\nolimits} + +\def\erf{\mathop{\rm erf}\nolimits} + +\def\zag#1#2{ + {{\hfill \left. {#1} \right|} + \over + {\left| {#2} \right. \hfill} + } +} + + +% these bitmaps are used by HyperDoc +\newdimen\commentWidth +\commentWidth=11pc +\newdimen\colGutterWidth +\colGutterWidth=1pc +\newdimen\baseLeftSkip +\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth + +\providecommand\ExitBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} + +\providecommand\ReturnBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} + +\providecommand\HelpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} + +\providecommand\UpBitmap% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} + +\providecommand{\tpd}[5]% +{{\setlength{\unitlength}{0.01in}% +\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} + +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 9: Axiom Compiler}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\section{Makefile} +This book is actually a literate program\cite{2} and can contain +executable source code. In particular, the Makefile for this book +is part of the source of the book and is included below. Axiom +uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +<<*>>= +PROJECT=bookvol9 +TANGLE=/usr/local/bin/NOTANGLE +WEAVE=/usr/local/bin/NOWEAVE +LATEX=/usr/bin/latex +MAKEINDEX=/usr/bin/makeindex + +all: + ${WEAVE} -t8 -delay ${PROJECT}.pamphlet >${PROJECT}.tex + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + ${MAKEINDEX} ${PROJECT}.idx + ${LATEX} ${PROJECT}.tex 2>/dev/null 1>/dev/null + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} Jenks, R.J. and Sutor, R.S. +``Axiom -- The Scientific Computation System'' +Springer-Verlag New York (1992) +ISBN 0-387-97855-0 +\bibitem{2} Knuth, Donald E., ``Literate Programming'' +Center for the Study of Language and Information +ISBN 0-937073-81-4 +Stanford CA (1992) +\bibitem{3} Page, William, ``The Axiom Wiki Website''\\ +{\bf http://wiki.axiom-developer.org} +\bibitem{4} Watt, Stephen, ``Aldor'',\\ +{\bf http://www.aldor.org} +\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', +Addison-Wesley, New York ISBN 0-201-52983-1 +\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for +Literate Programming''\\ +{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} +\end{thebibliography} +\printindex +\end{document} diff --git a/books/ps/23dcola.ps b/books/ps/23dcola.ps new file mode 100644 index 0000000..e4dc23e --- /dev/null +++ b/books/ps/23dcola.ps @@ -0,0 +1,347 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 244 0 244 psDrawLine + globalGC1 128 259 128 0 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 12 14 12 14 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 17 33 12 14 psDrawLine + globalGC1 16 32 3 3 0 360 17 33 psFillArc + globalGC1 22 51 17 33 psDrawLine + globalGC1 21 50 3 3 0 360 22 51 psFillArc + globalGC1 27 68 22 51 psDrawLine + globalGC1 26 67 3 3 0 360 27 68 psFillArc + globalGC1 32 84 27 68 psDrawLine + globalGC1 31 83 3 3 0 360 32 84 psFillArc + globalGC1 36 100 32 84 psDrawLine + globalGC1 35 99 3 3 0 360 36 100 psFillArc + globalGC1 41 115 36 100 psDrawLine + globalGC1 40 114 3 3 0 360 41 115 psFillArc + globalGC1 46 129 41 115 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 51 142 46 129 psDrawLine + globalGC1 50 141 3 3 0 360 51 142 psFillArc + globalGC1 56 154 51 142 psDrawLine + globalGC1 55 153 3 3 0 360 56 154 psFillArc + globalGC1 60 166 56 154 psDrawLine + globalGC1 59 165 3 3 0 360 60 166 psFillArc + globalGC1 65 177 60 166 psDrawLine + globalGC1 64 176 3 3 0 360 65 177 psFillArc + globalGC1 70 187 65 177 psDrawLine + globalGC1 69 186 3 3 0 360 70 187 psFillArc + globalGC1 75 196 70 187 psDrawLine + globalGC1 74 195 3 3 0 360 75 196 psFillArc + globalGC1 80 204 75 196 psDrawLine + globalGC1 79 203 3 3 0 360 80 204 psFillArc + globalGC1 84 212 80 204 psDrawLine + globalGC1 83 211 3 3 0 360 84 212 psFillArc + globalGC1 89 219 84 212 psDrawLine + globalGC1 88 218 3 3 0 360 89 219 psFillArc + globalGC1 94 225 89 219 psDrawLine + globalGC1 93 224 3 3 0 360 94 225 psFillArc + globalGC1 99 230 94 225 psDrawLine + globalGC1 98 229 3 3 0 360 99 230 psFillArc + globalGC1 104 234 99 230 psDrawLine + globalGC1 103 233 3 3 0 360 104 234 psFillArc + globalGC1 108 238 104 234 psDrawLine + globalGC1 107 237 3 3 0 360 108 238 psFillArc + globalGC1 113 241 108 238 psDrawLine + globalGC1 112 240 3 3 0 360 113 241 psFillArc + globalGC1 118 243 113 241 psDrawLine + globalGC1 117 242 3 3 0 360 118 243 psFillArc + globalGC1 123 244 118 243 psDrawLine + globalGC1 122 243 3 3 0 360 123 244 psFillArc + globalGC1 127 244 123 244 psDrawLine + globalGC1 126 243 3 3 0 360 127 244 psFillArc + globalGC1 132 244 127 244 psDrawLine + globalGC1 131 243 3 3 0 360 132 244 psFillArc + globalGC1 137 243 132 244 psDrawLine + globalGC1 136 242 3 3 0 360 137 243 psFillArc + globalGC1 142 241 137 243 psDrawLine + globalGC1 141 240 3 3 0 360 142 241 psFillArc + globalGC1 147 238 142 241 psDrawLine + globalGC1 146 237 3 3 0 360 147 238 psFillArc + globalGC1 151 234 147 238 psDrawLine + globalGC1 150 233 3 3 0 360 151 234 psFillArc + globalGC1 156 230 151 234 psDrawLine + globalGC1 155 229 3 3 0 360 156 230 psFillArc + globalGC1 161 225 156 230 psDrawLine + globalGC1 160 224 3 3 0 360 161 225 psFillArc + globalGC1 166 219 161 225 psDrawLine + globalGC1 165 218 3 3 0 360 166 219 psFillArc + globalGC1 171 212 166 219 psDrawLine + globalGC1 170 211 3 3 0 360 171 212 psFillArc + globalGC1 175 204 171 212 psDrawLine + globalGC1 174 203 3 3 0 360 175 204 psFillArc + globalGC1 180 196 175 204 psDrawLine + globalGC1 179 195 3 3 0 360 180 196 psFillArc + globalGC1 185 187 180 196 psDrawLine + globalGC1 184 186 3 3 0 360 185 187 psFillArc + globalGC1 190 177 185 187 psDrawLine + globalGC1 189 176 3 3 0 360 190 177 psFillArc + globalGC1 195 166 190 177 psDrawLine + globalGC1 194 165 3 3 0 360 195 166 psFillArc + globalGC1 199 154 195 166 psDrawLine + globalGC1 198 153 3 3 0 360 199 154 psFillArc + globalGC1 204 142 199 154 psDrawLine + globalGC1 203 141 3 3 0 360 204 142 psFillArc + globalGC1 209 129 204 142 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 214 115 209 129 psDrawLine + globalGC1 213 114 3 3 0 360 214 115 psFillArc + globalGC1 219 100 214 115 psDrawLine + globalGC1 218 99 3 3 0 360 219 100 psFillArc + globalGC1 223 84 219 100 psDrawLine + globalGC1 222 83 3 3 0 360 223 84 psFillArc + globalGC1 228 68 223 84 psDrawLine + globalGC1 227 67 3 3 0 360 228 68 psFillArc + globalGC1 233 51 228 68 psDrawLine + globalGC1 232 50 3 3 0 360 233 51 psFillArc + globalGC1 238 33 233 51 psDrawLine + globalGC1 237 32 3 3 0 360 238 33 psFillArc + globalGC1 243 14 238 33 psDrawLine + globalGC1 242 13 3 3 0 360 243 14 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/23dcolb.ps b/books/ps/23dcolb.ps new file mode 100644 index 0000000..e4dc23e --- /dev/null +++ b/books/ps/23dcolb.ps @@ -0,0 +1,347 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 244 0 244 psDrawLine + globalGC1 128 259 128 0 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 12 14 12 14 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 17 33 12 14 psDrawLine + globalGC1 16 32 3 3 0 360 17 33 psFillArc + globalGC1 22 51 17 33 psDrawLine + globalGC1 21 50 3 3 0 360 22 51 psFillArc + globalGC1 27 68 22 51 psDrawLine + globalGC1 26 67 3 3 0 360 27 68 psFillArc + globalGC1 32 84 27 68 psDrawLine + globalGC1 31 83 3 3 0 360 32 84 psFillArc + globalGC1 36 100 32 84 psDrawLine + globalGC1 35 99 3 3 0 360 36 100 psFillArc + globalGC1 41 115 36 100 psDrawLine + globalGC1 40 114 3 3 0 360 41 115 psFillArc + globalGC1 46 129 41 115 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 51 142 46 129 psDrawLine + globalGC1 50 141 3 3 0 360 51 142 psFillArc + globalGC1 56 154 51 142 psDrawLine + globalGC1 55 153 3 3 0 360 56 154 psFillArc + globalGC1 60 166 56 154 psDrawLine + globalGC1 59 165 3 3 0 360 60 166 psFillArc + globalGC1 65 177 60 166 psDrawLine + globalGC1 64 176 3 3 0 360 65 177 psFillArc + globalGC1 70 187 65 177 psDrawLine + globalGC1 69 186 3 3 0 360 70 187 psFillArc + globalGC1 75 196 70 187 psDrawLine + globalGC1 74 195 3 3 0 360 75 196 psFillArc + globalGC1 80 204 75 196 psDrawLine + globalGC1 79 203 3 3 0 360 80 204 psFillArc + globalGC1 84 212 80 204 psDrawLine + globalGC1 83 211 3 3 0 360 84 212 psFillArc + globalGC1 89 219 84 212 psDrawLine + globalGC1 88 218 3 3 0 360 89 219 psFillArc + globalGC1 94 225 89 219 psDrawLine + globalGC1 93 224 3 3 0 360 94 225 psFillArc + globalGC1 99 230 94 225 psDrawLine + globalGC1 98 229 3 3 0 360 99 230 psFillArc + globalGC1 104 234 99 230 psDrawLine + globalGC1 103 233 3 3 0 360 104 234 psFillArc + globalGC1 108 238 104 234 psDrawLine + globalGC1 107 237 3 3 0 360 108 238 psFillArc + globalGC1 113 241 108 238 psDrawLine + globalGC1 112 240 3 3 0 360 113 241 psFillArc + globalGC1 118 243 113 241 psDrawLine + globalGC1 117 242 3 3 0 360 118 243 psFillArc + globalGC1 123 244 118 243 psDrawLine + globalGC1 122 243 3 3 0 360 123 244 psFillArc + globalGC1 127 244 123 244 psDrawLine + globalGC1 126 243 3 3 0 360 127 244 psFillArc + globalGC1 132 244 127 244 psDrawLine + globalGC1 131 243 3 3 0 360 132 244 psFillArc + globalGC1 137 243 132 244 psDrawLine + globalGC1 136 242 3 3 0 360 137 243 psFillArc + globalGC1 142 241 137 243 psDrawLine + globalGC1 141 240 3 3 0 360 142 241 psFillArc + globalGC1 147 238 142 241 psDrawLine + globalGC1 146 237 3 3 0 360 147 238 psFillArc + globalGC1 151 234 147 238 psDrawLine + globalGC1 150 233 3 3 0 360 151 234 psFillArc + globalGC1 156 230 151 234 psDrawLine + globalGC1 155 229 3 3 0 360 156 230 psFillArc + globalGC1 161 225 156 230 psDrawLine + globalGC1 160 224 3 3 0 360 161 225 psFillArc + globalGC1 166 219 161 225 psDrawLine + globalGC1 165 218 3 3 0 360 166 219 psFillArc + globalGC1 171 212 166 219 psDrawLine + globalGC1 170 211 3 3 0 360 171 212 psFillArc + globalGC1 175 204 171 212 psDrawLine + globalGC1 174 203 3 3 0 360 175 204 psFillArc + globalGC1 180 196 175 204 psDrawLine + globalGC1 179 195 3 3 0 360 180 196 psFillArc + globalGC1 185 187 180 196 psDrawLine + globalGC1 184 186 3 3 0 360 185 187 psFillArc + globalGC1 190 177 185 187 psDrawLine + globalGC1 189 176 3 3 0 360 190 177 psFillArc + globalGC1 195 166 190 177 psDrawLine + globalGC1 194 165 3 3 0 360 195 166 psFillArc + globalGC1 199 154 195 166 psDrawLine + globalGC1 198 153 3 3 0 360 199 154 psFillArc + globalGC1 204 142 199 154 psDrawLine + globalGC1 203 141 3 3 0 360 204 142 psFillArc + globalGC1 209 129 204 142 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 214 115 209 129 psDrawLine + globalGC1 213 114 3 3 0 360 214 115 psFillArc + globalGC1 219 100 214 115 psDrawLine + globalGC1 218 99 3 3 0 360 219 100 psFillArc + globalGC1 223 84 219 100 psDrawLine + globalGC1 222 83 3 3 0 360 223 84 psFillArc + globalGC1 228 68 223 84 psDrawLine + globalGC1 227 67 3 3 0 360 228 68 psFillArc + globalGC1 233 51 228 68 psDrawLine + globalGC1 232 50 3 3 0 360 233 51 psFillArc + globalGC1 238 33 233 51 psDrawLine + globalGC1 237 32 3 3 0 360 238 33 psFillArc + globalGC1 243 14 238 33 psDrawLine + globalGC1 242 13 3 3 0 360 243 14 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/23dpal.ps b/books/ps/23dpal.ps new file mode 100644 index 0000000..e4dc23e --- /dev/null +++ b/books/ps/23dpal.ps @@ -0,0 +1,347 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 244 0 244 psDrawLine + globalGC1 128 259 128 0 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 12 14 12 14 psDrawLine + globalGC1 11 13 3 3 0 360 12 14 psFillArc + globalGC1 17 33 12 14 psDrawLine + globalGC1 16 32 3 3 0 360 17 33 psFillArc + globalGC1 22 51 17 33 psDrawLine + globalGC1 21 50 3 3 0 360 22 51 psFillArc + globalGC1 27 68 22 51 psDrawLine + globalGC1 26 67 3 3 0 360 27 68 psFillArc + globalGC1 32 84 27 68 psDrawLine + globalGC1 31 83 3 3 0 360 32 84 psFillArc + globalGC1 36 100 32 84 psDrawLine + globalGC1 35 99 3 3 0 360 36 100 psFillArc + globalGC1 41 115 36 100 psDrawLine + globalGC1 40 114 3 3 0 360 41 115 psFillArc + globalGC1 46 129 41 115 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 51 142 46 129 psDrawLine + globalGC1 50 141 3 3 0 360 51 142 psFillArc + globalGC1 56 154 51 142 psDrawLine + globalGC1 55 153 3 3 0 360 56 154 psFillArc + globalGC1 60 166 56 154 psDrawLine + globalGC1 59 165 3 3 0 360 60 166 psFillArc + globalGC1 65 177 60 166 psDrawLine + globalGC1 64 176 3 3 0 360 65 177 psFillArc + globalGC1 70 187 65 177 psDrawLine + globalGC1 69 186 3 3 0 360 70 187 psFillArc + globalGC1 75 196 70 187 psDrawLine + globalGC1 74 195 3 3 0 360 75 196 psFillArc + globalGC1 80 204 75 196 psDrawLine + globalGC1 79 203 3 3 0 360 80 204 psFillArc + globalGC1 84 212 80 204 psDrawLine + globalGC1 83 211 3 3 0 360 84 212 psFillArc + globalGC1 89 219 84 212 psDrawLine + globalGC1 88 218 3 3 0 360 89 219 psFillArc + globalGC1 94 225 89 219 psDrawLine + globalGC1 93 224 3 3 0 360 94 225 psFillArc + globalGC1 99 230 94 225 psDrawLine + globalGC1 98 229 3 3 0 360 99 230 psFillArc + globalGC1 104 234 99 230 psDrawLine + globalGC1 103 233 3 3 0 360 104 234 psFillArc + globalGC1 108 238 104 234 psDrawLine + globalGC1 107 237 3 3 0 360 108 238 psFillArc + globalGC1 113 241 108 238 psDrawLine + globalGC1 112 240 3 3 0 360 113 241 psFillArc + globalGC1 118 243 113 241 psDrawLine + globalGC1 117 242 3 3 0 360 118 243 psFillArc + globalGC1 123 244 118 243 psDrawLine + globalGC1 122 243 3 3 0 360 123 244 psFillArc + globalGC1 127 244 123 244 psDrawLine + globalGC1 126 243 3 3 0 360 127 244 psFillArc + globalGC1 132 244 127 244 psDrawLine + globalGC1 131 243 3 3 0 360 132 244 psFillArc + globalGC1 137 243 132 244 psDrawLine + globalGC1 136 242 3 3 0 360 137 243 psFillArc + globalGC1 142 241 137 243 psDrawLine + globalGC1 141 240 3 3 0 360 142 241 psFillArc + globalGC1 147 238 142 241 psDrawLine + globalGC1 146 237 3 3 0 360 147 238 psFillArc + globalGC1 151 234 147 238 psDrawLine + globalGC1 150 233 3 3 0 360 151 234 psFillArc + globalGC1 156 230 151 234 psDrawLine + globalGC1 155 229 3 3 0 360 156 230 psFillArc + globalGC1 161 225 156 230 psDrawLine + globalGC1 160 224 3 3 0 360 161 225 psFillArc + globalGC1 166 219 161 225 psDrawLine + globalGC1 165 218 3 3 0 360 166 219 psFillArc + globalGC1 171 212 166 219 psDrawLine + globalGC1 170 211 3 3 0 360 171 212 psFillArc + globalGC1 175 204 171 212 psDrawLine + globalGC1 174 203 3 3 0 360 175 204 psFillArc + globalGC1 180 196 175 204 psDrawLine + globalGC1 179 195 3 3 0 360 180 196 psFillArc + globalGC1 185 187 180 196 psDrawLine + globalGC1 184 186 3 3 0 360 185 187 psFillArc + globalGC1 190 177 185 187 psDrawLine + globalGC1 189 176 3 3 0 360 190 177 psFillArc + globalGC1 195 166 190 177 psDrawLine + globalGC1 194 165 3 3 0 360 195 166 psFillArc + globalGC1 199 154 195 166 psDrawLine + globalGC1 198 153 3 3 0 360 199 154 psFillArc + globalGC1 204 142 199 154 psDrawLine + globalGC1 203 141 3 3 0 360 204 142 psFillArc + globalGC1 209 129 204 142 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 214 115 209 129 psDrawLine + globalGC1 213 114 3 3 0 360 214 115 psFillArc + globalGC1 219 100 214 115 psDrawLine + globalGC1 218 99 3 3 0 360 219 100 psFillArc + globalGC1 223 84 219 100 psDrawLine + globalGC1 222 83 3 3 0 360 223 84 psFillArc + globalGC1 228 68 223 84 psDrawLine + globalGC1 227 67 3 3 0 360 228 68 psFillArc + globalGC1 233 51 228 68 psDrawLine + globalGC1 232 50 3 3 0 360 233 51 psFillArc + globalGC1 238 33 233 51 psDrawLine + globalGC1 237 32 3 3 0 360 238 33 psFillArc + globalGC1 243 14 238 33 psDrawLine + globalGC1 242 13 3 3 0 360 243 14 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2d1vara.ps b/books/ps/2d1vara.ps new file mode 100644 index 0000000..054f66a --- /dev/null +++ b/books/ps/2d1vara.ps @@ -0,0 +1,1535 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 12 259 12 0 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 12 129 12 129 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 17 129 12 129 psDrawLine + globalGC1 16 128 3 3 0 360 17 129 psFillArc + globalGC1 22 129 17 129 psDrawLine + globalGC1 21 128 3 3 0 360 22 129 psFillArc + globalGC1 27 129 22 129 psDrawLine + globalGC1 26 128 3 3 0 360 27 129 psFillArc + globalGC1 32 129 27 129 psDrawLine + globalGC1 31 128 3 3 0 360 32 129 psFillArc + globalGC1 36 129 32 129 psDrawLine + globalGC1 35 128 3 3 0 360 36 129 psFillArc + globalGC1 41 129 36 129 psDrawLine + globalGC1 40 128 3 3 0 360 41 129 psFillArc + globalGC1 43 130 41 129 psDrawLine + globalGC1 42 129 3 3 0 360 43 130 psFillArc + globalGC1 46 131 43 130 psDrawLine + globalGC1 45 130 3 3 0 360 46 131 psFillArc + globalGC1 47 131 46 131 psDrawLine + globalGC1 46 130 3 3 0 360 47 131 psFillArc + globalGC1 48 132 47 131 psDrawLine + globalGC1 47 131 3 3 0 360 48 132 psFillArc + globalGC1 50 133 48 132 psDrawLine + globalGC1 49 132 3 3 0 360 50 133 psFillArc + globalGC1 51 134 50 133 psDrawLine + globalGC1 50 133 3 3 0 360 51 134 psFillArc + globalGC1 53 139 51 134 psDrawLine + globalGC1 52 138 3 3 0 360 53 139 psFillArc + globalGC1 56 147 53 139 psDrawLine + globalGC1 55 146 3 3 0 360 56 147 psFillArc + globalGC1 58 161 56 147 psDrawLine + globalGC1 57 160 3 3 0 360 58 161 psFillArc + globalGC1 59 172 58 161 psDrawLine + globalGC1 58 171 3 3 0 360 59 172 psFillArc + globalGC1 60 186 59 172 psDrawLine + globalGC1 59 185 3 3 0 360 60 186 psFillArc + globalGC1 62 203 60 186 psDrawLine + globalGC1 61 202 3 3 0 360 62 203 psFillArc + globalGC1 63 221 62 203 psDrawLine + globalGC1 62 220 3 3 0 360 63 221 psFillArc + globalGC1 63 230 63 221 psDrawLine + globalGC1 62 229 3 3 0 360 63 230 psFillArc + globalGC1 64 237 63 230 psDrawLine + globalGC1 63 236 3 3 0 360 64 237 psFillArc + globalGC1 64 240 64 237 psDrawLine + globalGC1 63 239 3 3 0 360 64 240 psFillArc + globalGC1 65 241 64 240 psDrawLine + globalGC1 64 240 3 3 0 360 65 241 psFillArc + globalGC1 65 241 65 241 psDrawLine + globalGC1 64 240 3 3 0 360 65 241 psFillArc + globalGC1 65 241 65 241 psDrawLine + globalGC1 64 240 3 3 0 360 65 241 psFillArc + globalGC1 65 240 65 241 psDrawLine + globalGC1 64 239 3 3 0 360 65 240 psFillArc + globalGC1 65 239 65 240 psDrawLine + globalGC1 64 238 3 3 0 360 65 239 psFillArc + globalGC1 66 229 65 239 psDrawLine + globalGC1 65 228 3 3 0 360 66 229 psFillArc + globalGC1 66 209 66 229 psDrawLine + globalGC1 65 208 3 3 0 360 66 209 psFillArc + globalGC1 67 181 66 209 psDrawLine + globalGC1 66 180 3 3 0 360 67 181 psFillArc + globalGC1 67 167 67 181 psDrawLine + globalGC1 66 166 3 3 0 360 67 167 psFillArc + globalGC1 68 156 67 167 psDrawLine + globalGC1 67 155 3 3 0 360 68 156 psFillArc + globalGC1 68 153 68 156 psDrawLine + globalGC1 67 152 3 3 0 360 68 153 psFillArc + globalGC1 68 153 68 153 psDrawLine + globalGC1 67 152 3 3 0 360 68 153 psFillArc + globalGC1 68 153 68 153 psDrawLine + globalGC1 67 152 3 3 0 360 68 153 psFillArc + globalGC1 68 154 68 153 psDrawLine + globalGC1 67 153 3 3 0 360 68 154 psFillArc + globalGC1 68 155 68 154 psDrawLine + globalGC1 67 154 3 3 0 360 68 155 psFillArc + globalGC1 68 161 68 155 psDrawLine + globalGC1 67 160 3 3 0 360 68 161 psFillArc + globalGC1 68 182 68 161 psDrawLine + globalGC1 67 181 3 3 0 360 68 182 psFillArc + globalGC1 69 213 68 182 psDrawLine + globalGC1 68 212 3 3 0 360 69 213 psFillArc + globalGC1 69 229 69 213 psDrawLine + globalGC1 68 228 3 3 0 360 69 229 psFillArc + globalGC1 69 240 69 229 psDrawLine + globalGC1 68 239 3 3 0 360 69 240 psFillArc + globalGC1 69 243 69 240 psDrawLine + globalGC1 68 242 3 3 0 360 69 243 psFillArc + globalGC1 69 243 69 243 psDrawLine + globalGC1 68 242 3 3 0 360 69 243 psFillArc + globalGC1 69 244 69 243 psDrawLine + globalGC1 68 243 3 3 0 360 69 244 psFillArc + globalGC1 69 243 69 244 psDrawLine + globalGC1 68 242 3 3 0 360 69 243 psFillArc + globalGC1 69 242 69 243 psDrawLine + globalGC1 68 241 3 3 0 360 69 242 psFillArc + globalGC1 69 237 69 242 psDrawLine + globalGC1 68 236 3 3 0 360 69 237 psFillArc + globalGC1 69 219 69 237 psDrawLine + globalGC1 68 218 3 3 0 360 69 219 psFillArc + globalGC1 70 193 69 219 psDrawLine + globalGC1 69 192 3 3 0 360 70 193 psFillArc + globalGC1 70 167 70 193 psDrawLine + globalGC1 69 166 3 3 0 360 70 167 psFillArc + globalGC1 70 158 70 167 psDrawLine + globalGC1 69 157 3 3 0 360 70 158 psFillArc + globalGC1 70 155 70 158 psDrawLine + globalGC1 69 154 3 3 0 360 70 155 psFillArc + globalGC1 70 154 70 155 psDrawLine + globalGC1 69 153 3 3 0 360 70 154 psFillArc + globalGC1 70 154 70 154 psDrawLine + globalGC1 69 153 3 3 0 360 70 154 psFillArc + globalGC1 70 157 70 154 psDrawLine + globalGC1 69 156 3 3 0 360 70 157 psFillArc + globalGC1 70 167 70 157 psDrawLine + globalGC1 69 166 3 3 0 360 70 167 psFillArc + globalGC1 70 184 70 167 psDrawLine + globalGC1 69 183 3 3 0 360 70 184 psFillArc + globalGC1 70 206 70 184 psDrawLine + globalGC1 69 205 3 3 0 360 70 206 psFillArc + globalGC1 70 226 70 206 psDrawLine + globalGC1 69 225 3 3 0 360 70 226 psFillArc + globalGC1 70 241 70 226 psDrawLine + globalGC1 69 240 3 3 0 360 70 241 psFillArc + globalGC1 70 243 70 241 psDrawLine + globalGC1 69 242 3 3 0 360 70 243 psFillArc + globalGC1 70 244 70 243 psDrawLine + globalGC1 69 243 3 3 0 360 70 244 psFillArc + globalGC1 70 244 70 244 psDrawLine + globalGC1 69 243 3 3 0 360 70 244 psFillArc + globalGC1 70 243 70 244 psDrawLine + globalGC1 69 242 3 3 0 360 70 243 psFillArc + globalGC1 70 238 70 243 psDrawLine + globalGC1 69 237 3 3 0 360 70 238 psFillArc + globalGC1 71 229 70 238 psDrawLine + globalGC1 70 228 3 3 0 360 71 229 psFillArc + globalGC1 71 202 71 229 psDrawLine + globalGC1 70 201 3 3 0 360 71 202 psFillArc + globalGC1 71 172 71 202 psDrawLine + globalGC1 70 171 3 3 0 360 71 172 psFillArc + globalGC1 71 161 71 172 psDrawLine + globalGC1 70 160 3 3 0 360 71 161 psFillArc + globalGC1 71 157 71 161 psDrawLine + globalGC1 70 156 3 3 0 360 71 157 psFillArc + globalGC1 71 154 71 157 psDrawLine + globalGC1 70 153 3 3 0 360 71 154 psFillArc + globalGC1 71 154 71 154 psDrawLine + globalGC1 70 153 3 3 0 360 71 154 psFillArc + globalGC1 71 155 71 154 psDrawLine + globalGC1 70 154 3 3 0 360 71 155 psFillArc + globalGC1 71 164 71 155 psDrawLine + globalGC1 70 163 3 3 0 360 71 164 psFillArc + globalGC1 71 180 71 164 psDrawLine + globalGC1 70 179 3 3 0 360 71 180 psFillArc + globalGC1 71 200 71 180 psDrawLine + globalGC1 70 199 3 3 0 360 71 200 psFillArc + globalGC1 71 221 71 200 psDrawLine + globalGC1 70 220 3 3 0 360 71 221 psFillArc + globalGC1 71 237 71 221 psDrawLine + globalGC1 70 236 3 3 0 360 71 237 psFillArc + globalGC1 71 242 71 237 psDrawLine + globalGC1 70 241 3 3 0 360 71 242 psFillArc + globalGC1 71 244 71 242 psDrawLine + globalGC1 70 243 3 3 0 360 71 244 psFillArc + globalGC1 71 243 71 244 psDrawLine + globalGC1 70 242 3 3 0 360 71 243 psFillArc + globalGC1 71 238 71 243 psDrawLine + globalGC1 70 237 3 3 0 360 71 238 psFillArc + globalGC1 71 219 71 238 psDrawLine + globalGC1 70 218 3 3 0 360 71 219 psFillArc + globalGC1 71 193 71 219 psDrawLine + globalGC1 70 192 3 3 0 360 71 193 psFillArc + globalGC1 71 167 71 193 psDrawLine + globalGC1 70 166 3 3 0 360 71 167 psFillArc + globalGC1 71 159 71 167 psDrawLine + globalGC1 70 158 3 3 0 360 71 159 psFillArc + globalGC1 71 154 71 159 psDrawLine + globalGC1 70 153 3 3 0 360 71 154 psFillArc + globalGC1 71 155 71 154 psDrawLine + globalGC1 70 154 3 3 0 360 71 155 psFillArc + globalGC1 71 162 71 155 psDrawLine + globalGC1 70 161 3 3 0 360 71 162 psFillArc + globalGC1 71 189 71 162 psDrawLine + globalGC1 70 188 3 3 0 360 71 189 psFillArc + globalGC1 71 223 71 189 psDrawLine + globalGC1 70 222 3 3 0 360 71 223 psFillArc + globalGC1 71 236 71 223 psDrawLine + globalGC1 70 235 3 3 0 360 71 236 psFillArc + globalGC1 71 243 71 236 psDrawLine + globalGC1 70 242 3 3 0 360 71 243 psFillArc + globalGC1 71 243 71 243 psDrawLine + globalGC1 70 242 3 3 0 360 71 243 psFillArc + globalGC1 71 233 71 243 psDrawLine + globalGC1 70 232 3 3 0 360 71 233 psFillArc + globalGC1 71 216 71 233 psDrawLine + globalGC1 70 215 3 3 0 360 71 216 psFillArc + globalGC1 71 195 71 216 psDrawLine + globalGC1 70 194 3 3 0 360 71 195 psFillArc + globalGC1 71 174 71 195 psDrawLine + globalGC1 70 173 3 3 0 360 71 174 psFillArc + globalGC1 71 159 71 174 psDrawLine + globalGC1 70 158 3 3 0 360 71 159 psFillArc + globalGC1 71 154 71 159 psDrawLine + globalGC1 70 153 3 3 0 360 71 154 psFillArc + globalGC1 71 161 71 154 psDrawLine + globalGC1 70 160 3 3 0 360 71 161 psFillArc + globalGC1 71 180 71 161 psDrawLine + globalGC1 70 179 3 3 0 360 71 180 psFillArc + globalGC1 72 206 71 180 psDrawLine + globalGC1 71 205 3 3 0 360 72 206 psFillArc + globalGC1 72 230 72 206 psDrawLine + globalGC1 71 229 3 3 0 360 72 230 psFillArc + globalGC1 72 238 72 230 psDrawLine + globalGC1 71 237 3 3 0 360 72 238 psFillArc + globalGC1 72 243 72 238 psDrawLine + globalGC1 71 242 3 3 0 360 72 243 psFillArc + globalGC1 72 240 72 243 psDrawLine + globalGC1 71 239 3 3 0 360 72 240 psFillArc + globalGC1 72 219 72 240 psDrawLine + globalGC1 71 218 3 3 0 360 72 219 psFillArc + globalGC1 72 189 72 219 psDrawLine + globalGC1 71 188 3 3 0 360 72 189 psFillArc + globalGC1 72 163 72 189 psDrawLine + globalGC1 71 162 3 3 0 360 72 163 psFillArc + globalGC1 72 154 72 163 psDrawLine + globalGC1 71 153 3 3 0 360 72 154 psFillArc + globalGC1 72 171 72 154 psDrawLine + globalGC1 71 170 3 3 0 360 72 171 psFillArc + globalGC1 72 204 72 171 psDrawLine + globalGC1 71 203 3 3 0 360 72 204 psFillArc + globalGC1 72 236 72 204 psDrawLine + globalGC1 71 235 3 3 0 360 72 236 psFillArc + globalGC1 72 243 72 236 psDrawLine + globalGC1 71 242 3 3 0 360 72 243 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psFillArc + globalGC1 191 55 191 35 psDrawLine + globalGC1 190 54 3 3 0 360 191 55 psFillArc + globalGC1 191 77 191 55 psDrawLine + globalGC1 190 76 3 3 0 360 191 77 psFillArc + globalGC1 191 95 191 77 psDrawLine + globalGC1 190 94 3 3 0 360 191 95 psFillArc + globalGC1 191 101 191 95 psDrawLine + globalGC1 190 100 3 3 0 360 191 101 psFillArc + globalGC1 191 103 191 101 psDrawLine + globalGC1 190 102 3 3 0 360 191 103 psFillArc + globalGC1 191 104 191 103 psDrawLine + globalGC1 190 103 3 3 0 360 191 104 psFillArc + globalGC1 191 104 191 104 psDrawLine + globalGC1 190 103 3 3 0 360 191 104 psFillArc + globalGC1 191 103 191 104 psDrawLine + globalGC1 190 102 3 3 0 360 191 103 psFillArc + globalGC1 191 98 191 103 psDrawLine + globalGC1 190 97 3 3 0 360 191 98 psFillArc + globalGC1 191 77 191 98 psDrawLine + globalGC1 190 76 3 3 0 360 191 77 psFillArc + globalGC1 191 47 191 77 psDrawLine + globalGC1 190 46 3 3 0 360 191 47 psFillArc + globalGC1 191 21 191 47 psDrawLine + globalGC1 190 20 3 3 0 360 191 21 psFillArc + globalGC1 191 17 191 21 psDrawLine + globalGC1 190 16 3 3 0 360 191 17 psFillArc + globalGC1 191 14 191 17 psDrawLine + globalGC1 190 13 3 3 0 360 191 14 psFillArc + globalGC1 191 14 191 14 psDrawLine + globalGC1 190 13 3 3 0 360 191 14 psFillArc + globalGC1 191 15 191 14 psDrawLine + globalGC1 190 14 3 3 0 360 191 15 psFillArc + globalGC1 191 24 191 15 psDrawLine + globalGC1 190 23 3 3 0 360 191 24 psFillArc + globalGC1 191 40 191 24 psDrawLine + globalGC1 190 39 3 3 0 360 191 40 psFillArc + globalGC1 192 60 191 40 psDrawLine + globalGC1 191 59 3 3 0 360 192 60 psFillArc + globalGC1 192 81 192 60 psDrawLine + globalGC1 191 80 3 3 0 360 192 81 psFillArc + globalGC1 192 97 192 81 psDrawLine + globalGC1 191 96 3 3 0 360 192 97 psFillArc + globalGC1 192 102 192 97 psDrawLine + globalGC1 191 101 3 3 0 360 192 102 psFillArc + globalGC1 192 104 192 102 psDrawLine + globalGC1 191 103 3 3 0 360 192 104 psFillArc + globalGC1 192 103 192 104 psDrawLine + globalGC1 191 102 3 3 0 360 192 103 psFillArc + globalGC1 192 98 192 103 psDrawLine + globalGC1 191 97 3 3 0 360 192 98 psFillArc + globalGC1 192 79 192 98 psDrawLine + globalGC1 191 78 3 3 0 360 192 79 psFillArc + globalGC1 192 52 192 79 psDrawLine + globalGC1 191 51 3 3 0 360 192 52 psFillArc + globalGC1 192 27 192 52 psDrawLine + globalGC1 191 26 3 3 0 360 192 27 psFillArc + globalGC1 192 18 192 27 psDrawLine + globalGC1 191 17 3 3 0 360 192 18 psFillArc + globalGC1 192 14 192 18 psDrawLine + globalGC1 191 13 3 3 0 360 192 14 psFillArc + globalGC1 192 15 192 14 psDrawLine + globalGC1 191 14 3 3 0 360 192 15 psFillArc + globalGC1 192 22 192 15 psDrawLine + globalGC1 191 21 3 3 0 360 192 22 psFillArc + globalGC1 192 49 192 22 psDrawLine + globalGC1 191 48 3 3 0 360 192 49 psFillArc + globalGC1 192 83 192 49 psDrawLine + globalGC1 191 82 3 3 0 360 192 83 psFillArc + globalGC1 192 96 192 83 psDrawLine + globalGC1 191 95 3 3 0 360 192 96 psFillArc + globalGC1 192 103 192 96 psDrawLine + globalGC1 191 102 3 3 0 360 192 103 psFillArc + globalGC1 192 102 192 103 psDrawLine + globalGC1 191 101 3 3 0 360 192 102 psFillArc + globalGC1 192 93 192 102 psDrawLine + globalGC1 191 92 3 3 0 360 192 93 psFillArc + globalGC1 192 76 192 93 psDrawLine + globalGC1 191 75 3 3 0 360 192 76 psFillArc + globalGC1 192 55 192 76 psDrawLine + globalGC1 191 54 3 3 0 360 192 55 psFillArc + globalGC1 192 34 192 55 psDrawLine + globalGC1 191 33 3 3 0 360 192 34 psFillArc + globalGC1 192 19 192 34 psDrawLine + globalGC1 191 18 3 3 0 360 192 19 psFillArc + globalGC1 192 14 192 19 psDrawLine + globalGC1 191 13 3 3 0 360 192 14 psFillArc + globalGC1 192 22 192 14 psDrawLine + globalGC1 191 21 3 3 0 360 192 22 psFillArc + globalGC1 192 41 192 22 psDrawLine + globalGC1 191 40 3 3 0 360 192 41 psFillArc + globalGC1 192 66 192 41 psDrawLine + globalGC1 191 65 3 3 0 360 192 66 psFillArc + globalGC1 192 90 192 66 psDrawLine + globalGC1 191 89 3 3 0 360 192 90 psFillArc + globalGC1 192 103 192 90 psDrawLine + globalGC1 191 102 3 3 0 360 192 103 psFillArc + globalGC1 192 99 192 103 psDrawLine + globalGC1 191 98 3 3 0 360 192 99 psFillArc + globalGC1 192 79 192 99 psDrawLine + globalGC1 191 78 3 3 0 360 192 79 psFillArc + globalGC1 192 48 192 79 psDrawLine + globalGC1 191 47 3 3 0 360 192 48 psFillArc + globalGC1 192 22 192 48 psDrawLine + globalGC1 191 21 3 3 0 360 192 22 psFillArc + globalGC1 192 14 192 22 psDrawLine + globalGC1 191 13 3 3 0 360 192 14 psFillArc + globalGC1 192 31 192 14 psDrawLine + globalGC1 191 30 3 3 0 360 192 31 psFillArc + globalGC1 192 65 192 31 psDrawLine + globalGC1 191 64 3 3 0 360 192 65 psFillArc + globalGC1 192 96 192 65 psDrawLine + globalGC1 191 95 3 3 0 360 192 96 psFillArc + globalGC1 192 102 192 96 psDrawLine + globalGC1 191 101 3 3 0 360 192 102 psFillArc + globalGC1 192 76 192 102 psDrawLine + globalGC1 191 75 3 3 0 360 192 76 psFillArc + globalGC1 192 35 192 76 psDrawLine + globalGC1 191 34 3 3 0 360 192 35 psFillArc + globalGC1 192 14 192 35 psDrawLine + globalGC1 191 13 3 3 0 360 192 14 psFillArc + globalGC1 192 35 192 14 psDrawLine + globalGC1 191 34 3 3 0 360 192 35 psFillArc + globalGC1 192 82 192 35 psDrawLine + globalGC1 191 81 3 3 0 360 192 82 psFillArc + globalGC1 192 104 192 82 psDrawLine + globalGC1 191 103 3 3 0 360 192 104 psFillArc + globalGC1 192 72 192 104 psDrawLine + globalGC1 191 71 3 3 0 360 192 72 psFillArc + globalGC1 193 22 192 72 psDrawLine + globalGC1 192 21 3 3 0 360 193 22 psFillArc + globalGC1 193 25 193 22 psDrawLine + globalGC1 192 24 3 3 0 360 193 25 psFillArc + globalGC1 193 81 193 25 psDrawLine + globalGC1 192 80 3 3 0 360 193 81 psFillArc + globalGC1 193 100 193 81 psDrawLine + globalGC1 192 99 3 3 0 360 193 100 psFillArc + globalGC1 193 42 193 100 psDrawLine + globalGC1 192 41 3 3 0 360 193 42 psFillArc + globalGC1 193 18 193 42 psDrawLine + globalGC1 192 17 3 3 0 360 193 18 psFillArc + globalGC1 193 84 193 18 psDrawLine + globalGC1 192 83 3 3 0 360 193 84 psFillArc + globalGC1 193 89 193 84 psDrawLine + globalGC1 192 88 3 3 0 360 193 89 psFillArc + globalGC1 193 17 193 89 psDrawLine + globalGC1 192 16 3 3 0 360 193 17 psFillArc + globalGC1 193 23 193 17 psDrawLine + globalGC1 192 22 3 3 0 360 193 23 psFillArc + globalGC1 193 65 193 23 psDrawLine + globalGC1 192 64 3 3 0 360 193 65 psFillArc + globalGC1 193 91 193 65 psDrawLine + globalGC1 192 90 3 3 0 360 193 91 psFillArc + globalGC1 193 14 193 91 psDrawLine + globalGC1 192 13 3 3 0 360 193 14 psFillArc + globalGC1 193 96 193 14 psDrawLine + globalGC1 192 95 3 3 0 360 193 96 psFillArc + globalGC1 193 36 193 96 psDrawLine + globalGC1 192 35 3 3 0 360 193 36 psFillArc + globalGC1 193 73 193 36 psDrawLine + globalGC1 192 72 3 3 0 360 193 73 psFillArc + globalGC1 193 44 193 73 psDrawLine + globalGC1 192 43 3 3 0 360 193 44 psFillArc + globalGC1 193 84 193 44 psDrawLine + globalGC1 192 83 3 3 0 360 193 84 psFillArc + globalGC1 193 17 193 84 psDrawLine + globalGC1 192 16 3 3 0 360 193 17 psFillArc + globalGC1 193 98 193 17 psDrawLine + globalGC1 192 97 3 3 0 360 193 98 psFillArc + globalGC1 193 70 193 98 psDrawLine + globalGC1 192 69 3 3 0 360 193 70 psFillArc + globalGC1 193 18 193 70 psDrawLine + globalGC1 192 17 3 3 0 360 193 18 psFillArc + globalGC1 193 18 193 18 psDrawLine + globalGC1 192 17 3 3 0 360 193 18 psFillArc + globalGC1 193 25 193 18 psDrawLine + globalGC1 192 24 3 3 0 360 193 25 psFillArc + globalGC1 193 16 193 25 psDrawLine + globalGC1 192 15 3 3 0 360 193 16 psFillArc + globalGC1 193 34 193 16 psDrawLine + globalGC1 192 33 3 3 0 360 193 34 psFillArc + globalGC1 193 104 193 34 psDrawLine + globalGC1 192 103 3 3 0 360 193 104 psFillArc + globalGC1 193 19 193 104 psDrawLine + globalGC1 192 18 3 3 0 360 193 19 psFillArc + globalGC1 193 27 193 19 psDrawLine + globalGC1 192 26 3 3 0 360 193 27 psFillArc + globalGC1 193 14 193 27 psDrawLine + globalGC1 192 13 3 3 0 360 193 14 psFillArc + globalGC1 193 102 193 14 psDrawLine + globalGC1 192 101 3 3 0 360 193 102 psFillArc + globalGC1 193 100 193 102 psDrawLine + globalGC1 192 99 3 3 0 360 193 100 psFillArc + globalGC1 193 19 193 100 psDrawLine + globalGC1 192 18 3 3 0 360 193 19 psFillArc + globalGC1 193 89 193 19 psDrawLine + globalGC1 192 88 3 3 0 360 193 89 psFillArc + globalGC1 193 104 193 89 psDrawLine + globalGC1 192 103 3 3 0 360 193 104 psFillArc + globalGC1 193 79 193 104 psDrawLine + globalGC1 192 78 3 3 0 360 193 79 psFillArc + globalGC1 193 72 193 79 psDrawLine + globalGC1 192 71 3 3 0 360 193 72 psFillArc + globalGC1 193 45 193 72 psDrawLine + globalGC1 192 44 3 3 0 360 193 45 psFillArc + globalGC1 193 95 193 45 psDrawLine + globalGC1 192 94 3 3 0 360 193 95 psFillArc + globalGC1 193 14 193 95 psDrawLine + globalGC1 192 13 3 3 0 360 193 14 psFillArc + globalGC1 193 28 193 14 psDrawLine + globalGC1 192 27 3 3 0 360 193 28 psFillArc + globalGC1 193 16 193 28 psDrawLine + globalGC1 192 15 3 3 0 360 193 16 psFillArc + globalGC1 193 104 193 16 psDrawLine + globalGC1 192 103 3 3 0 360 193 104 psFillArc + globalGC1 193 14 193 104 psDrawLine + globalGC1 192 13 3 3 0 360 193 14 psFillArc + globalGC1 193 20 193 14 psDrawLine + globalGC1 192 19 3 3 0 360 193 20 psFillArc + globalGC1 193 41 193 20 psDrawLine + globalGC1 192 40 3 3 0 360 193 41 psFillArc + globalGC1 193 20 193 41 psDrawLine + globalGC1 192 19 3 3 0 360 193 20 psFillArc + globalGC1 193 80 193 20 psDrawLine + globalGC1 192 79 3 3 0 360 193 80 psFillArc + globalGC1 193 89 193 80 psDrawLine + globalGC1 192 88 3 3 0 360 193 89 psFillArc + globalGC1 193 31 193 89 psDrawLine + globalGC1 192 30 3 3 0 360 193 31 psFillArc + globalGC1 193 87 193 31 psDrawLine + globalGC1 192 86 3 3 0 360 193 87 psFillArc + globalGC1 193 57 193 87 psDrawLine + globalGC1 192 56 3 3 0 360 193 57 psFillArc + globalGC1 193 43 193 57 psDrawLine + globalGC1 192 42 3 3 0 360 193 43 psFillArc + globalGC1 193 85 193 43 psDrawLine + globalGC1 192 84 3 3 0 360 193 85 psFillArc + globalGC1 194 64 193 85 psDrawLine + globalGC1 193 63 3 3 0 360 194 64 psFillArc + globalGC1 194 104 194 64 psDrawLine + globalGC1 193 103 3 3 0 360 194 104 psFillArc + globalGC1 194 17 194 104 psDrawLine + globalGC1 193 16 3 3 0 360 194 17 psFillArc + globalGC1 194 97 194 17 psDrawLine + globalGC1 193 96 3 3 0 360 194 97 psFillArc + globalGC1 194 94 194 97 psDrawLine + globalGC1 193 93 3 3 0 360 194 94 psFillArc + globalGC1 194 92 194 94 psDrawLine + globalGC1 193 91 3 3 0 360 194 92 psFillArc + globalGC1 194 103 194 92 psDrawLine + globalGC1 193 102 3 3 0 360 194 103 psFillArc + globalGC1 194 59 194 103 psDrawLine + globalGC1 193 58 3 3 0 360 194 59 psFillArc + globalGC1 194 16 194 59 psDrawLine + globalGC1 193 15 3 3 0 360 194 16 psFillArc + globalGC1 194 96 194 16 psDrawLine + globalGC1 193 95 3 3 0 360 194 96 psFillArc + globalGC1 194 41 194 96 psDrawLine + globalGC1 193 40 3 3 0 360 194 41 psFillArc + globalGC1 194 65 194 41 psDrawLine + globalGC1 193 64 3 3 0 360 194 65 psFillArc + globalGC1 194 52 194 65 psDrawLine + globalGC1 193 51 3 3 0 360 194 52 psFillArc + globalGC1 194 75 194 52 psDrawLine + globalGC1 193 74 3 3 0 360 194 75 psFillArc + globalGC1 194 26 194 75 psDrawLine + globalGC1 193 25 3 3 0 360 194 26 psFillArc + globalGC1 194 103 194 26 psDrawLine + globalGC1 193 102 3 3 0 360 194 103 psFillArc + globalGC1 194 23 194 103 psDrawLine + globalGC1 193 22 3 3 0 360 194 23 psFillArc + globalGC1 194 58 194 23 psDrawLine + globalGC1 193 57 3 3 0 360 194 58 psFillArc + globalGC1 194 99 194 58 psDrawLine + globalGC1 193 98 3 3 0 360 194 99 psFillArc + globalGC1 194 26 194 99 psDrawLine + globalGC1 193 25 3 3 0 360 194 26 psFillArc + globalGC1 194 38 194 26 psDrawLine + globalGC1 193 37 3 3 0 360 194 38 psFillArc + globalGC1 194 101 194 38 psDrawLine + globalGC1 193 100 3 3 0 360 194 101 psFillArc + globalGC1 194 72 194 101 psDrawLine + globalGC1 193 71 3 3 0 360 194 72 psFillArc + globalGC1 194 16 194 72 psDrawLine + globalGC1 193 15 3 3 0 360 194 16 psFillArc + globalGC1 194 40 194 16 psDrawLine + globalGC1 193 39 3 3 0 360 194 40 psFillArc + globalGC1 194 95 194 40 psDrawLine + globalGC1 193 94 3 3 0 360 194 95 psFillArc + globalGC1 194 94 194 95 psDrawLine + globalGC1 193 93 3 3 0 360 194 94 psFillArc + globalGC1 194 43 194 94 psDrawLine + globalGC1 193 42 3 3 0 360 194 43 psFillArc + globalGC1 194 14 194 43 psDrawLine + globalGC1 193 13 3 3 0 360 194 14 psFillArc + globalGC1 194 39 194 14 psDrawLine + globalGC1 193 38 3 3 0 360 194 39 psFillArc + globalGC1 194 85 194 39 psDrawLine + globalGC1 193 84 3 3 0 360 194 85 psFillArc + globalGC1 194 104 194 85 psDrawLine + globalGC1 193 103 3 3 0 360 194 104 psFillArc + globalGC1 194 81 194 104 psDrawLine + globalGC1 193 80 3 3 0 360 194 81 psFillArc + globalGC1 194 40 194 81 psDrawLine + globalGC1 193 39 3 3 0 360 194 40 psFillArc + globalGC1 194 15 194 40 psDrawLine + globalGC1 193 14 3 3 0 360 194 15 psFillArc + globalGC1 194 23 194 15 psDrawLine + globalGC1 193 22 3 3 0 360 194 23 psFillArc + globalGC1 194 55 194 23 psDrawLine + globalGC1 193 54 3 3 0 360 194 55 psFillArc + globalGC1 194 88 194 55 psDrawLine + globalGC1 193 87 3 3 0 360 194 88 psFillArc + globalGC1 194 104 194 88 psDrawLine + globalGC1 193 103 3 3 0 360 194 104 psFillArc + globalGC1 194 95 194 104 psDrawLine + globalGC1 193 94 3 3 0 360 194 95 psFillArc + globalGC1 194 68 194 95 psDrawLine + globalGC1 193 67 3 3 0 360 194 68 psFillArc + globalGC1 194 38 194 68 psDrawLine + globalGC1 193 37 3 3 0 360 194 38 psFillArc + globalGC1 194 18 194 38 psDrawLine + globalGC1 193 17 3 3 0 360 194 18 psFillArc + globalGC1 194 15 194 18 psDrawLine + globalGC1 193 14 3 3 0 360 194 15 psFillArc + globalGC1 194 29 194 15 psDrawLine + globalGC1 193 28 3 3 0 360 194 29 psFillArc + globalGC1 194 53 194 29 psDrawLine + globalGC1 193 52 3 3 0 360 194 53 psFillArc + globalGC1 194 78 194 53 psDrawLine + globalGC1 193 77 3 3 0 360 194 78 psFillArc + globalGC1 194 97 194 78 psDrawLine + globalGC1 193 96 3 3 0 360 194 97 psFillArc + globalGC1 194 104 194 97 psDrawLine + globalGC1 193 103 3 3 0 360 194 104 psFillArc + globalGC1 194 98 194 104 psDrawLine + globalGC1 193 97 3 3 0 360 194 98 psFillArc + globalGC1 194 83 194 98 psDrawLine + globalGC1 193 82 3 3 0 360 194 83 psFillArc + globalGC1 194 62 194 83 psDrawLine + globalGC1 193 61 3 3 0 360 194 62 psFillArc + globalGC1 194 41 194 62 psDrawLine + globalGC1 193 40 3 3 0 360 194 41 psFillArc + globalGC1 195 24 194 41 psDrawLine + globalGC1 194 23 3 3 0 360 195 24 psFillArc + globalGC1 195 15 195 24 psDrawLine + globalGC1 194 14 3 3 0 360 195 15 psFillArc + globalGC1 195 15 195 15 psDrawLine + globalGC1 194 14 3 3 0 360 195 15 psFillArc + globalGC1 195 22 195 15 psDrawLine + globalGC1 194 21 3 3 0 360 195 22 psFillArc + globalGC1 195 35 195 22 psDrawLine + globalGC1 194 34 3 3 0 360 195 35 psFillArc + globalGC1 195 70 195 35 psDrawLine + globalGC1 194 69 3 3 0 360 195 70 psFillArc + globalGC1 195 97 195 70 psDrawLine + globalGC1 194 96 3 3 0 360 195 97 psFillArc + globalGC1 195 103 195 97 psDrawLine + globalGC1 194 102 3 3 0 360 195 103 psFillArc + globalGC1 195 104 195 103 psDrawLine + globalGC1 194 103 3 3 0 360 195 104 psFillArc + globalGC1 195 99 195 104 psDrawLine + globalGC1 194 98 3 3 0 360 195 99 psFillArc + globalGC1 195 90 195 99 psDrawLine + globalGC1 194 89 3 3 0 360 195 90 psFillArc + globalGC1 195 65 195 90 psDrawLine + globalGC1 194 64 3 3 0 360 195 65 psFillArc + globalGC1 195 38 195 65 psDrawLine + globalGC1 194 37 3 3 0 360 195 38 psFillArc + globalGC1 195 20 195 38 psDrawLine + globalGC1 194 19 3 3 0 360 195 20 psFillArc + globalGC1 195 15 195 20 psDrawLine + globalGC1 194 14 3 3 0 360 195 15 psFillArc + globalGC1 195 14 195 15 psDrawLine + globalGC1 194 13 3 3 0 360 195 14 psFillArc + globalGC1 195 16 195 14 psDrawLine + globalGC1 194 15 3 3 0 360 195 16 psFillArc + globalGC1 195 21 195 16 psDrawLine + globalGC1 194 20 3 3 0 360 195 21 psFillArc + globalGC1 195 38 195 21 psDrawLine + globalGC1 194 37 3 3 0 360 195 38 psFillArc + globalGC1 195 48 195 38 psDrawLine + globalGC1 194 47 3 3 0 360 195 48 psFillArc + globalGC1 195 59 195 48 psDrawLine + globalGC1 194 58 3 3 0 360 195 59 psFillArc + globalGC1 195 79 195 59 psDrawLine + globalGC1 194 78 3 3 0 360 195 79 psFillArc + globalGC1 195 94 195 79 psDrawLine + globalGC1 194 93 3 3 0 360 195 94 psFillArc + globalGC1 195 103 195 94 psDrawLine + globalGC1 194 102 3 3 0 360 195 103 psFillArc + globalGC1 195 104 195 103 psDrawLine + globalGC1 194 103 3 3 0 360 195 104 psFillArc + globalGC1 195 103 195 104 psDrawLine + globalGC1 194 102 3 3 0 360 195 103 psFillArc + globalGC1 195 101 195 103 psDrawLine + globalGC1 194 100 3 3 0 360 195 101 psFillArc + globalGC1 195 97 195 101 psDrawLine + globalGC1 194 96 3 3 0 360 195 97 psFillArc + globalGC1 195 86 195 97 psDrawLine + globalGC1 194 85 3 3 0 360 195 86 psFillArc + globalGC1 195 71 195 86 psDrawLine + globalGC1 194 70 3 3 0 360 195 71 psFillArc + globalGC1 195 55 195 71 psDrawLine + globalGC1 194 54 3 3 0 360 195 55 psFillArc + globalGC1 195 28 195 55 psDrawLine + globalGC1 194 27 3 3 0 360 195 28 psFillArc + globalGC1 195 20 195 28 psDrawLine + globalGC1 194 19 3 3 0 360 195 20 psFillArc + globalGC1 195 15 195 20 psDrawLine + globalGC1 194 14 3 3 0 360 195 15 psFillArc + globalGC1 195 14 195 15 psDrawLine + globalGC1 194 13 3 3 0 360 195 14 psFillArc + globalGC1 195 14 195 14 psDrawLine + globalGC1 194 13 3 3 0 360 195 14 psFillArc + globalGC1 196 15 195 14 psDrawLine + globalGC1 195 14 3 3 0 360 196 15 psFillArc + globalGC1 196 17 196 15 psDrawLine + globalGC1 195 16 3 3 0 360 196 17 psFillArc + globalGC1 196 32 196 17 psDrawLine + globalGC1 195 31 3 3 0 360 196 32 psFillArc + globalGC1 196 53 196 32 psDrawLine + globalGC1 195 52 3 3 0 360 196 53 psFillArc + globalGC1 196 74 196 53 psDrawLine + globalGC1 195 73 3 3 0 360 196 74 psFillArc + globalGC1 196 91 196 74 psDrawLine + globalGC1 195 90 3 3 0 360 196 91 psFillArc + globalGC1 196 101 196 91 psDrawLine + globalGC1 195 100 3 3 0 360 196 101 psFillArc + globalGC1 196 104 196 101 psDrawLine + globalGC1 195 103 3 3 0 360 196 104 psFillArc + globalGC1 196 104 196 104 psDrawLine + globalGC1 195 103 3 3 0 360 196 104 psFillArc + globalGC1 196 103 196 104 psDrawLine + globalGC1 195 102 3 3 0 360 196 103 psFillArc + globalGC1 196 100 196 103 psDrawLine + globalGC1 195 99 3 3 0 360 196 100 psFillArc + globalGC1 196 79 196 100 psDrawLine + globalGC1 195 78 3 3 0 360 196 79 psFillArc + globalGC1 196 51 196 79 psDrawLine + globalGC1 195 50 3 3 0 360 196 51 psFillArc + globalGC1 197 28 196 51 psDrawLine + globalGC1 196 27 3 3 0 360 197 28 psFillArc + globalGC1 197 16 197 28 psDrawLine + globalGC1 196 15 3 3 0 360 197 16 psFillArc + globalGC1 197 15 197 16 psDrawLine + globalGC1 196 14 3 3 0 360 197 15 psFillArc + globalGC1 197 14 197 15 psDrawLine + globalGC1 196 13 3 3 0 360 197 14 psFillArc + globalGC1 197 15 197 14 psDrawLine + globalGC1 196 14 3 3 0 360 197 15 psFillArc + globalGC1 197 15 197 15 psDrawLine + globalGC1 196 14 3 3 0 360 197 15 psFillArc + globalGC1 197 18 197 15 psDrawLine + globalGC1 196 17 3 3 0 360 197 18 psFillArc + globalGC1 197 23 197 18 psDrawLine + globalGC1 196 22 3 3 0 360 197 23 psFillArc + globalGC1 197 37 197 23 psDrawLine + globalGC1 196 36 3 3 0 360 197 37 psFillArc + globalGC1 197 68 197 37 psDrawLine + globalGC1 196 67 3 3 0 360 197 68 psFillArc + globalGC1 198 93 197 68 psDrawLine + globalGC1 197 92 3 3 0 360 198 93 psFillArc + globalGC1 198 100 198 93 psDrawLine + globalGC1 197 99 3 3 0 360 198 100 psFillArc + globalGC1 198 104 198 100 psDrawLine + globalGC1 197 103 3 3 0 360 198 104 psFillArc + globalGC1 198 105 198 104 psDrawLine + globalGC1 197 104 3 3 0 360 198 105 psFillArc + globalGC1 198 105 198 105 psDrawLine + globalGC1 197 104 3 3 0 360 198 105 psFillArc + globalGC1 198 105 198 105 psDrawLine + globalGC1 197 104 3 3 0 360 198 105 psFillArc + globalGC1 198 103 198 105 psDrawLine + globalGC1 197 102 3 3 0 360 198 103 psFillArc + globalGC1 199 81 198 103 psDrawLine + globalGC1 198 80 3 3 0 360 199 81 psFillArc + globalGC1 200 52 199 81 psDrawLine + globalGC1 199 51 3 3 0 360 200 52 psFillArc + globalGC1 200 31 200 52 psDrawLine + globalGC1 199 30 3 3 0 360 200 31 psFillArc + globalGC1 201 20 200 31 psDrawLine + globalGC1 200 19 3 3 0 360 201 20 psFillArc + globalGC1 201 18 201 20 psDrawLine + globalGC1 200 17 3 3 0 360 201 18 psFillArc + globalGC1 201 18 201 18 psDrawLine + globalGC1 200 17 3 3 0 360 201 18 psFillArc + globalGC1 201 17 201 18 psDrawLine + globalGC1 200 16 3 3 0 360 201 17 psFillArc + globalGC1 201 17 201 17 psDrawLine + globalGC1 200 16 3 3 0 360 201 17 psFillArc + globalGC1 202 18 201 17 psDrawLine + globalGC1 201 17 3 3 0 360 202 18 psFillArc + globalGC1 202 20 202 18 psDrawLine + globalGC1 201 19 3 3 0 360 202 20 psFillArc + globalGC1 203 36 202 20 psDrawLine + globalGC1 202 35 3 3 0 360 203 36 psFillArc + globalGC1 204 54 203 36 psDrawLine + globalGC1 203 53 3 3 0 360 204 54 psFillArc + globalGC1 207 85 204 54 psDrawLine + globalGC1 206 84 3 3 0 360 207 85 psFillArc + globalGC1 209 104 207 85 psDrawLine + globalGC1 208 103 3 3 0 360 209 104 psFillArc + globalGC1 212 115 209 104 psDrawLine + globalGC1 211 114 3 3 0 360 212 115 psFillArc + globalGC1 214 122 212 115 psDrawLine + globalGC1 213 121 3 3 0 360 214 122 psFillArc + globalGC1 215 124 214 122 psDrawLine + globalGC1 214 123 3 3 0 360 215 124 psFillArc + globalGC1 216 125 215 124 psDrawLine + globalGC1 215 124 3 3 0 360 216 125 psFillArc + globalGC1 218 126 216 125 psDrawLine + globalGC1 217 125 3 3 0 360 218 126 psFillArc + globalGC1 219 127 218 126 psDrawLine + globalGC1 218 126 3 3 0 360 219 127 psFillArc + globalGC1 221 128 219 127 psDrawLine + globalGC1 220 127 3 3 0 360 221 128 psFillArc + globalGC1 224 128 221 128 psDrawLine + globalGC1 223 127 3 3 0 360 224 128 psFillArc + globalGC1 228 129 224 128 psDrawLine + globalGC1 227 128 3 3 0 360 228 129 psFillArc + globalGC1 233 129 228 129 psDrawLine + globalGC1 232 128 3 3 0 360 233 129 psFillArc + globalGC1 238 129 233 129 psDrawLine + globalGC1 237 128 3 3 0 360 238 129 psFillArc + globalGC1 243 129 238 129 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2d1varb.ps b/books/ps/2d1varb.ps new file mode 100644 index 0000000..880f01f --- /dev/null +++ b/books/ps/2d1varb.ps @@ -0,0 +1,1527 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 -371 259 -371 0 psDrawLine + globalGC1 11 74 3 3 0 360 12 75 psFillArc + globalGC1 12 75 12 75 psDrawLine + globalGC1 11 74 3 3 0 360 12 75 psFillArc + globalGC1 17 62 12 75 psDrawLine + globalGC1 16 61 3 3 0 360 17 62 psFillArc + globalGC1 22 49 17 62 psDrawLine + globalGC1 21 48 3 3 0 360 22 49 psFillArc + globalGC1 24 42 22 49 psDrawLine + globalGC1 23 41 3 3 0 360 24 42 psFillArc + globalGC1 27 37 24 42 psDrawLine + globalGC1 26 36 3 3 0 360 27 37 psFillArc + globalGC1 28 35 27 37 psDrawLine + globalGC1 27 34 3 3 0 360 28 35 psFillArc + globalGC1 29 33 28 35 psDrawLine + globalGC1 28 32 3 3 0 360 29 33 psFillArc + globalGC1 30 32 29 33 psDrawLine + globalGC1 29 31 3 3 0 360 30 32 psFillArc + globalGC1 32 31 30 32 psDrawLine + globalGC1 31 30 3 3 0 360 32 31 psFillArc + globalGC1 32 31 32 31 psDrawLine + globalGC1 31 30 3 3 0 360 32 31 psFillArc + globalGC1 33 32 32 31 psDrawLine + globalGC1 32 31 3 3 0 360 33 32 psFillArc + globalGC1 33 33 33 32 psDrawLine + globalGC1 32 32 3 3 0 360 33 33 psFillArc + globalGC1 34 34 33 33 psDrawLine + globalGC1 33 33 3 3 0 360 34 34 psFillArc + globalGC1 35 38 34 34 psDrawLine + globalGC1 34 37 3 3 0 360 35 38 psFillArc + globalGC1 36 44 35 38 psDrawLine + globalGC1 35 43 3 3 0 360 36 44 psFillArc + globalGC1 39 66 36 44 psDrawLine + globalGC1 38 65 3 3 0 360 39 66 psFillArc + globalGC1 40 81 39 66 psDrawLine + globalGC1 39 80 3 3 0 360 40 81 psFillArc + globalGC1 41 97 40 81 psDrawLine + globalGC1 40 96 3 3 0 360 41 97 psFillArc + globalGC1 42 103 41 97 psDrawLine + globalGC1 41 102 3 3 0 360 42 103 psFillArc + globalGC1 42 106 42 103 psDrawLine + globalGC1 41 105 3 3 0 360 42 106 psFillArc + globalGC1 42 107 42 106 psDrawLine + globalGC1 41 106 3 3 0 360 42 107 psFillArc + globalGC1 43 107 42 107 psDrawLine + globalGC1 42 106 3 3 0 360 43 107 psFillArc + globalGC1 43 105 43 107 psDrawLine + globalGC1 42 104 3 3 0 360 43 105 psFillArc + globalGC1 43 96 43 105 psDrawLine + globalGC1 42 95 3 3 0 360 43 96 psFillArc + globalGC1 44 77 43 96 psDrawLine + globalGC1 43 76 3 3 0 360 44 77 psFillArc + globalGC1 45 49 44 77 psDrawLine + globalGC1 44 48 3 3 0 360 45 49 psFillArc + globalGC1 45 22 45 49 psDrawLine + globalGC1 44 21 3 3 0 360 45 22 psFillArc + globalGC1 45 18 45 22 psDrawLine + globalGC1 44 17 3 3 0 360 45 18 psFillArc + globalGC1 46 16 45 18 psDrawLine + globalGC1 45 15 3 3 0 360 46 16 psFillArc + globalGC1 46 15 46 16 psDrawLine + globalGC1 45 14 3 3 0 360 46 15 psFillArc + globalGC1 46 15 46 15 psDrawLine + globalGC1 45 14 3 3 0 360 46 15 psFillArc + globalGC1 46 15 46 15 psDrawLine + globalGC1 45 14 3 3 0 360 46 15 psFillArc + globalGC1 46 18 46 15 psDrawLine + globalGC1 45 17 3 3 0 360 46 18 psFillArc + globalGC1 46 35 46 18 psDrawLine + globalGC1 45 34 3 3 0 360 46 35 psFillArc + globalGC1 47 63 46 35 psDrawLine + globalGC1 46 62 3 3 0 360 47 63 psFillArc + globalGC1 47 93 47 63 psDrawLine + globalGC1 46 92 3 3 0 360 47 93 psFillArc + globalGC1 47 103 47 93 psDrawLine + globalGC1 46 102 3 3 0 360 47 103 psFillArc + globalGC1 47 104 47 103 psDrawLine + globalGC1 46 103 3 3 0 360 47 104 psFillArc + globalGC1 47 104 47 104 psDrawLine + globalGC1 46 103 3 3 0 360 47 104 psFillArc + globalGC1 47 104 47 104 psDrawLine + globalGC1 46 103 3 3 0 360 47 104 psFillArc + globalGC1 47 104 47 104 psDrawLine + globalGC1 46 103 3 3 0 360 47 104 psFillArc + globalGC1 47 101 47 104 psDrawLine + globalGC1 46 100 3 3 0 360 47 101 psFillArc + globalGC1 47 94 47 101 psDrawLine + globalGC1 46 93 3 3 0 360 47 94 psFillArc + globalGC1 47 74 47 94 psDrawLine + globalGC1 46 73 3 3 0 360 47 74 psFillArc + globalGC1 48 46 47 74 psDrawLine + globalGC1 47 45 3 3 0 360 48 46 psFillArc + globalGC1 48 22 48 46 psDrawLine + globalGC1 47 21 3 3 0 360 48 22 psFillArc + globalGC1 48 15 48 22 psDrawLine + globalGC1 47 14 3 3 0 360 48 15 psFillArc + globalGC1 48 14 48 15 psDrawLine + globalGC1 47 13 3 3 0 360 48 14 psFillArc + globalGC1 48 15 48 14 psDrawLine + globalGC1 47 14 3 3 0 360 48 15 psFillArc + globalGC1 48 17 48 15 psDrawLine + globalGC1 47 16 3 3 0 360 48 17 psFillArc + globalGC1 48 22 48 17 psDrawLine + globalGC1 47 21 3 3 0 360 48 22 psFillArc + globalGC1 48 36 48 22 psDrawLine + globalGC1 47 35 3 3 0 360 48 36 psFillArc + globalGC1 48 55 48 36 psDrawLine + globalGC1 47 54 3 3 0 360 48 55 psFillArc + globalGC1 48 77 48 55 psDrawLine + globalGC1 47 76 3 3 0 360 48 77 psFillArc + globalGC1 48 95 48 77 psDrawLine + globalGC1 47 94 3 3 0 360 48 95 psFillArc + globalGC1 48 101 48 95 psDrawLine + globalGC1 47 100 3 3 0 360 48 101 psFillArc + globalGC1 48 103 48 101 psDrawLine + globalGC1 47 102 3 3 0 360 48 103 psFillArc + globalGC1 48 104 48 103 psDrawLine + globalGC1 47 103 3 3 0 360 48 104 psFillArc + globalGC1 48 103 48 104 psDrawLine + globalGC1 47 102 3 3 0 360 48 103 psFillArc + globalGC1 48 98 48 103 psDrawLine + globalGC1 47 97 3 3 0 360 48 98 psFillArc + globalGC1 48 77 48 98 psDrawLine + globalGC1 47 76 3 3 0 360 48 77 psFillArc + globalGC1 49 46 48 77 psDrawLine + globalGC1 48 45 3 3 0 360 49 46 psFillArc + globalGC1 49 20 49 46 psDrawLine + globalGC1 48 19 3 3 0 360 49 20 psFillArc + globalGC1 49 17 49 20 psDrawLine + globalGC1 48 16 3 3 0 360 49 17 psFillArc + globalGC1 49 14 49 17 psDrawLine + globalGC1 48 13 3 3 0 360 49 14 psFillArc + globalGC1 49 14 49 14 psDrawLine + globalGC1 48 13 3 3 0 360 49 14 psFillArc + globalGC1 49 16 49 14 psDrawLine + globalGC1 48 15 3 3 0 360 49 16 psFillArc + globalGC1 49 24 49 16 psDrawLine + globalGC1 48 23 3 3 0 360 49 24 psFillArc + globalGC1 49 40 49 24 psDrawLine + globalGC1 48 39 3 3 0 360 49 40 psFillArc + globalGC1 49 60 49 40 psDrawLine + globalGC1 48 59 3 3 0 360 49 60 psFillArc + globalGC1 49 81 49 60 psDrawLine + globalGC1 48 80 3 3 0 360 49 81 psFillArc + globalGC1 49 97 49 81 psDrawLine + globalGC1 48 96 3 3 0 360 49 97 psFillArc + globalGC1 49 102 49 97 psDrawLine + globalGC1 48 101 3 3 0 360 49 102 psFillArc + globalGC1 49 104 49 102 psDrawLine + globalGC1 48 103 3 3 0 360 49 104 psFillArc + globalGC1 49 98 49 104 psDrawLine + globalGC1 48 97 3 3 0 360 49 98 psFillArc + globalGC1 49 79 49 98 psDrawLine + globalGC1 48 78 3 3 0 360 49 79 psFillArc + globalGC1 49 52 49 79 psDrawLine + globalGC1 48 51 3 3 0 360 49 52 psFillArc + globalGC1 49 27 49 52 psDrawLine + globalGC1 48 26 3 3 0 360 49 27 psFillArc + globalGC1 49 18 49 27 psDrawLine + globalGC1 48 17 3 3 0 360 49 18 psFillArc + globalGC1 49 14 49 18 psDrawLine + globalGC1 48 13 3 3 0 360 49 14 psFillArc + globalGC1 49 15 49 14 psDrawLine + globalGC1 48 14 3 3 0 360 49 15 psFillArc + globalGC1 49 22 49 15 psDrawLine + globalGC1 48 21 3 3 0 360 49 22 psFillArc + globalGC1 49 50 49 22 psDrawLine + globalGC1 48 49 3 3 0 360 49 50 psFillArc + globalGC1 49 84 49 50 psDrawLine + globalGC1 48 83 3 3 0 360 49 84 psFillArc + globalGC1 49 97 49 84 psDrawLine + globalGC1 48 96 3 3 0 360 49 97 psFillArc + globalGC1 49 103 49 97 psDrawLine + globalGC1 48 102 3 3 0 360 49 103 psFillArc + globalGC1 49 102 49 103 psDrawLine + globalGC1 48 101 3 3 0 360 49 102 psFillArc + globalGC1 49 92 49 102 psDrawLine + globalGC1 48 91 3 3 0 360 49 92 psFillArc + globalGC1 49 75 49 92 psDrawLine + globalGC1 48 74 3 3 0 360 49 75 psFillArc + globalGC1 49 54 49 75 psDrawLine + globalGC1 48 53 3 3 0 360 49 54 psFillArc + globalGC1 49 33 49 54 psDrawLine + globalGC1 48 32 3 3 0 360 49 33 psFillArc + globalGC1 49 18 49 33 psDrawLine + globalGC1 48 17 3 3 0 360 49 18 psFillArc + globalGC1 49 22 49 18 psDrawLine + globalGC1 48 21 3 3 0 360 49 22 psFillArc + globalGC1 49 41 49 22 psDrawLine + globalGC1 48 40 3 3 0 360 49 41 psFillArc + globalGC1 49 67 49 41 psDrawLine + globalGC1 48 66 3 3 0 360 49 67 psFillArc + globalGC1 49 91 49 67 psDrawLine + globalGC1 48 90 3 3 0 360 49 91 psFillArc + globalGC1 49 103 49 91 psDrawLine + globalGC1 48 102 3 3 0 360 49 103 psFillArc + globalGC1 49 99 49 103 psDrawLine + globalGC1 48 98 3 3 0 360 49 99 psFillArc + globalGC1 50 78 49 99 psDrawLine + globalGC1 49 77 3 3 0 360 50 78 psFillArc + globalGC1 50 47 50 78 psDrawLine + globalGC1 49 46 3 3 0 360 50 47 psFillArc + globalGC1 50 21 50 47 psDrawLine + globalGC1 49 20 3 3 0 360 50 21 psFillArc + globalGC1 50 14 50 21 psDrawLine + globalGC1 49 13 3 3 0 360 50 14 psFillArc + globalGC1 50 32 50 14 psDrawLine + globalGC1 49 31 3 3 0 360 50 32 psFillArc + globalGC1 50 66 50 32 psDrawLine + globalGC1 49 65 3 3 0 360 50 66 psFillArc + globalGC1 50 97 50 66 psDrawLine + globalGC1 49 96 3 3 0 360 50 97 psFillArc + globalGC1 50 102 50 97 psDrawLine + globalGC1 49 101 3 3 0 360 50 102 psFillArc + globalGC1 50 75 50 102 psDrawLine + globalGC1 49 74 3 3 0 360 50 75 psFillArc + globalGC1 50 34 50 75 psDrawLine + globalGC1 49 33 3 3 0 360 50 34 psFillArc + globalGC1 50 14 50 34 psDrawLine + globalGC1 49 13 3 3 0 360 50 14 psFillArc + globalGC1 50 37 50 14 psDrawLine + globalGC1 49 36 3 3 0 360 50 37 psFillArc + globalGC1 50 83 50 37 psDrawLine + globalGC1 49 82 3 3 0 360 50 83 psFillArc + globalGC1 50 104 50 83 psDrawLine + globalGC1 49 103 3 3 0 360 50 104 psFillArc + globalGC1 50 70 50 104 psDrawLine + globalGC1 49 69 3 3 0 360 50 70 psFillArc + globalGC1 50 21 50 70 psDrawLine + globalGC1 49 20 3 3 0 360 50 21 psFillArc + globalGC1 50 26 50 21 psDrawLine + globalGC1 49 25 3 3 0 360 50 26 psFillArc + globalGC1 50 83 50 26 psDrawLine + globalGC1 49 82 3 3 0 360 50 83 psFillArc + globalGC1 50 99 50 83 psDrawLine + globalGC1 49 98 3 3 0 360 50 99 psFillArc + globalGC1 50 40 50 99 psDrawLine + globalGC1 49 39 3 3 0 360 50 40 psFillArc + globalGC1 50 19 50 40 psDrawLine + globalGC1 49 18 3 3 0 360 50 19 psFillArc + globalGC1 50 86 50 19 psDrawLine + globalGC1 49 85 3 3 0 360 50 86 psFillArc + globalGC1 50 86 50 86 psDrawLine + globalGC1 49 85 3 3 0 360 50 86 psFillArc + globalGC1 50 16 50 86 psDrawLine + globalGC1 49 15 3 3 0 360 50 16 psFillArc + globalGC1 50 69 50 16 psDrawLine + globalGC1 49 68 3 3 0 360 50 69 psFillArc + globalGC1 50 88 50 69 psDrawLine + globalGC1 49 87 3 3 0 360 50 88 psFillArc + globalGC1 50 14 50 88 psDrawLine + globalGC1 49 13 3 3 0 360 50 14 psFillArc + globalGC1 50 32 50 14 psDrawLine + globalGC1 49 31 3 3 0 360 50 32 psFillArc + globalGC1 50 77 50 32 psDrawLine + globalGC1 49 76 3 3 0 360 50 77 psFillArc + globalGC1 50 39 50 77 psDrawLine + globalGC1 49 38 3 3 0 360 50 39 psFillArc + globalGC1 50 89 50 39 psDrawLine + globalGC1 49 88 3 3 0 360 50 89 psFillArc + globalGC1 50 15 50 89 psDrawLine + globalGC1 49 14 3 3 0 360 50 15 psFillArc + globalGC1 50 94 50 15 psDrawLine + globalGC1 49 93 3 3 0 360 50 94 psFillArc + globalGC1 50 78 50 94 psDrawLine + globalGC1 49 77 3 3 0 360 50 78 psFillArc + globalGC1 50 22 50 78 psDrawLine + globalGC1 49 21 3 3 0 360 50 22 psFillArc + globalGC1 50 15 50 22 psDrawLine + globalGC1 49 14 3 3 0 360 50 15 psFillArc + globalGC1 50 19 50 15 psDrawLine + globalGC1 49 18 3 3 0 360 50 19 psFillArc + globalGC1 50 14 50 19 psDrawLine + globalGC1 49 13 3 3 0 360 50 14 psFillArc + globalGC1 50 46 50 14 psDrawLine + globalGC1 49 45 3 3 0 360 50 46 psFillArc + globalGC1 50 99 50 46 psDrawLine + globalGC1 49 98 3 3 0 360 50 99 psFillArc + globalGC1 50 30 50 99 psDrawLine + globalGC1 49 29 3 3 0 360 50 30 psFillArc + globalGC1 50 16 50 30 psDrawLine + globalGC1 49 15 3 3 0 360 50 16 psFillArc + globalGC1 50 17 50 16 psDrawLine + globalGC1 49 16 3 3 0 360 50 17 psFillArc + globalGC1 50 101 50 17 psDrawLine + globalGC1 49 100 3 3 0 360 50 101 psFillArc + globalGC1 50 101 50 101 psDrawLine + globalGC1 49 100 3 3 0 360 50 101 psFillArc + globalGC1 50 52 50 101 psDrawLine + globalGC1 49 51 3 3 0 360 50 52 psFillArc + globalGC1 50 39 50 52 psDrawLine + globalGC1 49 38 3 3 0 360 50 39 psFillArc + globalGC1 50 61 50 39 psDrawLine + globalGC1 49 60 3 3 0 360 50 61 psFillArc + globalGC1 50 83 50 61 psDrawLine + globalGC1 49 82 3 3 0 360 50 83 psFillArc + globalGC1 50 50 50 83 psDrawLine + globalGC1 49 49 3 3 0 360 50 50 psFillArc + globalGC1 50 101 50 50 psDrawLine + globalGC1 49 100 3 3 0 360 50 101 psFillArc + globalGC1 50 55 50 101 psDrawLine + globalGC1 49 54 3 3 0 360 50 55 psFillArc + globalGC1 50 22 50 55 psDrawLine + globalGC1 49 21 3 3 0 360 50 22 psFillArc + globalGC1 51 18 50 22 psDrawLine + globalGC1 50 17 3 3 0 360 51 18 psFillArc + globalGC1 51 102 51 18 psDrawLine + globalGC1 50 101 3 3 0 360 51 102 psFillArc + globalGC1 51 78 51 102 psDrawLine + globalGC1 50 77 3 3 0 360 51 78 psFillArc + globalGC1 51 89 51 78 psDrawLine + globalGC1 50 88 3 3 0 360 51 89 psFillArc + globalGC1 51 40 51 89 psDrawLine + globalGC1 50 39 3 3 0 360 51 40 psFillArc + globalGC1 51 89 51 40 psDrawLine + globalGC1 50 88 3 3 0 360 51 89 psFillArc + globalGC1 51 102 51 89 psDrawLine + globalGC1 50 101 3 3 0 360 51 102 psFillArc + globalGC1 51 92 51 102 psDrawLine + globalGC1 50 91 3 3 0 360 51 92 psFillArc + globalGC1 51 14 51 92 psDrawLine + globalGC1 50 13 3 3 0 360 51 14 psFillArc + globalGC1 51 94 51 14 psDrawLine + globalGC1 50 93 3 3 0 360 51 94 psFillArc + globalGC1 51 80 51 94 psDrawLine + globalGC1 50 79 3 3 0 360 51 80 psFillArc + globalGC1 51 48 51 80 psDrawLine + globalGC1 50 47 3 3 0 360 51 48 psFillArc + globalGC1 51 88 51 48 psDrawLine + globalGC1 50 87 3 3 0 360 51 88 psFillArc + globalGC1 51 70 51 88 psDrawLine + globalGC1 50 69 3 3 0 360 51 70 psFillArc + globalGC1 51 64 51 70 psDrawLine + globalGC1 50 63 3 3 0 360 51 64 psFillArc + globalGC1 51 45 51 64 psDrawLine + globalGC1 50 44 3 3 0 360 51 45 psFillArc + globalGC1 51 99 51 45 psDrawLine + globalGC1 50 98 3 3 0 360 51 99 psFillArc + globalGC1 51 25 51 99 psDrawLine + globalGC1 50 24 3 3 0 360 51 25 psFillArc + globalGC1 51 102 51 25 psDrawLine + globalGC1 50 101 3 3 0 360 51 102 psFillArc + globalGC1 51 86 51 102 psDrawLine + globalGC1 50 85 3 3 0 360 51 86 psFillArc + globalGC1 51 85 51 86 psDrawLine + globalGC1 50 84 3 3 0 360 51 85 psFillArc + globalGC1 51 104 51 85 psDrawLine + globalGC1 50 103 3 3 0 360 51 104 psFillArc + globalGC1 51 66 51 104 psDrawLine + globalGC1 50 65 3 3 0 360 51 66 psFillArc + globalGC1 51 14 51 66 psDrawLine + globalGC1 50 13 3 3 0 360 51 14 psFillArc + globalGC1 51 92 51 14 psDrawLine + globalGC1 50 91 3 3 0 360 51 92 psFillArc + globalGC1 51 46 51 92 psDrawLine + globalGC1 50 45 3 3 0 360 51 46 psFillArc + globalGC1 51 60 51 46 psDrawLine + globalGC1 50 59 3 3 0 360 51 60 psFillArc + globalGC1 51 57 51 60 psDrawLine + globalGC1 50 56 3 3 0 360 51 57 psFillArc + globalGC1 51 71 51 57 psDrawLine + globalGC1 50 70 3 3 0 360 51 71 psFillArc + globalGC1 51 29 51 71 psDrawLine + globalGC1 50 28 3 3 0 360 51 29 psFillArc + globalGC1 51 103 51 29 psDrawLine + globalGC1 50 102 3 3 0 360 51 103 psFillArc + globalGC1 51 21 51 103 psDrawLine + globalGC1 50 20 3 3 0 360 51 21 psFillArc + globalGC1 51 61 51 21 psDrawLine + globalGC1 50 60 3 3 0 360 51 61 psFillArc + globalGC1 51 98 51 61 psDrawLine + globalGC1 50 97 3 3 0 360 51 98 psFillArc + globalGC1 51 24 51 98 psDrawLine + globalGC1 50 23 3 3 0 360 51 24 psFillArc + globalGC1 51 40 51 24 psDrawLine + globalGC1 50 39 3 3 0 360 51 40 psFillArc + globalGC1 51 102 51 40 psDrawLine + globalGC1 50 101 3 3 0 360 51 102 psFillArc + globalGC1 51 69 51 102 psDrawLine + globalGC1 50 68 3 3 0 360 51 69 psFillArc + globalGC1 51 15 51 69 psDrawLine + globalGC1 50 14 3 3 0 360 51 15 psFillArc + globalGC1 51 42 51 15 psDrawLine + globalGC1 50 41 3 3 0 360 51 42 psFillArc + globalGC1 51 96 51 42 psDrawLine + globalGC1 50 95 3 3 0 360 51 96 psFillArc + globalGC1 51 93 51 96 psDrawLine + globalGC1 50 92 3 3 0 360 51 93 psFillArc + globalGC1 51 41 51 93 psDrawLine + globalGC1 50 40 3 3 0 360 51 41 psFillArc + globalGC1 51 14 51 41 psDrawLine + globalGC1 50 13 3 3 0 360 51 14 psFillArc + globalGC1 51 40 51 14 psDrawLine + globalGC1 50 39 3 3 0 360 51 40 psFillArc + globalGC1 51 86 51 40 psDrawLine + globalGC1 50 85 3 3 0 360 51 86 psFillArc + globalGC1 51 104 51 86 psDrawLine + globalGC1 50 103 3 3 0 360 51 104 psFillArc + globalGC1 51 79 51 104 psDrawLine + globalGC1 50 78 3 3 0 360 51 79 psFillArc + globalGC1 51 38 51 79 psDrawLine + globalGC1 50 37 3 3 0 360 51 38 psFillArc + globalGC1 51 15 51 38 psDrawLine + globalGC1 50 14 3 3 0 360 51 15 psFillArc + globalGC1 51 24 51 15 psDrawLine + globalGC1 50 23 3 3 0 360 51 24 psFillArc + globalGC1 51 56 51 24 psDrawLine + globalGC1 50 55 3 3 0 360 51 56 psFillArc + globalGC1 51 89 51 56 psDrawLine + globalGC1 50 88 3 3 0 360 51 89 psFillArc + globalGC1 52 104 51 89 psDrawLine + globalGC1 51 103 3 3 0 360 52 104 psFillArc + globalGC1 52 94 52 104 psDrawLine + globalGC1 51 93 3 3 0 360 52 94 psFillArc + globalGC1 52 67 52 94 psDrawLine + globalGC1 51 66 3 3 0 360 52 67 psFillArc + globalGC1 52 37 52 67 psDrawLine + globalGC1 51 36 3 3 0 360 52 37 psFillArc + globalGC1 52 17 52 37 psDrawLine + globalGC1 51 16 3 3 0 360 52 17 psFillArc + globalGC1 52 15 52 17 psDrawLine + globalGC1 51 14 3 3 0 360 52 15 psFillArc + globalGC1 52 30 52 15 psDrawLine + globalGC1 51 29 3 3 0 360 52 30 psFillArc + globalGC1 52 54 52 30 psDrawLine + globalGC1 51 53 3 3 0 360 52 54 psFillArc + globalGC1 52 79 52 54 psDrawLine + globalGC1 51 78 3 3 0 360 52 79 psFillArc + globalGC1 52 98 52 79 psDrawLine + globalGC1 51 97 3 3 0 360 52 98 psFillArc + globalGC1 52 104 52 98 psDrawLine + globalGC1 51 103 3 3 0 360 52 104 psFillArc + globalGC1 52 98 52 104 psDrawLine + globalGC1 51 97 3 3 0 360 52 98 psFillArc + globalGC1 52 82 52 98 psDrawLine + globalGC1 51 81 3 3 0 360 52 82 psFillArc + globalGC1 52 61 52 82 psDrawLine + globalGC1 51 60 3 3 0 360 52 61 psFillArc + globalGC1 52 40 52 61 psDrawLine + globalGC1 51 39 3 3 0 360 52 40 psFillArc + globalGC1 52 24 52 40 psDrawLine + globalGC1 51 23 3 3 0 360 52 24 psFillArc + globalGC1 52 15 52 24 psDrawLine + globalGC1 51 14 3 3 0 360 52 15 psFillArc + globalGC1 52 15 52 15 psDrawLine + globalGC1 51 14 3 3 0 360 52 15 psFillArc + globalGC1 52 36 52 15 psDrawLine + globalGC1 51 35 3 3 0 360 52 36 psFillArc + globalGC1 52 70 52 36 psDrawLine + globalGC1 51 69 3 3 0 360 52 70 psFillArc + globalGC1 52 97 52 70 psDrawLine + globalGC1 51 96 3 3 0 360 52 97 psFillArc + globalGC1 52 103 52 97 psDrawLine + globalGC1 51 102 3 3 0 360 52 103 psFillArc + globalGC1 52 104 52 103 psDrawLine + globalGC1 51 103 3 3 0 360 52 104 psFillArc + globalGC1 52 99 52 104 psDrawLine + globalGC1 51 98 3 3 0 360 52 99 psFillArc + globalGC1 52 90 52 99 psDrawLine + globalGC1 51 89 3 3 0 360 52 90 psFillArc + globalGC1 52 64 52 90 psDrawLine + globalGC1 51 63 3 3 0 360 52 64 psFillArc + globalGC1 52 38 52 64 psDrawLine + globalGC1 51 37 3 3 0 360 52 38 psFillArc + globalGC1 52 19 52 38 psDrawLine + globalGC1 51 18 3 3 0 360 52 19 psFillArc + globalGC1 52 15 52 19 psDrawLine + globalGC1 51 14 3 3 0 360 52 15 psFillArc + globalGC1 52 14 52 15 psDrawLine + globalGC1 51 13 3 3 0 360 52 14 psFillArc + globalGC1 52 16 52 14 psDrawLine + globalGC1 51 15 3 3 0 360 52 16 psFillArc + globalGC1 52 21 52 16 psDrawLine + globalGC1 51 20 3 3 0 360 52 21 psFillArc + globalGC1 52 38 52 21 psDrawLine + globalGC1 51 37 3 3 0 360 52 38 psFillArc + globalGC1 52 59 52 38 psDrawLine + globalGC1 51 58 3 3 0 360 52 59 psFillArc + globalGC1 52 79 52 59 psDrawLine + globalGC1 51 78 3 3 0 360 52 79 psFillArc + globalGC1 52 95 52 79 psDrawLine + globalGC1 51 94 3 3 0 360 52 95 psFillArc + globalGC1 52 103 52 95 psDrawLine + globalGC1 51 102 3 3 0 360 52 103 psFillArc + globalGC1 52 104 52 103 psDrawLine + globalGC1 51 103 3 3 0 360 52 104 psFillArc + globalGC1 52 103 52 104 psDrawLine + globalGC1 51 102 3 3 0 360 52 103 psFillArc + globalGC1 52 101 52 103 psDrawLine + globalGC1 51 100 3 3 0 360 52 101 psFillArc + globalGC1 52 97 52 101 psDrawLine + globalGC1 51 96 3 3 0 360 52 97 psFillArc + globalGC1 53 85 52 97 psDrawLine + globalGC1 52 84 3 3 0 360 53 85 psFillArc + globalGC1 53 55 53 85 psDrawLine + globalGC1 52 54 3 3 0 360 53 55 psFillArc + globalGC1 53 28 53 55 psDrawLine + globalGC1 52 27 3 3 0 360 53 28 psFillArc + globalGC1 53 20 53 28 psDrawLine + globalGC1 52 19 3 3 0 360 53 20 psFillArc + globalGC1 53 15 53 20 psDrawLine + globalGC1 52 14 3 3 0 360 53 15 psFillArc + globalGC1 53 14 53 15 psDrawLine + globalGC1 52 13 3 3 0 360 53 14 psFillArc + globalGC1 53 14 53 14 psDrawLine + globalGC1 52 13 3 3 0 360 53 14 psFillArc + globalGC1 53 15 53 14 psDrawLine + globalGC1 52 14 3 3 0 360 53 15 psFillArc + globalGC1 53 17 53 15 psDrawLine + globalGC1 52 16 3 3 0 360 53 17 psFillArc + globalGC1 53 32 53 17 psDrawLine + globalGC1 52 31 3 3 0 360 53 32 psFillArc + globalGC1 53 53 53 32 psDrawLine + globalGC1 52 52 3 3 0 360 53 53 psFillArc + globalGC1 53 74 53 53 psDrawLine + globalGC1 52 73 3 3 0 360 53 74 psFillArc + globalGC1 53 91 53 74 psDrawLine + globalGC1 52 90 3 3 0 360 53 91 psFillArc + globalGC1 53 101 53 91 psDrawLine + globalGC1 52 100 3 3 0 360 53 101 psFillArc + globalGC1 53 104 53 101 psDrawLine + globalGC1 52 103 3 3 0 360 53 104 psFillArc + globalGC1 53 104 53 104 psDrawLine + globalGC1 52 103 3 3 0 360 53 104 psFillArc + globalGC1 53 104 53 104 psDrawLine + globalGC1 52 103 3 3 0 360 53 104 psFillArc + globalGC1 53 103 53 104 psDrawLine + globalGC1 52 102 3 3 0 360 53 103 psFillArc + globalGC1 53 100 53 103 psDrawLine + globalGC1 52 99 3 3 0 360 53 100 psFillArc + globalGC1 53 91 53 100 psDrawLine + globalGC1 52 90 3 3 0 360 53 91 psFillArc + globalGC1 54 65 53 91 psDrawLine + globalGC1 53 64 3 3 0 360 54 65 psFillArc + globalGC1 54 39 54 65 psDrawLine + globalGC1 53 38 3 3 0 360 54 39 psFillArc + globalGC1 54 21 54 39 psDrawLine + globalGC1 53 20 3 3 0 360 54 21 psFillArc + globalGC1 54 16 54 21 psDrawLine + globalGC1 53 15 3 3 0 360 54 16 psFillArc + globalGC1 54 15 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0 360 172 197 psFillArc + globalGC1 172 230 172 197 psDrawLine + globalGC1 171 229 3 3 0 360 172 230 psFillArc + globalGC1 172 244 172 230 psDrawLine + globalGC1 171 243 3 3 0 360 172 244 psFillArc + globalGC1 172 234 172 244 psDrawLine + globalGC1 171 233 3 3 0 360 172 234 psFillArc + globalGC1 172 206 172 234 psDrawLine + globalGC1 171 205 3 3 0 360 172 206 psFillArc + globalGC1 172 176 172 206 psDrawLine + globalGC1 171 175 3 3 0 360 172 176 psFillArc + globalGC1 172 157 172 176 psDrawLine + globalGC1 171 156 3 3 0 360 172 157 psFillArc + globalGC1 172 156 172 157 psDrawLine + globalGC1 171 155 3 3 0 360 172 156 psFillArc + globalGC1 172 170 172 156 psDrawLine + globalGC1 171 169 3 3 0 360 172 170 psFillArc + globalGC1 172 195 172 170 psDrawLine + globalGC1 171 194 3 3 0 360 172 195 psFillArc + globalGC1 172 220 172 195 psDrawLine + globalGC1 171 219 3 3 0 360 172 220 psFillArc + globalGC1 172 238 172 220 psDrawLine + globalGC1 171 237 3 3 0 360 172 238 psFillArc + globalGC1 172 244 172 238 psDrawLine + globalGC1 171 243 3 3 0 360 172 244 psFillArc + globalGC1 172 238 172 244 psDrawLine + globalGC1 171 237 3 3 0 360 172 238 psFillArc + globalGC1 172 222 172 238 psDrawLine + globalGC1 171 221 3 3 0 360 172 222 psFillArc + globalGC1 172 201 172 222 psDrawLine + globalGC1 171 200 3 3 0 360 172 201 psFillArc + globalGC1 172 180 172 201 psDrawLine + globalGC1 171 179 3 3 0 360 172 180 psFillArc + globalGC1 172 164 172 180 psDrawLine + globalGC1 171 163 3 3 0 360 172 164 psFillArc + globalGC1 172 155 172 164 psDrawLine + globalGC1 171 154 3 3 0 360 172 155 psFillArc + globalGC1 172 155 172 155 psDrawLine + globalGC1 171 154 3 3 0 360 172 155 psFillArc + globalGC1 173 176 172 155 psDrawLine + globalGC1 172 175 3 3 0 360 173 176 psFillArc + globalGC1 173 211 173 176 psDrawLine + globalGC1 172 210 3 3 0 360 173 211 psFillArc + globalGC1 173 237 173 211 psDrawLine + globalGC1 172 236 3 3 0 360 173 237 psFillArc + globalGC1 173 243 173 237 psDrawLine + globalGC1 172 242 3 3 0 360 173 243 psFillArc + globalGC1 173 244 173 243 psDrawLine + globalGC1 172 243 3 3 0 360 173 244 psFillArc + globalGC1 173 239 173 244 psDrawLine + globalGC1 172 238 3 3 0 360 173 239 psFillArc + globalGC1 173 230 173 239 psDrawLine + globalGC1 172 229 3 3 0 360 173 230 psFillArc + globalGC1 173 204 173 230 psDrawLine + globalGC1 172 203 3 3 0 360 173 204 psFillArc + globalGC1 173 178 173 204 psDrawLine + globalGC1 172 177 3 3 0 360 173 178 psFillArc + globalGC1 173 159 173 178 psDrawLine + globalGC1 172 158 3 3 0 360 173 159 psFillArc + globalGC1 173 155 173 159 psDrawLine + globalGC1 172 154 3 3 0 360 173 155 psFillArc + globalGC1 173 154 173 155 psDrawLine + globalGC1 172 153 3 3 0 360 173 154 psFillArc + globalGC1 173 156 173 154 psDrawLine + globalGC1 172 155 3 3 0 360 173 156 psFillArc + globalGC1 173 162 173 156 psDrawLine + globalGC1 172 161 3 3 0 360 173 162 psFillArc + globalGC1 173 178 173 162 psDrawLine + globalGC1 172 177 3 3 0 360 173 178 psFillArc + globalGC1 173 199 173 178 psDrawLine + globalGC1 172 198 3 3 0 360 173 199 psFillArc + globalGC1 173 219 173 199 psDrawLine + globalGC1 172 218 3 3 0 360 173 219 psFillArc + globalGC1 173 235 173 219 psDrawLine + globalGC1 172 234 3 3 0 360 173 235 psFillArc + globalGC1 173 240 173 235 psDrawLine + globalGC1 172 239 3 3 0 360 173 240 psFillArc + globalGC1 173 243 173 240 psDrawLine + globalGC1 172 242 3 3 0 360 173 243 psFillArc + globalGC1 173 244 173 243 psDrawLine + globalGC1 172 243 3 3 0 360 173 244 psFillArc + globalGC1 173 243 173 244 psDrawLine + globalGC1 172 242 3 3 0 360 173 243 psFillArc + globalGC1 173 241 173 243 psDrawLine + globalGC1 172 240 3 3 0 360 173 241 psFillArc + globalGC1 173 237 173 241 psDrawLine + globalGC1 172 236 3 3 0 360 173 237 psFillArc + globalGC1 173 211 173 237 psDrawLine + globalGC1 172 210 3 3 0 360 173 211 psFillArc + globalGC1 173 180 173 211 psDrawLine + globalGC1 172 179 3 3 0 360 173 180 psFillArc + globalGC1 173 159 173 180 psDrawLine + globalGC1 172 158 3 3 0 360 173 159 psFillArc + globalGC1 173 155 173 159 psDrawLine + globalGC1 172 154 3 3 0 360 173 155 psFillArc + globalGC1 173 154 173 155 psDrawLine + globalGC1 172 153 3 3 0 360 173 154 psFillArc + globalGC1 173 154 173 154 psDrawLine + globalGC1 172 153 3 3 0 360 173 154 psFillArc + globalGC1 173 155 173 154 psDrawLine + globalGC1 172 154 3 3 0 360 173 155 psFillArc + globalGC1 173 157 173 155 psDrawLine + globalGC1 172 156 3 3 0 360 173 157 psFillArc + globalGC1 173 163 173 157 psDrawLine + globalGC1 172 162 3 3 0 360 173 163 psFillArc + globalGC1 174 182 173 163 psDrawLine + globalGC1 173 181 3 3 0 360 174 182 psFillArc + globalGC1 174 204 174 182 psDrawLine + globalGC1 173 203 3 3 0 360 174 204 psFillArc + globalGC1 174 223 174 204 psDrawLine + globalGC1 173 222 3 3 0 360 174 223 psFillArc + globalGC1 174 237 174 223 psDrawLine + globalGC1 173 236 3 3 0 360 174 237 psFillArc + globalGC1 174 241 174 237 psDrawLine + globalGC1 173 240 3 3 0 360 174 241 psFillArc + globalGC1 174 242 174 241 psDrawLine + globalGC1 173 241 3 3 0 360 174 242 psFillArc + globalGC1 174 243 174 242 psDrawLine + globalGC1 173 242 3 3 0 360 174 243 psFillArc + globalGC1 174 244 174 243 psDrawLine + globalGC1 173 243 3 3 0 360 174 244 psFillArc + globalGC1 174 243 174 244 psDrawLine + globalGC1 173 242 3 3 0 360 174 243 psFillArc + globalGC1 174 236 174 243 psDrawLine + globalGC1 173 235 3 3 0 360 174 236 psFillArc + globalGC1 174 211 174 236 psDrawLine + globalGC1 173 210 3 3 0 360 174 211 psFillArc + globalGC1 174 184 174 211 psDrawLine + globalGC1 173 183 3 3 0 360 174 184 psFillArc + globalGC1 174 163 174 184 psDrawLine + globalGC1 173 162 3 3 0 360 174 163 psFillArc + globalGC1 175 157 174 163 psDrawLine + globalGC1 174 156 3 3 0 360 175 157 psFillArc + globalGC1 175 155 175 157 psDrawLine + globalGC1 174 154 3 3 0 360 175 155 psFillArc + globalGC1 175 154 175 155 psDrawLine + globalGC1 174 153 3 3 0 360 175 154 psFillArc + globalGC1 175 154 175 154 psDrawLine + globalGC1 174 153 3 3 0 360 175 154 psFillArc + globalGC1 175 154 175 154 psDrawLine + globalGC1 174 153 3 3 0 360 175 154 psFillArc + globalGC1 175 154 175 154 psDrawLine + globalGC1 174 153 3 3 0 360 175 154 psFillArc + globalGC1 175 156 175 154 psDrawLine + globalGC1 174 155 3 3 0 360 175 156 psFillArc + globalGC1 175 165 175 156 psDrawLine + globalGC1 174 164 3 3 0 360 175 165 psFillArc + globalGC1 175 196 175 165 psDrawLine + globalGC1 174 195 3 3 0 360 175 196 psFillArc + globalGC1 176 224 175 196 psDrawLine + globalGC1 175 223 3 3 0 360 176 224 psFillArc + globalGC1 176 240 176 224 psDrawLine + globalGC1 175 239 3 3 0 360 176 240 psFillArc + globalGC1 176 243 176 240 psDrawLine + globalGC1 175 242 3 3 0 360 176 243 psFillArc + globalGC1 176 243 176 243 psDrawLine + globalGC1 175 242 3 3 0 360 176 243 psFillArc + globalGC1 176 243 176 243 psDrawLine + globalGC1 175 242 3 3 0 360 176 243 psFillArc + globalGC1 176 240 176 243 psDrawLine + globalGC1 175 239 3 3 0 360 176 240 psFillArc + globalGC1 176 236 176 240 psDrawLine + globalGC1 175 235 3 3 0 360 176 236 psFillArc + globalGC1 177 223 176 236 psDrawLine + globalGC1 176 222 3 3 0 360 177 223 psFillArc + globalGC1 177 194 177 223 psDrawLine + globalGC1 176 193 3 3 0 360 177 194 psFillArc + globalGC1 178 170 177 194 psDrawLine + globalGC1 177 169 3 3 0 360 178 170 psFillArc + globalGC1 179 156 178 170 psDrawLine + globalGC1 178 155 3 3 0 360 179 156 psFillArc + globalGC1 179 153 179 156 psDrawLine + globalGC1 178 152 3 3 0 360 179 153 psFillArc + globalGC1 179 151 179 153 psDrawLine + globalGC1 178 150 3 3 0 360 179 151 psFillArc + globalGC1 179 151 179 151 psDrawLine + globalGC1 178 150 3 3 0 360 179 151 psFillArc + globalGC1 180 152 179 151 psDrawLine + globalGC1 179 151 3 3 0 360 180 152 psFillArc + globalGC1 180 155 180 152 psDrawLine + globalGC1 179 154 3 3 0 360 180 155 psFillArc + globalGC1 180 158 180 155 psDrawLine + globalGC1 179 157 3 3 0 360 180 158 psFillArc + globalGC1 182 173 180 158 psDrawLine + globalGC1 181 172 3 3 0 360 182 173 psFillArc + globalGC1 183 189 182 173 psDrawLine + globalGC1 182 188 3 3 0 360 183 189 psFillArc + globalGC1 185 212 183 189 psDrawLine + globalGC1 184 211 3 3 0 360 185 212 psFillArc + globalGC1 186 219 185 212 psDrawLine + globalGC1 185 218 3 3 0 360 186 219 psFillArc + globalGC1 188 223 186 219 psDrawLine + globalGC1 187 222 3 3 0 360 188 223 psFillArc + globalGC1 188 225 188 223 psDrawLine + globalGC1 187 224 3 3 0 360 188 225 psFillArc + globalGC1 189 226 188 225 psDrawLine + globalGC1 188 225 3 3 0 360 189 226 psFillArc + globalGC1 189 226 189 226 psDrawLine + globalGC1 188 225 3 3 0 360 189 226 psFillArc + globalGC1 190 227 189 226 psDrawLine + globalGC1 189 226 3 3 0 360 190 227 psFillArc + globalGC1 191 227 190 227 psDrawLine + globalGC1 190 226 3 3 0 360 191 227 psFillArc + globalGC1 192 226 191 227 psDrawLine + globalGC1 191 225 3 3 0 360 192 226 psFillArc + globalGC1 194 224 192 226 psDrawLine + globalGC1 193 223 3 3 0 360 194 224 psFillArc + globalGC1 195 222 194 224 psDrawLine + globalGC1 194 221 3 3 0 360 195 222 psFillArc + globalGC1 200 210 195 222 psDrawLine + globalGC1 199 209 3 3 0 360 200 210 psFillArc + globalGC1 204 197 200 210 psDrawLine + globalGC1 203 196 3 3 0 360 204 197 psFillArc + globalGC1 209 184 204 197 psDrawLine + globalGC1 208 183 3 3 0 360 209 184 psFillArc + globalGC1 214 172 209 184 psDrawLine + globalGC1 213 171 3 3 0 360 214 172 psFillArc + globalGC1 219 160 214 172 psDrawLine + globalGC1 218 159 3 3 0 360 219 160 psFillArc + globalGC1 224 148 219 160 psDrawLine + globalGC1 223 147 3 3 0 360 224 148 psFillArc + globalGC1 228 137 224 148 psDrawLine + globalGC1 227 136 3 3 0 360 228 137 psFillArc + globalGC1 233 125 228 137 psDrawLine + globalGC1 232 124 3 3 0 360 233 125 psFillArc + globalGC1 238 114 233 125 psDrawLine + globalGC1 237 113 3 3 0 360 238 114 psFillArc + globalGC1 243 102 238 114 psDrawLine + globalGC1 242 101 3 3 0 360 243 102 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2d1vard.ps b/books/ps/2d1vard.ps new file mode 100644 index 0000000..6ac18bb --- /dev/null +++ b/books/ps/2d1vard.ps @@ -0,0 +1,414 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 151 0 151 psDrawLine + globalGC1 6 303 6 0 psDrawLine + unitGC 62 153 62 149 psDrawLine + unitGC 50 166 (0.80) (window) psDrawIStr + unitGC 118 153 118 149 psDrawLine + unitGC 106 166 (1.60) (window) psDrawIStr + unitGC 174 153 174 149 psDrawLine + unitGC 162 166 (2.40) (window) psDrawIStr + unitGC 230 153 230 149 psDrawLine + unitGC 218 166 (3.20) (window) psDrawIStr + unitGC 286 153 286 149 psDrawLine + unitGC 274 166 (4.00) (window) psDrawIStr + unitGC 8 94 4 94 psDrawLine + unitGC -30 99 (2.40) (window) psDrawIStr + unitGC 8 37 4 37 psDrawLine + unitGC -30 42 (4.80) (window) psDrawIStr + unitGC 8 207 4 207 psDrawLine + unitGC -39 212 (-2.40) (window) psDrawIStr + unitGC 8 263 4 263 psDrawLine + unitGC -39 268 (-4.80) (window) psDrawIStr + globalGC1 5 290 3 3 0 360 6 291 psFillArc + globalGC1 6 291 6 291 psDrawLine + globalGC1 5 290 3 3 0 360 6 291 psFillArc + globalGC1 12 271 6 291 psDrawLine + globalGC1 11 270 3 3 0 360 12 271 psFillArc + globalGC1 18 252 12 271 psDrawLine + globalGC1 17 251 3 3 0 360 18 252 psFillArc + globalGC1 23 235 18 252 psDrawLine + globalGC1 22 234 3 3 0 360 23 235 psFillArc + globalGC1 29 220 23 235 psDrawLine + globalGC1 28 219 3 3 0 360 29 220 psFillArc + globalGC1 35 207 29 220 psDrawLine + globalGC1 34 206 3 3 0 360 35 207 psFillArc + globalGC1 41 195 35 207 psDrawLine + globalGC1 40 194 3 3 0 360 41 195 psFillArc + globalGC1 47 184 41 195 psDrawLine + globalGC1 46 183 3 3 0 360 47 184 psFillArc + globalGC1 53 175 47 184 psDrawLine + globalGC1 52 174 3 3 0 360 53 175 psFillArc + globalGC1 59 167 53 175 psDrawLine + globalGC1 58 166 3 3 0 360 59 167 psFillArc + globalGC1 64 161 59 167 psDrawLine + globalGC1 63 160 3 3 0 360 64 161 psFillArc + globalGC1 70 155 64 161 psDrawLine + globalGC1 69 154 3 3 0 360 70 155 psFillArc + globalGC1 76 151 70 155 psDrawLine + globalGC1 75 150 3 3 0 360 76 151 psFillArc + globalGC1 82 148 76 151 psDrawLine + globalGC1 81 147 3 3 0 360 82 148 psFillArc + globalGC1 88 145 82 148 psDrawLine + globalGC1 87 144 3 3 0 360 88 145 psFillArc + globalGC1 94 143 88 145 psDrawLine + globalGC1 93 142 3 3 0 360 94 143 psFillArc + globalGC1 99 142 94 143 psDrawLine + globalGC1 98 141 3 3 0 360 99 142 psFillArc + globalGC1 105 142 99 142 psDrawLine + globalGC1 104 141 3 3 0 360 105 142 psFillArc + globalGC1 111 142 105 142 psDrawLine + globalGC1 110 141 3 3 0 360 111 142 psFillArc + globalGC1 117 143 111 142 psDrawLine + globalGC1 116 142 3 3 0 360 117 143 psFillArc + globalGC1 123 144 117 143 psDrawLine + globalGC1 122 143 3 3 0 360 123 144 psFillArc + globalGC1 129 145 123 144 psDrawLine + globalGC1 128 144 3 3 0 360 129 145 psFillArc + globalGC1 135 147 129 145 psDrawLine + globalGC1 134 146 3 3 0 360 135 147 psFillArc + globalGC1 140 149 135 147 psDrawLine + globalGC1 139 148 3 3 0 360 140 149 psFillArc + globalGC1 146 151 140 149 psDrawLine + globalGC1 145 150 3 3 0 360 146 151 psFillArc + globalGC1 152 153 146 151 psDrawLine + globalGC1 151 152 3 3 0 360 152 153 psFillArc + globalGC1 158 155 152 153 psDrawLine + globalGC1 157 154 3 3 0 360 158 155 psFillArc + globalGC1 164 156 158 155 psDrawLine + globalGC1 163 155 3 3 0 360 164 156 psFillArc + globalGC1 170 158 164 156 psDrawLine + globalGC1 169 157 3 3 0 360 170 158 psFillArc + globalGC1 176 159 170 158 psDrawLine + globalGC1 175 158 3 3 0 360 176 159 psFillArc + globalGC1 181 160 176 159 psDrawLine + globalGC1 180 159 3 3 0 360 181 160 psFillArc + globalGC1 187 160 181 160 psDrawLine + globalGC1 186 159 3 3 0 360 187 160 psFillArc + globalGC1 193 160 187 160 psDrawLine + globalGC1 192 159 3 3 0 360 193 160 psFillArc + globalGC1 199 159 193 160 psDrawLine + globalGC1 198 158 3 3 0 360 199 159 psFillArc + globalGC1 205 157 199 159 psDrawLine + globalGC1 204 156 3 3 0 360 205 157 psFillArc + globalGC1 211 154 205 157 psDrawLine + globalGC1 210 153 3 3 0 360 211 154 psFillArc + globalGC1 217 151 211 154 psDrawLine + globalGC1 216 150 3 3 0 360 217 151 psFillArc + globalGC1 222 147 217 151 psDrawLine + globalGC1 221 146 3 3 0 360 222 147 psFillArc + globalGC1 228 141 222 147 psDrawLine + globalGC1 227 140 3 3 0 360 228 141 psFillArc + globalGC1 234 135 228 141 psDrawLine + globalGC1 233 134 3 3 0 360 234 135 psFillArc + globalGC1 240 127 234 135 psDrawLine + globalGC1 239 126 3 3 0 360 240 127 psFillArc + globalGC1 246 118 240 127 psDrawLine + globalGC1 245 117 3 3 0 360 246 118 psFillArc + globalGC1 252 107 246 118 psDrawLine + globalGC1 251 106 3 3 0 360 252 107 psFillArc + globalGC1 257 95 252 107 psDrawLine + globalGC1 256 94 3 3 0 360 257 95 psFillArc + globalGC1 263 82 257 95 psDrawLine + globalGC1 262 81 3 3 0 360 263 82 psFillArc + globalGC1 269 67 263 82 psDrawLine + globalGC1 268 66 3 3 0 360 269 67 psFillArc + globalGC1 275 50 269 67 psDrawLine + globalGC1 274 49 3 3 0 360 275 50 psFillArc + globalGC1 281 31 275 50 psDrawLine + globalGC1 280 30 3 3 0 360 281 31 psFillArc + globalGC1 287 11 281 31 psDrawLine + globalGC1 286 10 3 3 0 360 287 11 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dctrl.ps b/books/ps/2dctrl.ps new file mode 100644 index 0000000..1787bf4 --- /dev/null +++ b/books/ps/2dctrl.ps @@ -0,0 +1,722 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 127.588757 104.120003 484.411243 718.120003 +%%Creator: xwd2ps +%%CreationDate: Sun Dec 1 15:51:11 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 258, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 260, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 258 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 258 445 8 + [258 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +2.03 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +dbdb70 +2f4f4f +d4d8e8 +23238e +d3d3d3 +ff0000 +757780 +46474d +b4b8c5 +236b8e +238e6b +2f2f4f +600808 +931c1c 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+1e010100000100001e010000bc0100001f01010000001f010100bb0101001e010100000100001e010000bc0100001f01010000001f010100bb0101001e010100000100001e010000 +bc0100001f01010000000001ff000001ff000000 +pop pop setmatrix + +/#copies 1 def +grestore +%%Trailer diff --git a/books/ps/2doptad.ps b/books/ps/2doptad.ps new file mode 100644 index 0000000..51c9d7c --- /dev/null +++ b/books/ps/2doptad.ps @@ -0,0 +1,347 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 127 259 127 0 psDrawLine + globalGC1 11 147 3 3 0 360 12 148 psFillArc + globalGC1 12 148 12 148 psDrawLine + globalGC1 11 147 3 3 0 360 12 148 psFillArc + globalGC1 17 149 12 148 psDrawLine + globalGC1 16 148 3 3 0 360 17 149 psFillArc + globalGC1 22 150 17 149 psDrawLine + globalGC1 21 149 3 3 0 360 22 150 psFillArc + globalGC1 27 151 22 150 psDrawLine + globalGC1 26 150 3 3 0 360 27 151 psFillArc + globalGC1 32 152 27 151 psDrawLine + globalGC1 31 151 3 3 0 360 32 152 psFillArc + globalGC1 36 153 32 152 psDrawLine + globalGC1 35 152 3 3 0 360 36 153 psFillArc + globalGC1 41 154 36 153 psDrawLine + globalGC1 40 153 3 3 0 360 41 154 psFillArc + globalGC1 46 156 41 154 psDrawLine + globalGC1 45 155 3 3 0 360 46 156 psFillArc + globalGC1 51 157 46 156 psDrawLine + globalGC1 50 156 3 3 0 360 51 157 psFillArc + globalGC1 56 159 51 157 psDrawLine + globalGC1 55 158 3 3 0 360 56 159 psFillArc + globalGC1 60 161 56 159 psDrawLine + globalGC1 59 160 3 3 0 360 60 161 psFillArc + globalGC1 65 164 60 161 psDrawLine + globalGC1 64 163 3 3 0 360 65 164 psFillArc + globalGC1 70 167 65 164 psDrawLine + globalGC1 69 166 3 3 0 360 70 167 psFillArc + globalGC1 75 170 70 167 psDrawLine + globalGC1 74 169 3 3 0 360 75 170 psFillArc + globalGC1 80 174 75 170 psDrawLine + globalGC1 79 173 3 3 0 360 80 174 psFillArc + globalGC1 84 179 80 174 psDrawLine + globalGC1 83 178 3 3 0 360 84 179 psFillArc + globalGC1 89 184 84 179 psDrawLine + globalGC1 88 183 3 3 0 360 89 184 psFillArc + globalGC1 94 192 89 184 psDrawLine + globalGC1 93 191 3 3 0 360 94 192 psFillArc + globalGC1 99 201 94 192 psDrawLine + globalGC1 98 200 3 3 0 360 99 201 psFillArc + globalGC1 104 212 99 201 psDrawLine + globalGC1 103 211 3 3 0 360 104 212 psFillArc + globalGC1 108 227 104 212 psDrawLine + globalGC1 107 226 3 3 0 360 108 227 psFillArc + globalGC1 113 244 108 227 psDrawLine + globalGC1 112 243 3 3 0 360 113 244 psFillArc + globalGC1 118 243 113 244 psDrawLine + globalGC1 117 242 3 3 0 360 118 243 psFillArc + globalGC1 123 53 118 243 psDrawLine + globalGC1 122 52 3 3 0 360 123 53 psFillArc + globalGC1 127 35 123 53 psDrawLine + globalGC1 126 34 3 3 0 360 127 35 psFillArc + globalGC1 132 205 127 35 psDrawLine + globalGC1 131 204 3 3 0 360 132 205 psFillArc + globalGC1 137 15 132 205 psDrawLine + globalGC1 136 14 3 3 0 360 137 15 psFillArc + globalGC1 142 14 137 15 psDrawLine + globalGC1 141 13 3 3 0 360 142 14 psFillArc + globalGC1 147 31 142 14 psDrawLine + globalGC1 146 30 3 3 0 360 147 31 psFillArc + globalGC1 151 46 147 31 psDrawLine + globalGC1 150 45 3 3 0 360 151 46 psFillArc + globalGC1 156 57 151 46 psDrawLine + globalGC1 155 56 3 3 0 360 156 57 psFillArc + globalGC1 161 66 156 57 psDrawLine + globalGC1 160 65 3 3 0 360 161 66 psFillArc + globalGC1 166 74 161 66 psDrawLine + globalGC1 165 73 3 3 0 360 166 74 psFillArc + globalGC1 171 79 166 74 psDrawLine + globalGC1 170 78 3 3 0 360 171 79 psFillArc + globalGC1 175 84 171 79 psDrawLine + globalGC1 174 83 3 3 0 360 175 84 psFillArc + globalGC1 180 88 175 84 psDrawLine + globalGC1 179 87 3 3 0 360 180 88 psFillArc + globalGC1 185 91 180 88 psDrawLine + globalGC1 184 90 3 3 0 360 185 91 psFillArc + globalGC1 190 94 185 91 psDrawLine + globalGC1 189 93 3 3 0 360 190 94 psFillArc + globalGC1 195 97 190 94 psDrawLine + globalGC1 194 96 3 3 0 360 195 97 psFillArc + globalGC1 199 99 195 97 psDrawLine + globalGC1 198 98 3 3 0 360 199 99 psFillArc + globalGC1 204 100 199 99 psDrawLine + globalGC1 203 99 3 3 0 360 204 100 psFillArc + globalGC1 209 102 204 100 psDrawLine + globalGC1 208 101 3 3 0 360 209 102 psFillArc + globalGC1 214 104 209 102 psDrawLine + globalGC1 213 103 3 3 0 360 214 104 psFillArc + globalGC1 219 105 214 104 psDrawLine + globalGC1 218 104 3 3 0 360 219 105 psFillArc + globalGC1 223 106 219 105 psDrawLine + globalGC1 222 105 3 3 0 360 223 106 psFillArc + globalGC1 228 107 223 106 psDrawLine + globalGC1 227 106 3 3 0 360 228 107 psFillArc + globalGC1 233 108 228 107 psDrawLine + globalGC1 232 107 3 3 0 360 233 108 psFillArc + globalGC1 238 109 233 108 psDrawLine + globalGC1 237 108 3 3 0 360 238 109 psFillArc + globalGC1 243 110 238 109 psDrawLine + globalGC1 242 109 3 3 0 360 243 110 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptcp.ps b/books/ps/2doptcp.ps new file mode 100644 index 0000000..0900056 --- /dev/null +++ b/books/ps/2doptcp.ps @@ -0,0 +1,551 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 127 259 127 0 psDrawLine + globalGC1 213 243 3 3 0 360 214 244 psFillArc + globalGC1 214 244 214 244 psDrawLine + globalGC1 213 243 3 3 0 360 214 244 psFillArc + globalGC1 214 213 214 244 psDrawLine + globalGC1 213 212 3 3 0 360 214 213 psFillArc + globalGC1 214 192 214 213 psDrawLine + globalGC1 213 191 3 3 0 360 214 192 psFillArc + globalGC1 214 171 214 192 psDrawLine + globalGC1 213 170 3 3 0 360 214 171 psFillArc + globalGC1 214 160 214 171 psDrawLine + globalGC1 213 159 3 3 0 360 214 160 psFillArc + globalGC1 214 150 214 160 psDrawLine + globalGC1 213 149 3 3 0 360 214 150 psFillArc + globalGC1 214 145 214 150 psDrawLine + globalGC1 213 144 3 3 0 360 214 145 psFillArc + globalGC1 215 139 214 145 psDrawLine + globalGC1 214 138 3 3 0 360 215 139 psFillArc + globalGC1 215 137 215 139 psDrawLine + globalGC1 214 136 3 3 0 360 215 137 psFillArc + globalGC1 216 134 215 137 psDrawLine + globalGC1 215 133 3 3 0 360 216 134 psFillArc + globalGC1 216 133 216 134 psDrawLine + globalGC1 215 132 3 3 0 360 216 133 psFillArc + globalGC1 217 132 216 133 psDrawLine + globalGC1 216 131 3 3 0 360 217 132 psFillArc + globalGC1 219 131 217 132 psDrawLine + globalGC1 218 130 3 3 0 360 219 131 psFillArc + globalGC1 221 130 219 131 psDrawLine + globalGC1 220 129 3 3 0 360 221 130 psFillArc + globalGC1 223 130 221 130 psDrawLine + globalGC1 222 129 3 3 0 360 223 130 psFillArc + globalGC1 228 129 223 130 psDrawLine + globalGC1 227 128 3 3 0 360 228 129 psFillArc + globalGC1 233 129 228 129 psDrawLine + globalGC1 232 128 3 3 0 360 233 129 psFillArc + globalGC1 238 129 233 129 psDrawLine + globalGC1 237 128 3 3 0 360 238 129 psFillArc + globalGC1 243 129 238 129 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + globalGC1 155 243 3 3 0 360 156 244 psFillArc + globalGC1 156 244 156 244 psDrawLine + globalGC1 155 243 3 3 0 360 156 244 psFillArc + globalGC1 156 213 156 244 psDrawLine + globalGC1 155 212 3 3 0 360 156 213 psFillArc + globalGC1 156 192 156 213 psDrawLine + globalGC1 155 191 3 3 0 360 156 192 psFillArc + globalGC1 157 171 156 192 psDrawLine + globalGC1 156 170 3 3 0 360 157 171 psFillArc + globalGC1 157 160 157 171 psDrawLine + globalGC1 156 159 3 3 0 360 157 160 psFillArc + globalGC1 157 150 157 160 psDrawLine + globalGC1 156 149 3 3 0 360 157 150 psFillArc + globalGC1 157 145 157 150 psDrawLine + globalGC1 156 144 3 3 0 360 157 145 psFillArc + globalGC1 157 139 157 145 psDrawLine + globalGC1 156 138 3 3 0 360 157 139 psFillArc + globalGC1 157 137 157 139 psDrawLine + globalGC1 156 136 3 3 0 360 157 137 psFillArc + globalGC1 158 134 157 137 psDrawLine + globalGC1 157 133 3 3 0 360 158 134 psFillArc + globalGC1 159 133 158 134 psDrawLine + globalGC1 158 132 3 3 0 360 159 133 psFillArc + globalGC1 160 132 159 133 psDrawLine + globalGC1 159 131 3 3 0 360 160 132 psFillArc + globalGC1 161 131 160 132 psDrawLine + globalGC1 160 130 3 3 0 360 161 131 psFillArc + globalGC1 163 130 161 131 psDrawLine + globalGC1 162 129 3 3 0 360 163 130 psFillArc + globalGC1 166 130 163 130 psDrawLine + globalGC1 165 129 3 3 0 360 166 130 psFillArc + globalGC1 171 129 166 130 psDrawLine + globalGC1 170 128 3 3 0 360 171 129 psFillArc + globalGC1 175 129 171 129 psDrawLine + globalGC1 174 128 3 3 0 360 175 129 psFillArc + globalGC1 180 129 175 129 psDrawLine + globalGC1 179 128 3 3 0 360 180 129 psFillArc + globalGC1 185 129 180 129 psDrawLine + globalGC1 184 128 3 3 0 360 185 129 psFillArc + globalGC1 190 129 185 129 psDrawLine + globalGC1 189 128 3 3 0 360 190 129 psFillArc + globalGC1 195 129 190 129 psDrawLine + globalGC1 194 128 3 3 0 360 195 129 psFillArc + globalGC1 199 128 195 129 psDrawLine + globalGC1 198 127 3 3 0 360 199 128 psFillArc + globalGC1 204 128 199 128 psDrawLine + globalGC1 203 127 3 3 0 360 204 128 psFillArc + globalGC1 207 128 204 128 psDrawLine + globalGC1 206 127 3 3 0 360 207 128 psFillArc + globalGC1 209 127 207 128 psDrawLine + globalGC1 208 126 3 3 0 360 209 127 psFillArc + globalGC1 210 126 209 127 psDrawLine + globalGC1 209 125 3 3 0 360 210 126 psFillArc + globalGC1 211 125 210 126 psDrawLine + globalGC1 210 124 3 3 0 360 211 125 psFillArc + globalGC1 212 124 211 125 psDrawLine + globalGC1 211 123 3 3 0 360 212 124 psFillArc + globalGC1 213 121 212 124 psDrawLine + globalGC1 212 120 3 3 0 360 213 121 psFillArc + globalGC1 213 119 213 121 psDrawLine + globalGC1 212 118 3 3 0 360 213 119 psFillArc + globalGC1 213 113 213 119 psDrawLine + globalGC1 212 112 3 3 0 360 213 113 psFillArc + globalGC1 213 108 213 113 psDrawLine + globalGC1 212 107 3 3 0 360 213 108 psFillArc + globalGC1 214 98 213 108 psDrawLine + globalGC1 213 97 3 3 0 360 214 98 psFillArc + globalGC1 214 87 214 98 psDrawLine + globalGC1 213 86 3 3 0 360 214 87 psFillArc + globalGC1 214 66 214 87 psDrawLine + globalGC1 213 65 3 3 0 360 214 66 psFillArc + globalGC1 214 45 214 66 psDrawLine + globalGC1 213 44 3 3 0 360 214 45 psFillArc + globalGC1 214 14 214 45 psDrawLine + globalGC1 213 13 3 3 0 360 214 14 psFillArc + globalGC1 98 243 3 3 0 360 99 244 psFillArc + globalGC1 99 244 99 244 psDrawLine + globalGC1 98 243 3 3 0 360 99 244 psFillArc + globalGC1 99 213 99 244 psDrawLine + globalGC1 98 212 3 3 0 360 99 213 psFillArc + globalGC1 99 192 99 213 psDrawLine + globalGC1 98 191 3 3 0 360 99 192 psFillArc + globalGC1 99 171 99 192 psDrawLine + globalGC1 98 170 3 3 0 360 99 171 psFillArc + globalGC1 99 160 99 171 psDrawLine + globalGC1 98 159 3 3 0 360 99 160 psFillArc + globalGC1 99 150 99 160 psDrawLine + globalGC1 98 149 3 3 0 360 99 150 psFillArc + globalGC1 99 145 99 150 psDrawLine + globalGC1 98 144 3 3 0 360 99 145 psFillArc + globalGC1 100 139 99 145 psDrawLine + globalGC1 99 138 3 3 0 360 100 139 psFillArc + globalGC1 100 137 100 139 psDrawLine + globalGC1 99 136 3 3 0 360 100 137 psFillArc + globalGC1 101 134 100 137 psDrawLine + globalGC1 100 133 3 3 0 360 101 134 psFillArc + globalGC1 101 133 101 134 psDrawLine + globalGC1 100 132 3 3 0 360 101 133 psFillArc + globalGC1 102 132 101 133 psDrawLine + globalGC1 101 131 3 3 0 360 102 132 psFillArc + globalGC1 104 131 102 132 psDrawLine + globalGC1 103 130 3 3 0 360 104 131 psFillArc + globalGC1 106 130 104 131 psDrawLine + globalGC1 105 129 3 3 0 360 106 130 psFillArc + globalGC1 108 130 106 130 psDrawLine + globalGC1 107 129 3 3 0 360 108 130 psFillArc + globalGC1 113 129 108 130 psDrawLine + globalGC1 112 128 3 3 0 360 113 129 psFillArc + globalGC1 118 129 113 129 psDrawLine + globalGC1 117 128 3 3 0 360 118 129 psFillArc + globalGC1 123 129 118 129 psDrawLine + globalGC1 122 128 3 3 0 360 123 129 psFillArc + globalGC1 127 129 123 129 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 132 129 127 129 psDrawLine + globalGC1 131 128 3 3 0 360 132 129 psFillArc + globalGC1 137 129 132 129 psDrawLine + globalGC1 136 128 3 3 0 360 137 129 psFillArc + globalGC1 142 128 137 129 psDrawLine + globalGC1 141 127 3 3 0 360 142 128 psFillArc + globalGC1 147 128 142 128 psDrawLine + globalGC1 146 127 3 3 0 360 147 128 psFillArc + globalGC1 149 128 147 128 psDrawLine + globalGC1 148 127 3 3 0 360 149 128 psFillArc + globalGC1 151 127 149 128 psDrawLine + globalGC1 150 126 3 3 0 360 151 127 psFillArc + globalGC1 153 126 151 127 psDrawLine + globalGC1 152 125 3 3 0 360 153 126 psFillArc + globalGC1 154 125 153 126 psDrawLine + globalGC1 153 124 3 3 0 360 154 125 psFillArc + globalGC1 154 124 154 125 psDrawLine + globalGC1 153 123 3 3 0 360 154 124 psFillArc + globalGC1 155 121 154 124 psDrawLine + globalGC1 154 120 3 3 0 360 155 121 psFillArc + globalGC1 155 119 155 121 psDrawLine + globalGC1 154 118 3 3 0 360 155 119 psFillArc + globalGC1 156 113 155 119 psDrawLine + globalGC1 155 112 3 3 0 360 156 113 psFillArc + globalGC1 156 108 156 113 psDrawLine + globalGC1 155 107 3 3 0 360 156 108 psFillArc + globalGC1 156 98 156 108 psDrawLine + globalGC1 155 97 3 3 0 360 156 98 psFillArc + globalGC1 156 87 156 98 psDrawLine + globalGC1 155 86 3 3 0 360 156 87 psFillArc + globalGC1 156 66 156 87 psDrawLine + globalGC1 155 65 3 3 0 360 156 66 psFillArc + globalGC1 156 45 156 66 psDrawLine + globalGC1 155 44 3 3 0 360 156 45 psFillArc + globalGC1 156 14 156 45 psDrawLine + globalGC1 155 13 3 3 0 360 156 14 psFillArc + globalGC1 40 243 3 3 0 360 41 244 psFillArc + globalGC1 41 244 41 244 psDrawLine + globalGC1 40 243 3 3 0 360 41 244 psFillArc + globalGC1 41 213 41 244 psDrawLine + globalGC1 40 212 3 3 0 360 41 213 psFillArc + globalGC1 41 192 41 213 psDrawLine + globalGC1 40 191 3 3 0 360 41 192 psFillArc + globalGC1 41 171 41 192 psDrawLine + globalGC1 40 170 3 3 0 360 41 171 psFillArc + globalGC1 41 160 41 171 psDrawLine + globalGC1 40 159 3 3 0 360 41 160 psFillArc + globalGC1 42 150 41 160 psDrawLine + globalGC1 41 149 3 3 0 360 42 150 psFillArc + globalGC1 42 145 42 150 psDrawLine + globalGC1 41 144 3 3 0 360 42 145 psFillArc + globalGC1 42 139 42 145 psDrawLine + globalGC1 41 138 3 3 0 360 42 139 psFillArc + globalGC1 42 137 42 139 psDrawLine + globalGC1 41 136 3 3 0 360 42 137 psFillArc + globalGC1 43 134 42 137 psDrawLine + globalGC1 42 133 3 3 0 360 43 134 psFillArc + globalGC1 44 133 43 134 psDrawLine + globalGC1 43 132 3 3 0 360 44 133 psFillArc + globalGC1 45 132 44 133 psDrawLine + globalGC1 44 131 3 3 0 360 45 132 psFillArc + globalGC1 46 131 45 132 psDrawLine + globalGC1 45 130 3 3 0 360 46 131 psFillArc + globalGC1 48 130 46 131 psDrawLine + globalGC1 47 129 3 3 0 360 48 130 psFillArc + globalGC1 51 130 48 130 psDrawLine + globalGC1 50 129 3 3 0 360 51 130 psFillArc + globalGC1 56 129 51 130 psDrawLine + globalGC1 55 128 3 3 0 360 56 129 psFillArc + globalGC1 60 129 56 129 psDrawLine + globalGC1 59 128 3 3 0 360 60 129 psFillArc + globalGC1 65 129 60 129 psDrawLine + globalGC1 64 128 3 3 0 360 65 129 psFillArc + globalGC1 70 129 65 129 psDrawLine + globalGC1 69 128 3 3 0 360 70 129 psFillArc + globalGC1 75 129 70 129 psDrawLine + globalGC1 74 128 3 3 0 360 75 129 psFillArc + globalGC1 80 129 75 129 psDrawLine + globalGC1 79 128 3 3 0 360 80 129 psFillArc + globalGC1 84 128 80 129 psDrawLine + globalGC1 83 127 3 3 0 360 84 128 psFillArc + globalGC1 89 128 84 128 psDrawLine + globalGC1 88 127 3 3 0 360 89 128 psFillArc + globalGC1 92 128 89 128 psDrawLine + globalGC1 91 127 3 3 0 360 92 128 psFillArc + globalGC1 94 127 92 128 psDrawLine + globalGC1 93 126 3 3 0 360 94 127 psFillArc + globalGC1 95 126 94 127 psDrawLine + globalGC1 94 125 3 3 0 360 95 126 psFillArc + globalGC1 96 125 95 126 psDrawLine + globalGC1 95 124 3 3 0 360 96 125 psFillArc + globalGC1 97 124 96 125 psDrawLine + globalGC1 96 123 3 3 0 360 97 124 psFillArc + globalGC1 97 121 97 124 psDrawLine + globalGC1 96 120 3 3 0 360 97 121 psFillArc + globalGC1 98 119 97 121 psDrawLine + globalGC1 97 118 3 3 0 360 98 119 psFillArc + globalGC1 98 113 98 119 psDrawLine + globalGC1 97 112 3 3 0 360 98 113 psFillArc + globalGC1 98 108 98 113 psDrawLine + globalGC1 97 107 3 3 0 360 98 108 psFillArc + globalGC1 98 98 98 108 psDrawLine + globalGC1 97 97 3 3 0 360 98 98 psFillArc + globalGC1 98 87 98 98 psDrawLine + globalGC1 97 86 3 3 0 360 98 87 psFillArc + globalGC1 99 66 98 87 psDrawLine + globalGC1 98 65 3 3 0 360 99 66 psFillArc + globalGC1 99 45 99 66 psDrawLine + globalGC1 98 44 3 3 0 360 99 45 psFillArc + globalGC1 99 14 99 45 psDrawLine + globalGC1 98 13 3 3 0 360 99 14 psFillArc + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 12 129 12 129 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 17 129 12 129 psDrawLine + globalGC1 16 128 3 3 0 360 17 129 psFillArc + globalGC1 22 129 17 129 psDrawLine + globalGC1 21 128 3 3 0 360 22 129 psFillArc + globalGC1 27 128 22 129 psDrawLine + globalGC1 26 127 3 3 0 360 27 128 psFillArc + globalGC1 32 128 27 128 psDrawLine + globalGC1 31 127 3 3 0 360 32 128 psFillArc + globalGC1 34 128 32 128 psDrawLine + globalGC1 33 127 3 3 0 360 34 128 psFillArc + globalGC1 36 127 34 128 psDrawLine + globalGC1 35 126 3 3 0 360 36 127 psFillArc + globalGC1 38 126 36 127 psDrawLine + globalGC1 37 125 3 3 0 360 38 126 psFillArc + globalGC1 39 125 38 126 psDrawLine + globalGC1 38 124 3 3 0 360 39 125 psFillArc + globalGC1 39 124 39 125 psDrawLine + globalGC1 38 123 3 3 0 360 39 124 psFillArc + globalGC1 40 121 39 124 psDrawLine + globalGC1 39 120 3 3 0 360 40 121 psFillArc + globalGC1 40 119 40 121 psDrawLine + globalGC1 39 118 3 3 0 360 40 119 psFillArc + globalGC1 41 113 40 119 psDrawLine + globalGC1 40 112 3 3 0 360 41 113 psFillArc + globalGC1 41 108 41 113 psDrawLine + globalGC1 40 107 3 3 0 360 41 108 psFillArc + globalGC1 41 98 41 108 psDrawLine + globalGC1 40 97 3 3 0 360 41 98 psFillArc + globalGC1 41 87 41 98 psDrawLine + globalGC1 40 86 3 3 0 360 41 87 psFillArc + globalGC1 41 66 41 87 psDrawLine + globalGC1 40 65 3 3 0 360 41 66 psFillArc + globalGC1 41 45 41 66 psDrawLine + globalGC1 40 44 3 3 0 360 41 45 psFillArc + globalGC1 41 14 41 45 psDrawLine + globalGC1 40 13 3 3 0 360 41 14 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptcpr.ps b/books/ps/2doptcpr.ps new file mode 100644 index 0000000..35a3570 --- /dev/null +++ b/books/ps/2doptcpr.ps @@ -0,0 +1,434 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 151 0 151 psDrawLine + globalGC1 149 303 149 0 psDrawLine + unitGC 150 153 150 149 psDrawLine + unitGC 138 166 (0.00) (window) psDrawIStr + unitGC 190 153 190 149 psDrawLine + unitGC 178 166 (2.00) (window) psDrawIStr + unitGC 230 153 230 149 psDrawLine + unitGC 218 166 (4.00) (window) psDrawIStr + unitGC 270 153 270 149 psDrawLine + unitGC 258 166 (6.00) (window) psDrawIStr + unitGC 106 153 106 149 psDrawLine + unitGC 91 166 (-2.00) (window) psDrawIStr + unitGC 62 153 62 149 psDrawLine + unitGC 47 166 (-4.00) (window) psDrawIStr + unitGC 18 153 18 149 psDrawLine + unitGC 3 166 (-6.00) (window) psDrawIStr + unitGC 151 107 147 107 psDrawLine + unitGC 113 112 (1.00) (window) psDrawIStr + unitGC 151 63 147 63 psDrawLine + unitGC 113 68 (2.00) (window) psDrawIStr + unitGC 151 19 147 19 psDrawLine + unitGC 113 24 (3.00) (window) psDrawIStr + unitGC 151 195 147 195 psDrawLine + unitGC 104 200 (-1.00) (window) psDrawIStr + unitGC 151 238 147 238 psDrawLine + unitGC 104 243 (-2.00) (window) psDrawIStr + unitGC 151 281 147 281 psDrawLine + unitGC 104 286 (-3.00) (window) psDrawIStr + globalGC1 260 12 3 3 0 360 261 13 psFillArc + globalGC1 261 13 261 13 psDrawLine + globalGC1 260 12 3 3 0 360 261 13 psFillArc + globalGC1 262 36 261 13 psDrawLine + globalGC1 261 35 3 3 0 360 262 36 psFillArc + globalGC1 265 63 262 36 psDrawLine + globalGC1 264 62 3 3 0 360 265 63 psFillArc + globalGC1 270 89 265 63 psDrawLine + globalGC1 269 88 3 3 0 360 270 89 psFillArc + globalGC1 276 100 270 89 psDrawLine + globalGC1 275 99 3 3 0 360 276 100 psFillArc + globalGC1 282 105 276 100 psDrawLine + globalGC1 281 104 3 3 0 360 282 105 psFillArc + globalGC1 288 107 282 105 psDrawLine + globalGC1 287 106 3 3 0 360 288 107 psFillArc + globalGC1 190 288 3 3 0 360 191 289 psFillArc + globalGC1 191 289 191 289 psDrawLine + globalGC1 190 288 3 3 0 360 191 289 psFillArc + globalGC1 193 266 191 289 psDrawLine + globalGC1 192 265 3 3 0 360 193 266 psFillArc + globalGC1 196 239 193 266 psDrawLine + globalGC1 195 238 3 3 0 360 196 239 psFillArc + globalGC1 201 213 196 239 psDrawLine + globalGC1 200 212 3 3 0 360 201 213 psFillArc + globalGC1 207 202 201 213 psDrawLine + globalGC1 206 201 3 3 0 360 207 202 psFillArc + globalGC1 213 197 207 202 psDrawLine + globalGC1 212 196 3 3 0 360 213 197 psFillArc + globalGC1 219 195 213 197 psDrawLine + globalGC1 218 194 3 3 0 360 219 195 psFillArc + globalGC1 224 197 219 195 psDrawLine + globalGC1 223 196 3 3 0 360 224 197 psFillArc + globalGC1 230 202 224 197 psDrawLine + globalGC1 229 201 3 3 0 360 230 202 psFillArc + globalGC1 236 213 230 202 psDrawLine + globalGC1 235 212 3 3 0 360 236 213 psFillArc + globalGC1 242 239 236 213 psDrawLine + globalGC1 241 238 3 3 0 360 242 239 psFillArc + globalGC1 244 266 242 239 psDrawLine + globalGC1 243 265 3 3 0 360 244 266 psFillArc + globalGC1 246 289 244 266 psDrawLine + globalGC1 245 288 3 3 0 360 246 289 psFillArc + globalGC1 121 12 3 3 0 360 122 13 psFillArc + globalGC1 122 13 122 13 psDrawLine + globalGC1 121 12 3 3 0 360 122 13 psFillArc + globalGC1 124 36 122 13 psDrawLine + globalGC1 123 35 3 3 0 360 124 36 psFillArc + globalGC1 126 63 124 36 psDrawLine + globalGC1 125 62 3 3 0 360 126 63 psFillArc + globalGC1 132 89 126 63 psDrawLine + globalGC1 131 88 3 3 0 360 132 89 psFillArc + globalGC1 138 100 132 89 psDrawLine + globalGC1 137 99 3 3 0 360 138 100 psFillArc + globalGC1 144 105 138 100 psDrawLine + globalGC1 143 104 3 3 0 360 144 105 psFillArc + globalGC1 149 107 144 105 psDrawLine + globalGC1 148 106 3 3 0 360 149 107 psFillArc + globalGC1 155 105 149 107 psDrawLine + globalGC1 154 104 3 3 0 360 155 105 psFillArc + globalGC1 161 100 155 105 psDrawLine + globalGC1 160 99 3 3 0 360 161 100 psFillArc + globalGC1 167 89 161 100 psDrawLine + globalGC1 166 88 3 3 0 360 167 89 psFillArc + globalGC1 173 63 167 89 psDrawLine + globalGC1 172 62 3 3 0 360 173 63 psFillArc + globalGC1 175 36 173 63 psDrawLine + globalGC1 174 35 3 3 0 360 175 36 psFillArc + globalGC1 177 13 175 36 psDrawLine + globalGC1 176 12 3 3 0 360 177 13 psFillArc + globalGC1 52 288 3 3 0 360 53 289 psFillArc + globalGC1 53 289 53 289 psDrawLine + globalGC1 52 288 3 3 0 360 53 289 psFillArc + globalGC1 55 266 53 289 psDrawLine + globalGC1 54 265 3 3 0 360 55 266 psFillArc + globalGC1 57 239 55 266 psDrawLine + globalGC1 56 238 3 3 0 360 57 239 psFillArc + globalGC1 63 213 57 239 psDrawLine + globalGC1 62 212 3 3 0 360 63 213 psFillArc + globalGC1 69 202 63 213 psDrawLine + globalGC1 68 201 3 3 0 360 69 202 psFillArc + globalGC1 75 197 69 202 psDrawLine + globalGC1 74 196 3 3 0 360 75 197 psFillArc + globalGC1 80 195 75 197 psDrawLine + globalGC1 79 194 3 3 0 360 80 195 psFillArc + globalGC1 86 197 80 195 psDrawLine + globalGC1 85 196 3 3 0 360 86 197 psFillArc + globalGC1 92 202 86 197 psDrawLine + globalGC1 91 201 3 3 0 360 92 202 psFillArc + globalGC1 98 213 92 202 psDrawLine + globalGC1 97 212 3 3 0 360 98 213 psFillArc + globalGC1 103 239 98 213 psDrawLine + globalGC1 102 238 3 3 0 360 103 239 psFillArc + globalGC1 106 266 103 239 psDrawLine + globalGC1 105 265 3 3 0 360 106 266 psFillArc + globalGC1 108 289 106 266 psDrawLine + globalGC1 107 288 3 3 0 360 108 289 psFillArc + globalGC1 10 106 3 3 0 360 11 107 psFillArc + globalGC1 11 107 11 107 psDrawLine + globalGC1 10 106 3 3 0 360 11 107 psFillArc + globalGC1 17 105 11 107 psDrawLine + globalGC1 16 104 3 3 0 360 17 105 psFillArc + globalGC1 23 100 17 105 psDrawLine + globalGC1 22 99 3 3 0 360 23 100 psFillArc + globalGC1 29 89 23 100 psDrawLine + globalGC1 28 88 3 3 0 360 29 89 psFillArc + globalGC1 34 63 29 89 psDrawLine + globalGC1 33 62 3 3 0 360 34 63 psFillArc + globalGC1 37 36 34 63 psDrawLine + globalGC1 36 35 3 3 0 360 37 36 psFillArc + globalGC1 38 13 37 36 psDrawLine + globalGC1 37 12 3 3 0 360 38 13 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptcvc.ps b/books/ps/2doptcvc.ps new file mode 100644 index 0000000..d91c991 --- /dev/null +++ b/books/ps/2doptcvc.ps @@ -0,0 +1,363 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 127 259 127 0 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 12 129 12 129 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 17 144 12 129 psDrawLine + globalGC1 16 143 3 3 0 360 17 144 psFillArc + globalGC1 22 159 17 144 psDrawLine + globalGC1 21 158 3 3 0 360 22 159 psFillArc + globalGC1 27 173 22 159 psDrawLine + globalGC1 26 172 3 3 0 360 27 173 psFillArc + globalGC1 32 187 27 173 psDrawLine + globalGC1 31 186 3 3 0 360 32 187 psFillArc + globalGC1 36 199 32 187 psDrawLine + globalGC1 35 198 3 3 0 360 36 199 psFillArc + globalGC1 41 210 36 199 psDrawLine + globalGC1 40 209 3 3 0 360 41 210 psFillArc + globalGC1 46 220 41 210 psDrawLine + globalGC1 45 219 3 3 0 360 46 220 psFillArc + globalGC1 51 229 46 220 psDrawLine + globalGC1 50 228 3 3 0 360 51 229 psFillArc + globalGC1 56 235 51 229 psDrawLine + globalGC1 55 234 3 3 0 360 56 235 psFillArc + globalGC1 60 240 56 235 psDrawLine + globalGC1 59 239 3 3 0 360 60 240 psFillArc + globalGC1 63 242 60 240 psDrawLine + globalGC1 62 241 3 3 0 360 63 242 psFillArc + globalGC1 65 243 63 242 psDrawLine + globalGC1 64 242 3 3 0 360 65 243 psFillArc + globalGC1 68 244 65 243 psDrawLine + globalGC1 67 243 3 3 0 360 68 244 psFillArc + globalGC1 70 244 68 244 psDrawLine + globalGC1 69 243 3 3 0 360 70 244 psFillArc + globalGC1 72 244 70 244 psDrawLine + globalGC1 71 243 3 3 0 360 72 244 psFillArc + globalGC1 75 243 72 244 psDrawLine + globalGC1 74 242 3 3 0 360 75 243 psFillArc + globalGC1 77 242 75 243 psDrawLine + globalGC1 76 241 3 3 0 360 77 242 psFillArc + globalGC1 80 240 77 242 psDrawLine + globalGC1 79 239 3 3 0 360 80 240 psFillArc + globalGC1 84 235 80 240 psDrawLine + globalGC1 83 234 3 3 0 360 84 235 psFillArc + globalGC1 89 229 84 235 psDrawLine + globalGC1 88 228 3 3 0 360 89 229 psFillArc + globalGC1 94 220 89 229 psDrawLine + globalGC1 93 219 3 3 0 360 94 220 psFillArc + globalGC1 99 210 94 220 psDrawLine + globalGC1 98 209 3 3 0 360 99 210 psFillArc + globalGC1 104 199 99 210 psDrawLine + globalGC1 103 198 3 3 0 360 104 199 psFillArc + globalGC1 108 187 104 199 psDrawLine + globalGC1 107 186 3 3 0 360 108 187 psFillArc + globalGC1 113 173 108 187 psDrawLine + globalGC1 112 172 3 3 0 360 113 173 psFillArc + globalGC1 118 159 113 173 psDrawLine + globalGC1 117 158 3 3 0 360 118 159 psFillArc + globalGC1 123 144 118 159 psDrawLine + globalGC1 122 143 3 3 0 360 123 144 psFillArc + globalGC1 127 129 123 144 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 132 114 127 129 psDrawLine + globalGC1 131 113 3 3 0 360 132 114 psFillArc + globalGC1 137 99 132 114 psDrawLine + globalGC1 136 98 3 3 0 360 137 99 psFillArc + globalGC1 142 85 137 99 psDrawLine + globalGC1 141 84 3 3 0 360 142 85 psFillArc + globalGC1 147 71 142 85 psDrawLine + globalGC1 146 70 3 3 0 360 147 71 psFillArc + globalGC1 151 59 147 71 psDrawLine + globalGC1 150 58 3 3 0 360 151 59 psFillArc + globalGC1 156 48 151 59 psDrawLine + globalGC1 155 47 3 3 0 360 156 48 psFillArc + globalGC1 161 38 156 48 psDrawLine + globalGC1 160 37 3 3 0 360 161 38 psFillArc + globalGC1 166 29 161 38 psDrawLine + globalGC1 165 28 3 3 0 360 166 29 psFillArc + globalGC1 171 23 166 29 psDrawLine + globalGC1 170 22 3 3 0 360 171 23 psFillArc + globalGC1 175 18 171 23 psDrawLine + globalGC1 174 17 3 3 0 360 175 18 psFillArc + globalGC1 178 16 175 18 psDrawLine + globalGC1 177 15 3 3 0 360 178 16 psFillArc + globalGC1 180 15 178 16 psDrawLine + globalGC1 179 14 3 3 0 360 180 15 psFillArc + globalGC1 183 14 180 15 psDrawLine + globalGC1 182 13 3 3 0 360 183 14 psFillArc + globalGC1 185 14 183 14 psDrawLine + globalGC1 184 13 3 3 0 360 185 14 psFillArc + globalGC1 187 14 185 14 psDrawLine + globalGC1 186 13 3 3 0 360 187 14 psFillArc + globalGC1 190 15 187 14 psDrawLine + globalGC1 189 14 3 3 0 360 190 15 psFillArc + globalGC1 192 16 190 15 psDrawLine + globalGC1 191 15 3 3 0 360 192 16 psFillArc + globalGC1 195 18 192 16 psDrawLine + globalGC1 194 17 3 3 0 360 195 18 psFillArc + globalGC1 199 23 195 18 psDrawLine + globalGC1 198 22 3 3 0 360 199 23 psFillArc + globalGC1 204 29 199 23 psDrawLine + globalGC1 203 28 3 3 0 360 204 29 psFillArc + globalGC1 209 38 204 29 psDrawLine + globalGC1 208 37 3 3 0 360 209 38 psFillArc + globalGC1 214 48 209 38 psDrawLine + globalGC1 213 47 3 3 0 360 214 48 psFillArc + globalGC1 219 59 214 48 psDrawLine + globalGC1 218 58 3 3 0 360 219 59 psFillArc + globalGC1 223 71 219 59 psDrawLine + globalGC1 222 70 3 3 0 360 223 71 psFillArc + globalGC1 228 85 223 71 psDrawLine + globalGC1 227 84 3 3 0 360 228 85 psFillArc + globalGC1 233 99 228 85 psDrawLine + globalGC1 232 98 3 3 0 360 233 99 psFillArc + globalGC1 238 114 233 99 psDrawLine + globalGC1 237 113 3 3 0 360 238 114 psFillArc + globalGC1 243 129 238 114 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptplr.ps b/books/ps/2doptplr.ps new file mode 100644 index 0000000..553c212 --- /dev/null +++ b/books/ps/2doptplr.ps @@ -0,0 +1,795 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 141 0 141 psDrawLine + globalGC1 128 259 128 0 psDrawLine + globalGC1 127 140 3 3 0 360 128 141 psFillArc + globalGC1 128 141 128 141 psDrawLine + globalGC1 127 140 3 3 0 360 128 141 psFillArc + globalGC1 147 140 128 141 psDrawLine + globalGC1 146 139 3 3 0 360 147 140 psFillArc + globalGC1 166 138 147 140 psDrawLine + globalGC1 165 137 3 3 0 360 166 138 psFillArc + globalGC1 184 135 166 138 psDrawLine + globalGC1 183 134 3 3 0 360 184 135 psFillArc + globalGC1 200 131 184 135 psDrawLine + globalGC1 199 130 3 3 0 360 200 131 psFillArc + globalGC1 215 126 200 131 psDrawLine + globalGC1 214 125 3 3 0 360 215 126 psFillArc + globalGC1 226 120 215 126 psDrawLine + globalGC1 225 119 3 3 0 360 226 120 psFillArc + globalGC1 235 114 226 120 psDrawLine + globalGC1 234 113 3 3 0 360 235 114 psFillArc + globalGC1 238 112 235 114 psDrawLine + globalGC1 237 111 3 3 0 360 238 112 psFillArc + globalGC1 240 109 238 112 psDrawLine + globalGC1 239 108 3 3 0 360 240 109 psFillArc + globalGC1 242 106 240 109 psDrawLine + globalGC1 241 105 3 3 0 360 242 106 psFillArc + globalGC1 242 105 242 106 psDrawLine + globalGC1 241 104 3 3 0 360 242 105 psFillArc + globalGC1 243 104 242 105 psDrawLine + globalGC1 242 103 3 3 0 360 243 104 psFillArc + globalGC1 243 103 243 104 psDrawLine + globalGC1 242 102 3 3 0 360 243 103 psFillArc + globalGC1 243 102 243 103 psDrawLine + globalGC1 242 101 3 3 0 360 243 102 psFillArc + globalGC1 242 101 243 102 psDrawLine + globalGC1 241 100 3 3 0 360 242 101 psFillArc + globalGC1 242 100 242 101 psDrawLine + globalGC1 241 99 3 3 0 360 242 100 psFillArc + globalGC1 241 99 242 100 psDrawLine + globalGC1 240 98 3 3 0 360 241 99 psFillArc + globalGC1 240 98 241 99 psDrawLine + globalGC1 239 97 3 3 0 360 240 98 psFillArc + globalGC1 238 97 240 98 psDrawLine + globalGC1 237 96 3 3 0 360 238 97 psFillArc + globalGC1 235 96 238 97 psDrawLine + globalGC1 234 95 3 3 0 360 235 96 psFillArc + globalGC1 231 96 235 96 psDrawLine + globalGC1 230 95 3 3 0 360 231 96 psFillArc + globalGC1 221 96 231 96 psDrawLine + globalGC1 220 95 3 3 0 360 221 96 psFillArc + globalGC1 209 98 221 96 psDrawLine + globalGC1 208 97 3 3 0 360 209 98 psFillArc + globalGC1 180 109 209 98 psDrawLine + globalGC1 179 108 3 3 0 360 180 109 psFillArc + globalGC1 164 117 180 109 psDrawLine + globalGC1 163 116 3 3 0 360 164 117 psFillArc + globalGC1 147 127 164 117 psDrawLine + globalGC1 146 126 3 3 0 360 147 127 psFillArc + globalGC1 131 138 147 127 psDrawLine + globalGC1 130 137 3 3 0 360 131 138 psFillArc + globalGC1 115 151 131 138 psDrawLine + globalGC1 114 150 3 3 0 360 115 151 psFillArc + globalGC1 100 164 115 151 psDrawLine + globalGC1 99 163 3 3 0 360 100 164 psFillArc + globalGC1 87 178 100 164 psDrawLine + globalGC1 86 177 3 3 0 360 87 178 psFillArc + globalGC1 76 191 87 178 psDrawLine + globalGC1 75 190 3 3 0 360 76 191 psFillArc + globalGC1 67 204 76 191 psDrawLine + globalGC1 66 203 3 3 0 360 67 204 psFillArc + globalGC1 60 216 67 204 psDrawLine + globalGC1 59 215 3 3 0 360 60 216 psFillArc + globalGC1 56 226 60 216 psDrawLine + globalGC1 55 225 3 3 0 360 56 226 psFillArc + globalGC1 55 231 56 226 psDrawLine + globalGC1 54 230 3 3 0 360 55 231 psFillArc + globalGC1 54 234 55 231 psDrawLine + globalGC1 53 233 3 3 0 360 54 234 psFillArc + globalGC1 54 238 54 234 psDrawLine + globalGC1 53 237 3 3 0 360 54 238 psFillArc + globalGC1 54 239 54 238 psDrawLine + globalGC1 53 238 3 3 0 360 54 239 psFillArc + globalGC1 55 240 54 239 psDrawLine + globalGC1 54 239 3 3 0 360 55 240 psFillArc + globalGC1 55 241 55 240 psDrawLine + globalGC1 54 240 3 3 0 360 55 241 psFillArc + globalGC1 56 242 55 241 psDrawLine + globalGC1 55 241 3 3 0 360 56 242 psFillArc + globalGC1 56 243 56 242 psDrawLine + globalGC1 55 242 3 3 0 360 56 243 psFillArc + globalGC1 57 244 56 243 psDrawLine + globalGC1 56 243 3 3 0 360 57 244 psFillArc + globalGC1 58 244 57 244 psDrawLine + globalGC1 57 243 3 3 0 360 58 244 psFillArc + globalGC1 59 244 58 244 psDrawLine + globalGC1 58 243 3 3 0 360 59 244 psFillArc + globalGC1 60 244 59 244 psDrawLine + globalGC1 59 243 3 3 0 360 60 244 psFillArc + globalGC1 62 244 60 244 psDrawLine + globalGC1 61 243 3 3 0 360 62 244 psFillArc + globalGC1 65 243 62 244 psDrawLine + globalGC1 64 242 3 3 0 360 65 243 psFillArc + globalGC1 68 241 65 243 psDrawLine + globalGC1 67 240 3 3 0 360 68 241 psFillArc + globalGC1 75 236 68 241 psDrawLine + globalGC1 74 235 3 3 0 360 75 236 psFillArc + globalGC1 92 216 75 236 psDrawLine + globalGC1 91 215 3 3 0 360 92 216 psFillArc + globalGC1 101 202 92 216 psDrawLine + globalGC1 100 201 3 3 0 360 101 202 psFillArc + globalGC1 110 185 101 202 psDrawLine + globalGC1 109 184 3 3 0 360 110 185 psFillArc + globalGC1 118 168 110 185 psDrawLine + globalGC1 117 167 3 3 0 360 118 168 psFillArc + globalGC1 125 148 118 168 psDrawLine + globalGC1 124 147 3 3 0 360 125 148 psFillArc + globalGC1 131 129 125 148 psDrawLine + globalGC1 130 128 3 3 0 360 131 129 psFillArc + globalGC1 136 109 131 129 psDrawLine + globalGC1 135 108 3 3 0 360 136 109 psFillArc + globalGC1 139 90 136 109 psDrawLine + globalGC1 138 89 3 3 0 360 139 90 psFillArc + globalGC1 141 71 139 90 psDrawLine + globalGC1 140 70 3 3 0 360 141 71 psFillArc + globalGC1 140 41 141 71 psDrawLine + globalGC1 139 40 3 3 0 360 140 41 psFillArc + globalGC1 138 29 140 41 psDrawLine + globalGC1 137 28 3 3 0 360 138 29 psFillArc + globalGC1 137 25 138 29 psDrawLine + globalGC1 136 24 3 3 0 360 137 25 psFillArc + globalGC1 135 21 137 25 psDrawLine + globalGC1 134 20 3 3 0 360 135 21 psFillArc + globalGC1 133 18 135 21 psDrawLine + globalGC1 132 17 3 3 0 360 133 18 psFillArc + globalGC1 132 17 133 18 psDrawLine + globalGC1 131 16 3 3 0 360 132 17 psFillArc + globalGC1 131 16 132 17 psDrawLine + globalGC1 130 15 3 3 0 360 131 16 psFillArc + globalGC1 130 15 131 16 psDrawLine + globalGC1 129 14 3 3 0 360 130 15 psFillArc + globalGC1 129 14 130 15 psDrawLine + globalGC1 128 13 3 3 0 360 129 14 psFillArc + globalGC1 128 14 129 14 psDrawLine + globalGC1 127 13 3 3 0 360 128 14 psFillArc + globalGC1 128 14 128 14 psDrawLine + globalGC1 127 13 3 3 0 360 128 14 psFillArc + globalGC1 127 14 128 14 psDrawLine + globalGC1 126 13 3 3 0 360 127 14 psFillArc + globalGC1 126 14 127 14 psDrawLine + globalGC1 125 13 3 3 0 360 126 14 psFillArc + globalGC1 125 15 126 14 psDrawLine + globalGC1 124 14 3 3 0 360 125 15 psFillArc + globalGC1 124 16 125 15 psDrawLine + globalGC1 123 15 3 3 0 360 124 16 psFillArc + globalGC1 123 17 124 16 psDrawLine + globalGC1 122 16 3 3 0 360 123 17 psFillArc + globalGC1 122 18 123 17 psDrawLine + globalGC1 121 17 3 3 0 360 122 18 psFillArc + globalGC1 120 21 122 18 psDrawLine + globalGC1 119 20 3 3 0 360 120 21 psFillArc + globalGC1 118 25 120 21 psDrawLine + globalGC1 117 24 3 3 0 360 118 25 psFillArc + globalGC1 117 29 118 25 psDrawLine + globalGC1 116 28 3 3 0 360 117 29 psFillArc + globalGC1 115 41 117 29 psDrawLine + globalGC1 114 40 3 3 0 360 115 41 psFillArc + globalGC1 114 71 115 41 psDrawLine + globalGC1 113 70 3 3 0 360 114 71 psFillArc + globalGC1 116 90 114 71 psDrawLine + globalGC1 115 89 3 3 0 360 116 90 psFillArc + globalGC1 119 109 116 90 psDrawLine + globalGC1 118 108 3 3 0 360 119 109 psFillArc + globalGC1 124 129 119 109 psDrawLine + globalGC1 123 128 3 3 0 360 124 129 psFillArc + globalGC1 130 148 124 129 psDrawLine + globalGC1 129 147 3 3 0 360 130 148 psFillArc + globalGC1 137 168 130 148 psDrawLine + globalGC1 136 167 3 3 0 360 137 168 psFillArc + globalGC1 145 185 137 168 psDrawLine + globalGC1 144 184 3 3 0 360 145 185 psFillArc + globalGC1 154 202 145 185 psDrawLine + globalGC1 153 201 3 3 0 360 154 202 psFillArc + globalGC1 163 216 154 202 psDrawLine + globalGC1 162 215 3 3 0 360 163 216 psFillArc + globalGC1 172 227 163 216 psDrawLine + globalGC1 171 226 3 3 0 360 172 227 psFillArc + globalGC1 180 236 172 227 psDrawLine + globalGC1 179 235 3 3 0 360 180 236 psFillArc + globalGC1 187 241 180 236 psDrawLine + globalGC1 186 240 3 3 0 360 187 241 psFillArc + globalGC1 190 243 187 241 psDrawLine + globalGC1 189 242 3 3 0 360 190 243 psFillArc + globalGC1 193 244 190 243 psDrawLine + globalGC1 192 243 3 3 0 360 193 244 psFillArc + globalGC1 195 244 193 244 psDrawLine + globalGC1 194 243 3 3 0 360 195 244 psFillArc + globalGC1 196 244 195 244 psDrawLine + globalGC1 195 243 3 3 0 360 196 244 psFillArc + globalGC1 197 244 196 244 psDrawLine + globalGC1 196 243 3 3 0 360 197 244 psFillArc + globalGC1 198 244 197 244 psDrawLine + globalGC1 197 243 3 3 0 360 198 244 psFillArc + globalGC1 199 243 198 244 psDrawLine + globalGC1 198 242 3 3 0 360 199 243 psFillArc + globalGC1 199 242 199 243 psDrawLine + globalGC1 198 241 3 3 0 360 199 242 psFillArc + globalGC1 200 241 199 242 psDrawLine + globalGC1 199 240 3 3 0 360 200 241 psFillArc + globalGC1 200 240 200 241 psDrawLine + globalGC1 199 239 3 3 0 360 200 240 psFillArc + globalGC1 201 238 200 240 psDrawLine + globalGC1 200 237 3 3 0 360 201 238 psFillArc + globalGC1 201 234 201 238 psDrawLine + globalGC1 200 233 3 3 0 360 201 234 psFillArc + globalGC1 199 226 201 234 psDrawLine + globalGC1 198 225 3 3 0 360 199 226 psFillArc + globalGC1 188 204 199 226 psDrawLine + globalGC1 187 203 3 3 0 360 188 204 psFillArc + globalGC1 179 191 188 204 psDrawLine + globalGC1 178 190 3 3 0 360 179 191 psFillArc + globalGC1 168 178 179 191 psDrawLine + globalGC1 167 177 3 3 0 360 168 178 psFillArc + globalGC1 155 164 168 178 psDrawLine + globalGC1 154 163 3 3 0 360 155 164 psFillArc + globalGC1 140 151 155 164 psDrawLine + globalGC1 139 150 3 3 0 360 140 151 psFillArc + globalGC1 124 138 140 151 psDrawLine + globalGC1 123 137 3 3 0 360 124 138 psFillArc + globalGC1 108 127 124 138 psDrawLine + globalGC1 107 126 3 3 0 360 108 127 psFillArc + globalGC1 91 117 108 127 psDrawLine + globalGC1 90 116 3 3 0 360 91 117 psFillArc + globalGC1 75 109 91 117 psDrawLine + globalGC1 74 108 3 3 0 360 75 109 psFillArc + globalGC1 46 98 75 109 psDrawLine + globalGC1 45 97 3 3 0 360 46 98 psFillArc + globalGC1 34 96 46 98 psDrawLine + globalGC1 33 95 3 3 0 360 34 96 psFillArc + globalGC1 24 96 34 96 psDrawLine + globalGC1 23 95 3 3 0 360 24 96 psFillArc + globalGC1 20 96 24 96 psDrawLine + globalGC1 19 95 3 3 0 360 20 96 psFillArc + globalGC1 17 97 20 96 psDrawLine + globalGC1 16 96 3 3 0 360 17 97 psFillArc + globalGC1 16 98 17 97 psDrawLine + globalGC1 15 97 3 3 0 360 16 98 psFillArc + globalGC1 15 98 16 98 psDrawLine + globalGC1 14 97 3 3 0 360 15 98 psFillArc + globalGC1 14 99 15 98 psDrawLine + globalGC1 13 98 3 3 0 360 14 99 psFillArc + globalGC1 13 100 14 99 psDrawLine + globalGC1 12 99 3 3 0 360 13 100 psFillArc + globalGC1 13 101 13 100 psDrawLine + globalGC1 12 100 3 3 0 360 13 101 psFillArc + globalGC1 12 102 13 101 psDrawLine + globalGC1 11 101 3 3 0 360 12 102 psFillArc + globalGC1 12 103 12 102 psDrawLine + globalGC1 11 102 3 3 0 360 12 103 psFillArc + globalGC1 12 104 12 103 psDrawLine + globalGC1 11 103 3 3 0 360 12 104 psFillArc + globalGC1 13 106 12 104 psDrawLine + globalGC1 12 105 3 3 0 360 13 106 psFillArc + globalGC1 15 109 13 106 psDrawLine + globalGC1 14 108 3 3 0 360 15 109 psFillArc + globalGC1 20 114 15 109 psDrawLine + globalGC1 19 113 3 3 0 360 20 114 psFillArc + globalGC1 29 120 20 114 psDrawLine + globalGC1 28 119 3 3 0 360 29 120 psFillArc + globalGC1 55 131 29 120 psDrawLine + globalGC1 54 130 3 3 0 360 55 131 psFillArc + globalGC1 71 135 55 131 psDrawLine + globalGC1 70 134 3 3 0 360 71 135 psFillArc + globalGC1 89 138 71 135 psDrawLine + globalGC1 88 137 3 3 0 360 89 138 psFillArc + globalGC1 108 140 89 138 psDrawLine + globalGC1 107 139 3 3 0 360 108 140 psFillArc + globalGC1 128 141 108 140 psDrawLine + globalGC1 127 140 3 3 0 360 128 141 psFillArc + globalGC1 147 140 128 141 psDrawLine + globalGC1 146 139 3 3 0 360 147 140 psFillArc + globalGC1 166 138 147 140 psDrawLine + globalGC1 165 137 3 3 0 360 166 138 psFillArc + globalGC1 184 135 166 138 psDrawLine + globalGC1 183 134 3 3 0 360 184 135 psFillArc + globalGC1 200 131 184 135 psDrawLine + globalGC1 199 130 3 3 0 360 200 131 psFillArc + globalGC1 215 126 200 131 psDrawLine + globalGC1 214 125 3 3 0 360 215 126 psFillArc + globalGC1 226 120 215 126 psDrawLine + globalGC1 225 119 3 3 0 360 226 120 psFillArc + globalGC1 235 114 226 120 psDrawLine + globalGC1 234 113 3 3 0 360 235 114 psFillArc + globalGC1 238 112 235 114 psDrawLine + globalGC1 237 111 3 3 0 360 238 112 psFillArc + globalGC1 240 109 238 112 psDrawLine + globalGC1 239 108 3 3 0 360 240 109 psFillArc + globalGC1 242 106 240 109 psDrawLine + globalGC1 241 105 3 3 0 360 242 106 psFillArc + globalGC1 242 105 242 106 psDrawLine + globalGC1 241 104 3 3 0 360 242 105 psFillArc + globalGC1 243 104 242 105 psDrawLine + globalGC1 242 103 3 3 0 360 243 104 psFillArc + globalGC1 243 103 243 104 psDrawLine + globalGC1 242 102 3 3 0 360 243 103 psFillArc + globalGC1 243 102 243 103 psDrawLine + globalGC1 242 101 3 3 0 360 243 102 psFillArc + globalGC1 242 101 243 102 psDrawLine + globalGC1 241 100 3 3 0 360 242 101 psFillArc + globalGC1 242 100 242 101 psDrawLine + globalGC1 241 99 3 3 0 360 242 100 psFillArc + globalGC1 241 99 242 100 psDrawLine + globalGC1 240 98 3 3 0 360 241 99 psFillArc + globalGC1 240 98 241 99 psDrawLine + globalGC1 239 97 3 3 0 360 240 98 psFillArc + globalGC1 238 97 240 98 psDrawLine + globalGC1 237 96 3 3 0 360 238 97 psFillArc + globalGC1 235 96 238 97 psDrawLine + globalGC1 234 95 3 3 0 360 235 96 psFillArc + globalGC1 231 96 235 96 psDrawLine + globalGC1 230 95 3 3 0 360 231 96 psFillArc + globalGC1 221 96 231 96 psDrawLine + globalGC1 220 95 3 3 0 360 221 96 psFillArc + globalGC1 209 98 221 96 psDrawLine + globalGC1 208 97 3 3 0 360 209 98 psFillArc + globalGC1 180 109 209 98 psDrawLine + globalGC1 179 108 3 3 0 360 180 109 psFillArc + globalGC1 164 117 180 109 psDrawLine + globalGC1 163 116 3 3 0 360 164 117 psFillArc + globalGC1 147 127 164 117 psDrawLine + globalGC1 146 126 3 3 0 360 147 127 psFillArc + globalGC1 131 138 147 127 psDrawLine + globalGC1 130 137 3 3 0 360 131 138 psFillArc + globalGC1 115 151 131 138 psDrawLine + globalGC1 114 150 3 3 0 360 115 151 psFillArc + globalGC1 100 164 115 151 psDrawLine + globalGC1 99 163 3 3 0 360 100 164 psFillArc + globalGC1 87 178 100 164 psDrawLine + globalGC1 86 177 3 3 0 360 87 178 psFillArc + globalGC1 76 191 87 178 psDrawLine + globalGC1 75 190 3 3 0 360 76 191 psFillArc + globalGC1 67 204 76 191 psDrawLine + globalGC1 66 203 3 3 0 360 67 204 psFillArc + globalGC1 60 216 67 204 psDrawLine + globalGC1 59 215 3 3 0 360 60 216 psFillArc + globalGC1 56 226 60 216 psDrawLine + globalGC1 55 225 3 3 0 360 56 226 psFillArc + globalGC1 55 231 56 226 psDrawLine + globalGC1 54 230 3 3 0 360 55 231 psFillArc + globalGC1 54 234 55 231 psDrawLine + globalGC1 53 233 3 3 0 360 54 234 psFillArc + globalGC1 54 238 54 234 psDrawLine + globalGC1 53 237 3 3 0 360 54 238 psFillArc + globalGC1 54 239 54 238 psDrawLine + globalGC1 53 238 3 3 0 360 54 239 psFillArc + globalGC1 55 240 54 239 psDrawLine + globalGC1 54 239 3 3 0 360 55 240 psFillArc + globalGC1 55 241 55 240 psDrawLine + globalGC1 54 240 3 3 0 360 55 241 psFillArc + globalGC1 56 242 55 241 psDrawLine + globalGC1 55 241 3 3 0 360 56 242 psFillArc + globalGC1 56 243 56 242 psDrawLine + globalGC1 55 242 3 3 0 360 56 243 psFillArc + globalGC1 57 244 56 243 psDrawLine + globalGC1 56 243 3 3 0 360 57 244 psFillArc + globalGC1 58 244 57 244 psDrawLine + globalGC1 57 243 3 3 0 360 58 244 psFillArc + globalGC1 59 244 58 244 psDrawLine + globalGC1 58 243 3 3 0 360 59 244 psFillArc + globalGC1 60 244 59 244 psDrawLine + globalGC1 59 243 3 3 0 360 60 244 psFillArc + globalGC1 62 244 60 244 psDrawLine + globalGC1 61 243 3 3 0 360 62 244 psFillArc + globalGC1 65 243 62 244 psDrawLine + globalGC1 64 242 3 3 0 360 65 243 psFillArc + globalGC1 68 241 65 243 psDrawLine + globalGC1 67 240 3 3 0 360 68 241 psFillArc + globalGC1 75 236 68 241 psDrawLine + globalGC1 74 235 3 3 0 360 75 236 psFillArc + globalGC1 92 216 75 236 psDrawLine + globalGC1 91 215 3 3 0 360 92 216 psFillArc + globalGC1 101 202 92 216 psDrawLine + globalGC1 100 201 3 3 0 360 101 202 psFillArc + globalGC1 110 185 101 202 psDrawLine + globalGC1 109 184 3 3 0 360 110 185 psFillArc + globalGC1 118 168 110 185 psDrawLine + globalGC1 117 167 3 3 0 360 118 168 psFillArc + globalGC1 125 148 118 168 psDrawLine + globalGC1 124 147 3 3 0 360 125 148 psFillArc + globalGC1 131 129 125 148 psDrawLine + globalGC1 130 128 3 3 0 360 131 129 psFillArc + globalGC1 136 109 131 129 psDrawLine + globalGC1 135 108 3 3 0 360 136 109 psFillArc + globalGC1 139 90 136 109 psDrawLine + globalGC1 138 89 3 3 0 360 139 90 psFillArc + globalGC1 141 71 139 90 psDrawLine + globalGC1 140 70 3 3 0 360 141 71 psFillArc + globalGC1 140 41 141 71 psDrawLine + globalGC1 139 40 3 3 0 360 140 41 psFillArc + globalGC1 138 29 140 41 psDrawLine + globalGC1 137 28 3 3 0 360 138 29 psFillArc + globalGC1 137 25 138 29 psDrawLine + globalGC1 136 24 3 3 0 360 137 25 psFillArc + globalGC1 135 21 137 25 psDrawLine + globalGC1 134 20 3 3 0 360 135 21 psFillArc + globalGC1 133 18 135 21 psDrawLine + globalGC1 132 17 3 3 0 360 133 18 psFillArc + globalGC1 132 17 133 18 psDrawLine + globalGC1 131 16 3 3 0 360 132 17 psFillArc + globalGC1 131 16 132 17 psDrawLine + globalGC1 130 15 3 3 0 360 131 16 psFillArc + globalGC1 130 15 131 16 psDrawLine + globalGC1 129 14 3 3 0 360 130 15 psFillArc + globalGC1 129 14 130 15 psDrawLine + globalGC1 128 13 3 3 0 360 129 14 psFillArc + globalGC1 128 14 129 14 psDrawLine + globalGC1 127 13 3 3 0 360 128 14 psFillArc + globalGC1 128 14 128 14 psDrawLine + globalGC1 127 13 3 3 0 360 128 14 psFillArc + globalGC1 127 14 128 14 psDrawLine + globalGC1 126 13 3 3 0 360 127 14 psFillArc + globalGC1 126 14 127 14 psDrawLine + globalGC1 125 13 3 3 0 360 126 14 psFillArc + globalGC1 125 15 126 14 psDrawLine + globalGC1 124 14 3 3 0 360 125 15 psFillArc + globalGC1 124 16 125 15 psDrawLine + globalGC1 123 15 3 3 0 360 124 16 psFillArc + globalGC1 123 17 124 16 psDrawLine + globalGC1 122 16 3 3 0 360 123 17 psFillArc + globalGC1 122 18 123 17 psDrawLine + globalGC1 121 17 3 3 0 360 122 18 psFillArc + globalGC1 120 21 122 18 psDrawLine + globalGC1 119 20 3 3 0 360 120 21 psFillArc + globalGC1 118 25 120 21 psDrawLine + globalGC1 117 24 3 3 0 360 118 25 psFillArc + globalGC1 117 29 118 25 psDrawLine + globalGC1 116 28 3 3 0 360 117 29 psFillArc + globalGC1 115 41 117 29 psDrawLine + globalGC1 114 40 3 3 0 360 115 41 psFillArc + globalGC1 114 71 115 41 psDrawLine + globalGC1 113 70 3 3 0 360 114 71 psFillArc + globalGC1 116 90 114 71 psDrawLine + globalGC1 115 89 3 3 0 360 116 90 psFillArc + globalGC1 119 109 116 90 psDrawLine + globalGC1 118 108 3 3 0 360 119 109 psFillArc + globalGC1 124 129 119 109 psDrawLine + globalGC1 123 128 3 3 0 360 124 129 psFillArc + globalGC1 130 148 124 129 psDrawLine + globalGC1 129 147 3 3 0 360 130 148 psFillArc + globalGC1 137 168 130 148 psDrawLine + globalGC1 136 167 3 3 0 360 137 168 psFillArc + globalGC1 145 185 137 168 psDrawLine + globalGC1 144 184 3 3 0 360 145 185 psFillArc + globalGC1 154 202 145 185 psDrawLine + globalGC1 153 201 3 3 0 360 154 202 psFillArc + globalGC1 163 216 154 202 psDrawLine + globalGC1 162 215 3 3 0 360 163 216 psFillArc + globalGC1 172 227 163 216 psDrawLine + globalGC1 171 226 3 3 0 360 172 227 psFillArc + globalGC1 180 236 172 227 psDrawLine + globalGC1 179 235 3 3 0 360 180 236 psFillArc + globalGC1 187 241 180 236 psDrawLine + globalGC1 186 240 3 3 0 360 187 241 psFillArc + globalGC1 190 243 187 241 psDrawLine + globalGC1 189 242 3 3 0 360 190 243 psFillArc + globalGC1 193 244 190 243 psDrawLine + globalGC1 192 243 3 3 0 360 193 244 psFillArc + globalGC1 195 244 193 244 psDrawLine + globalGC1 194 243 3 3 0 360 195 244 psFillArc + globalGC1 196 244 195 244 psDrawLine + globalGC1 195 243 3 3 0 360 196 244 psFillArc + globalGC1 197 244 196 244 psDrawLine + globalGC1 196 243 3 3 0 360 197 244 psFillArc + globalGC1 198 244 197 244 psDrawLine + globalGC1 197 243 3 3 0 360 198 244 psFillArc + globalGC1 199 243 198 244 psDrawLine + globalGC1 198 242 3 3 0 360 199 243 psFillArc + globalGC1 199 242 199 243 psDrawLine + globalGC1 198 241 3 3 0 360 199 242 psFillArc + globalGC1 200 241 199 242 psDrawLine + globalGC1 199 240 3 3 0 360 200 241 psFillArc + globalGC1 200 240 200 241 psDrawLine + globalGC1 199 239 3 3 0 360 200 240 psFillArc + globalGC1 201 238 200 240 psDrawLine + globalGC1 200 237 3 3 0 360 201 238 psFillArc + globalGC1 201 234 201 238 psDrawLine + globalGC1 200 233 3 3 0 360 201 234 psFillArc + globalGC1 199 226 201 234 psDrawLine + globalGC1 198 225 3 3 0 360 199 226 psFillArc + globalGC1 188 204 199 226 psDrawLine + globalGC1 187 203 3 3 0 360 188 204 psFillArc + globalGC1 179 191 188 204 psDrawLine + globalGC1 178 190 3 3 0 360 179 191 psFillArc + globalGC1 168 178 179 191 psDrawLine + globalGC1 167 177 3 3 0 360 168 178 psFillArc + globalGC1 155 164 168 178 psDrawLine + globalGC1 154 163 3 3 0 360 155 164 psFillArc + globalGC1 140 151 155 164 psDrawLine + globalGC1 139 150 3 3 0 360 140 151 psFillArc + globalGC1 124 138 140 151 psDrawLine + globalGC1 123 137 3 3 0 360 124 138 psFillArc + globalGC1 108 127 124 138 psDrawLine + globalGC1 107 126 3 3 0 360 108 127 psFillArc + globalGC1 91 117 108 127 psDrawLine + globalGC1 90 116 3 3 0 360 91 117 psFillArc + globalGC1 75 109 91 117 psDrawLine + globalGC1 74 108 3 3 0 360 75 109 psFillArc + globalGC1 46 98 75 109 psDrawLine + globalGC1 45 97 3 3 0 360 46 98 psFillArc + globalGC1 34 96 46 98 psDrawLine + globalGC1 33 95 3 3 0 360 34 96 psFillArc + globalGC1 24 96 34 96 psDrawLine + globalGC1 23 95 3 3 0 360 24 96 psFillArc + globalGC1 20 96 24 96 psDrawLine + globalGC1 19 95 3 3 0 360 20 96 psFillArc + globalGC1 17 97 20 96 psDrawLine + globalGC1 16 96 3 3 0 360 17 97 psFillArc + globalGC1 16 98 17 97 psDrawLine + globalGC1 15 97 3 3 0 360 16 98 psFillArc + globalGC1 15 98 16 98 psDrawLine + globalGC1 14 97 3 3 0 360 15 98 psFillArc + globalGC1 14 99 15 98 psDrawLine + globalGC1 13 98 3 3 0 360 14 99 psFillArc + globalGC1 13 100 14 99 psDrawLine + globalGC1 12 99 3 3 0 360 13 100 psFillArc + globalGC1 13 101 13 100 psDrawLine + globalGC1 12 100 3 3 0 360 13 101 psFillArc + globalGC1 12 102 13 101 psDrawLine + globalGC1 11 101 3 3 0 360 12 102 psFillArc + globalGC1 12 103 12 102 psDrawLine + globalGC1 11 102 3 3 0 360 12 103 psFillArc + globalGC1 12 104 12 103 psDrawLine + globalGC1 11 103 3 3 0 360 12 104 psFillArc + globalGC1 13 106 12 104 psDrawLine + globalGC1 12 105 3 3 0 360 13 106 psFillArc + globalGC1 15 109 13 106 psDrawLine + globalGC1 14 108 3 3 0 360 15 109 psFillArc + globalGC1 20 114 15 109 psDrawLine + globalGC1 19 113 3 3 0 360 20 114 psFillArc + globalGC1 29 120 20 114 psDrawLine + globalGC1 28 119 3 3 0 360 29 120 psFillArc + globalGC1 55 131 29 120 psDrawLine + globalGC1 54 130 3 3 0 360 55 131 psFillArc + globalGC1 71 135 55 131 psDrawLine + globalGC1 70 134 3 3 0 360 71 135 psFillArc + globalGC1 89 138 71 135 psDrawLine + globalGC1 88 137 3 3 0 360 89 138 psFillArc + globalGC1 108 140 89 138 psDrawLine + globalGC1 107 139 3 3 0 360 108 140 psFillArc + globalGC1 128 141 108 140 psDrawLine + globalGC1 127 140 3 3 0 360 128 141 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptptc.ps b/books/ps/2doptptc.ps new file mode 100644 index 0000000..d91c991 --- /dev/null +++ b/books/ps/2doptptc.ps @@ -0,0 +1,363 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 127 259 127 0 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 12 129 12 129 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 17 144 12 129 psDrawLine + globalGC1 16 143 3 3 0 360 17 144 psFillArc + globalGC1 22 159 17 144 psDrawLine + globalGC1 21 158 3 3 0 360 22 159 psFillArc + globalGC1 27 173 22 159 psDrawLine + globalGC1 26 172 3 3 0 360 27 173 psFillArc + globalGC1 32 187 27 173 psDrawLine + globalGC1 31 186 3 3 0 360 32 187 psFillArc + globalGC1 36 199 32 187 psDrawLine + globalGC1 35 198 3 3 0 360 36 199 psFillArc + globalGC1 41 210 36 199 psDrawLine + globalGC1 40 209 3 3 0 360 41 210 psFillArc + globalGC1 46 220 41 210 psDrawLine + globalGC1 45 219 3 3 0 360 46 220 psFillArc + globalGC1 51 229 46 220 psDrawLine + globalGC1 50 228 3 3 0 360 51 229 psFillArc + globalGC1 56 235 51 229 psDrawLine + globalGC1 55 234 3 3 0 360 56 235 psFillArc + globalGC1 60 240 56 235 psDrawLine + globalGC1 59 239 3 3 0 360 60 240 psFillArc + globalGC1 63 242 60 240 psDrawLine + globalGC1 62 241 3 3 0 360 63 242 psFillArc + globalGC1 65 243 63 242 psDrawLine + globalGC1 64 242 3 3 0 360 65 243 psFillArc + globalGC1 68 244 65 243 psDrawLine + globalGC1 67 243 3 3 0 360 68 244 psFillArc + globalGC1 70 244 68 244 psDrawLine + globalGC1 69 243 3 3 0 360 70 244 psFillArc + globalGC1 72 244 70 244 psDrawLine + globalGC1 71 243 3 3 0 360 72 244 psFillArc + globalGC1 75 243 72 244 psDrawLine + globalGC1 74 242 3 3 0 360 75 243 psFillArc + globalGC1 77 242 75 243 psDrawLine + globalGC1 76 241 3 3 0 360 77 242 psFillArc + globalGC1 80 240 77 242 psDrawLine + globalGC1 79 239 3 3 0 360 80 240 psFillArc + globalGC1 84 235 80 240 psDrawLine + globalGC1 83 234 3 3 0 360 84 235 psFillArc + globalGC1 89 229 84 235 psDrawLine + globalGC1 88 228 3 3 0 360 89 229 psFillArc + globalGC1 94 220 89 229 psDrawLine + globalGC1 93 219 3 3 0 360 94 220 psFillArc + globalGC1 99 210 94 220 psDrawLine + globalGC1 98 209 3 3 0 360 99 210 psFillArc + globalGC1 104 199 99 210 psDrawLine + globalGC1 103 198 3 3 0 360 104 199 psFillArc + globalGC1 108 187 104 199 psDrawLine + globalGC1 107 186 3 3 0 360 108 187 psFillArc + globalGC1 113 173 108 187 psDrawLine + globalGC1 112 172 3 3 0 360 113 173 psFillArc + globalGC1 118 159 113 173 psDrawLine + globalGC1 117 158 3 3 0 360 118 159 psFillArc + globalGC1 123 144 118 159 psDrawLine + globalGC1 122 143 3 3 0 360 123 144 psFillArc + globalGC1 127 129 123 144 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 132 114 127 129 psDrawLine + globalGC1 131 113 3 3 0 360 132 114 psFillArc + globalGC1 137 99 132 114 psDrawLine + globalGC1 136 98 3 3 0 360 137 99 psFillArc + globalGC1 142 85 137 99 psDrawLine + globalGC1 141 84 3 3 0 360 142 85 psFillArc + globalGC1 147 71 142 85 psDrawLine + globalGC1 146 70 3 3 0 360 147 71 psFillArc + globalGC1 151 59 147 71 psDrawLine + globalGC1 150 58 3 3 0 360 151 59 psFillArc + globalGC1 156 48 151 59 psDrawLine + globalGC1 155 47 3 3 0 360 156 48 psFillArc + globalGC1 161 38 156 48 psDrawLine + globalGC1 160 37 3 3 0 360 161 38 psFillArc + globalGC1 166 29 161 38 psDrawLine + globalGC1 165 28 3 3 0 360 166 29 psFillArc + globalGC1 171 23 166 29 psDrawLine + globalGC1 170 22 3 3 0 360 171 23 psFillArc + globalGC1 175 18 171 23 psDrawLine + globalGC1 174 17 3 3 0 360 175 18 psFillArc + globalGC1 178 16 175 18 psDrawLine + globalGC1 177 15 3 3 0 360 178 16 psFillArc + globalGC1 180 15 178 16 psDrawLine + globalGC1 179 14 3 3 0 360 180 15 psFillArc + globalGC1 183 14 180 15 psDrawLine + globalGC1 182 13 3 3 0 360 183 14 psFillArc + globalGC1 185 14 183 14 psDrawLine + globalGC1 184 13 3 3 0 360 185 14 psFillArc + globalGC1 187 14 185 14 psDrawLine + globalGC1 186 13 3 3 0 360 187 14 psFillArc + globalGC1 190 15 187 14 psDrawLine + globalGC1 189 14 3 3 0 360 190 15 psFillArc + globalGC1 192 16 190 15 psDrawLine + globalGC1 191 15 3 3 0 360 192 16 psFillArc + globalGC1 195 18 192 16 psDrawLine + globalGC1 194 17 3 3 0 360 195 18 psFillArc + globalGC1 199 23 195 18 psDrawLine + globalGC1 198 22 3 3 0 360 199 23 psFillArc + globalGC1 204 29 199 23 psDrawLine + globalGC1 203 28 3 3 0 360 204 29 psFillArc + globalGC1 209 38 204 29 psDrawLine + globalGC1 208 37 3 3 0 360 209 38 psFillArc + globalGC1 214 48 209 38 psDrawLine + globalGC1 213 47 3 3 0 360 214 48 psFillArc + globalGC1 219 59 214 48 psDrawLine + globalGC1 218 58 3 3 0 360 219 59 psFillArc + globalGC1 223 71 219 59 psDrawLine + globalGC1 222 70 3 3 0 360 223 71 psFillArc + globalGC1 228 85 223 71 psDrawLine + globalGC1 227 84 3 3 0 360 228 85 psFillArc + globalGC1 233 99 228 85 psDrawLine + globalGC1 232 98 3 3 0 360 233 99 psFillArc + globalGC1 238 114 233 99 psDrawLine + globalGC1 237 113 3 3 0 360 238 114 psFillArc + globalGC1 243 129 238 114 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptrga.ps b/books/ps/2doptrga.ps new file mode 100644 index 0000000..c9026a7 --- /dev/null +++ b/books/ps/2doptrga.ps @@ -0,0 +1,433 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 92 0 92 psDrawLine + globalGC1 129 259 129 0 psDrawLine + unitGC 185 94 185 90 psDrawLine + unitGC 173 107 (1.00) (window) psDrawIStr + unitGC 241 94 241 90 psDrawLine + unitGC 229 107 (2.00) (window) psDrawIStr + unitGC 73 94 73 90 psDrawLine + unitGC 58 107 (-1.00) (window) psDrawIStr + unitGC 16 94 16 90 psDrawLine + unitGC 1 107 (-2.00) (window) psDrawIStr + unitGC 131 17 127 17 psDrawLine + unitGC 93 22 (1.00) (window) psDrawIStr + unitGC 131 166 127 166 psDrawLine + unitGC 84 171 (-1.00) (window) psDrawIStr + unitGC 131 240 127 240 psDrawLine + unitGC 84 245 (-2.00) (window) psDrawIStr + globalGC1 96 68 3 3 0 360 97 69 psFillArc + globalGC1 97 69 97 69 psDrawLine + globalGC1 96 68 3 3 0 360 97 69 psFillArc + globalGC1 90 71 97 69 psDrawLine + globalGC1 89 70 3 3 0 360 90 71 psFillArc + globalGC1 83 75 90 71 psDrawLine + globalGC1 82 74 3 3 0 360 83 75 psFillArc + globalGC1 77 84 83 75 psDrawLine + globalGC1 76 83 3 3 0 360 77 84 psFillArc + globalGC1 73 93 77 84 psDrawLine + globalGC1 72 92 3 3 0 360 73 93 psFillArc + globalGC1 70 102 73 93 psDrawLine + globalGC1 69 101 3 3 0 360 70 102 psFillArc + globalGC1 68 111 70 102 psDrawLine + globalGC1 67 110 3 3 0 360 68 111 psFillArc + globalGC1 67 120 68 111 psDrawLine + globalGC1 66 119 3 3 0 360 67 120 psFillArc + globalGC1 67 129 67 120 psDrawLine + globalGC1 66 128 3 3 0 360 67 129 psFillArc + globalGC1 67 129 67 129 psDrawLine + globalGC1 66 128 3 3 0 360 67 129 psFillArc + globalGC1 67 138 67 129 psDrawLine + globalGC1 66 137 3 3 0 360 67 138 psFillArc + globalGC1 68 147 67 138 psDrawLine + globalGC1 67 146 3 3 0 360 68 147 psFillArc + globalGC1 70 156 68 147 psDrawLine + globalGC1 69 155 3 3 0 360 70 156 psFillArc + globalGC1 73 165 70 156 psDrawLine + globalGC1 72 164 3 3 0 360 73 165 psFillArc + globalGC1 77 174 73 165 psDrawLine + globalGC1 76 173 3 3 0 360 77 174 psFillArc + globalGC1 83 183 77 174 psDrawLine + globalGC1 82 182 3 3 0 360 83 183 psFillArc + globalGC1 90 187 83 183 psDrawLine + globalGC1 89 186 3 3 0 360 90 187 psFillArc + globalGC1 97 189 90 187 psDrawLine + globalGC1 96 188 3 3 0 360 97 189 psFillArc + globalGC1 110 185 97 189 psDrawLine + globalGC1 109 184 3 3 0 360 110 185 psFillArc + globalGC1 116 180 110 185 psDrawLine + globalGC1 115 179 3 3 0 360 116 180 psFillArc + globalGC1 123 174 116 180 psDrawLine + globalGC1 122 173 3 3 0 360 123 174 psFillArc + globalGC1 130 165 123 174 psDrawLine + globalGC1 129 164 3 3 0 360 130 165 psFillArc + globalGC1 136 156 130 165 psDrawLine + globalGC1 135 155 3 3 0 360 136 156 psFillArc + globalGC1 140 147 136 156 psDrawLine + globalGC1 139 146 3 3 0 360 140 147 psFillArc + globalGC1 143 138 140 147 psDrawLine + globalGC1 142 137 3 3 0 360 143 138 psFillArc + globalGC1 144 129 143 138 psDrawLine + globalGC1 143 128 3 3 0 360 144 129 psFillArc + globalGC1 143 120 144 129 psDrawLine + globalGC1 142 119 3 3 0 360 143 120 psFillArc + globalGC1 141 111 143 120 psDrawLine + globalGC1 140 110 3 3 0 360 141 111 psFillArc + globalGC1 136 102 141 111 psDrawLine + globalGC1 135 101 3 3 0 360 136 102 psFillArc + globalGC1 131 93 136 102 psDrawLine + globalGC1 130 92 3 3 0 360 131 93 psFillArc + globalGC1 124 85 131 93 psDrawLine + globalGC1 123 84 3 3 0 360 124 85 psFillArc + globalGC1 117 78 124 85 psDrawLine + globalGC1 116 77 3 3 0 360 117 78 psFillArc + globalGC1 110 74 117 78 psDrawLine + globalGC1 109 73 3 3 0 360 110 74 psFillArc + globalGC1 104 70 110 74 psDrawLine + globalGC1 103 69 3 3 0 360 104 70 psFillArc + globalGC1 97 69 104 70 psDrawLine + globalGC1 96 68 3 3 0 360 97 69 psFillArc + globalGC1 214 16 3 3 0 360 215 17 psFillArc + globalGC1 215 17 215 17 psDrawLine + globalGC1 214 16 3 3 0 360 215 17 psFillArc + globalGC1 213 21 215 17 psDrawLine + globalGC1 212 20 3 3 0 360 213 21 psFillArc + globalGC1 209 30 213 21 psDrawLine + globalGC1 208 29 3 3 0 360 209 30 psFillArc + globalGC1 206 39 209 30 psDrawLine + globalGC1 205 38 3 3 0 360 206 39 psFillArc + globalGC1 202 48 206 39 psDrawLine + globalGC1 201 47 3 3 0 360 202 48 psFillArc + globalGC1 199 57 202 48 psDrawLine + globalGC1 198 56 3 3 0 360 199 57 psFillArc + globalGC1 195 66 199 57 psDrawLine + globalGC1 194 65 3 3 0 360 195 66 psFillArc + globalGC1 192 75 195 66 psDrawLine + globalGC1 191 74 3 3 0 360 192 75 psFillArc + globalGC1 188 84 192 75 psDrawLine + globalGC1 187 83 3 3 0 360 188 84 psFillArc + globalGC1 185 93 188 84 psDrawLine + globalGC1 184 92 3 3 0 360 185 93 psFillArc + globalGC1 182 102 185 93 psDrawLine + globalGC1 181 101 3 3 0 360 182 102 psFillArc + globalGC1 179 111 182 102 psDrawLine + globalGC1 178 110 3 3 0 360 179 111 psFillArc + globalGC1 177 120 179 111 psDrawLine + globalGC1 176 119 3 3 0 360 177 120 psFillArc + globalGC1 176 129 177 120 psDrawLine + globalGC1 175 128 3 3 0 360 176 129 psFillArc + globalGC1 178 142 176 129 psDrawLine + globalGC1 177 141 3 3 0 360 178 142 psFillArc + globalGC1 180 151 178 142 psDrawLine + globalGC1 179 150 3 3 0 360 180 151 psFillArc + globalGC1 183 160 180 151 psDrawLine + globalGC1 182 159 3 3 0 360 183 160 psFillArc + globalGC1 186 169 183 160 psDrawLine + globalGC1 185 168 3 3 0 360 186 169 psFillArc + globalGC1 190 178 186 169 psDrawLine + globalGC1 189 177 3 3 0 360 190 178 psFillArc + globalGC1 194 187 190 178 psDrawLine + globalGC1 193 186 3 3 0 360 194 187 psFillArc + globalGC1 197 196 194 187 psDrawLine + globalGC1 196 195 3 3 0 360 197 196 psFillArc + globalGC1 201 205 197 196 psDrawLine + globalGC1 200 204 3 3 0 360 201 205 psFillArc + globalGC1 204 214 201 205 psDrawLine + globalGC1 203 213 3 3 0 360 204 214 psFillArc + globalGC1 208 223 204 214 psDrawLine + globalGC1 207 222 3 3 0 360 208 223 psFillArc + globalGC1 211 232 208 223 psDrawLine + globalGC1 210 231 3 3 0 360 211 232 psFillArc + globalGC1 215 241 211 232 psDrawLine + globalGC1 214 240 3 3 0 360 215 241 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptrgb.ps b/books/ps/2doptrgb.ps new file mode 100644 index 0000000..21850e8 --- /dev/null +++ b/books/ps/2doptrgb.ps @@ -0,0 +1,420 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 128 259 128 0 psDrawLine + unitGC 128 131 128 127 psDrawLine + unitGC 116 144 (0.00) (window) psDrawIStr + unitGC 166 131 166 127 psDrawLine + unitGC 154 144 (0.50) (window) psDrawIStr + unitGC 204 131 204 127 psDrawLine + unitGC 192 144 (1.00) (window) psDrawIStr + unitGC 242 131 242 127 psDrawLine + unitGC 230 144 (1.50) (window) psDrawIStr + unitGC 89 131 89 127 psDrawLine + unitGC 74 144 (-0.50) (window) psDrawIStr + unitGC 50 131 50 127 psDrawLine + unitGC 35 144 (-1.00) (window) psDrawIStr + unitGC 11 131 11 127 psDrawLine + unitGC -4 144 (-1.50) (window) psDrawIStr + unitGC 130 90 126 90 psDrawLine + unitGC 92 95 (0.50) (window) psDrawIStr + unitGC 130 51 126 51 psDrawLine + unitGC 92 56 (1.00) (window) psDrawIStr + unitGC 130 12 126 12 psDrawLine + unitGC 92 17 (1.50) (window) psDrawIStr + unitGC 130 167 126 167 psDrawLine + unitGC 83 172 (-0.50) (window) psDrawIStr + unitGC 130 205 126 205 psDrawLine + unitGC 83 210 (-1.00) (window) psDrawIStr + unitGC 130 243 126 243 psDrawLine + unitGC 83 248 (-1.50) (window) psDrawIStr + globalGC1 127 51 3 3 0 360 128 52 psFillArc + globalGC1 128 52 128 52 psDrawLine + globalGC1 127 51 3 3 0 360 128 52 psFillArc + globalGC1 120 53 128 52 psDrawLine + globalGC1 119 52 3 3 0 360 120 53 psFillArc + globalGC1 111 54 120 53 psDrawLine + globalGC1 110 53 3 3 0 360 111 54 psFillArc + globalGC1 102 57 111 54 psDrawLine + globalGC1 101 56 3 3 0 360 102 57 psFillArc + globalGC1 93 61 102 57 psDrawLine + globalGC1 92 60 3 3 0 360 93 61 psFillArc + globalGC1 83 66 93 61 psDrawLine + globalGC1 82 65 3 3 0 360 83 66 psFillArc + globalGC1 74 74 83 66 psDrawLine + globalGC1 73 73 3 3 0 360 74 74 psFillArc + globalGC1 66 83 74 74 psDrawLine + globalGC1 65 82 3 3 0 360 66 83 psFillArc + globalGC1 60 92 66 83 psDrawLine + globalGC1 59 91 3 3 0 360 60 92 psFillArc + globalGC1 56 101 60 92 psDrawLine + globalGC1 55 100 3 3 0 360 56 101 psFillArc + globalGC1 53 111 56 101 psDrawLine + globalGC1 52 110 3 3 0 360 53 111 psFillArc + globalGC1 51 120 53 111 psDrawLine + globalGC1 50 119 3 3 0 360 51 120 psFillArc + globalGC1 51 129 51 120 psDrawLine + globalGC1 50 128 3 3 0 360 51 129 psFillArc + globalGC1 51 136 51 129 psDrawLine + globalGC1 50 135 3 3 0 360 51 136 psFillArc + globalGC1 52 145 51 136 psDrawLine + globalGC1 51 144 3 3 0 360 52 145 psFillArc + globalGC1 55 155 52 145 psDrawLine + globalGC1 54 154 3 3 0 360 55 155 psFillArc + globalGC1 59 164 55 155 psDrawLine + globalGC1 58 163 3 3 0 360 59 164 psFillArc + globalGC1 65 173 59 164 psDrawLine + globalGC1 64 172 3 3 0 360 65 173 psFillArc + globalGC1 72 182 65 173 psDrawLine + globalGC1 71 181 3 3 0 360 72 182 psFillArc + globalGC1 81 190 72 182 psDrawLine + globalGC1 80 189 3 3 0 360 81 190 psFillArc + globalGC1 91 196 81 190 psDrawLine + globalGC1 90 195 3 3 0 360 91 196 psFillArc + globalGC1 100 201 91 196 psDrawLine + globalGC1 99 200 3 3 0 360 100 201 psFillArc + globalGC1 109 204 100 201 psDrawLine + globalGC1 108 203 3 3 0 360 109 204 psFillArc + globalGC1 118 205 109 204 psDrawLine + globalGC1 117 204 3 3 0 360 118 205 psFillArc + globalGC1 128 206 118 205 psDrawLine + globalGC1 127 205 3 3 0 360 128 206 psFillArc + globalGC1 135 205 128 206 psDrawLine + globalGC1 134 204 3 3 0 360 135 205 psFillArc + globalGC1 144 204 135 205 psDrawLine + globalGC1 143 203 3 3 0 360 144 204 psFillArc + globalGC1 153 201 144 204 psDrawLine + globalGC1 152 200 3 3 0 360 153 201 psFillArc + globalGC1 162 197 153 201 psDrawLine + globalGC1 161 196 3 3 0 360 162 197 psFillArc + globalGC1 172 192 162 197 psDrawLine + globalGC1 171 191 3 3 0 360 172 192 psFillArc + globalGC1 181 184 172 192 psDrawLine + globalGC1 180 183 3 3 0 360 181 184 psFillArc + globalGC1 189 175 181 184 psDrawLine + globalGC1 188 174 3 3 0 360 189 175 psFillArc + globalGC1 195 166 189 175 psDrawLine + globalGC1 194 165 3 3 0 360 195 166 psFillArc + globalGC1 199 157 195 166 psDrawLine + globalGC1 198 156 3 3 0 360 199 157 psFillArc + globalGC1 202 147 199 157 psDrawLine + globalGC1 201 146 3 3 0 360 202 147 psFillArc + globalGC1 204 138 202 147 psDrawLine + globalGC1 203 137 3 3 0 360 204 138 psFillArc + globalGC1 204 129 204 138 psDrawLine + globalGC1 203 128 3 3 0 360 204 129 psFillArc + globalGC1 204 122 204 129 psDrawLine + globalGC1 203 121 3 3 0 360 204 122 psFillArc + globalGC1 203 113 204 122 psDrawLine + globalGC1 202 112 3 3 0 360 203 113 psFillArc + globalGC1 200 103 203 113 psDrawLine + globalGC1 199 102 3 3 0 360 200 103 psFillArc + globalGC1 196 94 200 103 psDrawLine + globalGC1 195 93 3 3 0 360 196 94 psFillArc + globalGC1 190 85 196 94 psDrawLine + globalGC1 189 84 3 3 0 360 190 85 psFillArc + globalGC1 183 76 190 85 psDrawLine + globalGC1 182 75 3 3 0 360 183 76 psFillArc + globalGC1 174 68 183 76 psDrawLine + globalGC1 173 67 3 3 0 360 174 68 psFillArc + globalGC1 164 62 174 68 psDrawLine + globalGC1 163 61 3 3 0 360 164 62 psFillArc + globalGC1 155 57 164 62 psDrawLine + globalGC1 154 56 3 3 0 360 155 57 psFillArc + globalGC1 146 54 155 57 psDrawLine + globalGC1 145 53 3 3 0 360 146 54 psFillArc + globalGC1 137 53 146 54 psDrawLine + globalGC1 136 52 3 3 0 360 137 53 psFillArc + globalGC1 128 52 137 53 psDrawLine + globalGC1 127 51 3 3 0 360 128 52 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptsc.ps b/books/ps/2doptsc.ps new file mode 100644 index 0000000..9e81b7b --- /dev/null +++ b/books/ps/2doptsc.ps @@ -0,0 +1,436 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 151 0 151 psDrawLine + globalGC1 154 303 154 0 psDrawLine + unitGC 198 153 198 149 psDrawLine + unitGC 186 166 (1.00) (window) psDrawIStr + unitGC 242 153 242 149 psDrawLine + unitGC 230 166 (2.00) (window) psDrawIStr + unitGC 286 153 286 149 psDrawLine + unitGC 274 166 (3.00) (window) psDrawIStr + unitGC 108 153 108 149 psDrawLine + unitGC 93 166 (-1.00) (window) psDrawIStr + unitGC 62 153 62 149 psDrawLine + unitGC 47 166 (-2.00) (window) psDrawIStr + unitGC 16 153 16 149 psDrawLine + unitGC 1 166 (-3.00) (window) psDrawIStr + unitGC 156 106 152 106 psDrawLine + unitGC 118 111 (1.00) (window) psDrawIStr + unitGC 156 61 152 61 psDrawLine + unitGC 118 66 (2.00) (window) psDrawIStr + unitGC 156 16 152 16 psDrawLine + unitGC 118 21 (3.00) (window) psDrawIStr + unitGC 156 195 152 195 psDrawLine + unitGC 109 200 (-1.00) (window) psDrawIStr + unitGC 156 239 152 239 psDrawLine + unitGC 109 244 (-2.00) (window) psDrawIStr + unitGC 156 283 152 283 psDrawLine + unitGC 109 288 (-3.00) (window) psDrawIStr + globalGC1 13 150 3 3 0 360 14 151 psFillArc + globalGC1 14 151 14 151 psDrawLine + globalGC1 13 150 3 3 0 360 14 151 psFillArc + globalGC1 20 157 14 151 psDrawLine + globalGC1 19 156 3 3 0 360 20 157 psFillArc + globalGC1 26 162 20 157 psDrawLine + globalGC1 25 161 3 3 0 360 26 162 psFillArc + globalGC1 32 168 26 162 psDrawLine + globalGC1 31 167 3 3 0 360 32 168 psFillArc + globalGC1 37 173 32 168 psDrawLine + globalGC1 36 172 3 3 0 360 37 173 psFillArc + globalGC1 43 178 37 173 psDrawLine + globalGC1 42 177 3 3 0 360 43 178 psFillArc + globalGC1 49 182 43 178 psDrawLine + globalGC1 48 181 3 3 0 360 49 182 psFillArc + globalGC1 55 186 49 182 psDrawLine + globalGC1 54 185 3 3 0 360 55 186 psFillArc + globalGC1 61 189 55 186 psDrawLine + globalGC1 60 188 3 3 0 360 61 189 psFillArc + globalGC1 66 192 61 189 psDrawLine + globalGC1 65 191 3 3 0 360 66 192 psFillArc + globalGC1 72 194 66 192 psDrawLine + globalGC1 71 193 3 3 0 360 72 194 psFillArc + globalGC1 75 195 72 194 psDrawLine + globalGC1 74 194 3 3 0 360 75 195 psFillArc + globalGC1 78 195 75 195 psDrawLine + globalGC1 77 194 3 3 0 360 78 195 psFillArc + globalGC1 81 195 78 195 psDrawLine + globalGC1 80 194 3 3 0 360 81 195 psFillArc + globalGC1 84 195 81 195 psDrawLine + globalGC1 83 194 3 3 0 360 84 195 psFillArc + globalGC1 87 195 84 195 psDrawLine + globalGC1 86 194 3 3 0 360 87 195 psFillArc + globalGC1 90 195 87 195 psDrawLine + globalGC1 89 194 3 3 0 360 90 195 psFillArc + globalGC1 93 195 90 195 psDrawLine + globalGC1 92 194 3 3 0 360 93 195 psFillArc + globalGC1 96 194 93 195 psDrawLine + globalGC1 95 193 3 3 0 360 96 194 psFillArc + globalGC1 101 192 96 194 psDrawLine + globalGC1 100 191 3 3 0 360 101 192 psFillArc + globalGC1 107 189 101 192 psDrawLine + globalGC1 106 188 3 3 0 360 107 189 psFillArc + globalGC1 113 186 107 189 psDrawLine + globalGC1 112 185 3 3 0 360 113 186 psFillArc + globalGC1 119 182 113 186 psDrawLine + globalGC1 118 181 3 3 0 360 119 182 psFillArc + globalGC1 125 178 119 182 psDrawLine + globalGC1 124 177 3 3 0 360 125 178 psFillArc + globalGC1 130 173 125 178 psDrawLine + globalGC1 129 172 3 3 0 360 130 173 psFillArc + globalGC1 136 168 130 173 psDrawLine + globalGC1 135 167 3 3 0 360 136 168 psFillArc + globalGC1 142 162 136 168 psDrawLine + globalGC1 141 161 3 3 0 360 142 162 psFillArc + globalGC1 148 157 142 162 psDrawLine + globalGC1 147 156 3 3 0 360 148 157 psFillArc + globalGC1 154 151 148 157 psDrawLine + globalGC1 153 150 3 3 0 360 154 151 psFillArc + globalGC1 159 145 154 151 psDrawLine + globalGC1 158 144 3 3 0 360 159 145 psFillArc + globalGC1 165 139 159 145 psDrawLine + globalGC1 164 138 3 3 0 360 165 139 psFillArc + globalGC1 171 134 165 139 psDrawLine + globalGC1 170 133 3 3 0 360 171 134 psFillArc + globalGC1 177 129 171 134 psDrawLine + globalGC1 176 128 3 3 0 360 177 129 psFillArc + globalGC1 183 124 177 129 psDrawLine + globalGC1 182 123 3 3 0 360 183 124 psFillArc + globalGC1 189 120 183 124 psDrawLine + globalGC1 188 119 3 3 0 360 189 120 psFillArc + globalGC1 194 116 189 120 psDrawLine + globalGC1 193 115 3 3 0 360 194 116 psFillArc + globalGC1 200 113 194 116 psDrawLine + globalGC1 199 112 3 3 0 360 200 113 psFillArc + globalGC1 206 110 200 113 psDrawLine + globalGC1 205 109 3 3 0 360 206 110 psFillArc + globalGC1 212 108 206 110 psDrawLine + globalGC1 211 107 3 3 0 360 212 108 psFillArc + globalGC1 215 107 212 108 psDrawLine + globalGC1 214 106 3 3 0 360 215 107 psFillArc + globalGC1 218 107 215 107 psDrawLine + globalGC1 217 106 3 3 0 360 218 107 psFillArc + globalGC1 221 107 218 107 psDrawLine + globalGC1 220 106 3 3 0 360 221 107 psFillArc + globalGC1 223 107 221 107 psDrawLine + globalGC1 222 106 3 3 0 360 223 107 psFillArc + globalGC1 226 107 223 107 psDrawLine + globalGC1 225 106 3 3 0 360 226 107 psFillArc + globalGC1 229 107 226 107 psDrawLine + globalGC1 228 106 3 3 0 360 229 107 psFillArc + globalGC1 232 107 229 107 psDrawLine + globalGC1 231 106 3 3 0 360 232 107 psFillArc + globalGC1 235 108 232 107 psDrawLine + globalGC1 234 107 3 3 0 360 235 108 psFillArc + globalGC1 241 110 235 108 psDrawLine + globalGC1 240 109 3 3 0 360 241 110 psFillArc + globalGC1 247 113 241 110 psDrawLine + globalGC1 246 112 3 3 0 360 247 113 psFillArc + globalGC1 252 116 247 113 psDrawLine + globalGC1 251 115 3 3 0 360 252 116 psFillArc + globalGC1 258 120 252 116 psDrawLine + globalGC1 257 119 3 3 0 360 258 120 psFillArc + globalGC1 264 124 258 120 psDrawLine + globalGC1 263 123 3 3 0 360 264 124 psFillArc + globalGC1 270 129 264 124 psDrawLine + globalGC1 269 128 3 3 0 360 270 129 psFillArc + globalGC1 276 134 270 129 psDrawLine + globalGC1 275 133 3 3 0 360 276 134 psFillArc + globalGC1 282 139 276 134 psDrawLine + globalGC1 281 138 3 3 0 360 282 139 psFillArc + globalGC1 287 145 282 139 psDrawLine + globalGC1 286 144 3 3 0 360 287 145 psFillArc + globalGC1 293 151 287 145 psDrawLine + globalGC1 292 150 3 3 0 360 293 151 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2doptut.ps b/books/ps/2doptut.ps new file mode 100644 index 0000000..303089b --- /dev/null +++ b/books/ps/2doptut.ps @@ -0,0 +1,644 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 128 259 128 0 psDrawLine + unitGC 128 131 128 127 psDrawLine + unitGC 116 144 (0.00) (window) psDrawIStr + unitGC 178 131 178 127 psDrawLine + unitGC 166 144 (4.00) (window) psDrawIStr + unitGC 228 131 228 127 psDrawLine + unitGC 216 144 (8.00) (window) psDrawIStr + unitGC 76 131 76 127 psDrawLine + unitGC 61 144 (-4.00) (window) psDrawIStr + unitGC 24 131 24 127 psDrawLine + unitGC 9 144 (-8.00) (window) psDrawIStr + unitGC 130 99 126 99 psDrawLine + unitGC 92 104 (2.00) (window) psDrawIStr + unitGC 130 69 126 69 psDrawLine + unitGC 92 74 (4.00) (window) psDrawIStr + unitGC 130 39 126 39 psDrawLine + unitGC 92 44 (6.00) (window) psDrawIStr + unitGC 130 9 126 9 psDrawLine + unitGC 92 14 (8.00) (window) psDrawIStr + unitGC 130 157 126 157 psDrawLine + unitGC 83 162 (-2.00) (window) psDrawIStr + unitGC 130 185 126 185 psDrawLine + unitGC 83 190 (-4.00) (window) psDrawIStr + unitGC 130 213 126 213 psDrawLine + unitGC 83 218 (-6.00) (window) psDrawIStr + unitGC 130 241 126 241 psDrawLine + unitGC 83 246 (-8.00) (window) psDrawIStr + globalGC1 127 128 3 3 0 360 128 129 psFillArc + globalGC1 128 129 128 129 psDrawLine + globalGC1 127 128 3 3 0 360 128 129 psFillArc + globalGC1 105 99 128 129 psDrawLine + globalGC1 104 98 3 3 0 360 105 99 psFillArc + globalGC1 83 71 105 99 psDrawLine + globalGC1 82 70 3 3 0 360 83 71 psFillArc + globalGC1 63 48 83 71 psDrawLine + globalGC1 62 47 3 3 0 360 63 48 psFillArc + globalGC1 46 29 63 48 psDrawLine + globalGC1 45 28 3 3 0 360 46 29 psFillArc + globalGC1 38 23 46 29 psDrawLine + globalGC1 37 22 3 3 0 360 38 23 psFillArc + globalGC1 32 18 38 23 psDrawLine + globalGC1 31 17 3 3 0 360 32 18 psFillArc + globalGC1 29 16 32 18 psDrawLine + globalGC1 28 15 3 3 0 360 29 16 psFillArc + globalGC1 26 15 29 16 psDrawLine + globalGC1 25 14 3 3 0 360 26 15 psFillArc + globalGC1 23 14 26 15 psDrawLine + globalGC1 22 13 3 3 0 360 23 14 psFillArc + globalGC1 21 14 23 14 psDrawLine + globalGC1 20 13 3 3 0 360 21 14 psFillArc + globalGC1 20 14 21 14 psDrawLine + globalGC1 19 13 3 3 0 360 20 14 psFillArc + globalGC1 19 14 20 14 psDrawLine + globalGC1 18 13 3 3 0 360 19 14 psFillArc + globalGC1 18 14 19 14 psDrawLine + globalGC1 17 13 3 3 0 360 18 14 psFillArc + globalGC1 17 15 18 14 psDrawLine + globalGC1 16 14 3 3 0 360 17 15 psFillArc + globalGC1 16 15 17 15 psDrawLine + globalGC1 15 14 3 3 0 360 16 15 psFillArc + globalGC1 16 16 16 15 psDrawLine + globalGC1 15 15 3 3 0 360 16 16 psFillArc + globalGC1 15 17 16 16 psDrawLine + globalGC1 14 16 3 3 0 360 15 17 psFillArc + globalGC1 15 18 15 17 psDrawLine + globalGC1 14 17 3 3 0 360 15 18 psFillArc + globalGC1 13 23 15 18 psDrawLine + globalGC1 12 22 3 3 0 360 13 23 psFillArc + globalGC1 12 29 13 23 psDrawLine + globalGC1 11 28 3 3 0 360 12 29 psFillArc + globalGC1 15 48 12 29 psDrawLine + globalGC1 14 47 3 3 0 360 15 48 psFillArc + globalGC1 21 71 15 48 psDrawLine + globalGC1 20 70 3 3 0 360 21 71 psFillArc + globalGC1 32 99 21 71 psDrawLine + globalGC1 31 98 3 3 0 360 32 99 psFillArc + globalGC1 46 129 32 99 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 63 159 46 129 psDrawLine + globalGC1 62 158 3 3 0 360 63 159 psFillArc + globalGC1 83 187 63 159 psDrawLine + globalGC1 82 186 3 3 0 360 83 187 psFillArc + globalGC1 105 210 83 187 psDrawLine + globalGC1 104 209 3 3 0 360 105 210 psFillArc + globalGC1 128 229 105 210 psDrawLine + globalGC1 127 228 3 3 0 360 128 229 psFillArc + globalGC1 139 235 128 229 psDrawLine + globalGC1 138 234 3 3 0 360 139 235 psFillArc + globalGC1 150 240 139 235 psDrawLine + globalGC1 149 239 3 3 0 360 150 240 psFillArc + globalGC1 161 243 150 240 psDrawLine + globalGC1 160 242 3 3 0 360 161 243 psFillArc + globalGC1 172 244 161 243 psDrawLine + globalGC1 171 243 3 3 0 360 172 244 psFillArc + globalGC1 182 243 172 244 psDrawLine + globalGC1 181 242 3 3 0 360 182 243 psFillArc + globalGC1 192 240 182 243 psDrawLine + globalGC1 191 239 3 3 0 360 192 240 psFillArc + globalGC1 201 235 192 240 psDrawLine + globalGC1 200 234 3 3 0 360 201 235 psFillArc + globalGC1 209 229 201 235 psDrawLine + globalGC1 208 228 3 3 0 360 209 229 psFillArc + globalGC1 223 210 209 229 psDrawLine + globalGC1 222 209 3 3 0 360 223 210 psFillArc + globalGC1 234 187 223 210 psDrawLine + globalGC1 233 186 3 3 0 360 234 187 psFillArc + globalGC1 240 159 234 187 psDrawLine + globalGC1 239 158 3 3 0 360 240 159 psFillArc + globalGC1 243 129 240 159 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + globalGC1 240 99 243 129 psDrawLine + globalGC1 239 98 3 3 0 360 240 99 psFillArc + globalGC1 234 71 240 99 psDrawLine + globalGC1 233 70 3 3 0 360 234 71 psFillArc + globalGC1 223 48 234 71 psDrawLine + globalGC1 222 47 3 3 0 360 223 48 psFillArc + globalGC1 217 38 223 48 psDrawLine + globalGC1 216 37 3 3 0 360 217 38 psFillArc + globalGC1 209 29 217 38 psDrawLine + globalGC1 208 28 3 3 0 360 209 29 psFillArc + globalGC1 201 23 209 29 psDrawLine + globalGC1 200 22 3 3 0 360 201 23 psFillArc + globalGC1 192 18 201 23 psDrawLine + globalGC1 191 17 3 3 0 360 192 18 psFillArc + globalGC1 182 15 192 18 psDrawLine + globalGC1 181 14 3 3 0 360 182 15 psFillArc + globalGC1 172 14 182 15 psDrawLine + globalGC1 171 13 3 3 0 360 172 14 psFillArc + globalGC1 161 15 172 14 psDrawLine + globalGC1 160 14 3 3 0 360 161 15 psFillArc + globalGC1 150 18 161 15 psDrawLine + globalGC1 149 17 3 3 0 360 150 18 psFillArc + globalGC1 128 29 150 18 psDrawLine + globalGC1 127 28 3 3 0 360 128 29 psFillArc + globalGC1 105 48 128 29 psDrawLine + globalGC1 104 47 3 3 0 360 105 48 psFillArc + globalGC1 83 71 105 48 psDrawLine + globalGC1 82 70 3 3 0 360 83 71 psFillArc + globalGC1 63 99 83 71 psDrawLine + globalGC1 62 98 3 3 0 360 63 99 psFillArc + globalGC1 46 129 63 99 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 32 159 46 129 psDrawLine + globalGC1 31 158 3 3 0 360 32 159 psFillArc + globalGC1 21 187 32 159 psDrawLine + globalGC1 20 186 3 3 0 360 21 187 psFillArc + globalGC1 15 210 21 187 psDrawLine + globalGC1 14 209 3 3 0 360 15 210 psFillArc + globalGC1 13 220 15 210 psDrawLine + globalGC1 12 219 3 3 0 360 13 220 psFillArc + globalGC1 12 229 13 220 psDrawLine + globalGC1 11 228 3 3 0 360 12 229 psFillArc + globalGC1 13 235 12 229 psDrawLine + globalGC1 12 234 3 3 0 360 13 235 psFillArc + globalGC1 14 238 13 235 psDrawLine + globalGC1 13 237 3 3 0 360 14 238 psFillArc + globalGC1 15 240 14 238 psDrawLine + globalGC1 14 239 3 3 0 360 15 240 psFillArc + globalGC1 15 241 15 240 psDrawLine + globalGC1 14 240 3 3 0 360 15 241 psFillArc + globalGC1 16 242 15 241 psDrawLine + globalGC1 15 241 3 3 0 360 16 242 psFillArc + globalGC1 16 243 16 242 psDrawLine + globalGC1 15 242 3 3 0 360 16 243 psFillArc + globalGC1 17 243 16 243 psDrawLine + globalGC1 16 242 3 3 0 360 17 243 psFillArc + globalGC1 18 244 17 243 psDrawLine + globalGC1 17 243 3 3 0 360 18 244 psFillArc + globalGC1 19 244 18 244 psDrawLine + globalGC1 18 243 3 3 0 360 19 244 psFillArc + globalGC1 20 244 19 244 psDrawLine + globalGC1 19 243 3 3 0 360 20 244 psFillArc + globalGC1 21 244 20 244 psDrawLine + globalGC1 20 243 3 3 0 360 21 244 psFillArc + globalGC1 23 244 21 244 psDrawLine + globalGC1 22 243 3 3 0 360 23 244 psFillArc + globalGC1 26 243 23 244 psDrawLine + globalGC1 25 242 3 3 0 360 26 243 psFillArc + globalGC1 32 240 26 243 psDrawLine + globalGC1 31 239 3 3 0 360 32 240 psFillArc + globalGC1 46 229 32 240 psDrawLine + globalGC1 45 228 3 3 0 360 46 229 psFillArc + globalGC1 63 210 46 229 psDrawLine + globalGC1 62 209 3 3 0 360 63 210 psFillArc + globalGC1 83 187 63 210 psDrawLine + globalGC1 82 186 3 3 0 360 83 187 psFillArc + globalGC1 105 159 83 187 psDrawLine + globalGC1 104 158 3 3 0 360 105 159 psFillArc + globalGC1 128 129 105 159 psDrawLine + globalGC1 127 128 3 3 0 360 128 129 psFillArc + globalGC1 150 99 128 129 psDrawLine + globalGC1 149 98 3 3 0 360 150 99 psFillArc + globalGC1 172 71 150 99 psDrawLine + globalGC1 171 70 3 3 0 360 172 71 psFillArc + globalGC1 192 48 172 71 psDrawLine + globalGC1 191 47 3 3 0 360 192 48 psFillArc + globalGC1 209 29 192 48 psDrawLine + globalGC1 208 28 3 3 0 360 209 29 psFillArc + globalGC1 217 23 209 29 psDrawLine + globalGC1 216 22 3 3 0 360 217 23 psFillArc + globalGC1 223 18 217 23 psDrawLine + globalGC1 222 17 3 3 0 360 223 18 psFillArc + globalGC1 226 16 223 18 psDrawLine + globalGC1 225 15 3 3 0 360 226 16 psFillArc + globalGC1 229 15 226 16 psDrawLine + globalGC1 228 14 3 3 0 360 229 15 psFillArc + globalGC1 232 14 229 15 psDrawLine + globalGC1 231 13 3 3 0 360 232 14 psFillArc + globalGC1 234 14 232 14 psDrawLine + globalGC1 233 13 3 3 0 360 234 14 psFillArc + globalGC1 235 14 234 14 psDrawLine + globalGC1 234 13 3 3 0 360 235 14 psFillArc + globalGC1 236 14 235 14 psDrawLine + globalGC1 235 13 3 3 0 360 236 14 psFillArc + globalGC1 237 14 236 14 psDrawLine + globalGC1 236 13 3 3 0 360 237 14 psFillArc + globalGC1 238 15 237 14 psDrawLine + globalGC1 237 14 3 3 0 360 238 15 psFillArc + globalGC1 239 15 238 15 psDrawLine + globalGC1 238 14 3 3 0 360 239 15 psFillArc + globalGC1 239 16 239 15 psDrawLine + globalGC1 238 15 3 3 0 360 239 16 psFillArc + globalGC1 240 17 239 16 psDrawLine + globalGC1 239 16 3 3 0 360 240 17 psFillArc + globalGC1 240 18 240 17 psDrawLine + globalGC1 239 17 3 3 0 360 240 18 psFillArc + globalGC1 242 23 240 18 psDrawLine + globalGC1 241 22 3 3 0 360 242 23 psFillArc + globalGC1 243 29 242 23 psDrawLine + globalGC1 242 28 3 3 0 360 243 29 psFillArc + globalGC1 240 48 243 29 psDrawLine + globalGC1 239 47 3 3 0 360 240 48 psFillArc + globalGC1 234 71 240 48 psDrawLine + globalGC1 233 70 3 3 0 360 234 71 psFillArc + globalGC1 223 99 234 71 psDrawLine + globalGC1 222 98 3 3 0 360 223 99 psFillArc + globalGC1 209 129 223 99 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 192 159 209 129 psDrawLine + globalGC1 191 158 3 3 0 360 192 159 psFillArc + globalGC1 172 187 192 159 psDrawLine + globalGC1 171 186 3 3 0 360 172 187 psFillArc + globalGC1 150 210 172 187 psDrawLine + globalGC1 149 209 3 3 0 360 150 210 psFillArc + globalGC1 128 229 150 210 psDrawLine + globalGC1 127 228 3 3 0 360 128 229 psFillArc + globalGC1 116 235 128 229 psDrawLine + globalGC1 115 234 3 3 0 360 116 235 psFillArc + globalGC1 105 240 116 235 psDrawLine + globalGC1 104 239 3 3 0 360 105 240 psFillArc + globalGC1 94 243 105 240 psDrawLine + globalGC1 93 242 3 3 0 360 94 243 psFillArc + globalGC1 83 244 94 243 psDrawLine + globalGC1 82 243 3 3 0 360 83 244 psFillArc + globalGC1 73 243 83 244 psDrawLine + globalGC1 72 242 3 3 0 360 73 243 psFillArc + globalGC1 63 240 73 243 psDrawLine + globalGC1 62 239 3 3 0 360 63 240 psFillArc + globalGC1 54 235 63 240 psDrawLine + globalGC1 53 234 3 3 0 360 54 235 psFillArc + globalGC1 46 229 54 235 psDrawLine + globalGC1 45 228 3 3 0 360 46 229 psFillArc + globalGC1 32 210 46 229 psDrawLine + globalGC1 31 209 3 3 0 360 32 210 psFillArc + globalGC1 21 187 32 210 psDrawLine + globalGC1 20 186 3 3 0 360 21 187 psFillArc + globalGC1 15 159 21 187 psDrawLine + globalGC1 14 158 3 3 0 360 15 159 psFillArc + globalGC1 12 129 15 159 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 15 99 12 129 psDrawLine + globalGC1 14 98 3 3 0 360 15 99 psFillArc + globalGC1 21 71 15 99 psDrawLine + globalGC1 20 70 3 3 0 360 21 71 psFillArc + globalGC1 32 48 21 71 psDrawLine + globalGC1 31 47 3 3 0 360 32 48 psFillArc + globalGC1 38 38 32 48 psDrawLine + globalGC1 37 37 3 3 0 360 38 38 psFillArc + globalGC1 46 29 38 38 psDrawLine + globalGC1 45 28 3 3 0 360 46 29 psFillArc + globalGC1 54 23 46 29 psDrawLine + globalGC1 53 22 3 3 0 360 54 23 psFillArc + globalGC1 63 18 54 23 psDrawLine + globalGC1 62 17 3 3 0 360 63 18 psFillArc + globalGC1 73 15 63 18 psDrawLine + globalGC1 72 14 3 3 0 360 73 15 psFillArc + globalGC1 83 14 73 15 psDrawLine + globalGC1 82 13 3 3 0 360 83 14 psFillArc + globalGC1 94 15 83 14 psDrawLine + globalGC1 93 14 3 3 0 360 94 15 psFillArc + globalGC1 105 18 94 15 psDrawLine + globalGC1 104 17 3 3 0 360 105 18 psFillArc + globalGC1 128 29 105 18 psDrawLine + globalGC1 127 28 3 3 0 360 128 29 psFillArc + globalGC1 150 48 128 29 psDrawLine + globalGC1 149 47 3 3 0 360 150 48 psFillArc + globalGC1 172 71 150 48 psDrawLine + globalGC1 171 70 3 3 0 360 172 71 psFillArc + globalGC1 192 99 172 71 psDrawLine + globalGC1 191 98 3 3 0 360 192 99 psFillArc + globalGC1 209 129 192 99 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 223 159 209 129 psDrawLine + globalGC1 222 158 3 3 0 360 223 159 psFillArc + globalGC1 234 187 223 159 psDrawLine + globalGC1 233 186 3 3 0 360 234 187 psFillArc + globalGC1 240 210 234 187 psDrawLine + globalGC1 239 209 3 3 0 360 240 210 psFillArc + globalGC1 242 220 240 210 psDrawLine + globalGC1 241 219 3 3 0 360 242 220 psFillArc + globalGC1 243 229 242 220 psDrawLine + globalGC1 242 228 3 3 0 360 243 229 psFillArc + globalGC1 242 235 243 229 psDrawLine + globalGC1 241 234 3 3 0 360 242 235 psFillArc + globalGC1 241 238 242 235 psDrawLine + globalGC1 240 237 3 3 0 360 241 238 psFillArc + globalGC1 240 240 241 238 psDrawLine + globalGC1 239 239 3 3 0 360 240 240 psFillArc + globalGC1 240 241 240 240 psDrawLine + globalGC1 239 240 3 3 0 360 240 241 psFillArc + globalGC1 239 242 240 241 psDrawLine + globalGC1 238 241 3 3 0 360 239 242 psFillArc + globalGC1 239 243 239 242 psDrawLine + globalGC1 238 242 3 3 0 360 239 243 psFillArc + globalGC1 238 243 239 243 psDrawLine + globalGC1 237 242 3 3 0 360 238 243 psFillArc + globalGC1 237 244 238 243 psDrawLine + globalGC1 236 243 3 3 0 360 237 244 psFillArc + globalGC1 236 244 237 244 psDrawLine + globalGC1 235 243 3 3 0 360 236 244 psFillArc + globalGC1 235 244 236 244 psDrawLine + globalGC1 234 243 3 3 0 360 235 244 psFillArc + globalGC1 234 244 235 244 psDrawLine + globalGC1 233 243 3 3 0 360 234 244 psFillArc + globalGC1 232 244 234 244 psDrawLine + globalGC1 231 243 3 3 0 360 232 244 psFillArc + globalGC1 229 243 232 244 psDrawLine + globalGC1 228 242 3 3 0 360 229 243 psFillArc + globalGC1 223 240 229 243 psDrawLine + globalGC1 222 239 3 3 0 360 223 240 psFillArc + globalGC1 209 229 223 240 psDrawLine + globalGC1 208 228 3 3 0 360 209 229 psFillArc + globalGC1 192 210 209 229 psDrawLine + globalGC1 191 209 3 3 0 360 192 210 psFillArc + globalGC1 172 187 192 210 psDrawLine + globalGC1 171 186 3 3 0 360 172 187 psFillArc + globalGC1 150 159 172 187 psDrawLine + globalGC1 149 158 3 3 0 360 150 159 psFillArc + globalGC1 128 129 150 159 psDrawLine + globalGC1 127 128 3 3 0 360 128 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dpaca.ps b/books/ps/2dpaca.ps new file mode 100644 index 0000000..0d906fb --- /dev/null +++ b/books/ps/2dpaca.ps @@ -0,0 +1,525 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 151 0 151 psDrawLine + globalGC1 39 303 39 0 psDrawLine + unitGC 88 153 88 149 psDrawLine + unitGC 76 166 (2.19) (window) psDrawIStr + unitGC 137 153 137 149 psDrawLine + unitGC 125 166 (4.37) (window) psDrawIStr + unitGC 186 153 186 149 psDrawLine + unitGC 174 166 (6.56) (window) psDrawIStr + unitGC 235 153 235 149 psDrawLine + unitGC 223 166 (8.74) (window) psDrawIStr + unitGC 284 153 284 149 psDrawLine + unitGC 269 166 (10.93) (window) psDrawIStr + unitGC 41 104 37 104 psDrawLine + unitGC 3 109 (2.39) (window) psDrawIStr + unitGC 41 57 37 57 psDrawLine + unitGC 3 62 (4.78) (window) psDrawIStr + unitGC 41 10 37 10 psDrawLine + unitGC 3 15 (7.18) (window) psDrawIStr + unitGC 41 197 37 197 psDrawLine + unitGC -6 202 (-2.39) (window) psDrawIStr + unitGC 41 243 37 243 psDrawLine + unitGC -6 248 (-4.78) (window) psDrawIStr + unitGC 41 289 37 289 psDrawLine + unitGC -6 294 (-7.18) (window) psDrawIStr + globalGC1 147 35 3 3 0 360 148 36 psFillArc + globalGC1 148 36 148 36 psDrawLine + globalGC1 147 35 3 3 0 360 148 36 psFillArc + globalGC1 138 36 148 36 psDrawLine + globalGC1 137 35 3 3 0 360 138 36 psFillArc + globalGC1 127 37 138 36 psDrawLine + globalGC1 126 36 3 3 0 360 127 37 psFillArc + globalGC1 116 39 127 37 psDrawLine + globalGC1 115 38 3 3 0 360 116 39 psFillArc + globalGC1 105 42 116 39 psDrawLine + globalGC1 104 41 3 3 0 360 105 42 psFillArc + globalGC1 94 46 105 42 psDrawLine + globalGC1 93 45 3 3 0 360 94 46 psFillArc + globalGC1 84 52 94 46 psDrawLine + globalGC1 83 51 3 3 0 360 84 52 psFillArc + globalGC1 73 58 84 52 psDrawLine + globalGC1 72 57 3 3 0 360 73 58 psFillArc + globalGC1 61 68 73 58 psDrawLine + globalGC1 60 67 3 3 0 360 61 68 psFillArc + globalGC1 53 77 61 68 psDrawLine + globalGC1 52 76 3 3 0 360 53 77 psFillArc + globalGC1 46 86 53 77 psDrawLine + globalGC1 45 85 3 3 0 360 46 86 psFillArc + globalGC1 41 95 46 86 psDrawLine + globalGC1 40 94 3 3 0 360 41 95 psFillArc + globalGC1 37 105 41 95 psDrawLine + globalGC1 36 104 3 3 0 360 37 105 psFillArc + globalGC1 35 114 37 105 psDrawLine + globalGC1 34 113 3 3 0 360 35 114 psFillArc + globalGC1 33 123 35 114 psDrawLine + globalGC1 32 122 3 3 0 360 33 123 psFillArc + globalGC1 33 132 33 123 psDrawLine + globalGC1 32 131 3 3 0 360 33 132 psFillArc + globalGC1 33 132 33 132 psDrawLine + globalGC1 32 131 3 3 0 360 33 132 psFillArc + globalGC1 33 142 33 132 psDrawLine + globalGC1 32 141 3 3 0 360 33 142 psFillArc + globalGC1 34 151 33 142 psDrawLine + globalGC1 33 150 3 3 0 360 34 151 psFillArc + globalGC1 34 151 34 151 psDrawLine + globalGC1 33 150 3 3 0 360 34 151 psFillArc + globalGC1 33 160 34 151 psDrawLine + globalGC1 32 159 3 3 0 360 33 160 psFillArc + globalGC1 33 170 33 160 psDrawLine + globalGC1 32 169 3 3 0 360 33 170 psFillArc + globalGC1 33 170 33 170 psDrawLine + globalGC1 32 169 3 3 0 360 33 170 psFillArc + globalGC1 33 179 33 170 psDrawLine + globalGC1 32 178 3 3 0 360 33 179 psFillArc + globalGC1 35 188 33 179 psDrawLine + globalGC1 34 187 3 3 0 360 35 188 psFillArc + globalGC1 37 198 35 188 psDrawLine + globalGC1 36 197 3 3 0 360 37 198 psFillArc + globalGC1 41 207 37 198 psDrawLine + globalGC1 40 206 3 3 0 360 41 207 psFillArc + globalGC1 46 216 41 207 psDrawLine + globalGC1 45 215 3 3 0 360 46 216 psFillArc + globalGC1 53 225 46 216 psDrawLine + globalGC1 52 224 3 3 0 360 53 225 psFillArc + globalGC1 62 235 53 225 psDrawLine + globalGC1 61 234 3 3 0 360 62 235 psFillArc + globalGC1 72 243 62 235 psDrawLine + globalGC1 71 242 3 3 0 360 72 243 psFillArc + globalGC1 83 250 72 243 psDrawLine + globalGC1 82 249 3 3 0 360 83 250 psFillArc + globalGC1 94 255 83 250 psDrawLine + globalGC1 93 254 3 3 0 360 94 255 psFillArc + globalGC1 105 260 94 255 psDrawLine + globalGC1 104 259 3 3 0 360 105 260 psFillArc + globalGC1 116 263 105 260 psDrawLine + globalGC1 115 262 3 3 0 360 116 263 psFillArc + globalGC1 126 265 116 263 psDrawLine + globalGC1 125 264 3 3 0 360 126 265 psFillArc + globalGC1 137 266 126 265 psDrawLine + globalGC1 136 265 3 3 0 360 137 266 psFillArc + globalGC1 148 266 137 266 psDrawLine + globalGC1 147 265 3 3 0 360 148 266 psFillArc + globalGC1 152 266 148 266 psDrawLine + globalGC1 151 265 3 3 0 360 152 266 psFillArc + globalGC1 163 266 152 266 psDrawLine + globalGC1 162 265 3 3 0 360 163 266 psFillArc + globalGC1 173 264 163 266 psDrawLine + globalGC1 172 263 3 3 0 360 173 264 psFillArc + globalGC1 184 262 173 264 psDrawLine + globalGC1 183 261 3 3 0 360 184 262 psFillArc + globalGC1 195 259 184 262 psDrawLine + globalGC1 194 258 3 3 0 360 195 259 psFillArc + globalGC1 206 254 195 259 psDrawLine + globalGC1 205 253 3 3 0 360 206 254 psFillArc + globalGC1 217 249 206 254 psDrawLine + globalGC1 216 248 3 3 0 360 217 249 psFillArc + globalGC1 227 242 217 249 psDrawLine + globalGC1 226 241 3 3 0 360 227 242 psFillArc + globalGC1 238 234 227 242 psDrawLine + globalGC1 237 233 3 3 0 360 238 234 psFillArc + globalGC1 248 225 238 234 psDrawLine + globalGC1 247 224 3 3 0 360 248 225 psFillArc + globalGC1 256 216 248 225 psDrawLine + globalGC1 255 215 3 3 0 360 256 216 psFillArc + globalGC1 263 207 256 216 psDrawLine + globalGC1 262 206 3 3 0 360 263 207 psFillArc + globalGC1 268 197 263 207 psDrawLine + globalGC1 267 196 3 3 0 360 268 197 psFillArc + globalGC1 272 188 268 197 psDrawLine + globalGC1 271 187 3 3 0 360 272 188 psFillArc + globalGC1 275 179 272 188 psDrawLine + globalGC1 274 178 3 3 0 360 275 179 psFillArc + globalGC1 277 170 275 179 psDrawLine + globalGC1 276 169 3 3 0 360 277 170 psFillArc + globalGC1 279 160 277 170 psDrawLine + globalGC1 278 159 3 3 0 360 279 160 psFillArc + globalGC1 279 151 279 160 psDrawLine + globalGC1 278 150 3 3 0 360 279 151 psFillArc + globalGC1 279 148 279 151 psDrawLine + globalGC1 278 147 3 3 0 360 279 148 psFillArc + globalGC1 278 139 279 148 psDrawLine + globalGC1 277 138 3 3 0 360 278 139 psFillArc + globalGC1 277 129 278 139 psDrawLine + globalGC1 276 128 3 3 0 360 277 129 psFillArc + globalGC1 274 120 277 129 psDrawLine + globalGC1 273 119 3 3 0 360 274 120 psFillArc + globalGC1 271 111 274 120 psDrawLine + globalGC1 270 110 3 3 0 360 271 111 psFillArc + globalGC1 266 102 271 111 psDrawLine + globalGC1 265 101 3 3 0 360 266 102 psFillArc + globalGC1 261 92 266 102 psDrawLine + globalGC1 260 91 3 3 0 360 261 92 psFillArc + globalGC1 254 83 261 92 psDrawLine + globalGC1 253 82 3 3 0 360 254 83 psFillArc + globalGC1 245 74 254 83 psDrawLine + globalGC1 244 73 3 3 0 360 245 74 psFillArc + globalGC1 234 65 245 74 psDrawLine + globalGC1 233 64 3 3 0 360 234 65 psFillArc + globalGC1 224 57 234 65 psDrawLine + globalGC1 223 56 3 3 0 360 224 57 psFillArc + globalGC1 213 51 224 57 psDrawLine + globalGC1 212 50 3 3 0 360 213 51 psFillArc + globalGC1 202 46 213 51 psDrawLine + globalGC1 201 45 3 3 0 360 202 46 psFillArc + globalGC1 191 42 202 46 psDrawLine + globalGC1 190 41 3 3 0 360 191 42 psFillArc + globalGC1 180 39 191 42 psDrawLine + globalGC1 179 38 3 3 0 360 180 39 psFillArc + globalGC1 170 37 180 39 psDrawLine + globalGC1 169 36 3 3 0 360 170 37 psFillArc + globalGC1 159 36 170 37 psDrawLine + globalGC1 158 35 3 3 0 360 159 36 psFillArc + globalGC1 148 36 159 36 psDrawLine + globalGC1 147 35 3 3 0 360 148 36 psFillArc + globalGC1 103 111 3 3 0 360 104 112 psFillArc + globalGC1 104 112 104 112 psDrawLine + globalGC1 103 111 3 3 0 360 104 112 psFillArc + globalGC1 100 112 104 112 psDrawLine + globalGC1 99 111 3 3 0 360 100 112 psFillArc + globalGC1 90 114 100 112 psDrawLine + globalGC1 89 113 3 3 0 360 90 114 psFillArc + globalGC1 79 117 90 114 psDrawLine + globalGC1 78 116 3 3 0 360 79 117 psFillArc + globalGC1 68 123 79 117 psDrawLine + globalGC1 67 122 3 3 0 360 68 123 psFillArc + globalGC1 58 132 68 123 psDrawLine + globalGC1 57 131 3 3 0 360 58 132 psFillArc + globalGC1 52 142 58 132 psDrawLine + globalGC1 51 141 3 3 0 360 52 142 psFillArc + globalGC1 50 151 52 142 psDrawLine + globalGC1 49 150 3 3 0 360 50 151 psFillArc + globalGC1 50 155 50 151 psDrawLine + globalGC1 49 154 3 3 0 360 50 155 psFillArc + globalGC1 53 164 50 155 psDrawLine + globalGC1 52 163 3 3 0 360 53 164 psFillArc + globalGC1 61 173 53 164 psDrawLine + globalGC1 60 172 3 3 0 360 61 173 psFillArc + globalGC1 72 181 61 173 psDrawLine + globalGC1 71 180 3 3 0 360 72 181 psFillArc + globalGC1 83 186 72 181 psDrawLine + globalGC1 82 185 3 3 0 360 83 186 psFillArc + globalGC1 93 189 83 186 psDrawLine + globalGC1 92 188 3 3 0 360 93 189 psFillArc + globalGC1 104 190 93 189 psDrawLine + globalGC1 103 189 3 3 0 360 104 190 psFillArc + globalGC1 106 190 104 190 psDrawLine + globalGC1 105 189 3 3 0 360 106 190 psFillArc + globalGC1 117 189 106 190 psDrawLine + globalGC1 116 188 3 3 0 360 117 189 psFillArc + globalGC1 127 185 117 189 psDrawLine + globalGC1 126 184 3 3 0 360 127 185 psFillArc + globalGC1 138 179 127 185 psDrawLine + globalGC1 137 178 3 3 0 360 138 179 psFillArc + globalGC1 147 170 138 179 psDrawLine + globalGC1 146 169 3 3 0 360 147 170 psFillArc + globalGC1 151 160 147 170 psDrawLine + globalGC1 150 159 3 3 0 360 151 160 psFillArc + globalGC1 153 151 151 160 psDrawLine + globalGC1 152 150 3 3 0 360 153 151 psFillArc + globalGC1 152 142 153 151 psDrawLine + globalGC1 151 141 3 3 0 360 152 142 psFillArc + globalGC1 147 133 152 142 psDrawLine + globalGC1 146 132 3 3 0 360 147 133 psFillArc + globalGC1 137 122 147 133 psDrawLine + globalGC1 136 121 3 3 0 360 137 122 psFillArc + globalGC1 126 116 137 122 psDrawLine + globalGC1 125 115 3 3 0 360 126 116 psFillArc + globalGC1 115 113 126 116 psDrawLine + globalGC1 114 112 3 3 0 360 115 113 psFillArc + globalGC1 104 112 115 113 psDrawLine + globalGC1 103 111 3 3 0 360 104 112 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dppca.ps b/books/ps/2dppca.ps new file mode 100644 index 0000000..1e5bcd7 --- /dev/null +++ b/books/ps/2dppca.ps @@ -0,0 +1,1051 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 127 259 127 0 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 127 129 127 129 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 128 125 127 129 psDrawLine + globalGC1 127 124 3 3 0 360 128 125 psFillArc + globalGC1 129 116 128 125 psDrawLine + globalGC1 128 115 3 3 0 360 129 116 psFillArc + globalGC1 130 102 129 116 psDrawLine + globalGC1 129 101 3 3 0 360 130 102 psFillArc + globalGC1 133 82 130 102 psDrawLine + globalGC1 132 81 3 3 0 360 133 82 psFillArc + globalGC1 136 60 133 82 psDrawLine + globalGC1 135 59 3 3 0 360 136 60 psFillArc + globalGC1 141 39 136 60 psDrawLine + globalGC1 140 38 3 3 0 360 141 39 psFillArc + globalGC1 144 30 141 39 psDrawLine + globalGC1 143 29 3 3 0 360 144 30 psFillArc + globalGC1 147 23 144 30 psDrawLine + globalGC1 146 22 3 3 0 360 147 23 psFillArc + globalGC1 148 20 147 23 psDrawLine + globalGC1 147 19 3 3 0 360 148 20 psFillArc + globalGC1 150 17 148 20 psDrawLine + globalGC1 149 16 3 3 0 360 150 17 psFillArc + globalGC1 152 16 150 17 psDrawLine + globalGC1 151 15 3 3 0 360 152 16 psFillArc + globalGC1 153 15 152 16 psDrawLine + globalGC1 152 14 3 3 0 360 153 15 psFillArc + globalGC1 154 14 153 15 psDrawLine + globalGC1 153 13 3 3 0 360 154 14 psFillArc + globalGC1 155 14 154 14 psDrawLine + globalGC1 154 13 3 3 0 360 155 14 psFillArc + globalGC1 156 14 155 14 psDrawLine + globalGC1 155 13 3 3 0 360 156 14 psFillArc + globalGC1 157 14 156 14 psDrawLine + globalGC1 156 13 3 3 0 360 157 14 psFillArc + globalGC1 158 14 157 14 psDrawLine + globalGC1 157 13 3 3 0 360 158 14 psFillArc + globalGC1 159 14 158 14 psDrawLine + globalGC1 158 13 3 3 0 360 159 14 psFillArc + globalGC1 160 15 159 14 psDrawLine + globalGC1 159 14 3 3 0 360 160 15 psFillArc + globalGC1 162 16 160 15 psDrawLine + globalGC1 161 15 3 3 0 360 162 16 psFillArc + globalGC1 166 20 162 16 psDrawLine + globalGC1 165 19 3 3 0 360 166 20 psFillArc + globalGC1 171 27 166 20 psDrawLine + globalGC1 170 26 3 3 0 360 171 27 psFillArc + globalGC1 180 46 171 27 psDrawLine + globalGC1 179 45 3 3 0 360 180 46 psFillArc + globalGC1 190 70 180 46 psDrawLine + globalGC1 189 69 3 3 0 360 190 70 psFillArc + globalGC1 200 94 190 70 psDrawLine + globalGC1 199 93 3 3 0 360 200 94 psFillArc + globalGC1 209 114 200 94 psDrawLine + globalGC1 208 113 3 3 0 360 209 114 psFillArc + globalGC1 214 123 209 114 psDrawLine + globalGC1 213 122 3 3 0 360 214 123 psFillArc + globalGC1 218 129 214 123 psDrawLine + globalGC1 217 128 3 3 0 360 218 129 psFillArc + globalGC1 223 133 218 129 psDrawLine + globalGC1 222 132 3 3 0 360 223 133 psFillArc + globalGC1 225 135 223 133 psDrawLine + globalGC1 224 134 3 3 0 360 225 135 psFillArc + globalGC1 226 136 225 135 psDrawLine + globalGC1 225 135 3 3 0 360 226 136 psFillArc + globalGC1 228 136 226 136 psDrawLine + globalGC1 227 135 3 3 0 360 228 136 psFillArc + globalGC1 230 137 228 136 psDrawLine + globalGC1 229 136 3 3 0 360 230 137 psFillArc + globalGC1 232 136 230 137 psDrawLine + globalGC1 231 135 3 3 0 360 232 136 psFillArc + globalGC1 233 136 232 136 psDrawLine + globalGC1 232 135 3 3 0 360 233 136 psFillArc + globalGC1 235 135 233 136 psDrawLine + globalGC1 234 134 3 3 0 360 235 135 psFillArc + globalGC1 236 133 235 135 psDrawLine + globalGC1 235 132 3 3 0 360 236 133 psFillArc + globalGC1 238 130 236 133 psDrawLine + globalGC1 237 129 3 3 0 360 238 130 psFillArc + globalGC1 242 123 238 130 psDrawLine + globalGC1 241 122 3 3 0 360 242 123 psFillArc + globalGC1 242 120 242 123 psDrawLine + globalGC1 241 119 3 3 0 360 242 120 psFillArc + globalGC1 243 118 242 120 psDrawLine + globalGC1 242 117 3 3 0 360 243 118 psFillArc + globalGC1 243 117 243 118 psDrawLine + globalGC1 242 116 3 3 0 360 243 117 psFillArc + globalGC1 243 116 243 117 psDrawLine + globalGC1 242 115 3 3 0 360 243 116 psFillArc + globalGC1 242 116 243 116 psDrawLine + globalGC1 241 115 3 3 0 360 242 116 psFillArc + globalGC1 242 115 242 116 psDrawLine + globalGC1 241 114 3 3 0 360 242 115 psFillArc + globalGC1 242 115 242 115 psDrawLine + globalGC1 241 114 3 3 0 360 242 115 psFillArc + globalGC1 242 115 242 115 psDrawLine + globalGC1 241 114 3 3 0 360 242 115 psFillArc + globalGC1 242 116 242 115 psDrawLine + globalGC1 241 115 3 3 0 360 242 116 psFillArc + globalGC1 241 116 242 116 psDrawLine + globalGC1 240 115 3 3 0 360 241 116 psFillArc + globalGC1 240 117 241 116 psDrawLine + globalGC1 239 116 3 3 0 360 240 117 psFillArc + globalGC1 238 120 240 117 psDrawLine + globalGC1 237 119 3 3 0 360 238 120 psFillArc + globalGC1 232 129 238 120 psDrawLine + globalGC1 231 128 3 3 0 360 232 129 psFillArc + globalGC1 224 142 232 129 psDrawLine + globalGC1 223 141 3 3 0 360 224 142 psFillArc + globalGC1 214 156 224 142 psDrawLine + globalGC1 213 155 3 3 0 360 214 156 psFillArc + globalGC1 209 162 214 156 psDrawLine + globalGC1 208 161 3 3 0 360 209 162 psFillArc + globalGC1 203 168 209 162 psDrawLine + globalGC1 202 167 3 3 0 360 203 168 psFillArc + globalGC1 196 171 203 168 psDrawLine + globalGC1 195 170 3 3 0 360 196 171 psFillArc + globalGC1 193 173 196 171 psDrawLine + globalGC1 192 172 3 3 0 360 193 173 psFillArc + globalGC1 189 174 193 173 psDrawLine + globalGC1 188 173 3 3 0 360 189 174 psFillArc + globalGC1 185 174 189 174 psDrawLine + globalGC1 184 173 3 3 0 360 185 174 psFillArc + globalGC1 182 174 185 174 psDrawLine + globalGC1 181 173 3 3 0 360 182 174 psFillArc + globalGC1 174 172 182 174 psDrawLine + globalGC1 173 171 3 3 0 360 174 172 psFillArc + globalGC1 167 168 174 172 psDrawLine + globalGC1 166 167 3 3 0 360 167 168 psFillArc + globalGC1 159 163 167 168 psDrawLine + globalGC1 158 162 3 3 0 360 159 163 psFillArc + globalGC1 143 147 159 163 psDrawLine + globalGC1 142 146 3 3 0 360 143 147 psFillArc + globalGC1 127 129 143 147 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 112 111 127 129 psDrawLine + globalGC1 111 110 3 3 0 360 112 111 psFillArc + globalGC1 105 104 112 111 psDrawLine + globalGC1 104 103 3 3 0 360 105 104 psFillArc + globalGC1 98 98 105 104 psDrawLine + globalGC1 97 97 3 3 0 360 98 98 psFillArc + globalGC1 95 96 98 98 psDrawLine + globalGC1 94 95 3 3 0 360 95 96 psFillArc + globalGC1 92 94 95 96 psDrawLine + globalGC1 91 93 3 3 0 360 92 94 psFillArc + globalGC1 89 93 92 94 psDrawLine + globalGC1 88 92 3 3 0 360 89 93 psFillArc + globalGC1 86 92 89 93 psDrawLine + globalGC1 85 91 3 3 0 360 86 92 psFillArc + globalGC1 83 92 86 92 psDrawLine + globalGC1 82 91 3 3 0 360 83 92 psFillArc + globalGC1 80 92 83 92 psDrawLine + globalGC1 79 91 3 3 0 360 80 92 psFillArc + globalGC1 78 93 80 92 psDrawLine + globalGC1 77 92 3 3 0 360 78 93 psFillArc + globalGC1 75 95 78 93 psDrawLine + globalGC1 74 94 3 3 0 360 75 95 psFillArc + globalGC1 70 99 75 95 psDrawLine + globalGC1 69 98 3 3 0 360 70 99 psFillArc + globalGC1 66 105 70 99 psDrawLine + globalGC1 65 104 3 3 0 360 66 105 psFillArc + globalGC1 60 122 66 105 psDrawLine + globalGC1 59 121 3 3 0 360 60 122 psFillArc + globalGC1 57 140 60 122 psDrawLine + globalGC1 56 139 3 3 0 360 57 140 psFillArc + globalGC1 56 157 57 140 psDrawLine + globalGC1 55 156 3 3 0 360 56 157 psFillArc + globalGC1 56 164 56 157 psDrawLine + globalGC1 55 163 3 3 0 360 56 164 psFillArc + globalGC1 58 170 56 164 psDrawLine + globalGC1 57 169 3 3 0 360 58 170 psFillArc + globalGC1 58 172 58 170 psDrawLine + globalGC1 57 171 3 3 0 360 58 172 psFillArc + globalGC1 59 173 58 172 psDrawLine + globalGC1 58 172 3 3 0 360 59 173 psFillArc + globalGC1 61 174 59 173 psDrawLine + globalGC1 60 173 3 3 0 360 61 174 psFillArc + globalGC1 62 175 61 174 psDrawLine + globalGC1 61 174 3 3 0 360 62 175 psFillArc + globalGC1 63 175 62 175 psDrawLine + globalGC1 62 174 3 3 0 360 63 175 psFillArc + globalGC1 65 175 63 175 psDrawLine + globalGC1 64 174 3 3 0 360 65 175 psFillArc + globalGC1 67 174 65 175 psDrawLine + globalGC1 66 173 3 3 0 360 67 174 psFillArc + globalGC1 69 173 67 174 psDrawLine + globalGC1 68 172 3 3 0 360 69 173 psFillArc + globalGC1 73 169 69 173 psDrawLine + globalGC1 72 168 3 3 0 360 73 169 psFillArc + globalGC1 78 165 73 169 psDrawLine + globalGC1 77 164 3 3 0 360 78 165 psFillArc + globalGC1 88 153 78 165 psDrawLine + globalGC1 87 152 3 3 0 360 88 153 psFillArc + globalGC1 101 141 88 153 psDrawLine + globalGC1 100 140 3 3 0 360 101 141 psFillArc + globalGC1 107 136 101 141 psDrawLine + globalGC1 106 135 3 3 0 360 107 136 psFillArc + globalGC1 114 132 107 136 psDrawLine + globalGC1 113 131 3 3 0 360 114 132 psFillArc + globalGC1 121 130 114 132 psDrawLine + globalGC1 120 129 3 3 0 360 121 130 psFillArc + globalGC1 124 129 121 130 psDrawLine + globalGC1 123 128 3 3 0 360 124 129 psFillArc + globalGC1 127 129 124 129 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 131 129 127 129 psDrawLine + globalGC1 130 128 3 3 0 360 131 129 psFillArc + globalGC1 134 130 131 129 psDrawLine + globalGC1 133 129 3 3 0 360 134 130 psFillArc + globalGC1 141 132 134 130 psDrawLine + globalGC1 140 131 3 3 0 360 141 132 psFillArc + globalGC1 148 136 141 132 psDrawLine + globalGC1 147 135 3 3 0 360 148 136 psFillArc + globalGC1 154 141 148 136 psDrawLine + globalGC1 153 140 3 3 0 360 154 141 psFillArc + globalGC1 167 153 154 141 psDrawLine + globalGC1 166 152 3 3 0 360 167 153 psFillArc + globalGC1 177 165 167 153 psDrawLine + globalGC1 176 164 3 3 0 360 177 165 psFillArc + globalGC1 182 169 177 165 psDrawLine + globalGC1 181 168 3 3 0 360 182 169 psFillArc + globalGC1 186 173 182 169 psDrawLine + globalGC1 185 172 3 3 0 360 186 173 psFillArc + globalGC1 188 174 186 173 psDrawLine + globalGC1 187 173 3 3 0 360 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123 psFillArc + globalGC1 32 125 30 123 psDrawLine + globalGC1 31 124 3 3 0 360 32 125 psFillArc + globalGC1 37 129 32 125 psDrawLine + globalGC1 36 128 3 3 0 360 37 129 psFillArc + globalGC1 46 144 37 129 psDrawLine + globalGC1 45 143 3 3 0 360 46 144 psFillArc + globalGC1 55 164 46 144 psDrawLine + globalGC1 54 163 3 3 0 360 55 164 psFillArc + globalGC1 65 188 55 164 psDrawLine + globalGC1 64 187 3 3 0 360 65 188 psFillArc + globalGC1 75 212 65 188 psDrawLine + globalGC1 74 211 3 3 0 360 75 212 psFillArc + globalGC1 84 231 75 212 psDrawLine + globalGC1 83 230 3 3 0 360 84 231 psFillArc + globalGC1 89 238 84 231 psDrawLine + globalGC1 88 237 3 3 0 360 89 238 psFillArc + globalGC1 91 240 89 238 psDrawLine + globalGC1 90 239 3 3 0 360 91 240 psFillArc + globalGC1 93 242 91 240 psDrawLine + globalGC1 92 241 3 3 0 360 93 242 psFillArc + globalGC1 95 243 93 242 psDrawLine + globalGC1 94 242 3 3 0 360 95 243 psFillArc + globalGC1 96 244 95 243 psDrawLine + globalGC1 95 243 3 3 0 360 96 244 psFillArc + globalGC1 97 244 96 244 psDrawLine + globalGC1 96 243 3 3 0 360 97 244 psFillArc + globalGC1 98 244 97 244 psDrawLine + globalGC1 97 243 3 3 0 360 98 244 psFillArc + globalGC1 99 244 98 244 psDrawLine + globalGC1 98 243 3 3 0 360 99 244 psFillArc + globalGC1 100 244 99 244 psDrawLine + globalGC1 99 243 3 3 0 360 100 244 psFillArc + globalGC1 101 244 100 244 psDrawLine + globalGC1 100 243 3 3 0 360 101 244 psFillArc + globalGC1 102 243 101 244 psDrawLine + globalGC1 101 242 3 3 0 360 102 243 psFillArc + globalGC1 103 242 102 243 psDrawLine + globalGC1 102 241 3 3 0 360 103 242 psFillArc + globalGC1 105 241 103 242 psDrawLine + globalGC1 104 240 3 3 0 360 105 241 psFillArc + globalGC1 108 235 105 241 psDrawLine + globalGC1 107 234 3 3 0 360 108 235 psFillArc + globalGC1 111 228 108 235 psDrawLine + globalGC1 110 227 3 3 0 360 111 228 psFillArc + globalGC1 114 219 111 228 psDrawLine + globalGC1 113 218 3 3 0 360 114 219 psFillArc + globalGC1 119 198 114 219 psDrawLine + globalGC1 118 197 3 3 0 360 119 198 psFillArc + globalGC1 122 176 119 198 psDrawLine + globalGC1 121 175 3 3 0 360 122 176 psFillArc + globalGC1 125 156 122 176 psDrawLine + globalGC1 124 155 3 3 0 360 125 156 psFillArc + globalGC1 127 133 125 156 psDrawLine + globalGC1 126 132 3 3 0 360 127 133 psFillArc + globalGC1 127 129 127 133 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dppcb.ps b/books/ps/2dppcb.ps new file mode 100644 index 0000000..12ffb78 --- /dev/null +++ b/books/ps/2dppcb.ps @@ -0,0 +1,414 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 151 0 151 psDrawLine + globalGC1 150 303 150 0 psDrawLine + unitGC 150 153 150 149 psDrawLine + unitGC 138 166 (0.00) (window) psDrawIStr + unitGC 207 153 207 149 psDrawLine + unitGC 195 166 (0.40) (window) psDrawIStr + unitGC 264 153 264 149 psDrawLine + unitGC 252 166 (0.80) (window) psDrawIStr + unitGC 92 153 92 149 psDrawLine + unitGC 77 166 (-0.40) (window) psDrawIStr + unitGC 34 153 34 149 psDrawLine + unitGC 19 166 (-0.80) (window) psDrawIStr + unitGC 152 93 148 93 psDrawLine + unitGC 114 98 (0.40) (window) psDrawIStr + unitGC 152 35 148 35 psDrawLine + unitGC 114 40 (0.80) (window) psDrawIStr + unitGC 152 209 148 209 psDrawLine + unitGC 105 214 (-0.40) (window) psDrawIStr + unitGC 152 266 148 266 psDrawLine + unitGC 105 271 (-0.80) (window) psDrawIStr + globalGC1 293 150 3 3 0 360 294 151 psFillArc + globalGC1 294 151 294 151 psDrawLine + globalGC1 293 150 3 3 0 360 294 151 psFillArc + globalGC1 293 132 294 151 psDrawLine + globalGC1 292 131 3 3 0 360 293 132 psFillArc + globalGC1 289 114 293 132 psDrawLine + globalGC1 288 113 3 3 0 360 289 114 psFillArc + globalGC1 283 96 289 114 psDrawLine + globalGC1 282 95 3 3 0 360 283 96 psFillArc + globalGC1 275 79 283 96 psDrawLine + globalGC1 274 78 3 3 0 360 275 79 psFillArc + globalGC1 264 63 275 79 psDrawLine + globalGC1 263 62 3 3 0 360 264 63 psFillArc + globalGC1 252 49 264 63 psDrawLine + globalGC1 251 48 3 3 0 360 252 49 psFillArc + globalGC1 238 36 252 49 psDrawLine + globalGC1 237 35 3 3 0 360 238 36 psFillArc + globalGC1 222 26 238 36 psDrawLine + globalGC1 221 25 3 3 0 360 222 26 psFillArc + globalGC1 205 17 222 26 psDrawLine + globalGC1 204 16 3 3 0 360 205 17 psFillArc + globalGC1 187 11 205 17 psDrawLine + globalGC1 186 10 3 3 0 360 187 11 psFillArc + globalGC1 168 8 187 11 psDrawLine + globalGC1 167 7 3 3 0 360 168 8 psFillArc + globalGC1 149 6 168 8 psDrawLine + globalGC1 148 5 3 3 0 360 149 6 psFillArc + globalGC1 131 8 149 6 psDrawLine + globalGC1 130 7 3 3 0 360 131 8 psFillArc + globalGC1 112 11 131 8 psDrawLine + globalGC1 111 10 3 3 0 360 112 11 psFillArc + globalGC1 94 17 112 11 psDrawLine + globalGC1 93 16 3 3 0 360 94 17 psFillArc + globalGC1 77 26 94 17 psDrawLine + globalGC1 76 25 3 3 0 360 77 26 psFillArc + globalGC1 61 36 77 26 psDrawLine + globalGC1 60 35 3 3 0 360 61 36 psFillArc + globalGC1 47 49 61 36 psDrawLine + globalGC1 46 48 3 3 0 360 47 49 psFillArc + globalGC1 35 63 47 49 psDrawLine + globalGC1 34 62 3 3 0 360 35 63 psFillArc + globalGC1 24 79 35 63 psDrawLine + globalGC1 23 78 3 3 0 360 24 79 psFillArc + globalGC1 16 96 24 79 psDrawLine + globalGC1 15 95 3 3 0 360 16 96 psFillArc + globalGC1 10 114 16 96 psDrawLine + globalGC1 9 113 3 3 0 360 10 114 psFillArc + globalGC1 6 132 10 114 psDrawLine + globalGC1 5 131 3 3 0 360 6 132 psFillArc + globalGC1 5 151 6 132 psDrawLine + globalGC1 4 150 3 3 0 360 5 151 psFillArc + globalGC1 6 170 5 151 psDrawLine + globalGC1 5 169 3 3 0 360 6 170 psFillArc + globalGC1 10 188 6 170 psDrawLine + globalGC1 9 187 3 3 0 360 10 188 psFillArc + globalGC1 16 206 10 188 psDrawLine + globalGC1 15 205 3 3 0 360 16 206 psFillArc + globalGC1 24 223 16 206 psDrawLine + globalGC1 23 222 3 3 0 360 24 223 psFillArc + globalGC1 35 239 24 223 psDrawLine + globalGC1 34 238 3 3 0 360 35 239 psFillArc + globalGC1 47 253 35 239 psDrawLine + globalGC1 46 252 3 3 0 360 47 253 psFillArc + globalGC1 61 266 47 253 psDrawLine + globalGC1 60 265 3 3 0 360 61 266 psFillArc + globalGC1 77 276 61 266 psDrawLine + globalGC1 76 275 3 3 0 360 77 276 psFillArc + globalGC1 94 285 77 276 psDrawLine + globalGC1 93 284 3 3 0 360 94 285 psFillArc + globalGC1 112 291 94 285 psDrawLine + globalGC1 111 290 3 3 0 360 112 291 psFillArc + globalGC1 131 294 112 291 psDrawLine + globalGC1 130 293 3 3 0 360 131 294 psFillArc + globalGC1 149 296 131 294 psDrawLine + globalGC1 148 295 3 3 0 360 149 296 psFillArc + globalGC1 168 294 149 296 psDrawLine + globalGC1 167 293 3 3 0 360 168 294 psFillArc + globalGC1 187 291 168 294 psDrawLine + globalGC1 186 290 3 3 0 360 187 291 psFillArc + globalGC1 205 285 187 291 psDrawLine + globalGC1 204 284 3 3 0 360 205 285 psFillArc + globalGC1 222 276 205 285 psDrawLine + globalGC1 221 275 3 3 0 360 222 276 psFillArc + globalGC1 238 266 222 276 psDrawLine + globalGC1 237 265 3 3 0 360 238 266 psFillArc + globalGC1 252 253 238 266 psDrawLine + globalGC1 251 252 3 3 0 360 252 253 psFillArc + globalGC1 264 239 252 253 psDrawLine + globalGC1 263 238 3 3 0 360 264 239 psFillArc + globalGC1 275 223 264 239 psDrawLine + globalGC1 274 222 3 3 0 360 275 223 psFillArc + globalGC1 283 206 275 223 psDrawLine + globalGC1 282 205 3 3 0 360 283 206 psFillArc + globalGC1 289 188 283 206 psDrawLine + globalGC1 288 187 3 3 0 360 289 188 psFillArc + globalGC1 293 170 289 188 psDrawLine + globalGC1 292 169 3 3 0 360 293 170 psFillArc + globalGC1 294 151 293 170 psDrawLine + globalGC1 293 150 3 3 0 360 294 151 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dppcc.ps b/books/ps/2dppcc.ps new file mode 100644 index 0000000..6f86ea9 --- /dev/null +++ b/books/ps/2dppcc.ps @@ -0,0 +1,424 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 280 0 280 psDrawLine + globalGC1 7 303 7 0 psDrawLine + unitGC 61 282 61 278 psDrawLine + unitGC 49 295 (0.20) (window) psDrawIStr + unitGC 115 282 115 278 psDrawLine + unitGC 103 295 (0.40) (window) psDrawIStr + unitGC 169 282 169 278 psDrawLine + unitGC 157 295 (0.60) (window) psDrawIStr + unitGC 223 282 223 278 psDrawLine + unitGC 211 295 (0.80) (window) psDrawIStr + unitGC 277 282 277 278 psDrawLine + unitGC 265 295 (1.00) (window) psDrawIStr + unitGC 9 226 5 226 psDrawLine + unitGC -29 231 (0.20) (window) psDrawIStr + unitGC 9 172 5 172 psDrawLine + unitGC -29 177 (0.40) (window) psDrawIStr + unitGC 9 118 5 118 psDrawLine + unitGC -29 123 (0.60) (window) psDrawIStr + unitGC 9 64 5 64 psDrawLine + unitGC -29 69 (0.80) (window) psDrawIStr + unitGC 9 10 5 10 psDrawLine + unitGC -29 15 (1.00) (window) psDrawIStr + globalGC1 6 279 3 3 0 360 7 280 psFillArc + globalGC1 7 280 7 280 psDrawLine + globalGC1 6 279 3 3 0 360 7 280 psFillArc + globalGC1 20 262 7 280 psDrawLine + globalGC1 19 261 3 3 0 360 20 262 psFillArc + globalGC1 33 245 20 262 psDrawLine + globalGC1 32 244 3 3 0 360 33 245 psFillArc + globalGC1 46 227 33 245 psDrawLine + globalGC1 45 226 3 3 0 360 46 227 psFillArc + globalGC1 59 210 46 227 psDrawLine + globalGC1 58 209 3 3 0 360 59 210 psFillArc + globalGC1 72 193 59 210 psDrawLine + globalGC1 71 192 3 3 0 360 72 193 psFillArc + globalGC1 85 177 72 193 psDrawLine + globalGC1 84 176 3 3 0 360 85 177 psFillArc + globalGC1 98 161 85 177 psDrawLine + globalGC1 97 160 3 3 0 360 98 161 psFillArc + globalGC1 110 145 98 161 psDrawLine + globalGC1 109 144 3 3 0 360 110 145 psFillArc + globalGC1 122 130 110 145 psDrawLine + globalGC1 121 129 3 3 0 360 122 130 psFillArc + globalGC1 134 116 122 130 psDrawLine + globalGC1 133 115 3 3 0 360 134 116 psFillArc + globalGC1 146 102 134 116 psDrawLine + globalGC1 145 101 3 3 0 360 146 102 psFillArc + globalGC1 157 89 146 102 psDrawLine + globalGC1 156 88 3 3 0 360 157 89 psFillArc + globalGC1 168 77 157 89 psDrawLine + globalGC1 167 76 3 3 0 360 168 77 psFillArc + globalGC1 178 66 168 77 psDrawLine + globalGC1 177 65 3 3 0 360 178 66 psFillArc + globalGC1 188 56 178 66 psDrawLine + globalGC1 187 55 3 3 0 360 188 56 psFillArc + globalGC1 198 46 188 56 psDrawLine + globalGC1 197 45 3 3 0 360 198 46 psFillArc + globalGC1 207 38 198 46 psDrawLine + globalGC1 206 37 3 3 0 360 207 38 psFillArc + globalGC1 215 31 207 38 psDrawLine + globalGC1 214 30 3 3 0 360 215 31 psFillArc + globalGC1 224 24 215 31 psDrawLine + globalGC1 223 23 3 3 0 360 224 24 psFillArc + globalGC1 231 19 224 24 psDrawLine + globalGC1 230 18 3 3 0 360 231 19 psFillArc + globalGC1 238 15 231 19 psDrawLine + globalGC1 237 14 3 3 0 360 238 15 psFillArc + globalGC1 245 12 238 15 psDrawLine + globalGC1 244 11 3 3 0 360 245 12 psFillArc + globalGC1 251 11 245 12 psDrawLine + globalGC1 250 10 3 3 0 360 251 11 psFillArc + globalGC1 256 10 251 11 psDrawLine + globalGC1 255 9 3 3 0 360 256 10 psFillArc + globalGC1 259 10 256 10 psDrawLine + globalGC1 258 9 3 3 0 360 259 10 psFillArc + globalGC1 261 11 259 10 psDrawLine + globalGC1 260 10 3 3 0 360 261 11 psFillArc + globalGC1 263 11 261 11 psDrawLine + globalGC1 262 10 3 3 0 360 263 11 psFillArc + globalGC1 265 12 263 11 psDrawLine + globalGC1 264 11 3 3 0 360 265 12 psFillArc + globalGC1 267 14 265 12 psDrawLine + globalGC1 266 13 3 3 0 360 267 14 psFillArc + globalGC1 269 15 267 14 psDrawLine + globalGC1 268 14 3 3 0 360 269 15 psFillArc + globalGC1 270 17 269 15 psDrawLine + globalGC1 269 16 3 3 0 360 270 17 psFillArc + globalGC1 272 19 270 17 psDrawLine + globalGC1 271 18 3 3 0 360 272 19 psFillArc + globalGC1 274 24 272 19 psDrawLine + globalGC1 273 23 3 3 0 360 274 24 psFillArc + globalGC1 275 31 274 24 psDrawLine + globalGC1 274 30 3 3 0 360 275 31 psFillArc + globalGC1 276 38 275 31 psDrawLine + globalGC1 275 37 3 3 0 360 276 38 psFillArc + globalGC1 277 46 276 38 psDrawLine + globalGC1 276 45 3 3 0 360 277 46 psFillArc + globalGC1 276 56 277 46 psDrawLine + globalGC1 275 55 3 3 0 360 276 56 psFillArc + globalGC1 275 66 276 56 psDrawLine + globalGC1 274 65 3 3 0 360 275 66 psFillArc + globalGC1 274 77 275 66 psDrawLine + globalGC1 273 76 3 3 0 360 274 77 psFillArc + globalGC1 272 89 274 77 psDrawLine + globalGC1 271 88 3 3 0 360 272 89 psFillArc + globalGC1 269 102 272 89 psDrawLine + globalGC1 268 101 3 3 0 360 269 102 psFillArc + globalGC1 265 116 269 102 psDrawLine + globalGC1 264 115 3 3 0 360 265 116 psFillArc + globalGC1 261 130 265 116 psDrawLine + globalGC1 260 129 3 3 0 360 261 130 psFillArc + globalGC1 256 145 261 130 psDrawLine + globalGC1 255 144 3 3 0 360 256 145 psFillArc + globalGC1 251 161 256 145 psDrawLine + globalGC1 250 160 3 3 0 360 251 161 psFillArc + globalGC1 245 177 251 161 psDrawLine + globalGC1 244 176 3 3 0 360 245 177 psFillArc + globalGC1 238 193 245 177 psDrawLine + globalGC1 237 192 3 3 0 360 238 193 psFillArc + globalGC1 231 210 238 193 psDrawLine + globalGC1 230 209 3 3 0 360 231 210 psFillArc + globalGC1 224 227 231 210 psDrawLine + globalGC1 223 226 3 3 0 360 224 227 psFillArc + globalGC1 215 245 224 227 psDrawLine + globalGC1 214 244 3 3 0 360 215 245 psFillArc + globalGC1 207 262 215 245 psDrawLine + globalGC1 206 261 3 3 0 360 207 262 psFillArc + globalGC1 198 280 207 262 psDrawLine + globalGC1 197 279 3 3 0 360 198 280 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/2dppce.ps b/books/ps/2dppce.ps new file mode 100644 index 0000000..a54a6ca --- /dev/null +++ b/books/ps/2dppce.ps @@ -0,0 +1,571 @@ +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 24 259 256 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 256 129 0 129 psDrawLine + globalGC1 128 259 128 0 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 127 129 127 129 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 105 99 127 129 psDrawLine + globalGC1 104 98 3 3 0 360 105 99 psFillArc + globalGC1 83 71 105 99 psDrawLine + globalGC1 82 70 3 3 0 360 83 71 psFillArc + globalGC1 63 48 83 71 psDrawLine + globalGC1 62 47 3 3 0 360 63 48 psFillArc + globalGC1 46 29 63 48 psDrawLine + globalGC1 45 28 3 3 0 360 46 29 psFillArc + globalGC1 38 23 46 29 psDrawLine + globalGC1 37 22 3 3 0 360 38 23 psFillArc + globalGC1 32 18 38 23 psDrawLine + globalGC1 31 17 3 3 0 360 32 18 psFillArc + globalGC1 29 16 32 18 psDrawLine + globalGC1 28 15 3 3 0 360 29 16 psFillArc + globalGC1 26 15 29 16 psDrawLine + globalGC1 25 14 3 3 0 360 26 15 psFillArc + globalGC1 23 14 26 15 psDrawLine + globalGC1 22 13 3 3 0 360 23 14 psFillArc + globalGC1 21 14 23 14 psDrawLine + globalGC1 20 13 3 3 0 360 21 14 psFillArc + globalGC1 20 14 21 14 psDrawLine + globalGC1 19 13 3 3 0 360 20 14 psFillArc + globalGC1 19 14 20 14 psDrawLine + globalGC1 18 13 3 3 0 360 19 14 psFillArc + globalGC1 18 14 19 14 psDrawLine + globalGC1 17 13 3 3 0 360 18 14 psFillArc + globalGC1 17 15 18 14 psDrawLine + globalGC1 16 14 3 3 0 360 17 15 psFillArc + globalGC1 16 15 17 15 psDrawLine + globalGC1 15 14 3 3 0 360 16 15 psFillArc + globalGC1 16 16 16 15 psDrawLine + globalGC1 15 15 3 3 0 360 16 16 psFillArc + globalGC1 15 17 16 16 psDrawLine + globalGC1 14 16 3 3 0 360 15 17 psFillArc + globalGC1 15 18 15 17 psDrawLine + globalGC1 14 17 3 3 0 360 15 18 psFillArc + globalGC1 13 23 15 18 psDrawLine + globalGC1 12 22 3 3 0 360 13 23 psFillArc + globalGC1 12 29 13 23 psDrawLine + globalGC1 11 28 3 3 0 360 12 29 psFillArc + globalGC1 15 48 12 29 psDrawLine + globalGC1 14 47 3 3 0 360 15 48 psFillArc + globalGC1 21 71 15 48 psDrawLine + globalGC1 20 70 3 3 0 360 21 71 psFillArc + globalGC1 32 99 21 71 psDrawLine + globalGC1 31 98 3 3 0 360 32 99 psFillArc + globalGC1 46 129 32 99 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 63 159 46 129 psDrawLine + globalGC1 62 158 3 3 0 360 63 159 psFillArc + globalGC1 83 187 63 159 psDrawLine + globalGC1 82 186 3 3 0 360 83 187 psFillArc + globalGC1 105 210 83 187 psDrawLine + globalGC1 104 209 3 3 0 360 105 210 psFillArc + globalGC1 127 229 105 210 psDrawLine + globalGC1 126 228 3 3 0 360 127 229 psFillArc + globalGC1 139 235 127 229 psDrawLine + globalGC1 138 234 3 3 0 360 139 235 psFillArc + globalGC1 150 240 139 235 psDrawLine + globalGC1 149 239 3 3 0 360 150 240 psFillArc + globalGC1 161 243 150 240 psDrawLine + globalGC1 160 242 3 3 0 360 161 243 psFillArc + globalGC1 172 244 161 243 psDrawLine + globalGC1 171 243 3 3 0 360 172 244 psFillArc + globalGC1 182 243 172 244 psDrawLine + globalGC1 181 242 3 3 0 360 182 243 psFillArc + globalGC1 192 240 182 243 psDrawLine + globalGC1 191 239 3 3 0 360 192 240 psFillArc + globalGC1 201 235 192 240 psDrawLine + globalGC1 200 234 3 3 0 360 201 235 psFillArc + globalGC1 209 229 201 235 psDrawLine + globalGC1 208 228 3 3 0 360 209 229 psFillArc + globalGC1 223 210 209 229 psDrawLine + globalGC1 222 209 3 3 0 360 223 210 psFillArc + globalGC1 234 187 223 210 psDrawLine + globalGC1 233 186 3 3 0 360 234 187 psFillArc + globalGC1 240 159 234 187 psDrawLine + globalGC1 239 158 3 3 0 360 240 159 psFillArc + globalGC1 243 129 240 159 psDrawLine + globalGC1 242 128 3 3 0 360 243 129 psFillArc + globalGC1 240 99 243 129 psDrawLine + globalGC1 239 98 3 3 0 360 240 99 psFillArc + globalGC1 234 71 240 99 psDrawLine + globalGC1 233 70 3 3 0 360 234 71 psFillArc + globalGC1 223 48 234 71 psDrawLine + globalGC1 222 47 3 3 0 360 223 48 psFillArc + globalGC1 217 38 223 48 psDrawLine + globalGC1 216 37 3 3 0 360 217 38 psFillArc + globalGC1 209 29 217 38 psDrawLine + globalGC1 208 28 3 3 0 360 209 29 psFillArc + globalGC1 201 23 209 29 psDrawLine + globalGC1 200 22 3 3 0 360 201 23 psFillArc + globalGC1 192 18 201 23 psDrawLine + globalGC1 191 17 3 3 0 360 192 18 psFillArc + globalGC1 182 15 192 18 psDrawLine + globalGC1 181 14 3 3 0 360 182 15 psFillArc + globalGC1 172 14 182 15 psDrawLine + globalGC1 171 13 3 3 0 360 172 14 psFillArc + globalGC1 161 15 172 14 psDrawLine + globalGC1 160 14 3 3 0 360 161 15 psFillArc + globalGC1 150 18 161 15 psDrawLine + globalGC1 149 17 3 3 0 360 150 18 psFillArc + globalGC1 128 29 150 18 psDrawLine + globalGC1 127 28 3 3 0 360 128 29 psFillArc + globalGC1 105 48 128 29 psDrawLine + globalGC1 104 47 3 3 0 360 105 48 psFillArc + globalGC1 83 71 105 48 psDrawLine + globalGC1 82 70 3 3 0 360 83 71 psFillArc + globalGC1 63 99 83 71 psDrawLine + globalGC1 62 98 3 3 0 360 63 99 psFillArc + globalGC1 46 129 63 99 psDrawLine + globalGC1 45 128 3 3 0 360 46 129 psFillArc + globalGC1 32 159 46 129 psDrawLine + globalGC1 31 158 3 3 0 360 32 159 psFillArc + globalGC1 21 187 32 159 psDrawLine + globalGC1 20 186 3 3 0 360 21 187 psFillArc + globalGC1 15 210 21 187 psDrawLine + globalGC1 14 209 3 3 0 360 15 210 psFillArc + globalGC1 13 220 15 210 psDrawLine + globalGC1 12 219 3 3 0 360 13 220 psFillArc + globalGC1 12 229 13 220 psDrawLine + globalGC1 11 228 3 3 0 360 12 229 psFillArc + globalGC1 13 235 12 229 psDrawLine + globalGC1 12 234 3 3 0 360 13 235 psFillArc + globalGC1 14 238 13 235 psDrawLine + globalGC1 13 237 3 3 0 360 14 238 psFillArc + globalGC1 15 240 14 238 psDrawLine + globalGC1 14 239 3 3 0 360 15 240 psFillArc + globalGC1 15 241 15 240 psDrawLine + globalGC1 14 240 3 3 0 360 15 241 psFillArc + globalGC1 16 242 15 241 psDrawLine + globalGC1 15 241 3 3 0 360 16 242 psFillArc + globalGC1 16 243 16 242 psDrawLine + globalGC1 15 242 3 3 0 360 16 243 psFillArc + globalGC1 17 243 16 243 psDrawLine + globalGC1 16 242 3 3 0 360 17 243 psFillArc + globalGC1 18 244 17 243 psDrawLine + globalGC1 17 243 3 3 0 360 18 244 psFillArc + globalGC1 19 244 18 244 psDrawLine + globalGC1 18 243 3 3 0 360 19 244 psFillArc + globalGC1 20 244 19 244 psDrawLine + globalGC1 19 243 3 3 0 360 20 244 psFillArc + globalGC1 21 244 20 244 psDrawLine + globalGC1 20 243 3 3 0 360 21 244 psFillArc + globalGC1 23 244 21 244 psDrawLine + globalGC1 22 243 3 3 0 360 23 244 psFillArc + globalGC1 26 243 23 244 psDrawLine + globalGC1 25 242 3 3 0 360 26 243 psFillArc + globalGC1 32 240 26 243 psDrawLine + globalGC1 31 239 3 3 0 360 32 240 psFillArc + globalGC1 46 229 32 240 psDrawLine + globalGC1 45 228 3 3 0 360 46 229 psFillArc + globalGC1 63 210 46 229 psDrawLine + globalGC1 62 209 3 3 0 360 63 210 psFillArc + globalGC1 83 187 63 210 psDrawLine + globalGC1 82 186 3 3 0 360 83 187 psFillArc + globalGC1 105 159 83 187 psDrawLine + globalGC1 104 158 3 3 0 360 105 159 psFillArc + globalGC1 127 129 105 159 psDrawLine + globalGC1 126 128 3 3 0 360 127 129 psFillArc + globalGC1 150 99 127 129 psDrawLine + globalGC1 149 98 3 3 0 360 150 99 psFillArc + globalGC1 172 71 150 99 psDrawLine + globalGC1 171 70 3 3 0 360 172 71 psFillArc + globalGC1 192 48 172 71 psDrawLine + globalGC1 191 47 3 3 0 360 192 48 psFillArc + globalGC1 209 29 192 48 psDrawLine + globalGC1 208 28 3 3 0 360 209 29 psFillArc + globalGC1 217 23 209 29 psDrawLine + globalGC1 216 22 3 3 0 360 217 23 psFillArc + globalGC1 223 18 217 23 psDrawLine + globalGC1 222 17 3 3 0 360 223 18 psFillArc + globalGC1 226 16 223 18 psDrawLine + globalGC1 225 15 3 3 0 360 226 16 psFillArc + globalGC1 229 15 226 16 psDrawLine + globalGC1 228 14 3 3 0 360 229 15 psFillArc + globalGC1 232 14 229 15 psDrawLine + globalGC1 231 13 3 3 0 360 232 14 psFillArc + globalGC1 234 14 232 14 psDrawLine + globalGC1 233 13 3 3 0 360 234 14 psFillArc + globalGC1 235 14 234 14 psDrawLine + globalGC1 234 13 3 3 0 360 235 14 psFillArc + globalGC1 236 14 235 14 psDrawLine + globalGC1 235 13 3 3 0 360 236 14 psFillArc + globalGC1 237 14 236 14 psDrawLine + globalGC1 236 13 3 3 0 360 237 14 psFillArc + globalGC1 238 15 237 14 psDrawLine + globalGC1 237 14 3 3 0 360 238 15 psFillArc + globalGC1 239 15 238 15 psDrawLine + globalGC1 238 14 3 3 0 360 239 15 psFillArc + globalGC1 239 16 239 15 psDrawLine + globalGC1 238 15 3 3 0 360 239 16 psFillArc + globalGC1 240 17 239 16 psDrawLine + globalGC1 239 16 3 3 0 360 240 17 psFillArc + globalGC1 240 18 240 17 psDrawLine + globalGC1 239 17 3 3 0 360 240 18 psFillArc + globalGC1 242 23 240 18 psDrawLine + globalGC1 241 22 3 3 0 360 242 23 psFillArc + globalGC1 243 29 242 23 psDrawLine + globalGC1 242 28 3 3 0 360 243 29 psFillArc + globalGC1 240 48 243 29 psDrawLine + globalGC1 239 47 3 3 0 360 240 48 psFillArc + globalGC1 234 71 240 48 psDrawLine + globalGC1 233 70 3 3 0 360 234 71 psFillArc + globalGC1 223 99 234 71 psDrawLine + globalGC1 222 98 3 3 0 360 223 99 psFillArc + globalGC1 209 129 223 99 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 192 159 209 129 psDrawLine + globalGC1 191 158 3 3 0 360 192 159 psFillArc + globalGC1 172 187 192 159 psDrawLine + globalGC1 171 186 3 3 0 360 172 187 psFillArc + globalGC1 150 210 172 187 psDrawLine + globalGC1 149 209 3 3 0 360 150 210 psFillArc + globalGC1 128 229 150 210 psDrawLine + globalGC1 127 228 3 3 0 360 128 229 psFillArc + globalGC1 116 235 128 229 psDrawLine + globalGC1 115 234 3 3 0 360 116 235 psFillArc + globalGC1 105 240 116 235 psDrawLine + globalGC1 104 239 3 3 0 360 105 240 psFillArc + globalGC1 94 243 105 240 psDrawLine + globalGC1 93 242 3 3 0 360 94 243 psFillArc + globalGC1 83 244 94 243 psDrawLine + globalGC1 82 243 3 3 0 360 83 244 psFillArc + globalGC1 73 243 83 244 psDrawLine + globalGC1 72 242 3 3 0 360 73 243 psFillArc + globalGC1 63 240 73 243 psDrawLine + globalGC1 62 239 3 3 0 360 63 240 psFillArc + globalGC1 54 235 63 240 psDrawLine + globalGC1 53 234 3 3 0 360 54 235 psFillArc + globalGC1 46 229 54 235 psDrawLine + globalGC1 45 228 3 3 0 360 46 229 psFillArc + globalGC1 32 210 46 229 psDrawLine + globalGC1 31 209 3 3 0 360 32 210 psFillArc + globalGC1 21 187 32 210 psDrawLine + globalGC1 20 186 3 3 0 360 21 187 psFillArc + globalGC1 15 159 21 187 psDrawLine + globalGC1 14 158 3 3 0 360 15 159 psFillArc + globalGC1 12 129 15 159 psDrawLine + globalGC1 11 128 3 3 0 360 12 129 psFillArc + globalGC1 15 99 12 129 psDrawLine + globalGC1 14 98 3 3 0 360 15 99 psFillArc + globalGC1 21 71 15 99 psDrawLine + globalGC1 20 70 3 3 0 360 21 71 psFillArc + globalGC1 32 48 21 71 psDrawLine + globalGC1 31 47 3 3 0 360 32 48 psFillArc + globalGC1 38 38 32 48 psDrawLine + globalGC1 37 37 3 3 0 360 38 38 psFillArc + globalGC1 46 29 38 38 psDrawLine + globalGC1 45 28 3 3 0 360 46 29 psFillArc + globalGC1 54 23 46 29 psDrawLine + globalGC1 53 22 3 3 0 360 54 23 psFillArc + globalGC1 63 18 54 23 psDrawLine + globalGC1 62 17 3 3 0 360 63 18 psFillArc + globalGC1 73 15 63 18 psDrawLine + globalGC1 72 14 3 3 0 360 73 15 psFillArc + globalGC1 83 14 73 15 psDrawLine + globalGC1 82 13 3 3 0 360 83 14 psFillArc + globalGC1 94 15 83 14 psDrawLine + globalGC1 93 14 3 3 0 360 94 15 psFillArc + globalGC1 105 18 94 15 psDrawLine + globalGC1 104 17 3 3 0 360 105 18 psFillArc + globalGC1 127 29 105 18 psDrawLine + globalGC1 126 28 3 3 0 360 127 29 psFillArc + globalGC1 150 48 127 29 psDrawLine + globalGC1 149 47 3 3 0 360 150 48 psFillArc + globalGC1 172 71 150 48 psDrawLine + globalGC1 171 70 3 3 0 360 172 71 psFillArc + globalGC1 192 99 172 71 psDrawLine + globalGC1 191 98 3 3 0 360 192 99 psFillArc + globalGC1 209 129 192 99 psDrawLine + globalGC1 208 128 3 3 0 360 209 129 psFillArc + globalGC1 223 159 209 129 psDrawLine + globalGC1 222 158 3 3 0 360 223 159 psFillArc + globalGC1 234 187 223 159 psDrawLine + globalGC1 233 186 3 3 0 360 234 187 psFillArc + globalGC1 240 210 234 187 psDrawLine + globalGC1 239 209 3 3 0 360 240 210 psFillArc + globalGC1 242 220 240 210 psDrawLine + globalGC1 241 219 3 3 0 360 242 220 psFillArc + globalGC1 243 229 242 220 psDrawLine + globalGC1 242 228 3 3 0 360 243 229 psFillArc + globalGC1 242 235 243 229 psDrawLine + globalGC1 241 234 3 3 0 360 242 235 psFillArc + globalGC1 241 238 242 235 psDrawLine + globalGC1 240 237 3 3 0 360 241 238 psFillArc + globalGC1 240 240 241 238 psDrawLine + globalGC1 239 239 3 3 0 360 240 240 psFillArc + globalGC1 240 241 240 240 psDrawLine + globalGC1 239 240 3 3 0 360 240 241 psFillArc + globalGC1 239 242 240 241 psDrawLine + globalGC1 238 241 3 3 0 360 239 242 psFillArc + globalGC1 239 243 239 242 psDrawLine + globalGC1 238 242 3 3 0 360 239 243 psFillArc + globalGC1 238 243 239 243 psDrawLine + globalGC1 237 242 3 3 0 360 238 243 psFillArc + globalGC1 237 244 238 243 psDrawLine + globalGC1 236 243 3 3 0 360 237 244 psFillArc + globalGC1 236 244 237 244 psDrawLine + globalGC1 235 243 3 3 0 360 236 244 psFillArc + globalGC1 235 244 236 244 psDrawLine + globalGC1 234 243 3 3 0 360 235 244 psFillArc + globalGC1 234 244 235 244 psDrawLine + globalGC1 233 243 3 3 0 360 234 244 psFillArc + globalGC1 232 244 234 244 psDrawLine + globalGC1 231 243 3 3 0 360 232 244 psFillArc + globalGC1 229 243 232 244 psDrawLine + globalGC1 228 242 3 3 0 360 229 243 psFillArc + globalGC1 223 240 229 243 psDrawLine + globalGC1 222 239 3 3 0 360 223 240 psFillArc + globalGC1 209 229 223 240 psDrawLine + globalGC1 208 228 3 3 0 360 209 229 psFillArc + globalGC1 192 210 209 229 psDrawLine + globalGC1 191 209 3 3 0 360 192 210 psFillArc + globalGC1 172 187 192 210 psDrawLine + globalGC1 171 186 3 3 0 360 172 187 psFillArc + globalGC1 150 159 172 187 psDrawLine + globalGC1 149 158 3 3 0 360 150 159 psFillArc + globalGC1 128 129 150 159 psDrawLine + globalGC1 127 128 3 3 0 360 128 129 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3d2vara.ps b/books/ps/3d2vara.ps new file mode 100644 index 0000000..b4fc170 --- /dev/null +++ b/books/ps/3d2vara.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state 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222 + 128 234 + 3 135 222 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3d2varb.ps b/books/ps/3d2varb.ps new file mode 100644 index 0000000..3e42ed1 --- /dev/null +++ b/books/ps/3d2varb.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 52 221 227 97 psDrawLine + globGC (X) 47 226 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 247 221 72 97 psDrawLine + globGC (Y) 252 226 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 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281 psFillPoly + renderGC 0.000000 setForeground + renderGC + 143 276 + 150 282 + 3 156 281 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dbuilda.ps b/books/ps/3dbuilda.ps new file mode 100644 index 0000000..cd42f38 --- /dev/null +++ b/books/ps/3dbuilda.ps @@ -0,0 +1,444 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 8 262 257 94 psDrawLine + globGC (X) 3 267 psDrawStr + globalGC1 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componentGC 0.000000 setForeground + componentGC 192 140 207 149 psDrawLine + componentGC 0.000000 setForeground + componentGC 190 188 192 140 psDrawLine + componentGC 0.000000 setForeground + componentGC 190 182 190 188 psDrawLine + componentGC 0.000000 setForeground + componentGC 205 192 190 182 psDrawLine + componentGC 0.000000 setForeground + componentGC 60 157 66 211 psDrawLine + componentGC 0.000000 setForeground + componentGC 67 153 60 157 psDrawLine + componentGC 0.000000 setForeground + componentGC 95 192 67 153 psDrawLine + componentGC 0.000000 setForeground + componentGC 89 196 95 192 psDrawLine + componentGC 0.000000 setForeground + componentGC 80 184 89 196 psDrawLine + componentGC 0.000000 setForeground + componentGC 70 190 80 184 psDrawLine + componentGC 0.000000 setForeground + componentGC 70 184 70 190 psDrawLine + componentGC 0.000000 setForeground + componentGC 76 180 70 184 psDrawLine + componentGC 0.000000 setForeground + componentGC 68 169 76 180 psDrawLine + componentGC 0.000000 setForeground + componentGC 72 207 68 169 psDrawLine + componentGC 0.000000 setForeground + componentGC 66 211 72 207 psDrawLine + componentGC 0.000000 setForeground + componentGC 230 157 225 211 psDrawLine + componentGC 0.000000 setForeground + componentGC 237 161 230 157 psDrawLine + componentGC 0.000000 setForeground + componentGC 246 206 237 161 psDrawLine + componentGC 0.000000 setForeground + componentGC 263 177 246 206 psDrawLine + componentGC 0.000000 setForeground + componentGC 270 182 263 177 psDrawLine + componentGC 0.000000 setForeground + componentGC 263 236 270 182 psDrawLine + componentGC 0.000000 setForeground + componentGC 256 232 263 236 psDrawLine + componentGC 0.000000 setForeground + componentGC 260 199 256 232 psDrawLine + componentGC 0.000000 setForeground + componentGC 248 217 260 199 psDrawLine + componentGC 0.000000 setForeground + componentGC 241 213 248 217 psDrawLine + componentGC 0.000000 setForeground + componentGC 234 183 241 213 psDrawLine + componentGC 0.000000 setForeground + componentGC 231 215 234 183 psDrawLine + componentGC 0.000000 setForeground + componentGC 225 211 231 215 psDrawLine + componentGC 0.000000 setForeground + componentGC 20 182 28 236 psDrawLine + componentGC 0.000000 setForeground + componentGC 28 177 20 182 psDrawLine + componentGC 0.000000 setForeground + componentGC 52 205 28 177 psDrawLine + componentGC 0.000000 setForeground + componentGC 47 165 52 205 psDrawLine + componentGC 0.000000 setForeground + componentGC 54 161 47 165 psDrawLine + componentGC 0.000000 setForeground + componentGC 59 215 54 161 psDrawLine + componentGC 0.000000 setForeground + componentGC 53 220 59 215 psDrawLine + componentGC 0.000000 setForeground + componentGC 29 193 53 220 psDrawLine + componentGC 0.000000 setForeground + componentGC 34 232 29 193 psDrawLine + componentGC 0.000000 setForeground + componentGC 28 236 34 232 psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dbuildb.ps b/books/ps/3dbuildb.ps new file mode 100644 index 0000000..8ab0c30 --- /dev/null +++ b/books/ps/3dbuildb.ps @@ -0,0 +1,444 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 69 199 213 97 psDrawLine + globGC (X) 64 204 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 230 199 86 97 psDrawLine + globGC (Y) 235 204 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 57 149 203 psDrawLine + globGC (Z) 154 52 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.750000 setForeground + renderGC + 149 123 + 43 195 + 256 195 + 4 149 123 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 123 + 43 195 + 256 195 + 4 149 123 psDrawLines + renderGC 0.437500 setForeground + renderGC + 149 123 + 8 71 + 43 195 + 4 149 123 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 123 + 8 71 + 43 195 + 4 149 123 psDrawLines + renderGC 0.500000 setForeground + renderGC + 149 4 + 8 71 + 149 123 + 4 149 4 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 4 + 8 71 + 149 123 + 4 149 4 psDrawLines + renderGC 0.687500 setForeground + renderGC + 256 195 + 149 4 + 149 123 + 4 256 195 psFillPoly + renderGC 0.000000 setForeground + renderGC + 256 195 + 149 4 + 149 123 + 4 256 195 psDrawLines + renderGC 0.687500 setForeground + renderGC + 256 195 + 43 195 + 149 298 + 4 256 195 psFillPoly + renderGC 0.000000 setForeground + renderGC + 256 195 + 43 195 + 149 298 + 4 256 195 psDrawLines + renderGC 0.750000 setForeground + renderGC + 291 71 + 149 4 + 256 195 + 4 291 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 291 71 + 149 4 + 256 195 + 4 291 71 psDrawLines + renderGC 0.625000 setForeground + renderGC + 149 298 + 291 71 + 256 195 + 4 149 298 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 298 + 291 71 + 256 195 + 4 149 298 psDrawLines + renderGC 0.812500 setForeground + renderGC + 43 195 + 149 180 + 149 298 + 4 43 195 psFillPoly + renderGC 0.000000 setForeground + renderGC + 43 195 + 149 180 + 149 298 + 4 43 195 psDrawLines + renderGC 0.625000 setForeground + renderGC + 8 71 + 149 180 + 43 195 + 4 8 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 8 71 + 149 180 + 43 195 + 4 8 71 psDrawLines + renderGC 0.500000 setForeground + renderGC + 149 4 + 149 180 + 8 71 + 4 149 4 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 4 + 149 180 + 8 71 + 4 149 4 psDrawLines + renderGC 0.687500 setForeground + renderGC + 291 71 + 149 180 + 149 4 + 4 291 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 291 71 + 149 180 + 149 4 + 4 291 71 psDrawLines + renderGC 0.687500 setForeground + renderGC + 149 180 + 291 71 + 149 298 + 4 149 180 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 180 + 291 71 + 149 298 + 4 149 180 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dctrl.ps b/books/ps/3dctrl.ps new file mode 100644 index 0000000..96d543d --- /dev/null +++ b/books/ps/3dctrl.ps @@ -0,0 +1,770 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 83.579775 104.120003 528.420225 718.120003 +%%Creator: xwd2ps +%%CreationDate: Sun Dec 1 17:24:32 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 322, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 324, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 322 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 322 445 8 + [322 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +2.53 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +dbdb70 +2f4f4f +d4d8e8 +757780 +46474d +b4b8c5 +236b8e +0000ff +238e6b +2f2f4f +600808 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+ +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDraw: +% psDraw +% vlist[0].y +% vlist[0].x +% n +% ... +% vlist[n].y +% vlist[n].x +% graphics-context dictionary +% to draw lines connecting points in vlist[0] to vlist[n] + +/psDraw + { gsave + newpath + yVal moveto %% set currentpoint + 1 sub { %% loop to draw lines. + yVal lineto + } repeat + begin installGC stroke end %% draw in foreground color + grestore } + def + +% operand stack configuration in order to use psDrawFilled: +% psDrawFilled +% vlist[0].y +% vlist[0].x +% n +% ... +% vlist[n].y +% vlist[n].x +% graphics-context dictionary +% this draws lines connecting all the points and fills the +% region with background color (default: 1, or white). + +/psDrawFilled + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 6 263 259 72 psDrawLine + globGC (X) 1 268 psDrawStr + globalGC1 108.000000 setForeground + globGC 108.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 286 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-0,0 +1,647 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 83.579775 104.120003 528.420225 718.120003 +%%Creator: xwd2ps +%%CreationDate: Sun Dec 1 17:24:46 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 322, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 324, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 322 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 322 445 8 + [322 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +2.53 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +dbdb70 +2f4f4f +d4d8e8 +757780 +46474d +b4b8c5 +236b8e +0000ff +238e6b +2f2f4f +600808 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+%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 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+ renderGC + 122 176 + 127 137 + 3 140 138 psDrawLines + renderGC 0.437500 setForeground + renderGC + 178 136 + 169 177 + 165 138 + 4 178 136 psFillPoly + renderGC 0.000000 setForeground + renderGC + 169 177 + 165 138 + 3 178 136 psDrawLines + renderGC 0.375000 setForeground + renderGC + 153 178 + 165 138 + 169 177 + 4 153 178 psFillPoly + renderGC 0.000000 setForeground + renderGC + 165 138 + 169 177 + 3 153 178 psDrawLines + renderGC 0.312500 setForeground + renderGC + 138 178 + 153 138 + 153 178 + 4 138 178 psFillPoly + renderGC 0.000000 setForeground + renderGC + 153 138 + 153 178 + 3 138 178 psDrawLines + renderGC 0.375000 setForeground + renderGC + 153 138 + 138 178 + 140 138 + 4 153 138 psFillPoly + renderGC 0.000000 setForeground + renderGC + 138 178 + 140 138 + 3 153 138 psDrawLines + renderGC 0.437500 setForeground + renderGC + 165 138 + 153 178 + 153 138 + 4 165 138 psFillPoly + renderGC 0.000000 setForeground + renderGC + 153 178 + 153 138 + 3 165 138 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dmult1b.ps b/books/ps/3dmult1b.ps new file mode 100644 index 0000000..f563436 --- /dev/null +++ b/books/ps/3dmult1b.ps @@ -0,0 +1,24489 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 14 259 256 88 psDrawLine + globGC (X) 9 264 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 285 259 42 88 psDrawLine + globGC (Y) 290 264 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 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setForeground + renderGC + 168 247 + 149 266 + 170 265 + 4 168 247 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 266 + 170 265 + 3 168 247 psDrawLines + renderGC 0.687500 setForeground + renderGC + 149 248 + 128 265 + 149 266 + 4 149 248 psFillPoly + renderGC 0.000000 setForeground + renderGC + 128 265 + 149 266 + 3 149 248 psDrawLines + renderGC 0.625000 setForeground + renderGC + 149 282 + 170 265 + 149 266 + 4 149 282 psFillPoly + renderGC 0.000000 setForeground + renderGC + 170 265 + 149 266 + 3 149 282 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3doptcf1.ps b/books/ps/3doptcf1.ps new file mode 100644 index 0000000..d042a0a --- /dev/null +++ b/books/ps/3doptcf1.ps @@ -0,0 +1,14580 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 21 243 251 81 psDrawLine + globGC (X) 16 248 psDrawStr + globalGC1 108.000000 setForeground + globGC 108.000000 setForeground + globalGC1 0.000000 setForeground + 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+%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context 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4 149 204 psFillPoly + renderGC 0.687500 setForeground + renderGC + 177 203 + 163 204 + 170 215 + 4 177 203 psFillPoly + renderGC 0.500000 setForeground + renderGC + 136 204 + 122 203 + 129 215 + 4 136 204 psFillPoly + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3doptcf3.ps b/books/ps/3doptcf3.ps new file mode 100644 index 0000000..c1e25ca --- /dev/null +++ b/books/ps/3doptcf3.ps @@ -0,0 +1,9024 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC 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setForeground + renderGC + 156 199 + 143 199 + 149 204 + 4 156 199 psFillPoly + renderGC 0.562500 setForeground + renderGC + 156 217 + 170 215 + 163 204 + 4 156 217 psFillPoly + renderGC 0.625000 setForeground + renderGC + 143 217 + 156 217 + 149 204 + 4 143 217 psFillPoly + renderGC 0.562500 setForeground + renderGC + 163 204 + 149 204 + 156 217 + 4 163 204 psFillPoly + renderGC 0.687500 setForeground + renderGC + 129 215 + 143 217 + 136 204 + 4 129 215 psFillPoly + renderGC 0.562500 setForeground + renderGC + 149 204 + 136 204 + 143 217 + 4 149 204 psFillPoly + renderGC 0.562500 setForeground + renderGC + 177 203 + 163 204 + 170 215 + 4 177 203 psFillPoly + renderGC 0.687500 setForeground + renderGC + 136 204 + 122 203 + 129 215 + 4 136 204 psFillPoly + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3doptcrd.ps b/books/ps/3doptcrd.ps new file mode 100644 index 0000000..01d0b91 --- /dev/null +++ b/books/ps/3doptcrd.ps @@ -0,0 +1,16041 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 23 244 250 83 psDrawLine + globGC (X) 18 249 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 277 244 49 83 psDrawLine + globGC (Y) 282 249 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 20 150 251 psDrawLine + globGC (Z) 155 15 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + 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--------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 16 255 255 85 psDrawLine + globGC (X) 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psDrawLines + renderGC 0.750000 setForeground + renderGC + 177 252 + 212 267 + 192 273 + 4 177 252 psFillPoly + renderGC 0.000000 setForeground + renderGC + 212 267 + 192 273 + 3 177 252 psDrawLines + renderGC 0.437500 setForeground + renderGC + 135 255 + 120 218 + 135 220 + 4 135 255 psFillPoly + renderGC 0.000000 setForeground + renderGC + 120 218 + 135 220 + 3 135 255 psDrawLines + renderGC 0.812500 setForeground + renderGC + 135 255 + 150 278 + 128 277 + 4 135 255 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 278 + 128 277 + 3 135 255 psDrawLines + renderGC 0.625000 setForeground + renderGC + 150 278 + 135 255 + 150 255 + 4 150 278 psFillPoly + renderGC 0.000000 setForeground + renderGC + 135 255 + 150 255 + 3 150 278 psDrawLines + renderGC 0.687500 setForeground + renderGC + 135 220 + 150 255 + 135 255 + 4 135 220 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 255 + 135 255 + 3 135 220 psDrawLines + renderGC 0.625000 setForeground + renderGC + 171 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psFillPoly + renderGC 0.000000 setForeground + renderGC + 217 203 + 206 209 + 3 231 260 psDrawLines + renderGC 0.750000 setForeground + renderGC + 217 203 + 231 260 + 247 250 + 4 217 203 psFillPoly + renderGC 0.000000 setForeground + renderGC + 231 260 + 247 250 + 3 217 203 psDrawLines + renderGC 0.500000 setForeground + renderGC + 93 209 + 52 250 + 68 260 + 4 93 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 52 250 + 68 260 + 3 93 209 psDrawLines + renderGC 0.312500 setForeground + renderGC + 68 260 + 106 214 + 93 209 + 4 68 260 psFillPoly + renderGC 0.000000 setForeground + renderGC + 106 214 + 93 209 + 3 68 260 psDrawLines + renderGC 0.562500 setForeground + renderGC + 212 267 + 206 209 + 193 214 + 4 212 267 psFillPoly + renderGC 0.000000 setForeground + renderGC + 206 209 + 193 214 + 3 212 267 psDrawLines + renderGC 0.687500 setForeground + renderGC + 206 209 + 212 267 + 231 260 + 4 206 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 212 267 + 231 260 + 3 206 209 psDrawLines + renderGC 0.500000 setForeground + renderGC + 106 214 + 68 260 + 87 267 + 4 106 214 psFillPoly + renderGC 0.000000 setForeground + renderGC + 68 260 + 87 267 + 3 106 214 psDrawLines + renderGC 0.312500 setForeground + renderGC + 87 267 + 120 218 + 106 214 + 4 87 267 psFillPoly + renderGC 0.000000 setForeground + renderGC + 120 218 + 106 214 + 3 87 267 psDrawLines + renderGC 0.500000 setForeground + renderGC + 192 273 + 193 214 + 179 218 + 4 192 273 psFillPoly + renderGC 0.000000 setForeground + renderGC + 193 214 + 179 218 + 3 192 273 psDrawLines + renderGC 0.625000 setForeground + renderGC + 193 214 + 192 273 + 212 267 + 4 193 214 psFillPoly + renderGC 0.000000 setForeground + renderGC + 192 273 + 212 267 + 3 193 214 psDrawLines + renderGC 0.500000 setForeground + renderGC + 120 218 + 87 267 + 107 273 + 4 120 218 psFillPoly + renderGC 0.000000 setForeground + renderGC + 87 267 + 107 273 + 3 120 218 psDrawLines + renderGC 0.375000 setForeground + renderGC + 107 273 + 135 220 + 120 218 + 4 107 273 psFillPoly + renderGC 0.000000 setForeground + renderGC + 135 220 + 120 218 + 3 107 273 psDrawLines + renderGC 0.437500 setForeground + renderGC + 171 277 + 179 218 + 164 220 + 4 171 277 psFillPoly + renderGC 0.000000 setForeground + renderGC + 179 218 + 164 220 + 3 171 277 psDrawLines + renderGC 0.625000 setForeground + renderGC + 179 218 + 171 277 + 192 273 + 4 179 218 psFillPoly + renderGC 0.000000 setForeground + renderGC + 171 277 + 192 273 + 3 179 218 psDrawLines + renderGC 0.562500 setForeground + renderGC + 135 220 + 107 273 + 128 277 + 4 135 220 psFillPoly + renderGC 0.000000 setForeground + renderGC + 107 273 + 128 277 + 3 135 220 psDrawLines + renderGC 0.375000 setForeground + renderGC + 128 277 + 150 221 + 135 220 + 4 128 277 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 221 + 135 220 + 3 128 277 psDrawLines + renderGC 0.437500 setForeground + renderGC + 150 278 + 164 220 + 150 221 + 4 150 278 psFillPoly + renderGC 0.000000 setForeground + renderGC + 164 220 + 150 221 + 3 150 278 psDrawLines + renderGC 0.562500 setForeground + renderGC + 164 220 + 150 278 + 171 277 + 4 164 220 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 278 + 171 277 + 3 164 220 psDrawLines + renderGC 0.562500 setForeground + renderGC + 150 221 + 128 277 + 150 278 + 4 150 221 psFillPoly + renderGC 0.000000 setForeground + renderGC + 128 277 + 150 278 + 3 150 221 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3doptrad.ps b/books/ps/3doptrad.ps new file mode 100644 index 0000000..2f97b46 --- /dev/null +++ b/books/ps/3doptrad.ps @@ -0,0 +1,8748 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 16 255 255 86 psDrawLine + globGC (X) 11 260 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 284 255 44 86 psDrawLine + globGC (Y) 289 260 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 19 150 262 psDrawLine + globGC (Z) 155 14 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.625000 setForeground + renderGC + 150 72 + 165 94 + 150 93 + 4 150 72 psFillPoly + renderGC 0.000000 setForeground + renderGC + 165 94 + 150 93 + 3 150 72 psDrawLines + renderGC 0.562500 setForeground + renderGC + 162 113 + 150 93 + 165 94 + 4 162 113 psFillPoly + 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+ 129 252 + 150 224 + 133 223 + 4 129 252 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 224 + 133 223 + 3 129 252 psDrawLines + renderGC 0.500000 setForeground + renderGC + 150 253 + 166 223 + 150 224 + 4 150 253 psFillPoly + renderGC 0.000000 setForeground + renderGC + 166 223 + 150 224 + 3 150 253 psDrawLines + renderGC 0.500000 setForeground + renderGC + 166 223 + 150 253 + 170 252 + 4 166 223 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 253 + 170 252 + 3 166 223 psDrawLines + renderGC 0.500000 setForeground + renderGC + 150 224 + 129 252 + 150 253 + 4 150 224 psFillPoly + renderGC 0.000000 setForeground + renderGC + 129 252 + 150 253 + 3 150 224 psDrawLines + renderGC 0.375000 setForeground + renderGC + 150 278 + 170 252 + 150 253 + 4 150 278 psFillPoly + renderGC 0.000000 setForeground + renderGC + 170 252 + 150 253 + 3 150 278 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3doptsty.ps b/books/ps/3doptsty.ps new file mode 100644 index 0000000..635fcce --- /dev/null +++ b/books/ps/3doptsty.ps @@ -0,0 +1,1761 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 144.385410 559.000000 677.854595 +%%Creator: xwd2ps +%%CreationDate: Thu Mar 26 15:55:23 1992 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 422, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 445 8 + [422 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +4 inch 4 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +0000ff +ff00ff +00ff00 +ffff00 +00ffff +ff0000 +23238e +d3d3d3 +d4d8e8 +757780 +46474d +b4b8c5 +600808 +931c1c +bf3030 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 422, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 445 8 + [422 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +4 inch 4 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +0000ff +ff00ff +00ff00 +ffff00 +00ffff +ff0000 +23238e +d3d3d3 +d4d8e8 +757780 +46474d +b4b8c5 +600808 +931c1c +bf3030 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+%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 48 205 230 76 psDrawLine + globGC (X) 43 210 psDrawStr + globalGC1 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psDrawLines + renderGC 0.500000 setForeground + renderGC + 143 215 + 130 216 + 136 223 + 4 143 215 psFillPoly + renderGC 0.000000 setForeground + renderGC + 130 216 + 136 223 + 3 143 215 psDrawLines + renderGC 0.500000 setForeground + renderGC + 149 224 + 162 223 + 156 215 + 4 149 224 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 223 + 156 215 + 3 149 224 psDrawLines + renderGC 0.562500 setForeground + renderGC + 136 223 + 149 224 + 143 215 + 4 136 223 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 224 + 143 215 + 3 136 223 psDrawLines + renderGC 0.500000 setForeground + renderGC + 156 215 + 143 215 + 149 224 + 4 156 215 psFillPoly + renderGC 0.000000 setForeground + renderGC + 143 215 + 149 224 + 3 156 215 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dpsa.ps b/books/ps/3dpsa.ps new file mode 100644 index 0000000..a96c258 --- /dev/null +++ b/books/ps/3dpsa.ps @@ -0,0 +1,15744 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 18 252 254 85 psDrawLine + globGC (X) 13 257 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 282 252 46 85 psDrawLine + globGC (Y) 287 257 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 19 150 259 psDrawLine + globGC (Z) 155 14 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + 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setForeground + renderGC + 170 260 + 136 277 + 174 277 + 4 170 260 psFillPoly + renderGC 0.000000 setForeground + renderGC + 136 277 + 174 277 + 3 170 260 psDrawLines + renderGC 0.750000 setForeground + renderGC + 136 277 + 170 260 + 138 259 + 4 136 277 psFillPoly + renderGC 0.000000 setForeground + renderGC + 170 260 + 138 259 + 3 136 277 psDrawLines + renderGC 0.750000 setForeground + renderGC + 174 277 + 138 259 + 170 260 + 4 174 277 psFillPoly + renderGC 0.000000 setForeground + renderGC + 138 259 + 170 260 + 3 174 277 psDrawLines + renderGC 0.750000 setForeground + renderGC + 138 259 + 174 277 + 136 277 + 4 138 259 psFillPoly + renderGC 0.000000 setForeground + renderGC + 174 277 + 136 277 + 3 138 259 psDrawLines + renderGC 0.750000 setForeground + renderGC + 138 259 + 99 270 + 136 277 + 4 138 259 psFillPoly + renderGC 0.000000 setForeground + renderGC + 99 270 + 136 277 + 3 138 259 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dpsb.ps b/books/ps/3dpsb.ps new file mode 100644 index 0000000..e169d0b --- /dev/null +++ b/books/ps/3dpsb.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 29 235 245 82 psDrawLine + globGC (X) 24 240 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 270 235 54 82 psDrawLine + globGC (Y) 275 240 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 22 149 241 psDrawLine + 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psDrawLines + renderGC 0.687500 setForeground + renderGC + 196 193 + 187 231 + 184 200 + 4 196 193 psFillPoly + renderGC 0.000000 setForeground + renderGC + 187 231 + 184 200 + 3 196 193 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dpsca.ps b/books/ps/3dpsca.ps new file mode 100644 index 0000000..634410d --- /dev/null +++ b/books/ps/3dpsca.ps @@ -0,0 +1,446 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 18 256 253 90 psDrawLine + globGC (X) 13 261 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 281 256 46 90 psDrawLine + globGC (Y) 286 261 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 25 150 263 psDrawLine + globGC (Z) 155 20 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC 0.000000 setForeground + componentGC 151 292 138 292 psDrawLine + componentGC 0.000000 setForeground + componentGC 165 288 151 292 psDrawLine + componentGC 0.000000 setForeground + componentGC 177 282 165 288 psDrawLine + componentGC 0.000000 setForeground + componentGC 187 274 177 282 psDrawLine + componentGC 0.000000 setForeground + componentGC 195 264 187 274 psDrawLine + componentGC 0.000000 setForeground + componentGC 200 253 195 264 psDrawLine + componentGC 0.000000 setForeground + componentGC 202 241 200 253 psDrawLine + componentGC 0.000000 setForeground + componentGC 200 229 202 241 psDrawLine + componentGC 0.000000 setForeground + componentGC 195 218 200 229 psDrawLine + componentGC 0.000000 setForeground + componentGC 188 208 195 218 psDrawLine + componentGC 0.000000 setForeground + componentGC 178 200 188 208 psDrawLine + componentGC 0.000000 setForeground + componentGC 167 194 178 200 psDrawLine + componentGC 0.000000 setForeground + componentGC 155 190 167 194 psDrawLine + componentGC 0.000000 setForeground + componentGC 143 189 155 190 psDrawLine + componentGC 0.000000 setForeground + componentGC 130 190 143 189 psDrawLine + componentGC 0.000000 setForeground + componentGC 119 192 130 190 psDrawLine + componentGC 0.000000 setForeground + componentGC 109 197 119 192 psDrawLine + componentGC 0.000000 setForeground + componentGC 102 204 109 197 psDrawLine + componentGC 0.000000 setForeground + componentGC 97 211 102 204 psDrawLine + componentGC 0.000000 setForeground + componentGC 95 220 97 211 psDrawLine + componentGC 0.000000 setForeground + componentGC 97 228 95 220 psDrawLine + componentGC 0.000000 setForeground + componentGC 102 236 97 228 psDrawLine + componentGC 0.000000 setForeground + componentGC 111 243 102 236 psDrawLine + componentGC 0.000000 setForeground + componentGC 122 248 111 243 psDrawLine + componentGC 0.000000 setForeground + componentGC 135 250 122 248 psDrawLine + componentGC 0.000000 setForeground + componentGC 149 250 135 250 psDrawLine + componentGC 0.000000 setForeground + componentGC 164 247 149 250 psDrawLine + componentGC 0.000000 setForeground + componentGC 177 241 164 247 psDrawLine + componentGC 0.000000 setForeground + componentGC 189 232 177 241 psDrawLine + componentGC 0.000000 setForeground + componentGC 198 222 189 232 psDrawLine + componentGC 0.000000 setForeground + componentGC 203 210 198 222 psDrawLine + componentGC 0.000000 setForeground + componentGC 205 198 203 210 psDrawLine + componentGC 0.000000 setForeground + componentGC 204 186 205 198 psDrawLine + componentGC 0.000000 setForeground + componentGC 199 175 204 186 psDrawLine + componentGC 0.000000 setForeground + componentGC 192 164 199 175 psDrawLine + componentGC 0.000000 setForeground + componentGC 182 156 192 164 psDrawLine + componentGC 0.000000 setForeground + componentGC 170 149 182 156 psDrawLine + componentGC 0.000000 setForeground + componentGC 157 145 170 149 psDrawLine + componentGC 0.000000 setForeground + componentGC 144 143 157 145 psDrawLine + componentGC 0.000000 setForeground + componentGC 131 143 144 143 psDrawLine + componentGC 0.000000 setForeground + componentGC 118 145 131 143 psDrawLine + componentGC 0.000000 setForeground + componentGC 108 150 118 145 psDrawLine + componentGC 0.000000 setForeground + componentGC 99 156 108 150 psDrawLine + componentGC 0.000000 setForeground + componentGC 94 163 99 156 psDrawLine + componentGC 0.000000 setForeground + componentGC 92 171 94 163 psDrawLine + componentGC 0.000000 setForeground + componentGC 93 180 92 171 psDrawLine + componentGC 0.000000 setForeground + componentGC 98 187 93 180 psDrawLine + componentGC 0.000000 setForeground + componentGC 106 194 98 187 psDrawLine + componentGC 0.000000 setForeground + componentGC 118 199 106 194 psDrawLine + componentGC 0.000000 setForeground + componentGC 132 201 118 199 psDrawLine + componentGC 0.000000 setForeground + componentGC 147 201 132 201 psDrawLine + componentGC 0.000000 setForeground + componentGC 163 198 147 201 psDrawLine + componentGC 0.000000 setForeground + componentGC 178 192 163 198 psDrawLine + componentGC 0.000000 setForeground + componentGC 190 184 178 192 psDrawLine + componentGC 0.000000 setForeground + componentGC 200 173 190 184 psDrawLine + componentGC 0.000000 setForeground + componentGC 207 161 200 173 psDrawLine + componentGC 0.000000 setForeground + componentGC 210 149 207 161 psDrawLine + componentGC 0.000000 setForeground + componentGC 209 136 210 149 psDrawLine + componentGC 0.000000 setForeground + componentGC 204 125 209 136 psDrawLine + componentGC 0.000000 setForeground + componentGC 196 114 204 125 psDrawLine + componentGC 0.000000 setForeground + componentGC 186 105 196 114 psDrawLine + componentGC 0.000000 setForeground + componentGC 173 98 186 105 psDrawLine + componentGC 0.000000 setForeground + componentGC 160 93 173 98 psDrawLine + componentGC 0.000000 setForeground + componentGC 145 90 160 93 psDrawLine + componentGC 0.000000 setForeground + componentGC 131 90 145 90 psDrawLine + componentGC 0.000000 setForeground + componentGC 118 91 131 90 psDrawLine + componentGC 0.000000 setForeground + componentGC 106 95 118 91 psDrawLine + componentGC 0.000000 setForeground + componentGC 97 101 106 95 psDrawLine + componentGC 0.000000 setForeground + componentGC 90 107 97 101 psDrawLine + componentGC 0.000000 setForeground + componentGC 87 115 90 107 psDrawLine + componentGC 0.000000 setForeground + componentGC 88 123 87 115 psDrawLine + componentGC 0.000000 setForeground + componentGC 93 131 88 123 psDrawLine + componentGC 0.000000 setForeground + componentGC 101 137 93 131 psDrawLine + componentGC 0.000000 setForeground + componentGC 114 142 101 137 psDrawLine + componentGC 0.000000 setForeground + componentGC 128 144 114 142 psDrawLine + componentGC 0.000000 setForeground + componentGC 145 144 128 144 psDrawLine + componentGC 0.000000 setForeground + componentGC 162 141 145 144 psDrawLine + componentGC 0.000000 setForeground + componentGC 178 135 162 141 psDrawLine + componentGC 0.000000 setForeground + componentGC 192 127 178 135 psDrawLine + componentGC 0.000000 setForeground + componentGC 203 116 192 127 psDrawLine + componentGC 0.000000 setForeground + componentGC 211 104 203 116 psDrawLine + componentGC 0.000000 setForeground + componentGC 214 91 211 104 psDrawLine + componentGC 0.000000 setForeground + componentGC 214 79 214 91 psDrawLine + componentGC 0.000000 setForeground + componentGC 209 66 214 79 psDrawLine + componentGC 0.000000 setForeground + componentGC 201 55 209 66 psDrawLine + componentGC 0.000000 setForeground + componentGC 190 46 201 55 psDrawLine + componentGC 0.000000 setForeground + componentGC 177 38 190 46 psDrawLine + componentGC 0.000000 setForeground + componentGC 163 33 177 38 psDrawLine + componentGC 0.000000 setForeground + componentGC 147 29 163 33 psDrawLine + componentGC 0.000000 setForeground + componentGC 132 28 147 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 117 29 132 28 psDrawLine + componentGC 0.000000 setForeground + componentGC 104 32 117 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 93 37 104 32 psDrawLine + componentGC 0.000000 setForeground + componentGC 86 43 93 37 psDrawLine + componentGC 0.000000 setForeground + componentGC 82 50 86 43 psDrawLine + componentGC 0.000000 setForeground + componentGC 82 57 82 50 psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/3dpscb.ps b/books/ps/3dpscb.ps new file mode 100644 index 0000000..61c1a60 --- /dev/null +++ b/books/ps/3dpscb.ps @@ -0,0 +1,1142 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 19 254 258 88 psDrawLine + globGC (X) 14 259 psDrawStr + globalGC1 107.000000 setForeground + globGC 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0000000..8e158b6 --- /dev/null +++ b/books/ps/3dvolume.ps @@ -0,0 +1,682 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 83.579775 104.120003 528.420225 718.120003 +%%Creator: xwd2ps +%%CreationDate: Sun Dec 1 18:05:05 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 322, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 324, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 322 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 322 445 8 + [322 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +2.53 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +dbdb70 +2f4f4f +d4d8e8 +757780 +46474d +b4b8c5 +236b8e +0000ff +238e6b +2f2f4f +600808 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+%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 47 213 234 84 psDrawLine + globGC (X) 42 218 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 252 213 65 84 psDrawLine + globGC (Y) 257 218 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 32 150 225 psDrawLine + globGC (Z) 155 27 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC 0.000000 setForeground + componentGC 150 182 150 142 psDrawLine + componentGC 0.000000 setForeground + componentGC 155 171 150 182 psDrawLine + componentGC 0.000000 setForeground + componentGC 144 171 150 182 psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/arrowr.ps b/books/ps/arrowr.ps new file mode 100644 index 0000000..e356449 --- /dev/null +++ b/books/ps/arrowr.ps @@ -0,0 +1,260 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 255 132 16 187 psDrawLine + globGC (X) 260 127 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 114 95 190 251 psDrawLine + globGC (Y) 109 90 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 140 46 148 244 psDrawLine + globGC (Z) 135 41 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC 0.000000 setForeground + componentGC 204 6 144 157 psDrawLine + componentGC 0.000000 setForeground + componentGC 193 24 204 6 psDrawLine + componentGC 0.000000 setForeground + componentGC 199 25 204 6 psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/atan-1.epsi b/books/ps/atan-1.epsi new file mode 100644 index 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+%%EndDocument +%%Trailer +cleartomark countdictstack exch sub { end } repeat restore +%%EOF diff --git a/books/ps/atan-1.ps b/books/ps/atan-1.ps new file mode 100644 index 0000000..efb311a --- /dev/null +++ b/books/ps/atan-1.ps @@ -0,0 +1,16391 @@ +%-------------------------- postscript draw commands --------------------------% +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + 0 0 3 + 1072693248 0 /frameDict makeDict + + gsave % save graphics state for clipping path + + 0 260 257 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 175 84 65 188 psDrawLine + globGC (X) 180 79 psDrawStr + globalGC1 219.000000 setForeground + globGC 219.000000 setForeground + globalGC1 0.000000 setForeground + globGC 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setForeground + renderGC + 76 128 + 69 128 + 73 130 + 4 76 128 psFillPoly + renderGC 0.000000 setForeground + renderGC + 69 128 + 73 130 + 3 76 128 psDrawLines + renderGC 0.625000 setForeground + renderGC + 76 128 + 77 74 + 73 71 + 4 76 128 psFillPoly + renderGC 0.000000 setForeground + renderGC + 77 74 + 73 71 + 3 76 128 psDrawLines + renderGC 0.437500 setForeground + renderGC + 77 74 + 76 128 + 80 130 + 4 77 74 psFillPoly + renderGC 0.000000 setForeground + renderGC + 76 128 + 80 130 + 3 77 74 psDrawLines + renderGC 0.562500 setForeground + renderGC + 73 130 + 80 130 + 76 128 + 4 73 130 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 130 + 76 128 + 3 73 130 psDrawLines + renderGC 0.562500 setForeground + renderGC + 65 131 + 73 130 + 69 128 + 4 65 131 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 130 + 69 128 + 3 65 131 psDrawLines + renderGC 0.500000 setForeground + renderGC + 80 130 + 73 130 + 76 132 + 4 80 130 psFillPoly + renderGC 0.000000 setForeground 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b/books/ps/basic2d.ps @@ -0,0 +1,1375 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 128.262184 559.000000 693.977821 +%%Creator: xwd2ps +%%CreationDate: Mon Dec 2 10:32:59 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +matrix currentmatrix +3.125 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +dbdb70 +2f4f4f +d4d8e8 +757780 +46474d +b4b8c5 +236b8e +0000ff +238e6b +2f2f4f +600808 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-------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 31 232 219 101 psDrawLine + globGC (X) 26 237 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 238 232 56 101 psDrawLine + globGC (Y) 243 237 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 134 49 138 236 psDrawLine + globGC (Z) 129 44 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.500000 setForeground + renderGC + 135 137 + 140 138 + 137 136 + 4 135 137 psFillPoly + renderGC 0.000000 setForeground + renderGC + 140 138 + 137 136 + 3 135 137 psDrawLines + renderGC 0.625000 setForeground + renderGC + 142 68 + 137 136 + 140 138 + 4 142 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 136 + 140 138 + 3 142 68 psDrawLines + renderGC 0.812500 setForeground + renderGC + 137 136 + 142 68 + 139 67 + 4 137 136 psFillPoly + renderGC 0.000000 setForeground + renderGC + 142 68 + 139 67 + 3 137 136 psDrawLines + renderGC 0.500000 setForeground + renderGC + 133 139 + 138 139 + 135 137 + 4 133 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 138 139 + 135 137 + 3 133 139 psDrawLines + renderGC 0.500000 setForeground + renderGC + 140 138 + 135 137 + 138 139 + 4 140 138 psFillPoly + renderGC 0.000000 setForeground + renderGC + 135 137 + 138 139 + 3 140 138 psDrawLines + renderGC 0.250000 setForeground + renderGC + 138 139 + 143 137 + 140 138 + 4 138 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 143 137 + 140 138 + 3 138 139 psDrawLines + renderGC 0.687500 setForeground + renderGC + 145 70 + 140 138 + 143 137 + 4 145 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 140 138 + 143 137 + 3 145 70 psDrawLines + renderGC 0.812500 setForeground + renderGC + 140 138 + 145 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00000000000000000000000000000000000000000000000000000000000000000000000000000000 +% 0000000000000000000000000000000000 +%%EndImage +%%EndPreview +save countdictstack mark newpath /showpage {} def /setpagedevice {pop} def +%%EndProlog +%%Page: 1 1 +%%BeginDocument: bessintr.ps +%%EndDocument +%%Trailer +cleartomark countdictstack exch sub { end } repeat restore +%%EOF diff --git a/books/ps/bessintr.ps b/books/ps/bessintr.ps new file mode 100644 index 0000000..b3fbcc3 --- /dev/null +++ b/books/ps/bessintr.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 0 300 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 51 196 227 71 psDrawLine + globGC (X) 46 201 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 h-root.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.919405 inch translate +matrix currentmatrix +4.500000 inch 3.581190 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 49 206 227 87 psDrawLine + globGC (X) 44 211 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 249 206 71 87 psDrawLine + globGC (Y) 254 211 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 27 149 218 psDrawLine + globGC (Z) 154 22 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.687500 setForeground + renderGC + 146 112 + 152 112 + 149 158 + 4 146 112 psFillPoly + renderGC 0.000000 setForeground + renderGC + 152 112 + 149 158 + 3 146 112 psDrawLines + renderGC 0.812500 setForeground + renderGC + 156 79 + 149 158 + 152 112 + 4 156 79 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 158 + 152 112 + 3 156 79 psDrawLines + renderGC 0.812500 setForeground + renderGC + 149 158 + 156 79 + 152 132 + 4 149 158 psFillPoly + renderGC 0.000000 setForeground + renderGC + 156 79 + 152 132 + 3 149 158 psDrawLines + renderGC 0.812500 setForeground + renderGC + 149 158 + 142 79 + 146 112 + 4 149 158 psFillPoly + renderGC 0.000000 setForeground + renderGC + 142 79 + 146 112 + 3 149 158 psDrawLines + renderGC 0.812500 setForeground + renderGC + 142 79 + 149 158 + 146 132 + 4 142 79 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 158 + 146 132 + 3 142 79 psDrawLines + renderGC 0.437500 setForeground + renderGC + 152 132 + 146 132 + 149 158 + 4 152 132 psFillPoly + renderGC 0.000000 setForeground + renderGC + 146 132 + 149 158 + 3 152 132 psDrawLines + renderGC 0.687500 setForeground + renderGC + 166 122 + 175 54 + 169 121 + 4 166 122 psFillPoly + renderGC 0.000000 setForeground + renderGC + 175 54 + 169 121 + 3 166 122 psDrawLines + renderGC 0.812500 setForeground + renderGC + 169 121 + 164 84 + 166 122 + 4 169 121 psFillPoly + renderGC 0.000000 setForeground + renderGC + 164 84 + 166 122 + 3 169 121 psDrawLines + renderGC 0.687500 setForeground + renderGC + 164 84 + 169 121 + 167 81 + 4 164 84 psFillPoly 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165 + 43 164 + 39 162 + 4 35 165 psFillPoly + renderGC 0.000000 setForeground + renderGC + 43 164 + 39 162 + 3 35 165 psDrawLines + renderGC 0.500000 setForeground + renderGC + 47 162 + 39 162 + 43 164 + 4 47 162 psFillPoly + renderGC 0.000000 setForeground + renderGC + 39 162 + 43 164 + 3 47 162 psDrawLines + renderGC 0.437500 setForeground + renderGC + 194 167 + 202 165 + 198 162 + 4 194 167 psFillPoly + renderGC 0.000000 setForeground + renderGC + 202 165 + 198 162 + 3 194 167 psDrawLines + renderGC 0.500000 setForeground + renderGC + 206 162 + 198 162 + 202 165 + 4 206 162 psFillPoly + renderGC 0.000000 setForeground + renderGC + 198 162 + 202 165 + 3 206 162 psDrawLines + renderGC 0.750000 setForeground + renderGC + 96 165 + 104 167 + 100 162 + 4 96 165 psFillPoly + renderGC 0.000000 setForeground + renderGC + 104 167 + 100 162 + 3 96 165 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbon1.ps b/books/ps/ribbon1.ps new file mode 100644 index 0000000..ed2e084 --- /dev/null +++ b/books/ps/ribbon1.ps @@ -0,0 +1,1900 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 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91 + 215 89 + 212 87 + 209 85 + 206 83 + 203 82 + 200 80 + 197 78 + 194 76 + 191 74 + 188 73 + 185 71 + 182 69 + 179 67 + 176 66 + 173 64 + 28 170 62 psDrawLines + componentGC + 248 126 + 244 124 + 241 122 + 238 120 + 234 118 + 231 116 + 228 114 + 225 112 + 221 110 + 218 108 + 215 106 + 212 104 + 209 102 + 206 100 + 203 98 + 200 96 + 197 95 + 194 93 + 191 91 + 188 89 + 185 87 + 182 85 + 179 84 + 176 82 + 173 80 + 170 78 + 167 76 + 28 165 75 psDrawLines + componentGC + 241 139 + 238 136 + 234 134 + 231 132 + 228 130 + 225 128 + 221 126 + 218 124 + 215 122 + 212 120 + 209 118 + 206 116 + 203 114 + 200 112 + 197 110 + 194 108 + 191 106 + 188 104 + 185 103 + 182 101 + 179 99 + 176 97 + 173 95 + 170 93 + 167 92 + 165 90 + 162 88 + 28 159 86 psDrawLines + componentGC + 235 150 + 231 148 + 228 146 + 225 143 + 222 141 + 218 139 + 215 137 + 212 135 + 209 133 + 206 131 + 203 129 + 200 127 + 197 125 + 194 123 + 191 121 + 188 119 + 185 117 + 182 115 + 179 113 + 176 111 + 173 110 + 170 108 + 167 106 + 165 104 + 162 102 + 159 100 + 156 99 + 28 154 97 psDrawLines + componentGC + 228 160 + 225 158 + 222 156 + 219 154 + 216 152 + 212 149 + 209 147 + 206 145 + 203 143 + 200 141 + 197 139 + 194 137 + 191 135 + 188 133 + 185 131 + 182 129 + 179 127 + 176 125 + 173 123 + 170 121 + 168 119 + 165 117 + 162 116 + 159 114 + 156 112 + 154 110 + 151 108 + 28 148 106 psDrawLines + componentGC + 222 170 + 219 167 + 216 165 + 213 163 + 210 161 + 206 159 + 203 157 + 200 154 + 197 152 + 194 150 + 191 148 + 188 146 + 185 144 + 182 142 + 179 140 + 176 138 + 173 136 + 171 134 + 168 132 + 165 130 + 162 128 + 159 126 + 156 124 + 154 123 + 151 121 + 148 119 + 145 117 + 28 143 115 psDrawLines + componentGC + 216 178 + 213 176 + 210 174 + 207 172 + 204 169 + 201 167 + 197 165 + 194 163 + 191 161 + 188 159 + 185 157 + 182 154 + 179 152 + 176 150 + 174 148 + 171 146 + 168 144 + 165 142 + 162 140 + 159 138 + 156 136 + 154 134 + 151 132 + 148 130 + 145 129 + 143 127 + 140 125 + 28 137 123 psDrawLines + componentGC + 210 186 + 207 184 + 204 181 + 201 179 + 198 177 + 195 175 + 192 173 + 189 170 + 186 168 + 183 166 + 180 164 + 177 162 + 174 160 + 171 158 + 168 156 + 165 154 + 162 152 + 159 149 + 156 147 + 154 145 + 151 143 + 148 142 + 145 140 + 143 138 + 140 136 + 137 134 + 135 132 + 28 132 130 psDrawLines + componentGC + 205 193 + 201 191 + 198 188 + 195 186 + 192 184 + 189 182 + 186 179 + 183 177 + 180 175 + 177 173 + 174 171 + 171 169 + 168 166 + 165 164 + 162 162 + 159 160 + 157 158 + 154 156 + 151 154 + 148 152 + 145 150 + 143 148 + 140 146 + 137 144 + 134 142 + 132 140 + 129 138 + 28 126 136 psDrawLines + componentGC + 199 199 + 196 197 + 193 195 + 189 192 + 186 190 + 183 188 + 180 185 + 177 183 + 174 181 + 171 179 + 168 177 + 165 174 + 162 172 + 160 170 + 157 168 + 154 166 + 151 164 + 148 162 + 145 160 + 142 158 + 140 156 + 137 154 + 134 152 + 132 150 + 129 148 + 126 146 + 123 144 + 28 121 142 psDrawLines + componentGC + 193 205 + 190 202 + 187 200 + 184 198 + 181 195 + 178 193 + 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92 170 + 89 167 + 86 165 + 83 163 + 81 161 + 28 78 159 psDrawLines + componentGC + 151 223 + 148 220 + 145 218 + 141 215 + 138 212 + 135 210 + 132 207 + 129 205 + 126 202 + 123 200 + 120 197 + 116 195 + 113 192 + 110 190 + 108 188 + 105 185 + 102 183 + 99 180 + 96 178 + 93 176 + 90 174 + 87 171 + 85 169 + 82 167 + 79 165 + 76 162 + 74 160 + 28 71 158 psDrawLines + componentGC + 144 222 + 141 220 + 138 217 + 135 214 + 131 212 + 128 209 + 125 206 + 122 204 + 119 201 + 116 199 + 113 196 + 109 194 + 106 191 + 103 189 + 100 186 + 97 184 + 94 182 + 92 179 + 89 177 + 86 175 + 83 172 + 80 170 + 77 168 + 74 165 + 72 163 + 69 161 + 66 159 + 28 63 156 psDrawLines + componentGC + 138 221 + 134 218 + 131 215 + 128 213 + 124 210 + 121 207 + 118 205 + 115 202 + 111 199 + 108 197 + 105 194 + 102 192 + 99 189 + 96 187 + 93 184 + 90 182 + 87 179 + 84 177 + 81 175 + 78 172 + 75 170 + 72 168 + 70 165 + 67 163 + 64 161 + 61 158 + 58 156 + 28 56 154 psDrawLines + componentGC + 130 219 + 127 216 + 123 213 + 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45 155 + 42 152 + 39 150 + 36 147 + 33 145 + 30 142 + 27 140 + 24 137 + 21 135 + 28 19 132 psDrawLines + componentGC + 96 197 + 92 194 + 89 191 + 85 188 + 82 185 + 78 182 + 74 179 + 71 177 + 68 174 + 64 171 + 61 168 + 57 165 + 54 162 + 51 160 + 47 157 + 44 154 + 41 152 + 38 149 + 35 146 + 32 144 + 28 141 + 25 139 + 22 136 + 19 134 + 16 131 + 13 129 + 10 126 + 28 7 124 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbon2.ps b/books/ps/ribbon2.ps new file mode 100644 index 0000000..203629d --- /dev/null +++ b/books/ps/ribbon2.ps @@ -0,0 +1,418 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 17 215 192 94 psDrawLine + globGC (X) 12 220 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 202 215 42 94 psDrawLine + globGC (Y) 207 220 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 108 53 119 219 psDrawLine + globGC (Z) 103 48 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC + 14 124 + 25 133 + 34 140 + 44 145 + 52 150 + 60 153 + 67 155 + 75 157 + 81 158 + 88 158 + 94 157 + 100 155 + 106 153 + 111 150 + 117 146 + 122 141 + 127 136 + 133 130 + 138 124 + 143 116 + 148 108 + 153 99 + 159 89 + 164 78 + 169 66 + 175 53 + 180 38 + 28 186 23 psDrawLines + componentGC + 99 195 + 108 202 + 116 207 + 124 211 + 131 215 + 138 217 + 145 218 + 151 219 + 157 219 + 163 217 + 169 216 + 175 213 + 180 210 + 186 206 + 191 202 + 197 196 + 202 190 + 208 184 + 213 176 + 219 168 + 225 159 + 231 149 + 237 138 + 243 127 + 250 114 + 256 100 + 264 85 + 28 271 68 psDrawLines + componentGC + 271 68 + 2 186 23 psDrawLines + componentGC + 264 85 + 2 180 38 psDrawLines + componentGC + 256 100 + 2 175 53 psDrawLines + componentGC + 250 114 + 2 169 66 psDrawLines + componentGC + 243 127 + 2 164 78 psDrawLines + componentGC + 237 138 + 2 159 89 psDrawLines + componentGC + 231 149 + 2 153 99 psDrawLines + componentGC + 225 159 + 2 148 108 psDrawLines + componentGC + 219 168 + 2 143 116 psDrawLines + componentGC + 213 176 + 2 138 124 psDrawLines + componentGC + 208 184 + 2 133 130 psDrawLines + componentGC + 202 190 + 2 127 136 psDrawLines + componentGC + 197 196 + 2 122 141 psDrawLines + componentGC + 191 202 + 2 117 146 psDrawLines + componentGC + 186 206 + 2 111 150 psDrawLines + componentGC + 180 210 + 2 106 153 psDrawLines + componentGC + 175 213 + 2 100 155 psDrawLines + componentGC + 169 216 + 2 94 157 psDrawLines + componentGC + 163 217 + 2 88 158 psDrawLines + componentGC + 157 219 + 2 81 158 psDrawLines + componentGC + 151 219 + 2 75 157 psDrawLines + componentGC + 145 218 + 2 67 155 psDrawLines + componentGC + 138 217 + 2 60 153 psDrawLines + componentGC + 131 215 + 2 52 150 psDrawLines + componentGC + 124 211 + 2 44 145 psDrawLines + componentGC + 116 207 + 2 34 140 psDrawLines + componentGC + 108 202 + 2 25 133 psDrawLines + componentGC + 99 195 + 2 14 124 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbon2r.ps b/books/ps/ribbon2r.ps new file mode 100644 index 0000000..d198150 --- /dev/null +++ b/books/ps/ribbon2r.ps @@ -0,0 +1,418 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 12 196 287 196 psDrawLine + globGC (X) 7 201 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 212 150 186 psDrawLine + globGC (Y) 155 217 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 59 150 334 psDrawLine + globGC (Z) 155 54 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC + 32 79 + 40 95 + 49 111 + 58 125 + 67 138 + 75 150 + 84 160 + 93 169 + 102 177 + 110 183 + 119 189 + 128 192 + 136 195 + 145 196 + 154 196 + 163 195 + 171 192 + 180 189 + 189 183 + 197 177 + 206 169 + 215 160 + 224 150 + 232 138 + 241 125 + 250 111 + 259 95 + 28 267 79 psDrawLines + componentGC + 6 66 + 16 86 + 27 105 + 38 122 + 48 138 + 59 153 + 70 165 + 80 176 + 91 186 + 102 194 + 112 200 + 123 205 + 134 208 + 144 209 + 155 209 + 165 208 + 176 205 + 187 200 + 197 194 + 208 186 + 219 176 + 229 165 + 240 153 + 251 138 + 261 122 + 272 105 + 283 86 + 28 293 66 psDrawLines + componentGC + 293 66 + 2 267 79 psDrawLines + componentGC + 283 86 + 2 259 95 psDrawLines + componentGC + 272 105 + 2 250 111 psDrawLines + componentGC + 261 122 + 2 241 125 psDrawLines + componentGC + 251 138 + 2 232 138 psDrawLines + componentGC + 240 153 + 2 224 150 psDrawLines + componentGC + 229 165 + 2 215 160 psDrawLines + componentGC + 219 176 + 2 206 169 psDrawLines + componentGC + 208 186 + 2 197 177 psDrawLines + componentGC + 197 194 + 2 189 183 psDrawLines + componentGC + 187 200 + 2 180 189 psDrawLines + componentGC + 176 205 + 2 171 192 psDrawLines + componentGC + 165 208 + 2 163 195 psDrawLines + componentGC + 155 209 + 2 154 196 psDrawLines + componentGC + 144 209 + 2 145 196 psDrawLines + componentGC + 134 208 + 2 136 195 psDrawLines + componentGC + 123 205 + 2 128 192 psDrawLines + componentGC + 112 200 + 2 119 189 psDrawLines + componentGC + 102 194 + 2 110 183 psDrawLines + componentGC + 91 186 + 2 102 177 psDrawLines + componentGC + 80 176 + 2 93 169 psDrawLines + componentGC + 70 165 + 2 84 160 psDrawLines + componentGC + 59 153 + 2 75 150 psDrawLines + componentGC + 48 138 + 2 67 138 psDrawLines + componentGC + 38 122 + 2 58 125 psDrawLines + componentGC + 27 105 + 2 49 111 psDrawLines + componentGC + 16 86 + 2 40 95 psDrawLines + componentGC + 6 66 + 2 32 79 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbons.ps b/books/ps/ribbons.ps new file mode 100644 index 0000000..e4a3e34 --- /dev/null +++ b/books/ps/ribbons.ps @@ -0,0 +1,560 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 11 180 180 81 psDrawLine + globGC (X) 6 185 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 182 180 42 81 psDrawLine + globGC (Y) 187 185 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 93 27 113 198 psDrawLine + globGC (Z) 88 22 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC + 3 88 + 14 98 + 24 106 + 34 113 + 42 119 + 51 123 + 58 127 + 66 129 + 72 131 + 79 132 + 85 132 + 91 131 + 97 129 + 102 127 + 107 124 + 112 120 + 117 115 + 122 110 + 127 104 + 132 97 + 137 89 + 141 81 + 146 71 + 151 61 + 155 49 + 160 37 + 165 23 + 28 170 8 psDrawLines + componentGC + 75 144 + 85 153 + 95 161 + 103 167 + 111 172 + 119 175 + 126 178 + 132 180 + 139 181 + 145 181 + 151 180 + 156 179 + 162 176 + 167 173 + 173 170 + 178 165 + 183 160 + 188 154 + 193 147 + 199 139 + 204 131 + 209 122 + 215 111 + 220 100 + 226 88 + 232 74 + 238 60 + 28 245 44 psDrawLines + componentGC + 245 44 + 2 170 8 psDrawLines + componentGC + 238 60 + 2 165 23 psDrawLines + componentGC + 232 74 + 2 160 37 psDrawLines + componentGC + 226 88 + 2 155 49 psDrawLines + componentGC + 220 100 + 2 151 61 psDrawLines + componentGC + 215 111 + 2 146 71 psDrawLines + componentGC + 209 122 + 2 141 81 psDrawLines + componentGC + 204 131 + 2 137 89 psDrawLines + componentGC + 199 139 + 2 132 97 psDrawLines + componentGC + 193 147 + 2 127 104 psDrawLines + componentGC + 188 154 + 2 122 110 psDrawLines + componentGC + 183 160 + 2 117 115 psDrawLines + componentGC + 178 165 + 2 112 120 psDrawLines + componentGC + 173 170 + 2 107 124 psDrawLines + componentGC + 167 173 + 2 102 127 psDrawLines + componentGC + 162 176 + 2 97 129 psDrawLines + componentGC + 156 179 + 2 91 131 psDrawLines + componentGC + 151 180 + 2 85 132 psDrawLines + componentGC + 145 181 + 2 79 132 psDrawLines + componentGC + 139 181 + 2 72 131 psDrawLines + componentGC + 132 180 + 2 66 129 psDrawLines + componentGC + 126 178 + 2 58 127 psDrawLines + componentGC + 119 175 + 2 51 123 psDrawLines + componentGC + 111 172 + 2 42 119 psDrawLines + componentGC + 103 167 + 2 34 113 psDrawLines + componentGC + 95 161 + 2 24 106 psDrawLines + componentGC + 85 153 + 2 14 98 psDrawLines + componentGC + 75 144 + 2 3 88 psDrawLines + componentGC + 75 144 + 86 159 + 96 170 + 105 178 + 113 184 + 120 187 + 127 189 + 133 189 + 139 188 + 145 186 + 151 184 + 156 181 + 162 177 + 167 173 + 173 170 + 178 166 + 183 163 + 188 160 + 193 158 + 198 156 + 202 156 + 207 157 + 211 159 + 215 162 + 219 167 + 222 172 + 225 180 + 28 228 188 psDrawLines + componentGC + 170 217 + 178 230 + 185 240 + 191 246 + 197 250 + 203 252 + 209 253 + 214 252 + 220 250 + 226 247 + 231 243 + 236 239 + 242 235 + 247 230 + 252 226 + 258 221 + 263 217 + 267 213 + 272 210 + 276 208 + 280 207 + 284 207 + 287 208 + 290 211 + 292 215 + 294 220 + 295 226 + 28 296 234 psDrawLines + componentGC + 296 234 + 2 228 188 psDrawLines + componentGC + 295 226 + 2 225 180 psDrawLines + componentGC + 294 220 + 2 222 172 psDrawLines + componentGC + 292 215 + 2 219 167 psDrawLines + componentGC + 290 211 + 2 215 162 psDrawLines + componentGC + 287 208 + 2 211 159 psDrawLines + componentGC + 284 207 + 2 207 157 psDrawLines + componentGC + 280 207 + 2 202 156 psDrawLines + componentGC + 276 208 + 2 198 156 psDrawLines + componentGC + 272 210 + 2 193 158 psDrawLines + componentGC + 267 213 + 2 188 160 psDrawLines + componentGC + 263 217 + 2 183 163 psDrawLines + componentGC + 258 221 + 2 178 166 psDrawLines + componentGC + 252 226 + 2 173 170 psDrawLines + componentGC + 247 230 + 2 167 173 psDrawLines + componentGC + 242 235 + 2 162 177 psDrawLines + componentGC + 236 239 + 2 156 181 psDrawLines + componentGC + 231 243 + 2 151 184 psDrawLines + componentGC + 226 247 + 2 145 186 psDrawLines + componentGC + 220 250 + 2 139 188 psDrawLines + componentGC + 214 252 + 2 133 189 psDrawLines + componentGC + 209 253 + 2 127 189 psDrawLines + componentGC + 203 252 + 2 120 187 psDrawLines + componentGC + 197 250 + 2 113 184 psDrawLines + componentGC + 191 246 + 2 105 178 psDrawLines + componentGC + 185 240 + 2 96 170 psDrawLines + componentGC + 178 230 + 2 86 159 psDrawLines + componentGC + 170 217 + 2 75 144 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbons2.ps b/books/ps/ribbons2.ps new file mode 100644 index 0000000..d09dd45 --- /dev/null +++ b/books/ps/ribbons2.ps @@ -0,0 +1,1488 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 11 180 180 81 psDrawLine + globGC (X) 6 185 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 182 180 42 81 psDrawLine + globGC (Y) 187 185 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 93 27 113 198 psDrawLine + globGC (Z) 88 22 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.562500 setForeground + renderGC + 225 180 + 296 234 + 228 188 + 4 225 180 psFillPoly + renderGC 0.000000 setForeground + renderGC + 296 234 + 228 188 + 3 225 180 psDrawLines + renderGC 0.687500 setForeground + renderGC + 222 172 + 295 226 + 225 180 + 4 222 172 psFillPoly + renderGC 0.000000 setForeground + renderGC + 295 226 + 225 180 + 3 222 172 psDrawLines + renderGC 0.625000 setForeground + renderGC + 296 234 + 225 180 + 295 226 + 4 296 234 psFillPoly + renderGC 0.000000 setForeground + renderGC + 225 180 + 295 226 + 3 296 234 psDrawLines + renderGC 0.750000 setForeground + renderGC + 219 167 + 294 220 + 222 172 + 4 219 167 psFillPoly + renderGC 0.000000 setForeground + renderGC + 294 220 + 222 172 + 3 219 167 psDrawLines + renderGC 0.687500 setForeground + renderGC + 295 226 + 222 172 + 294 220 + 4 295 226 psFillPoly + renderGC 0.000000 setForeground + renderGC + 222 172 + 294 220 + 3 295 226 psDrawLines + renderGC 0.812500 setForeground + renderGC + 215 162 + 292 215 + 219 167 + 4 215 162 psFillPoly + renderGC 0.000000 setForeground + renderGC + 292 215 + 219 167 + 3 215 162 psDrawLines + renderGC 0.750000 setForeground + renderGC + 294 220 + 219 167 + 292 215 + 4 294 220 psFillPoly + renderGC 0.000000 setForeground + renderGC + 219 167 + 292 215 + 3 294 220 psDrawLines + renderGC 0.812500 setForeground + renderGC + 211 159 + 290 211 + 215 162 + 4 211 159 psFillPoly + renderGC 0.000000 setForeground + renderGC + 290 211 + 215 162 + 3 211 159 psDrawLines + renderGC 0.812500 setForeground + renderGC + 292 215 + 215 162 + 290 211 + 4 292 215 psFillPoly + renderGC 0.000000 setForeground + renderGC + 215 162 + 290 211 + 3 292 215 psDrawLines + renderGC 0.562500 setForeground + renderGC + 122 110 + 193 147 + 127 104 + 4 122 110 psFillPoly + renderGC 0.000000 setForeground + renderGC + 193 147 + 127 104 + 3 122 110 psDrawLines + renderGC 0.562500 setForeground + renderGC + 199 139 + 127 104 + 193 147 + 4 199 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 127 104 + 193 147 + 3 199 139 psDrawLines + renderGC 0.562500 setForeground + renderGC + 127 104 + 199 139 + 132 97 + 4 127 104 psFillPoly + renderGC 0.000000 setForeground + renderGC + 199 139 + 132 97 + 3 127 104 psDrawLines + renderGC 0.562500 setForeground + renderGC + 204 131 + 132 97 + 199 139 + 4 204 131 psFillPoly + renderGC 0.000000 setForeground + renderGC + 132 97 + 199 139 + 3 204 131 psDrawLines + renderGC 0.562500 setForeground + renderGC + 132 97 + 204 131 + 137 89 + 4 132 97 psFillPoly + renderGC 0.000000 setForeground + renderGC + 204 131 + 137 89 + 3 132 97 psDrawLines + renderGC 0.625000 setForeground + renderGC + 117 115 + 188 154 + 122 110 + 4 117 115 psFillPoly + renderGC 0.000000 setForeground + renderGC + 188 154 + 122 110 + 3 117 115 psDrawLines + renderGC 0.562500 setForeground + renderGC + 193 147 + 122 110 + 188 154 + 4 193 147 psFillPoly + renderGC 0.000000 setForeground + renderGC + 122 110 + 188 154 + 3 193 147 psDrawLines + renderGC 0.562500 setForeground + renderGC + 209 122 + 137 89 + 204 131 + 4 209 122 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 89 + 204 131 + 3 209 122 psDrawLines + renderGC 0.500000 setForeground + renderGC + 137 89 + 209 122 + 141 81 + 4 137 89 psFillPoly + renderGC 0.000000 setForeground + renderGC + 209 122 + 141 81 + 3 137 89 psDrawLines + renderGC 0.625000 setForeground + renderGC + 112 120 + 183 160 + 117 115 + 4 112 120 psFillPoly + renderGC 0.000000 setForeground + renderGC + 183 160 + 117 115 + 3 112 120 psDrawLines + renderGC 0.625000 setForeground + renderGC + 188 154 + 117 115 + 183 160 + 4 188 154 psFillPoly + renderGC 0.000000 setForeground + renderGC + 117 115 + 183 160 + 3 188 154 psDrawLines + renderGC 0.437500 setForeground + renderGC + 215 111 + 141 81 + 209 122 + 4 215 111 psFillPoly + renderGC 0.000000 setForeground + renderGC + 141 81 + 209 122 + 3 215 111 psDrawLines + renderGC 0.437500 setForeground + renderGC + 141 81 + 215 111 + 146 71 + 4 141 81 psFillPoly + renderGC 0.000000 setForeground + renderGC + 215 111 + 146 71 + 3 141 81 psDrawLines + renderGC 0.625000 setForeground + renderGC + 107 124 + 178 165 + 112 120 + 4 107 124 psFillPoly + renderGC 0.000000 setForeground + renderGC + 178 165 + 112 120 + 3 107 124 psDrawLines + renderGC 0.625000 setForeground + renderGC + 183 160 + 112 120 + 178 165 + 4 183 160 psFillPoly + renderGC 0.000000 setForeground + renderGC + 112 120 + 178 165 + 3 183 160 psDrawLines + renderGC 0.312500 setForeground + renderGC + 220 100 + 146 71 + 215 111 + 4 220 100 psFillPoly + renderGC 0.000000 setForeground + renderGC + 146 71 + 215 111 + 3 220 100 psDrawLines + renderGC 0.312500 setForeground + renderGC + 146 71 + 220 100 + 151 61 + 4 146 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 220 100 + 151 61 + 3 146 71 psDrawLines + renderGC 0.687500 setForeground + renderGC + 102 127 + 173 170 + 107 124 + 4 102 127 psFillPoly + renderGC 0.000000 setForeground + renderGC + 173 170 + 107 124 + 3 102 127 psDrawLines + renderGC 0.625000 setForeground + renderGC + 178 165 + 107 124 + 173 170 + 4 178 165 psFillPoly + renderGC 0.000000 setForeground + renderGC + 107 124 + 173 170 + 3 178 165 psDrawLines + renderGC 0.250000 setForeground + renderGC + 226 88 + 151 61 + 220 100 + 4 226 88 psFillPoly + renderGC 0.000000 setForeground + renderGC + 151 61 + 220 100 + 3 226 88 psDrawLines + renderGC 0.187500 setForeground + renderGC + 151 61 + 226 88 + 155 49 + 4 151 61 psFillPoly + renderGC 0.000000 setForeground + renderGC + 226 88 + 155 49 + 3 151 61 psDrawLines + renderGC 0.875000 setForeground + renderGC + 207 157 + 287 208 + 211 159 + 4 207 157 psFillPoly + renderGC 0.000000 setForeground + renderGC + 287 208 + 211 159 + 3 207 157 psDrawLines + renderGC 0.812500 setForeground + renderGC + 290 211 + 211 159 + 287 208 + 4 290 211 psFillPoly + renderGC 0.000000 setForeground + renderGC + 211 159 + 287 208 + 3 290 211 psDrawLines + renderGC 0.687500 setForeground + renderGC + 97 129 + 167 173 + 102 127 + 4 97 129 psFillPoly + renderGC 0.000000 setForeground + renderGC + 167 173 + 102 127 + 3 97 129 psDrawLines + renderGC 0.687500 setForeground + renderGC + 173 170 + 102 127 + 167 173 + 4 173 170 psFillPoly + renderGC 0.000000 setForeground + renderGC + 102 127 + 167 173 + 3 173 170 psDrawLines + renderGC 0.250000 setForeground + renderGC + 232 74 + 155 49 + 226 88 + 4 232 74 psFillPoly + renderGC 0.000000 setForeground + renderGC + 155 49 + 226 88 + 3 232 74 psDrawLines + renderGC 0.250000 setForeground + renderGC + 155 49 + 232 74 + 160 37 + 4 155 49 psFillPoly + renderGC 0.000000 setForeground + renderGC + 232 74 + 160 37 + 3 155 49 psDrawLines + renderGC 0.687500 setForeground + renderGC + 91 131 + 162 176 + 97 129 + 4 91 131 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 176 + 97 129 + 3 91 131 psDrawLines + renderGC 0.687500 setForeground + renderGC + 167 173 + 97 129 + 162 176 + 4 167 173 psFillPoly + renderGC 0.000000 setForeground + renderGC + 97 129 + 162 176 + 3 167 173 psDrawLines + renderGC 0.312500 setForeground + renderGC + 238 60 + 160 37 + 232 74 + 4 238 60 psFillPoly + renderGC 0.000000 setForeground + renderGC + 160 37 + 232 74 + 3 238 60 psDrawLines + renderGC 0.312500 setForeground + renderGC + 160 37 + 238 60 + 165 23 + 4 160 37 psFillPoly + renderGC 0.000000 setForeground + renderGC + 238 60 + 165 23 + 3 160 37 psDrawLines + renderGC 0.812500 setForeground + renderGC + 202 156 + 284 207 + 207 157 + 4 202 156 psFillPoly + renderGC 0.000000 setForeground + renderGC + 284 207 + 207 157 + 3 202 156 psDrawLines + renderGC 0.875000 setForeground + renderGC + 287 208 + 207 157 + 284 207 + 4 287 208 psFillPoly + renderGC 0.000000 setForeground + renderGC + 207 157 + 284 207 + 3 287 208 psDrawLines + renderGC 0.750000 setForeground + renderGC + 85 132 + 156 179 + 91 131 + 4 85 132 psFillPoly + renderGC 0.000000 setForeground + renderGC + 156 179 + 91 131 + 3 85 132 psDrawLines + renderGC 0.687500 setForeground + renderGC + 162 176 + 91 131 + 156 179 + 4 162 176 psFillPoly + renderGC 0.000000 setForeground + renderGC + 91 131 + 156 179 + 3 162 176 psDrawLines + renderGC 0.375000 setForeground + renderGC + 245 44 + 165 23 + 238 60 + 4 245 44 psFillPoly + renderGC 0.000000 setForeground + renderGC + 165 23 + 238 60 + 3 245 44 psDrawLines + renderGC 0.437500 setForeground + renderGC + 165 23 + 245 44 + 170 8 + 4 165 23 psFillPoly + renderGC 0.000000 setForeground + renderGC + 245 44 + 170 8 + 3 165 23 psDrawLines + renderGC 0.750000 setForeground + renderGC + 79 132 + 151 180 + 85 132 + 4 79 132 psFillPoly + renderGC 0.000000 setForeground + renderGC + 151 180 + 85 132 + 3 79 132 psDrawLines + renderGC 0.750000 setForeground + renderGC + 156 179 + 85 132 + 151 180 + 4 156 179 psFillPoly + renderGC 0.000000 setForeground + renderGC + 85 132 + 151 180 + 3 156 179 psDrawLines + renderGC 0.750000 setForeground + renderGC + 198 156 + 280 207 + 202 156 + 4 198 156 psFillPoly + renderGC 0.000000 setForeground + renderGC + 280 207 + 202 156 + 3 198 156 psDrawLines + renderGC 0.812500 setForeground + renderGC + 284 207 + 202 156 + 280 207 + 4 284 207 psFillPoly + renderGC 0.000000 setForeground + renderGC + 202 156 + 280 207 + 3 284 207 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psDrawLines + renderGC 0.562500 setForeground + renderGC + 103 167 + 24 106 + 95 161 + 4 103 167 psFillPoly + renderGC 0.000000 setForeground + renderGC + 24 106 + 95 161 + 3 103 167 psDrawLines + renderGC 0.750000 setForeground + renderGC + 113 184 + 203 252 + 120 187 + 4 113 184 psFillPoly + renderGC 0.000000 setForeground + renderGC + 203 252 + 120 187 + 3 113 184 psDrawLines + renderGC 0.812500 setForeground + renderGC + 209 253 + 120 187 + 203 252 + 4 209 253 psFillPoly + renderGC 0.000000 setForeground + renderGC + 120 187 + 203 252 + 3 209 253 psDrawLines + renderGC 0.625000 setForeground + renderGC + 3 88 + 85 153 + 14 98 + 4 3 88 psFillPoly + renderGC 0.000000 setForeground + renderGC + 85 153 + 14 98 + 3 3 88 psDrawLines + renderGC 0.625000 setForeground + renderGC + 95 161 + 14 98 + 85 153 + 4 95 161 psFillPoly + renderGC 0.000000 setForeground + renderGC + 14 98 + 85 153 + 3 95 161 psDrawLines + renderGC 0.687500 setForeground + renderGC + 105 178 + 197 250 + 113 184 + 4 105 178 psFillPoly + renderGC 0.000000 setForeground + renderGC + 197 250 + 113 184 + 3 105 178 psDrawLines + renderGC 0.750000 setForeground + renderGC + 203 252 + 113 184 + 197 250 + 4 203 252 psFillPoly + renderGC 0.000000 setForeground + renderGC + 113 184 + 197 250 + 3 203 252 psDrawLines + renderGC 0.625000 setForeground + renderGC + 85 153 + 3 88 + 75 144 + 4 85 153 psFillPoly + renderGC 0.000000 setForeground + renderGC + 3 88 + 75 144 + 3 85 153 psDrawLines + renderGC 0.562500 setForeground + renderGC + 96 170 + 191 246 + 105 178 + 4 96 170 psFillPoly + renderGC 0.000000 setForeground + renderGC + 191 246 + 105 178 + 3 96 170 psDrawLines + renderGC 0.625000 setForeground + renderGC + 197 250 + 105 178 + 191 246 + 4 197 250 psFillPoly + renderGC 0.000000 setForeground + renderGC + 105 178 + 191 246 + 3 197 250 psDrawLines + renderGC 0.500000 setForeground + renderGC + 86 159 + 185 240 + 96 170 + 4 86 159 psFillPoly + renderGC 0.000000 setForeground + renderGC + 185 240 + 96 170 + 3 86 159 psDrawLines + renderGC 0.500000 setForeground + renderGC + 191 246 + 96 170 + 185 240 + 4 191 246 psFillPoly + renderGC 0.000000 setForeground + renderGC + 96 170 + 185 240 + 3 191 246 psDrawLines + renderGC 0.625000 setForeground + renderGC + 75 144 + 178 230 + 86 159 + 4 75 144 psFillPoly + renderGC 0.000000 setForeground + renderGC + 178 230 + 86 159 + 3 75 144 psDrawLines + renderGC 0.562500 setForeground + renderGC + 185 240 + 86 159 + 178 230 + 4 185 240 psFillPoly + renderGC 0.000000 setForeground + renderGC + 86 159 + 178 230 + 3 185 240 psDrawLines + renderGC 0.625000 setForeground + renderGC + 178 230 + 75 144 + 170 217 + 4 178 230 psFillPoly + renderGC 0.000000 setForeground + renderGC + 75 144 + 170 217 + 3 178 230 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbons2b.ps b/books/ps/ribbons2b.ps new file mode 100644 index 0000000..f0898dc --- /dev/null +++ b/books/ps/ribbons2b.ps @@ -0,0 +1,1516 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 62 115 213 91 psDrawLine + globGC (X) 57 120 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 170 141 122 75 psDrawLine + globGC (Y) 175 146 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 140 23 144 167 psDrawLine + globGC (Z) 135 18 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 trashGC setLineAttributes + trashGC 0.000000 setForeground + 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renderGC 0.000000 setForeground + renderGC + 124 155 + 102 110 + 3 95 100 psDrawLines + renderGC 0.625000 setForeground + renderGC + 132 164 + 102 110 + 124 155 + 4 132 164 psFillPoly + renderGC 0.000000 setForeground + renderGC + 102 110 + 124 155 + 3 132 164 psDrawLines + renderGC 0.500000 setForeground + renderGC + 86 86 + 116 143 + 95 100 + 4 86 86 psFillPoly + renderGC 0.000000 setForeground + renderGC + 116 143 + 95 100 + 3 86 86 psDrawLines + renderGC 0.500000 setForeground + renderGC + 124 155 + 95 100 + 116 143 + 4 124 155 psFillPoly + renderGC 0.000000 setForeground + renderGC + 95 100 + 116 143 + 3 124 155 psDrawLines + renderGC 0.625000 setForeground + renderGC + 78 69 + 106 127 + 86 86 + 4 78 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 106 127 + 86 86 + 3 78 69 psDrawLines + renderGC 0.562500 setForeground + renderGC + 116 143 + 86 86 + 106 127 + 4 116 143 psFillPoly + renderGC 0.000000 setForeground + renderGC + 86 86 + 106 127 + 3 116 143 psDrawLines + renderGC 0.625000 setForeground + renderGC + 106 127 + 78 69 + 96 108 + 4 106 127 psFillPoly + renderGC 0.000000 setForeground + renderGC + 78 69 + 96 108 + 3 106 127 psDrawLines + 1 0 0 trashGC setLineAttributes + trashGC 0.000000 setForeground + trashGC + 96 108 + 291 78 + 3 266 226 psDrawLines + trashGC + 96 108 + 112 269 + 3 82 171 psDrawLines + trashGC + 96 108 + 65 43 + 3 207 29 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/ribbons5.ps b/books/ps/ribbons5.ps new file mode 100644 index 0000000..0175046 --- /dev/null +++ b/books/ps/ribbons5.ps @@ -0,0 +1,3298 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 -422 157 420 32 psDrawLine + globGC (X) -427 162 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 215 206 79 20 psDrawLine + globGC (Y) 220 211 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 92 -639 130 324 psDrawLine + globGC (Z) 87 -644 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 trashGC setLineAttributes + trashGC 0.000000 setForeground + trashGC + 299 67 + 186 1 + 3 181 123 psDrawLines + trashGC + 41 15 + 59 147 + 3 181 123 psDrawLines + trashGC + 99 277 + 268 228 + 3 181 123 psDrawLines + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.625000 setForeground + renderGC + 177 120 + 194 139 + 181 123 + 4 177 120 psFillPoly + renderGC 0.000000 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b/books/ps/rose-1.ps @@ -0,0 +1,1230 @@ +%-------------------------- postscript draw commands --------------------------% +%!IBM Personal Pageprinter (4216) Adapter Program V1.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + 0 0 3 + 1072693248 0 /frameDict makeDict + 0 0 3 + 1072693248 0 /frameDict makeDict + + gsave % save graphics state for clipping path + + 24 240 262 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 262 118 0 118 psDrawLine + globalGC1 131 240 131 0 psDrawLine + unitGC 131 120 131 116 psDrawLine + unitGC 119 133 (0.00) (window) psDrawIStr + unitGC 175 120 175 116 psDrawLine + unitGC 163 133 (0.40) (window) psDrawIStr + unitGC 219 120 219 116 psDrawLine + unitGC 207 133 (0.80) (window) psDrawIStr + unitGC 86 120 86 116 psDrawLine + unitGC 71 133 (-0.40) (window) psDrawIStr + unitGC 41 120 41 116 psDrawLine + unitGC 26 133 (-0.80) (window) psDrawIStr + unitGC 133 73 129 73 psDrawLine + unitGC 95 78 (0.39) (window) psDrawIStr + unitGC 133 28 129 28 psDrawLine + unitGC 95 33 (0.79) (window) psDrawIStr + unitGC 133 163 129 163 psDrawLine + unitGC 86 168 (-0.39) (window) psDrawIStr + unitGC 133 207 129 207 psDrawLine + unitGC 86 212 (-0.79) (window) psDrawIStr + globalGC1 240 117 3 3 0 360 241 118 psFillArc + globalGC1 241 118 241 118 psDrawLine + globalGC1 240 117 3 3 0 360 241 118 psFillArc + globalGC1 240 104 241 118 psDrawLine + globalGC1 239 103 3 3 0 360 240 104 psFillArc + globalGC1 236 90 240 104 psDrawLine + globalGC1 235 89 3 3 0 360 236 90 psFillArc + globalGC1 230 77 236 90 psDrawLine + globalGC1 229 76 3 3 0 360 230 77 psFillArc + globalGC1 221 65 230 77 psDrawLine + globalGC1 220 64 3 3 0 360 221 65 psFillArc + globalGC1 211 56 221 65 psDrawLine + globalGC1 210 55 3 3 0 360 211 56 psFillArc + globalGC1 199 48 211 56 psDrawLine + globalGC1 198 47 3 3 0 360 199 48 psFillArc + globalGC1 187 44 199 48 psDrawLine + globalGC1 186 43 3 3 0 360 187 44 psFillArc + globalGC1 174 41 187 44 psDrawLine + globalGC1 173 40 3 3 0 360 174 41 psFillArc + globalGC1 162 42 174 41 psDrawLine + globalGC1 161 41 3 3 0 360 162 42 psFillArc + globalGC1 150 44 162 42 psDrawLine + globalGC1 149 43 3 3 0 360 150 44 psFillArc + globalGC1 139 49 150 44 psDrawLine + globalGC1 138 48 3 3 0 360 139 49 psFillArc + globalGC1 131 56 139 49 psDrawLine + globalGC1 130 55 3 3 0 360 131 56 psFillArc + globalGC1 123 64 131 56 psDrawLine + globalGC1 122 63 3 3 0 360 123 64 psFillArc + globalGC1 119 74 123 64 psDrawLine + globalGC1 118 73 3 3 0 360 119 74 psFillArc + globalGC1 116 84 119 74 psDrawLine + globalGC1 115 83 3 3 0 360 116 84 psFillArc + globalGC1 116 93 116 84 psDrawLine + globalGC1 115 92 3 3 0 360 116 93 psFillArc + globalGC1 118 103 116 93 psDrawLine + globalGC1 117 102 3 3 0 360 118 103 psFillArc + globalGC1 123 111 118 103 psDrawLine + globalGC1 122 110 3 3 0 360 123 111 psFillArc + globalGC1 126 114 123 111 psDrawLine + globalGC1 125 113 3 3 0 360 126 114 psFillArc + globalGC1 129 117 126 114 psDrawLine + globalGC1 128 116 3 3 0 360 129 117 psFillArc + globalGC1 133 120 129 117 psDrawLine + globalGC1 132 119 3 3 0 360 133 120 psFillArc + globalGC1 137 122 133 120 psDrawLine + globalGC1 136 121 3 3 0 360 137 122 psFillArc + globalGC1 141 124 137 122 psDrawLine + globalGC1 140 123 3 3 0 360 141 124 psFillArc + globalGC1 145 125 141 124 psDrawLine + globalGC1 144 124 3 3 0 360 145 125 psFillArc + globalGC1 150 125 145 125 psDrawLine + globalGC1 149 124 3 3 0 360 150 125 psFillArc + globalGC1 155 125 150 125 psDrawLine + globalGC1 154 124 3 3 0 360 155 125 psFillArc + globalGC1 159 124 155 125 psDrawLine + globalGC1 158 123 3 3 0 360 159 124 psFillArc + globalGC1 164 123 159 124 psDrawLine + globalGC1 163 122 3 3 0 360 164 123 psFillArc + globalGC1 169 121 164 123 psDrawLine + globalGC1 168 120 3 3 0 360 169 121 psFillArc + globalGC1 173 118 169 121 psDrawLine + globalGC1 172 117 3 3 0 360 173 118 psFillArc + globalGC1 181 112 173 118 psDrawLine + globalGC1 180 111 3 3 0 360 181 112 psFillArc + globalGC1 187 103 181 112 psDrawLine + globalGC1 186 102 3 3 0 360 187 103 psFillArc + globalGC1 191 93 187 103 psDrawLine + globalGC1 190 92 3 3 0 360 191 93 psFillArc + globalGC1 194 81 191 93 psDrawLine + globalGC1 193 80 3 3 0 360 194 81 psFillArc + globalGC1 194 69 194 81 psDrawLine + globalGC1 193 68 3 3 0 360 194 69 psFillArc + globalGC1 191 57 194 69 psDrawLine + globalGC1 190 56 3 3 0 360 191 57 psFillArc + globalGC1 186 45 191 57 psDrawLine + globalGC1 185 44 3 3 0 360 186 45 psFillArc + globalGC1 178 34 186 45 psDrawLine + globalGC1 177 33 3 3 0 360 178 34 psFillArc + globalGC1 169 25 178 34 psDrawLine + globalGC1 168 24 3 3 0 360 169 25 psFillArc + globalGC1 157 17 169 25 psDrawLine + globalGC1 156 16 3 3 0 360 157 17 psFillArc + globalGC1 144 11 157 17 psDrawLine + globalGC1 143 10 3 3 0 360 144 11 psFillArc + globalGC1 131 8 144 11 psDrawLine + globalGC1 130 7 3 3 0 360 131 8 psFillArc + globalGC1 116 8 131 8 psDrawLine + globalGC1 115 7 3 3 0 360 116 8 psFillArc + globalGC1 102 10 116 8 psDrawLine + globalGC1 101 9 3 3 0 360 102 10 psFillArc + globalGC1 88 15 102 10 psDrawLine + globalGC1 87 14 3 3 0 360 88 15 psFillArc + globalGC1 76 22 88 15 psDrawLine + globalGC1 75 21 3 3 0 360 76 22 psFillArc + globalGC1 65 31 76 22 psDrawLine + globalGC1 64 30 3 3 0 360 65 31 psFillArc + globalGC1 56 42 65 31 psDrawLine + globalGC1 55 41 3 3 0 360 56 42 psFillArc + globalGC1 49 55 56 42 psDrawLine + globalGC1 48 54 3 3 0 360 49 55 psFillArc + globalGC1 45 68 49 55 psDrawLine + globalGC1 44 67 3 3 0 360 45 68 psFillArc + globalGC1 43 82 45 68 psDrawLine + globalGC1 42 81 3 3 0 360 43 82 psFillArc + globalGC1 44 95 43 82 psDrawLine + globalGC1 43 94 3 3 0 360 44 95 psFillArc + globalGC1 47 107 44 95 psDrawLine + globalGC1 46 106 3 3 0 360 47 107 psFillArc + globalGC1 52 118 47 107 psDrawLine + globalGC1 51 117 3 3 0 360 52 118 psFillArc + globalGC1 60 128 52 118 psDrawLine + globalGC1 59 127 3 3 0 360 60 128 psFillArc + globalGC1 68 135 60 128 psDrawLine + globalGC1 67 134 3 3 0 360 68 135 psFillArc + globalGC1 78 141 68 135 psDrawLine + globalGC1 77 140 3 3 0 360 78 141 psFillArc + globalGC1 83 142 78 141 psDrawLine + globalGC1 82 141 3 3 0 360 83 142 psFillArc + globalGC1 88 143 83 142 psDrawLine + globalGC1 87 142 3 3 0 360 88 143 psFillArc + globalGC1 93 144 88 143 psDrawLine + globalGC1 92 143 3 3 0 360 93 144 psFillArc + globalGC1 98 144 93 144 psDrawLine + globalGC1 97 143 3 3 0 360 98 144 psFillArc + globalGC1 103 143 98 144 psDrawLine + globalGC1 102 142 3 3 0 360 103 143 psFillArc + globalGC1 108 142 103 143 psDrawLine + globalGC1 107 141 3 3 0 360 108 142 psFillArc + globalGC1 112 140 108 142 psDrawLine + globalGC1 111 139 3 3 0 360 112 140 psFillArc + globalGC1 116 137 112 140 psDrawLine + globalGC1 115 136 3 3 0 360 116 137 psFillArc + globalGC1 120 134 116 137 psDrawLine + globalGC1 119 133 3 3 0 360 120 134 psFillArc + globalGC1 123 131 120 134 psDrawLine + globalGC1 122 130 3 3 0 360 123 131 psFillArc + globalGC1 126 128 123 131 psDrawLine + globalGC1 125 127 3 3 0 360 126 128 psFillArc + globalGC1 128 124 126 128 psDrawLine + globalGC1 127 123 3 3 0 360 128 124 psFillArc + globalGC1 131 115 128 124 psDrawLine + globalGC1 130 114 3 3 0 360 131 115 psFillArc + globalGC1 132 106 131 115 psDrawLine + globalGC1 131 105 3 3 0 360 132 106 psFillArc + globalGC1 131 97 132 106 psDrawLine + globalGC1 130 96 3 3 0 360 131 97 psFillArc + globalGC1 127 88 131 97 psDrawLine + globalGC1 126 87 3 3 0 360 127 88 psFillArc + globalGC1 120 80 127 88 psDrawLine + globalGC1 119 79 3 3 0 360 120 80 psFillArc + globalGC1 112 74 120 80 psDrawLine + globalGC1 111 73 3 3 0 360 112 74 psFillArc + globalGC1 108 72 112 74 psDrawLine + globalGC1 107 71 3 3 0 360 108 72 psFillArc + globalGC1 103 70 108 72 psDrawLine + globalGC1 102 69 3 3 0 360 103 70 psFillArc + globalGC1 98 69 103 70 psDrawLine + globalGC1 97 68 3 3 0 360 98 69 psFillArc + globalGC1 92 68 98 69 psDrawLine + globalGC1 91 67 3 3 0 360 92 68 psFillArc + globalGC1 81 68 92 68 psDrawLine + globalGC1 80 67 3 3 0 360 81 68 psFillArc + globalGC1 69 71 81 68 psDrawLine + globalGC1 68 70 3 3 0 360 69 71 psFillArc + globalGC1 58 76 69 71 psDrawLine + globalGC1 57 75 3 3 0 360 58 76 psFillArc + globalGC1 48 84 58 76 psDrawLine + globalGC1 47 83 3 3 0 360 48 84 psFillArc + globalGC1 40 94 48 84 psDrawLine + globalGC1 39 93 3 3 0 360 40 94 psFillArc + globalGC1 33 105 40 94 psDrawLine + globalGC1 32 104 3 3 0 360 33 105 psFillArc + globalGC1 28 118 33 105 psDrawLine + globalGC1 27 117 3 3 0 360 28 118 psFillArc + globalGC1 26 132 28 118 psDrawLine + globalGC1 25 131 3 3 0 360 26 132 psFillArc + globalGC1 26 147 26 132 psDrawLine + globalGC1 25 146 3 3 0 360 26 147 psFillArc + globalGC1 29 161 26 147 psDrawLine + globalGC1 28 160 3 3 0 360 29 161 psFillArc + globalGC1 35 175 29 161 psDrawLine + globalGC1 34 174 3 3 0 360 35 175 psFillArc + globalGC1 43 187 35 175 psDrawLine + globalGC1 42 186 3 3 0 360 43 187 psFillArc + globalGC1 53 198 43 187 psDrawLine + globalGC1 52 197 3 3 0 360 53 198 psFillArc + globalGC1 64 206 53 198 psDrawLine + globalGC1 63 205 3 3 0 360 64 206 psFillArc + globalGC1 77 212 64 206 psDrawLine + globalGC1 76 211 3 3 0 360 77 212 psFillArc + globalGC1 91 216 77 212 psDrawLine + globalGC1 90 215 3 3 0 360 91 216 psFillArc + globalGC1 104 217 91 216 psDrawLine + globalGC1 103 216 3 3 0 360 104 217 psFillArc + globalGC1 118 216 104 217 psDrawLine + globalGC1 117 215 3 3 0 360 118 216 psFillArc + globalGC1 131 212 118 216 psDrawLine + globalGC1 130 211 3 3 0 360 131 212 psFillArc + globalGC1 142 206 131 212 psDrawLine + globalGC1 141 205 3 3 0 360 142 206 psFillArc + globalGC1 151 198 142 206 psDrawLine + globalGC1 150 197 3 3 0 360 151 198 psFillArc + globalGC1 159 188 151 198 psDrawLine + globalGC1 158 187 3 3 0 360 159 188 psFillArc + globalGC1 164 178 159 188 psDrawLine + globalGC1 163 177 3 3 0 360 164 178 psFillArc + globalGC1 167 167 164 178 psDrawLine + globalGC1 166 166 3 3 0 360 167 167 psFillArc + globalGC1 167 156 167 167 psDrawLine + globalGC1 166 155 3 3 0 360 167 156 psFillArc + globalGC1 165 146 167 156 psDrawLine + globalGC1 164 145 3 3 0 360 165 146 psFillArc + globalGC1 161 137 165 146 psDrawLine + globalGC1 160 136 3 3 0 360 161 137 psFillArc + globalGC1 155 129 161 137 psDrawLine + globalGC1 154 128 3 3 0 360 155 129 psFillArc + globalGC1 152 126 155 129 psDrawLine + globalGC1 151 125 3 3 0 360 152 126 psFillArc + globalGC1 148 123 152 126 psDrawLine + globalGC1 147 122 3 3 0 360 148 123 psFillArc + globalGC1 144 121 148 123 psDrawLine + globalGC1 143 120 3 3 0 360 144 121 psFillArc + globalGC1 139 120 144 121 psDrawLine + globalGC1 138 119 3 3 0 360 139 120 psFillArc + globalGC1 135 119 139 120 psDrawLine + globalGC1 134 118 3 3 0 360 135 119 psFillArc + globalGC1 130 118 135 119 psDrawLine + globalGC1 129 117 3 3 0 360 130 118 psFillArc + globalGC1 126 119 130 118 psDrawLine + globalGC1 125 118 3 3 0 360 126 119 psFillArc + globalGC1 122 120 126 119 psDrawLine + globalGC1 121 119 3 3 0 360 122 120 psFillArc + globalGC1 117 121 122 120 psDrawLine + globalGC1 116 120 3 3 0 360 117 121 psFillArc + globalGC1 113 123 117 121 psDrawLine + globalGC1 112 122 3 3 0 360 113 123 psFillArc + globalGC1 109 126 113 123 psDrawLine + globalGC1 108 125 3 3 0 360 109 126 psFillArc + globalGC1 106 129 109 126 psDrawLine + globalGC1 105 128 3 3 0 360 106 129 psFillArc + globalGC1 100 137 106 129 psDrawLine + globalGC1 99 136 3 3 0 360 100 137 psFillArc + globalGC1 96 146 100 137 psDrawLine + globalGC1 95 145 3 3 0 360 96 146 psFillArc + globalGC1 94 156 96 146 psDrawLine + globalGC1 93 155 3 3 0 360 94 156 psFillArc + globalGC1 94 167 94 156 psDrawLine + globalGC1 93 166 3 3 0 360 94 167 psFillArc + globalGC1 97 178 94 167 psDrawLine + globalGC1 96 177 3 3 0 360 97 178 psFillArc + globalGC1 102 188 97 178 psDrawLine + globalGC1 101 187 3 3 0 360 102 188 psFillArc + globalGC1 110 198 102 188 psDrawLine + globalGC1 109 197 3 3 0 360 110 198 psFillArc + globalGC1 119 206 110 198 psDrawLine + globalGC1 118 205 3 3 0 360 119 206 psFillArc + globalGC1 131 212 119 206 psDrawLine + globalGC1 130 211 3 3 0 360 131 212 psFillArc + globalGC1 143 216 131 212 psDrawLine + globalGC1 142 215 3 3 0 360 143 216 psFillArc + globalGC1 157 217 143 216 psDrawLine + globalGC1 156 216 3 3 0 360 157 217 psFillArc + globalGC1 170 216 157 217 psDrawLine + globalGC1 169 215 3 3 0 360 170 216 psFillArc + globalGC1 184 212 170 216 psDrawLine + globalGC1 183 211 3 3 0 360 184 212 psFillArc + globalGC1 197 206 184 212 psDrawLine + globalGC1 196 205 3 3 0 360 197 206 psFillArc + globalGC1 208 198 197 206 psDrawLine + globalGC1 207 197 3 3 0 360 208 198 psFillArc + globalGC1 218 187 208 198 psDrawLine + globalGC1 217 186 3 3 0 360 218 187 psFillArc + globalGC1 226 175 218 187 psDrawLine + globalGC1 225 174 3 3 0 360 226 175 psFillArc + globalGC1 232 161 226 175 psDrawLine + globalGC1 231 160 3 3 0 360 232 161 psFillArc + globalGC1 235 147 232 161 psDrawLine + globalGC1 234 146 3 3 0 360 235 147 psFillArc + globalGC1 235 132 235 147 psDrawLine + globalGC1 234 131 3 3 0 360 235 132 psFillArc + globalGC1 233 118 235 132 psDrawLine + globalGC1 232 117 3 3 0 360 233 118 psFillArc + globalGC1 228 105 233 118 psDrawLine + globalGC1 227 104 3 3 0 360 228 105 psFillArc + globalGC1 221 94 228 105 psDrawLine + globalGC1 220 93 3 3 0 360 221 94 psFillArc + globalGC1 213 84 221 94 psDrawLine + globalGC1 212 83 3 3 0 360 213 84 psFillArc + globalGC1 203 76 213 84 psDrawLine + globalGC1 202 75 3 3 0 360 203 76 psFillArc + globalGC1 192 71 203 76 psDrawLine + globalGC1 191 70 3 3 0 360 192 71 psFillArc + globalGC1 180 68 192 71 psDrawLine + globalGC1 179 67 3 3 0 360 180 68 psFillArc + globalGC1 174 68 180 68 psDrawLine + globalGC1 173 67 3 3 0 360 174 68 psFillArc + globalGC1 169 68 174 68 psDrawLine + globalGC1 168 67 3 3 0 360 169 68 psFillArc + globalGC1 163 69 169 68 psDrawLine + globalGC1 162 68 3 3 0 360 163 69 psFillArc + globalGC1 158 70 163 69 psDrawLine + globalGC1 157 69 3 3 0 360 158 70 psFillArc + globalGC1 149 74 158 70 psDrawLine + globalGC1 148 73 3 3 0 360 149 74 psFillArc + globalGC1 141 80 149 74 psDrawLine + globalGC1 140 79 3 3 0 360 141 80 psFillArc + globalGC1 134 88 141 80 psDrawLine + globalGC1 133 87 3 3 0 360 134 88 psFillArc + globalGC1 130 97 134 88 psDrawLine + globalGC1 129 96 3 3 0 360 130 97 psFillArc + globalGC1 129 106 130 97 psDrawLine + globalGC1 128 105 3 3 0 360 129 106 psFillArc + globalGC1 130 115 129 106 psDrawLine + globalGC1 129 114 3 3 0 360 130 115 psFillArc + globalGC1 133 124 130 115 psDrawLine + globalGC1 132 123 3 3 0 360 133 124 psFillArc + globalGC1 135 128 133 124 psDrawLine + globalGC1 134 127 3 3 0 360 135 128 psFillArc + globalGC1 138 131 135 128 psDrawLine + globalGC1 137 130 3 3 0 360 138 131 psFillArc + globalGC1 141 134 138 131 psDrawLine + globalGC1 140 133 3 3 0 360 141 134 psFillArc + globalGC1 145 137 141 134 psDrawLine + globalGC1 144 136 3 3 0 360 145 137 psFillArc + globalGC1 149 140 145 137 psDrawLine + globalGC1 148 139 3 3 0 360 149 140 psFillArc + globalGC1 153 142 149 140 psDrawLine + globalGC1 152 141 3 3 0 360 153 142 psFillArc + globalGC1 158 143 153 142 psDrawLine + globalGC1 157 142 3 3 0 360 158 143 psFillArc + globalGC1 163 144 158 143 psDrawLine + globalGC1 162 143 3 3 0 360 163 144 psFillArc + globalGC1 168 144 163 144 psDrawLine + globalGC1 167 143 3 3 0 360 168 144 psFillArc + globalGC1 173 143 168 144 psDrawLine + globalGC1 172 142 3 3 0 360 173 143 psFillArc + globalGC1 178 142 173 143 psDrawLine + globalGC1 177 141 3 3 0 360 178 142 psFillArc + globalGC1 183 141 178 142 psDrawLine + globalGC1 182 140 3 3 0 360 183 141 psFillArc + globalGC1 193 135 183 141 psDrawLine + globalGC1 192 134 3 3 0 360 193 135 psFillArc + globalGC1 201 128 193 135 psDrawLine + globalGC1 200 127 3 3 0 360 201 128 psFillArc + globalGC1 209 118 201 128 psDrawLine + globalGC1 208 117 3 3 0 360 209 118 psFillArc + globalGC1 214 107 209 118 psDrawLine + globalGC1 213 106 3 3 0 360 214 107 psFillArc + globalGC1 217 95 214 107 psDrawLine + globalGC1 216 94 3 3 0 360 217 95 psFillArc + globalGC1 218 82 217 95 psDrawLine + globalGC1 217 81 3 3 0 360 218 82 psFillArc + globalGC1 216 68 218 82 psDrawLine + globalGC1 215 67 3 3 0 360 216 68 psFillArc + globalGC1 212 55 216 68 psDrawLine + globalGC1 211 54 3 3 0 360 212 55 psFillArc + globalGC1 205 42 212 55 psDrawLine + globalGC1 204 41 3 3 0 360 205 42 psFillArc + globalGC1 196 31 205 42 psDrawLine + globalGC1 195 30 3 3 0 360 196 31 psFillArc + globalGC1 185 22 196 31 psDrawLine + globalGC1 184 21 3 3 0 360 185 22 psFillArc + globalGC1 173 15 185 22 psDrawLine + globalGC1 172 14 3 3 0 360 173 15 psFillArc + globalGC1 159 10 173 15 psDrawLine + globalGC1 158 9 3 3 0 360 159 10 psFillArc + globalGC1 145 8 159 10 psDrawLine + globalGC1 144 7 3 3 0 360 145 8 psFillArc + globalGC1 131 8 145 8 psDrawLine + globalGC1 130 7 3 3 0 360 131 8 psFillArc + globalGC1 117 11 131 8 psDrawLine + globalGC1 116 10 3 3 0 360 117 11 psFillArc + globalGC1 104 17 117 11 psDrawLine + globalGC1 103 16 3 3 0 360 104 17 psFillArc + globalGC1 92 25 104 17 psDrawLine + globalGC1 91 24 3 3 0 360 92 25 psFillArc + globalGC1 83 34 92 25 psDrawLine + globalGC1 82 33 3 3 0 360 83 34 psFillArc + globalGC1 75 45 83 34 psDrawLine + globalGC1 74 44 3 3 0 360 75 45 psFillArc + globalGC1 70 57 75 45 psDrawLine + globalGC1 69 56 3 3 0 360 70 57 psFillArc + globalGC1 67 69 70 57 psDrawLine + globalGC1 66 68 3 3 0 360 67 69 psFillArc + globalGC1 67 81 67 69 psDrawLine + globalGC1 66 80 3 3 0 360 67 81 psFillArc + globalGC1 70 93 67 81 psDrawLine + globalGC1 69 92 3 3 0 360 70 93 psFillArc + globalGC1 74 103 70 93 psDrawLine + globalGC1 73 102 3 3 0 360 74 103 psFillArc + globalGC1 80 112 74 103 psDrawLine + globalGC1 79 111 3 3 0 360 80 112 psFillArc + globalGC1 88 118 80 112 psDrawLine + globalGC1 87 117 3 3 0 360 88 118 psFillArc + globalGC1 92 121 88 118 psDrawLine + globalGC1 91 120 3 3 0 360 92 121 psFillArc + globalGC1 97 123 92 121 psDrawLine + globalGC1 96 122 3 3 0 360 97 123 psFillArc + globalGC1 102 124 97 123 psDrawLine + globalGC1 101 123 3 3 0 360 102 124 psFillArc + globalGC1 106 125 102 124 psDrawLine + globalGC1 105 124 3 3 0 360 106 125 psFillArc + globalGC1 111 125 106 125 psDrawLine + globalGC1 110 124 3 3 0 360 111 125 psFillArc + globalGC1 116 125 111 125 psDrawLine + globalGC1 115 124 3 3 0 360 116 125 psFillArc + globalGC1 120 124 116 125 psDrawLine + globalGC1 119 123 3 3 0 360 120 124 psFillArc + globalGC1 124 122 120 124 psDrawLine + globalGC1 123 121 3 3 0 360 124 122 psFillArc + globalGC1 128 120 124 122 psDrawLine + globalGC1 127 119 3 3 0 360 128 120 psFillArc + globalGC1 132 117 128 120 psDrawLine + globalGC1 131 116 3 3 0 360 132 117 psFillArc + globalGC1 135 114 132 117 psDrawLine + globalGC1 134 113 3 3 0 360 135 114 psFillArc + globalGC1 138 111 135 114 psDrawLine + globalGC1 137 110 3 3 0 360 138 111 psFillArc + globalGC1 143 103 138 111 psDrawLine + globalGC1 142 102 3 3 0 360 143 103 psFillArc + globalGC1 145 93 143 103 psDrawLine + globalGC1 144 92 3 3 0 360 145 93 psFillArc + globalGC1 145 84 145 93 psDrawLine + globalGC1 144 83 3 3 0 360 145 84 psFillArc + globalGC1 142 74 145 84 psDrawLine + globalGC1 141 73 3 3 0 360 142 74 psFillArc + globalGC1 138 64 142 74 psDrawLine + globalGC1 137 63 3 3 0 360 138 64 psFillArc + globalGC1 131 56 138 64 psDrawLine + globalGC1 130 55 3 3 0 360 131 56 psFillArc + globalGC1 122 49 131 56 psDrawLine + globalGC1 121 48 3 3 0 360 122 49 psFillArc + globalGC1 111 44 122 49 psDrawLine + globalGC1 110 43 3 3 0 360 111 44 psFillArc + globalGC1 99 42 111 44 psDrawLine + globalGC1 98 41 3 3 0 360 99 42 psFillArc + globalGC1 87 41 99 42 psDrawLine + globalGC1 86 40 3 3 0 360 87 41 psFillArc + globalGC1 74 44 87 41 psDrawLine + globalGC1 73 43 3 3 0 360 74 44 psFillArc + globalGC1 62 48 74 44 psDrawLine + globalGC1 61 47 3 3 0 360 62 48 psFillArc + globalGC1 50 56 62 48 psDrawLine + globalGC1 49 55 3 3 0 360 50 56 psFillArc + globalGC1 40 65 50 56 psDrawLine + globalGC1 39 64 3 3 0 360 40 65 psFillArc + globalGC1 31 77 40 65 psDrawLine + globalGC1 30 76 3 3 0 360 31 77 psFillArc + globalGC1 25 90 31 77 psDrawLine + globalGC1 24 89 3 3 0 360 25 90 psFillArc + globalGC1 21 104 25 90 psDrawLine + globalGC1 20 103 3 3 0 360 21 104 psFillArc + globalGC1 20 118 21 104 psDrawLine + globalGC1 19 117 3 3 0 360 20 118 psFillArc + globalGC1 21 133 20 118 psDrawLine + globalGC1 20 132 3 3 0 360 21 133 psFillArc + globalGC1 25 147 21 133 psDrawLine + globalGC1 24 146 3 3 0 360 25 147 psFillArc + globalGC1 31 160 25 147 psDrawLine + globalGC1 30 159 3 3 0 360 31 160 psFillArc + globalGC1 40 172 31 160 psDrawLine + globalGC1 39 171 3 3 0 360 40 172 psFillArc + globalGC1 50 181 40 172 psDrawLine + globalGC1 49 180 3 3 0 360 50 181 psFillArc + globalGC1 62 189 50 181 psDrawLine + globalGC1 61 188 3 3 0 360 62 189 psFillArc + globalGC1 74 193 62 189 psDrawLine + globalGC1 73 192 3 3 0 360 74 193 psFillArc + globalGC1 87 196 74 193 psDrawLine + globalGC1 86 195 3 3 0 360 87 196 psFillArc + globalGC1 99 195 87 196 psDrawLine + globalGC1 98 194 3 3 0 360 99 195 psFillArc + globalGC1 111 193 99 195 psDrawLine + globalGC1 110 192 3 3 0 360 111 193 psFillArc + globalGC1 122 188 111 193 psDrawLine + globalGC1 121 187 3 3 0 360 122 188 psFillArc + globalGC1 130 181 122 188 psDrawLine + globalGC1 129 180 3 3 0 360 130 181 psFillArc + globalGC1 138 173 130 181 psDrawLine + globalGC1 137 172 3 3 0 360 138 173 psFillArc + globalGC1 142 163 138 173 psDrawLine + globalGC1 141 162 3 3 0 360 142 163 psFillArc + globalGC1 145 153 142 163 psDrawLine + globalGC1 144 152 3 3 0 360 145 153 psFillArc + globalGC1 145 144 145 153 psDrawLine + globalGC1 144 143 3 3 0 360 145 144 psFillArc + globalGC1 143 134 145 144 psDrawLine + globalGC1 142 133 3 3 0 360 143 134 psFillArc + globalGC1 138 126 143 134 psDrawLine + globalGC1 137 125 3 3 0 360 138 126 psFillArc + globalGC1 135 123 138 126 psDrawLine + globalGC1 134 122 3 3 0 360 135 123 psFillArc + globalGC1 132 120 135 123 psDrawLine + globalGC1 131 119 3 3 0 360 132 120 psFillArc + globalGC1 128 117 132 120 psDrawLine + globalGC1 127 116 3 3 0 360 128 117 psFillArc + globalGC1 124 115 128 117 psDrawLine + globalGC1 123 114 3 3 0 360 124 115 psFillArc + globalGC1 120 113 124 115 psDrawLine + globalGC1 119 112 3 3 0 360 120 113 psFillArc + globalGC1 116 112 120 113 psDrawLine + globalGC1 115 111 3 3 0 360 116 112 psFillArc + globalGC1 111 112 116 112 psDrawLine + globalGC1 110 111 3 3 0 360 111 112 psFillArc + globalGC1 106 112 111 112 psDrawLine + globalGC1 105 111 3 3 0 360 106 112 psFillArc + globalGC1 102 113 106 112 psDrawLine + globalGC1 101 112 3 3 0 360 102 113 psFillArc + globalGC1 97 114 102 113 psDrawLine + globalGC1 96 113 3 3 0 360 97 114 psFillArc + globalGC1 92 116 97 114 psDrawLine + globalGC1 91 115 3 3 0 360 92 116 psFillArc + globalGC1 88 118 92 116 psDrawLine + globalGC1 87 117 3 3 0 360 88 118 psFillArc + globalGC1 80 125 88 118 psDrawLine + globalGC1 79 124 3 3 0 360 80 125 psFillArc + globalGC1 74 134 80 125 psDrawLine + globalGC1 73 133 3 3 0 360 74 134 psFillArc + globalGC1 70 144 74 134 psDrawLine + globalGC1 69 143 3 3 0 360 70 144 psFillArc + globalGC1 67 156 70 144 psDrawLine + globalGC1 66 155 3 3 0 360 67 156 psFillArc + globalGC1 67 168 67 156 psDrawLine + globalGC1 66 167 3 3 0 360 67 168 psFillArc + globalGC1 70 180 67 168 psDrawLine + globalGC1 69 179 3 3 0 360 70 180 psFillArc + globalGC1 75 192 70 180 psDrawLine + globalGC1 74 191 3 3 0 360 75 192 psFillArc + globalGC1 83 203 75 192 psDrawLine + globalGC1 82 202 3 3 0 360 83 203 psFillArc + globalGC1 92 212 83 203 psDrawLine + globalGC1 91 211 3 3 0 360 92 212 psFillArc + globalGC1 104 220 92 212 psDrawLine + globalGC1 103 219 3 3 0 360 104 220 psFillArc + globalGC1 117 226 104 220 psDrawLine + globalGC1 116 225 3 3 0 360 117 226 psFillArc + globalGC1 130 229 117 226 psDrawLine + globalGC1 129 228 3 3 0 360 130 229 psFillArc + globalGC1 145 229 130 229 psDrawLine + globalGC1 144 228 3 3 0 360 145 229 psFillArc + globalGC1 159 227 145 229 psDrawLine 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+JcC<$JcC<$bl<1~> +JcC<$JcC<$bl<1~> +JcC<$JcC<$bl<1~> +JcC<$JcC<$bl<1~> +JcC<$JcC<$bl<1~> +%%EndData +showpage +%%Trailer +end +%%EOF diff --git a/books/ps/up.ps b/books/ps/up.ps new file mode 100644 index 0000000..81706d0 --- /dev/null +++ b/books/ps/up.ps @@ -0,0 +1,382 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 284.120003 559.000000 538.120003 +%%Creator: /u/jonms/XtoPS/xwd2ps +%%CreationDate: Tue Oct 8 20:20:28 1991 +%%Title: standard input +%%EndComments +% /u/jonms/XtoPS/xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% /u/jonms/XtoPS/xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 60, height = 30, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 60, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 30 +% starting column = 1 +% ending column = 60 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 60 30 8 + [60 0 0 -30 0 30] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 3.960000 inch translate +matrix currentmatrix +.5 inch .16 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +dbdb70 +2f4f4f +23238e +d3d3d3 +ff0000 +000000 +004900 +004955 +0049aa +0049ff 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+3b003b003b003b003b003b003b003b003b001c0000011d001a0004011b0018000801190016000c01170014001001150012001401130010001801110015000e01160015000e011600 +15000e01160015000e01160015000e0116003b003b003b003b003b003b003b003b003b00 +pop pop setmatrix + +/#copies 1 def +%grestore +grestore +%%Trailer diff --git a/books/ps/vectorroot.ps b/books/ps/vectorroot.ps new file mode 100644 index 0000000..a6ad7d6 --- /dev/null +++ b/books/ps/vectorroot.ps @@ -0,0 +1,4310 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 57 203 225 84 psDrawLine + globGC (X) 52 208 psDrawStr + globalGC1 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+%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 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psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/wd-atanz.ps b/books/ps/wd-atanz.ps new file mode 100644 index 0000000..e40ff3a --- /dev/null +++ b/books/ps/wd-atanz.ps @@ -0,0 +1,963 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 192.347462 284.120003 419.652538 538.120003 +%%Creator: xwd2ps +%%CreationDate: Sat Dec 7 19:08:40 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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+%%Trailer diff --git a/books/ps/wd-bessi3.ps b/books/ps/wd-bessi3.ps new file mode 100644 index 0000000..e960c65 --- /dev/null +++ b/books/ps/wd-bessi3.ps @@ -0,0 +1,1472 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 192.347462 284.120003 419.652538 538.120003 +%%Creator: xwd2ps +%%CreationDate: Sat Dec 7 19:19:34 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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b/books/ps/wd-bessi3s.ps new file mode 100644 index 0000000..3e1a072 --- /dev/null +++ b/books/ps/wd-bessi3s.ps @@ -0,0 +1,1048 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 144.385410 559.000000 677.854595 +%%Creator: xwd2ps +%%CreationDate: Fri Jun 28 18:46:23 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by leonardo.watson.ibm.com:smwatt (Stephen Watt) +% Information from XWD rasterfile header: +% width = 422, height = 445, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 445 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 445 8 + [422 0 0 -445 0 445] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.019242 inch translate +matrix currentmatrix +3.5 inch 3.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +0000ff +d8d8bf +5f9f9f +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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+/#copies 1 def +grestore +%%Trailer diff --git a/books/ps/wd-beta.ps b/books/ps/wd-beta.ps new file mode 100644 index 0000000..4c158d7 --- /dev/null +++ b/books/ps/wd-beta.ps @@ -0,0 +1,1420 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 192.347462 284.120003 419.652538 538.120003 +%%Creator: xwd2ps +%%CreationDate: Sat Dec 7 19:13:02 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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+ff015f01001100121e01011101121e0100110012ff015f01001100121e01011101121e0100110012ff015f01001100121e01011101121e0100110012ff015f01001100121e010111 +01121e0100110012ff015f01001100121e01011101121e0100110012ff015f01001100121e01011101121e0100110012ff015f01001100121e01011101121f110012ff1160110012 +20110012ff11a411 +pop pop setmatrix + +/#copies 1 def +grestore +%%Trailer diff --git a/books/ps/wd-expz.ps b/books/ps/wd-expz.ps new file mode 100644 index 0000000..54961b6 --- /dev/null +++ b/books/ps/wd-expz.ps @@ -0,0 +1,818 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 192.108269 284.120003 419.891714 538.120003 +%%Creator: xwd2ps +%%CreationDate: Sat Dec 7 19:06:21 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 471, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 471 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 471 8 + [422 0 0 -471 0 471] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.682059 inch 3.960000 inch translate +matrix currentmatrix +3.135881 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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a/books/ps/wd-gammaz.ps b/books/ps/wd-gammaz.ps new file mode 100644 index 0000000..5734b2c --- /dev/null +++ b/books/ps/wd-gammaz.ps @@ -0,0 +1,1079 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 192.347462 284.120003 419.652538 538.120003 +%%Creator: xwd2ps +%%CreationDate: Sat Dec 7 19:11:24 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -h3.5 +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 422, height = 472, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 424, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 472 +% starting column = 1 +% ending column = 422 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 422 472 8 + [422 0 0 -472 0 472] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.685381 inch 3.960000 inch translate +matrix currentmatrix +3.129237 inch 3.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +236b8e +b3b3b3 +d8d8bf +dbdb70 +3299cc +ffff00 +238e23 +0000ff 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diff --git a/books/ps/bouquet.ps b/books/ps/bouquet.ps new file mode 100644 index 0000000..a8538f4 --- /dev/null +++ b/books/ps/bouquet.ps @@ -0,0 +1,326 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 53 231 227 100 psDrawLine + globGC (X) 48 236 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 246 231 72 100 psDrawLine + globGC (Y) 251 236 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 63 150 227 psDrawLine + globGC (Z) 155 58 psDrawStr + globalGC1 106.000000 setForeground + globGC 106.000000 setForeground + 1 0 0 componentGC setLineAttributes + componentGC 0.000000 setForeground + componentGC 0.000000 setForeground + componentGC 55 141 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 77 142 55 141 psDrawLine + componentGC 0.000000 setForeground + componentGC 72 139 55 141 psDrawLine + componentGC 0.000000 setForeground + componentGC 113 168 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 125 162 113 168 psDrawLine + componentGC 0.000000 setForeground + componentGC 118 160 113 168 psDrawLine + componentGC 0.000000 setForeground + componentGC 186 168 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 181 160 186 168 psDrawLine + componentGC 0.000000 setForeground + componentGC 174 162 186 168 psDrawLine + componentGC 0.000000 setForeground + componentGC 244 141 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 227 139 244 141 psDrawLine + componentGC 0.000000 setForeground + componentGC 222 142 244 141 psDrawLine + componentGC 0.000000 setForeground + componentGC 270 100 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 246 105 270 100 psDrawLine + componentGC 0.000000 setForeground + componentGC 245 110 270 100 psDrawLine + componentGC 0.000000 setForeground + componentGC 260 60 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 239 72 260 60 psDrawLine + componentGC 0.000000 setForeground + componentGC 241 76 260 60 psDrawLine + componentGC 0.000000 setForeground + componentGC 226 29 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 210 47 226 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 215 50 226 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 176 14 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 169 34 176 14 psDrawLine + componentGC 0.000000 setForeground + componentGC 175 35 176 14 psDrawLine + componentGC 0.000000 setForeground + componentGC 123 14 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 124 35 123 14 psDrawLine + componentGC 0.000000 setForeground + componentGC 130 34 123 14 psDrawLine + componentGC 0.000000 setForeground + componentGC 73 29 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 84 50 73 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 89 47 73 29 psDrawLine + componentGC 0.000000 setForeground + componentGC 39 60 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 58 76 39 60 psDrawLine + componentGC 0.000000 setForeground + componentGC 60 72 39 60 psDrawLine + componentGC 0.000000 setForeground + componentGC 30 100 150 158 psDrawLine + componentGC 0.000000 setForeground + componentGC 54 110 30 100 psDrawLine + componentGC 0.000000 setForeground + componentGC 53 105 30 100 psDrawLine + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/cartcoord.ps b/books/ps/cartcoord.ps new file mode 100644 index 0000000..48dca14 --- /dev/null +++ b/books/ps/cartcoord.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 10 259 263 95 psDrawLine + globGC (X) 5 264 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 286 269 40 88 psDrawLine + globGC (Y) 291 274 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 156 20 144 279 psDrawLine + globGC (Z) 161 15 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.562500 setForeground + renderGC + 150 170 + 149 167 + 149 169 + 4 150 170 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 167 + 149 169 + 3 150 170 psDrawLines + renderGC 0.562500 setForeground + renderGC + 151 171 + 150 168 + 150 170 + 4 151 171 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 168 + 150 170 + 3 151 171 psDrawLines + renderGC 0.562500 setForeground + renderGC + 149 167 + 150 170 + 150 168 + 4 149 167 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 170 + 150 168 + 3 149 167 psDrawLines + renderGC 0.562500 setForeground + renderGC + 150 168 + 149 164 + 149 167 + 4 150 168 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 164 + 149 167 + 3 150 168 psDrawLines + renderGC 0.625000 setForeground + renderGC + 152 173 + 151 169 + 151 171 + 4 152 173 psFillPoly + renderGC 0.000000 setForeground + renderGC + 151 169 + 151 171 + 3 152 173 psDrawLines + renderGC 0.562500 setForeground + renderGC + 150 168 + 151 171 + 151 169 + 4 150 168 psFillPoly + renderGC 0.000000 setForeground + renderGC + 151 171 + 151 169 + 3 150 168 psDrawLines + renderGC 0.625000 setForeground + renderGC + 151 169 + 151 165 + 150 168 + 4 151 169 psFillPoly + renderGC 0.000000 setForeground + renderGC + 151 165 + 150 168 + 3 151 169 psDrawLines + renderGC 0.562500 setForeground + renderGC + 149 164 + 150 168 + 151 165 + 4 149 164 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 168 + 151 165 + 3 149 164 psDrawLines + renderGC 0.625000 setForeground + renderGC + 152 174 + 152 170 + 152 173 + 4 152 174 psFillPoly + renderGC 0.000000 setForeground + renderGC + 152 170 + 152 173 + 3 152 174 psDrawLines + renderGC 0.625000 setForeground + renderGC + 151 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setForeground + renderGC + 73 224 + 81 215 + 81 217 + 4 73 224 psFillPoly + renderGC 0.000000 setForeground + renderGC + 81 215 + 81 217 + 3 73 224 psDrawLines + renderGC 0.625000 setForeground + renderGC + 81 217 + 73 227 + 73 224 + 4 81 217 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 227 + 73 224 + 3 81 217 psDrawLines + renderGC 0.625000 setForeground + renderGC + 102 177 + 95 186 + 95 183 + 4 102 177 psFillPoly + renderGC 0.000000 setForeground + renderGC + 95 186 + 95 183 + 3 102 177 psDrawLines + renderGC 0.500000 setForeground + renderGC + 81 210 + 88 200 + 88 203 + 4 81 210 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 200 + 88 203 + 3 81 210 psDrawLines + renderGC 0.437500 setForeground + renderGC + 88 203 + 81 212 + 81 210 + 4 88 203 psFillPoly + renderGC 0.000000 setForeground + renderGC + 81 212 + 81 210 + 3 88 203 psDrawLines + renderGC 0.687500 setForeground + renderGC + 65 237 + 73 227 + 73 229 + 4 65 237 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 227 + 73 229 + 3 65 237 psDrawLines + renderGC 0.750000 setForeground + renderGC + 73 229 + 65 239 + 65 237 + 4 73 229 psFillPoly + renderGC 0.000000 setForeground + renderGC + 65 239 + 65 237 + 3 73 229 psDrawLines + renderGC 0.562500 setForeground + renderGC + 88 195 + 95 186 + 95 188 + 4 88 195 psFillPoly + renderGC 0.000000 setForeground + renderGC + 95 186 + 95 188 + 3 88 195 psDrawLines + renderGC 0.562500 setForeground + renderGC + 95 188 + 88 197 + 88 195 + 4 95 188 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 197 + 88 195 + 3 95 188 psDrawLines + renderGC 0.500000 setForeground + renderGC + 73 222 + 81 212 + 81 215 + 4 73 222 psFillPoly + renderGC 0.000000 setForeground + renderGC + 81 212 + 81 215 + 3 73 222 psDrawLines + renderGC 0.562500 setForeground + renderGC + 81 215 + 73 224 + 73 222 + 4 81 215 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 224 + 73 222 + 3 81 215 psDrawLines + renderGC 0.500000 setForeground + renderGC + 80 207 + 88 197 + 88 200 + 4 80 207 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 197 + 88 200 + 3 80 207 psDrawLines + renderGC 0.500000 setForeground + renderGC + 88 200 + 81 210 + 80 207 + 4 88 200 psFillPoly + renderGC 0.000000 setForeground + renderGC + 81 210 + 80 207 + 3 88 200 psDrawLines + renderGC 0.625000 setForeground + renderGC + 64 234 + 73 224 + 73 227 + 4 64 234 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 224 + 73 227 + 3 64 234 psDrawLines + renderGC 0.687500 setForeground + renderGC + 73 227 + 65 237 + 64 234 + 4 73 227 psFillPoly + renderGC 0.000000 setForeground + renderGC + 65 237 + 64 234 + 3 73 227 psDrawLines + renderGC 0.625000 setForeground + renderGC + 88 192 + 95 183 + 95 186 + 4 88 192 psFillPoly + renderGC 0.000000 setForeground + renderGC + 95 183 + 95 186 + 3 88 192 psDrawLines + renderGC 0.562500 setForeground + renderGC + 95 186 + 88 195 + 88 192 + 4 95 186 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 195 + 88 192 + 3 95 186 psDrawLines + renderGC 0.437500 setForeground + renderGC + 73 219 + 81 210 + 81 212 + 4 73 219 psFillPoly + renderGC 0.000000 setForeground + renderGC + 81 210 + 81 212 + 3 73 219 psDrawLines + renderGC 0.500000 setForeground + renderGC + 81 212 + 73 222 + 73 219 + 4 81 212 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 222 + 73 219 + 3 81 212 psDrawLines + renderGC 0.562500 setForeground + renderGC + 80 204 + 88 195 + 88 197 + 4 80 204 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 195 + 88 197 + 3 80 204 psDrawLines + renderGC 0.500000 setForeground + renderGC + 88 197 + 80 207 + 80 204 + 4 88 197 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 207 + 80 204 + 3 88 197 psDrawLines + renderGC 0.562500 setForeground + renderGC + 64 232 + 73 222 + 73 224 + 4 64 232 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 222 + 73 224 + 3 64 232 psDrawLines + renderGC 0.625000 setForeground + renderGC + 73 224 + 64 234 + 64 232 + 4 73 224 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 234 + 64 232 + 3 73 224 psDrawLines + renderGC 0.625000 setForeground + renderGC + 95 183 + 88 192 + 88 190 + 4 95 183 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 192 + 88 190 + 3 95 183 psDrawLines + renderGC 0.500000 setForeground + renderGC + 72 217 + 80 207 + 81 210 + 4 72 217 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 207 + 81 210 + 3 72 217 psDrawLines + renderGC 0.437500 setForeground + renderGC + 81 210 + 73 219 + 72 217 + 4 81 210 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 219 + 72 217 + 3 81 210 psDrawLines + renderGC 0.562500 setForeground + renderGC + 80 202 + 88 192 + 88 195 + 4 80 202 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 192 + 88 195 + 3 80 202 psDrawLines + renderGC 0.562500 setForeground + renderGC + 88 195 + 80 204 + 80 202 + 4 88 195 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 204 + 80 202 + 3 88 195 psDrawLines + renderGC 0.500000 setForeground + renderGC + 64 230 + 73 219 + 73 222 + 4 64 230 psFillPoly + renderGC 0.000000 setForeground + renderGC + 73 219 + 73 222 + 3 64 230 psDrawLines + renderGC 0.562500 setForeground + renderGC + 73 222 + 64 232 + 64 230 + 4 73 222 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 232 + 64 230 + 3 73 222 psDrawLines + renderGC 0.500000 setForeground + renderGC + 72 214 + 80 204 + 80 207 + 4 72 214 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 204 + 80 207 + 3 72 214 psDrawLines + renderGC 0.500000 setForeground + renderGC + 80 207 + 72 217 + 72 214 + 4 80 207 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 217 + 72 214 + 3 80 207 psDrawLines + renderGC 0.625000 setForeground + renderGC + 80 199 + 88 190 + 88 192 + 4 80 199 psFillPoly + renderGC 0.000000 setForeground + renderGC + 88 190 + 88 192 + 3 80 199 psDrawLines + renderGC 0.562500 setForeground + renderGC + 88 192 + 80 202 + 80 199 + 4 88 192 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 202 + 80 199 + 3 88 192 psDrawLines + renderGC 0.437500 setForeground + renderGC + 64 227 + 72 217 + 73 219 + 4 64 227 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 217 + 73 219 + 3 64 227 psDrawLines + renderGC 0.500000 setForeground + renderGC + 73 219 + 64 230 + 64 227 + 4 73 219 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 230 + 64 227 + 3 73 219 psDrawLines + renderGC 0.562500 setForeground + renderGC + 72 212 + 80 202 + 80 204 + 4 72 212 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 202 + 80 204 + 3 72 212 psDrawLines + renderGC 0.500000 setForeground + renderGC + 80 204 + 72 214 + 72 212 + 4 80 204 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 214 + 72 212 + 3 80 204 psDrawLines + renderGC 0.625000 setForeground + renderGC + 88 190 + 80 199 + 80 197 + 4 88 190 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 199 + 80 197 + 3 88 190 psDrawLines + renderGC 0.500000 setForeground + renderGC + 64 225 + 72 214 + 72 217 + 4 64 225 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 214 + 72 217 + 3 64 225 psDrawLines + renderGC 0.437500 setForeground + renderGC + 72 217 + 64 227 + 64 225 + 4 72 217 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 227 + 64 225 + 3 72 217 psDrawLines + renderGC 0.562500 setForeground + renderGC + 72 209 + 80 199 + 80 202 + 4 72 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 199 + 80 202 + 3 72 209 psDrawLines + renderGC 0.562500 setForeground + renderGC + 80 202 + 72 212 + 72 209 + 4 80 202 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 212 + 72 209 + 3 80 202 psDrawLines + renderGC 0.500000 setForeground + renderGC + 64 222 + 72 212 + 72 214 + 4 64 222 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 212 + 72 214 + 3 64 222 psDrawLines + renderGC 0.500000 setForeground + renderGC + 72 214 + 64 225 + 64 222 + 4 72 214 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 225 + 64 222 + 3 72 214 psDrawLines + renderGC 0.625000 setForeground + renderGC + 72 207 + 80 197 + 80 199 + 4 72 207 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 197 + 80 199 + 3 72 207 psDrawLines + renderGC 0.562500 setForeground + renderGC + 80 199 + 72 209 + 72 207 + 4 80 199 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 209 + 72 207 + 3 80 199 psDrawLines + renderGC 0.562500 setForeground + renderGC + 64 219 + 72 209 + 72 212 + 4 64 219 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 209 + 72 212 + 3 64 219 psDrawLines + renderGC 0.500000 setForeground + renderGC + 72 212 + 64 222 + 64 219 + 4 72 212 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 222 + 64 219 + 3 72 212 psDrawLines + renderGC 0.625000 setForeground + renderGC + 80 197 + 72 207 + 72 204 + 4 80 197 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 207 + 72 204 + 3 80 197 psDrawLines + renderGC 0.562500 setForeground + renderGC + 63 217 + 72 207 + 72 209 + 4 63 217 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 207 + 72 209 + 3 63 217 psDrawLines + renderGC 0.562500 setForeground + renderGC + 72 209 + 64 219 + 63 217 + 4 72 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 64 219 + 63 217 + 3 72 209 psDrawLines + renderGC 0.625000 setForeground + renderGC + 63 214 + 72 204 + 72 207 + 4 63 214 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 204 + 72 207 + 3 63 214 psDrawLines + renderGC 0.562500 setForeground + renderGC + 72 207 + 63 217 + 63 214 + 4 72 207 psFillPoly + renderGC 0.000000 setForeground + renderGC + 63 217 + 63 214 + 3 72 207 psDrawLines + renderGC 0.625000 setForeground + renderGC + 72 204 + 63 214 + 63 212 + 4 72 204 psFillPoly + renderGC 0.000000 setForeground + renderGC + 63 214 + 63 212 + 3 72 204 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/clipgamma.ps b/books/ps/clipgamma.ps new file mode 100644 index 0000000..50d5976 --- /dev/null +++ b/books/ps/clipgamma.ps @@ -0,0 +1,55300 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 30 228 244 77 psDrawLine + globGC (X) 25 233 psDrawStr + globalGC1 177.000000 setForeground + globGC 177.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 269 228 55 77 psDrawLine + globGC (Y) 274 233 psDrawStr + globalGC1 177.000000 setForeground + globGC 177.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 18 149 234 psDrawLine + globGC (Z) 154 13 psDrawStr + globalGC1 177.000000 setForeground + globGC 177.000000 setForeground + 0 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.625000 setForeground + renderGC + 196 11 + 196 113 + 187 192 + 4 196 11 psFillPoly + renderGC 0.000000 setForeground + renderGC + 196 113 + 187 192 + 3 196 11 psDrawLines + renderGC 0.625000 setForeground + renderGC + 195 148 + 187 192 + 196 113 + 4 195 148 psFillPoly + renderGC 0.000000 setForeground + renderGC + 187 192 + 196 113 + 3 195 148 psDrawLines + renderGC 0.562500 setForeground + renderGC + 187 192 + 195 148 + 190 178 + 4 187 192 psFillPoly + renderGC 0.000000 setForeground + renderGC + 195 148 + 190 178 + 3 187 192 psDrawLines + renderGC 0.562500 setForeground + renderGC + 187 192 + 191 51 + 196 11 + 4 187 192 psFillPoly + renderGC 0.000000 setForeground + renderGC + 191 51 + 196 11 + 3 187 192 psDrawLines + renderGC 0.500000 setForeground + renderGC + 191 51 + 187 192 + 190 109 + 4 191 51 psFillPoly + renderGC 0.000000 setForeground + renderGC + 187 192 + 190 109 + 3 191 51 psDrawLines + renderGC 0.437500 setForeground + renderGC + 190 178 + 190 109 + 187 192 + 4 190 178 psFillPoly + renderGC 0.000000 setForeground + renderGC + 190 109 + 187 192 + 3 190 178 psDrawLines + renderGC 0.812500 setForeground + renderGC + 198 139 + 190 178 + 195 148 + 4 198 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 190 178 + 195 148 + 3 198 139 psDrawLines + renderGC 0.625000 setForeground + renderGC + 190 178 + 198 139 + 195 145 + 4 190 178 psFillPoly + renderGC 0.000000 setForeground + renderGC + 198 139 + 195 145 + 3 190 178 psDrawLines + renderGC 0.625000 setForeground + renderGC + 190 109 + 190 178 + 190 145 + 4 190 109 psFillPoly + renderGC 0.000000 setForeground + renderGC + 190 178 + 190 145 + 3 190 109 psDrawLines + renderGC 0.312500 setForeground + renderGC + 195 145 + 190 145 + 190 178 + 4 195 145 psFillPoly + renderGC 0.000000 setForeground + renderGC + 190 145 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renderGC + 169 128 + 162 170 + 164 207 + 4 169 128 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 170 + 164 207 + 3 169 128 psDrawLines + renderGC 0.750000 setForeground + renderGC + 164 211 + 169 211 + 166 209 + 4 164 211 psFillPoly + renderGC 0.000000 setForeground + renderGC + 169 211 + 166 209 + 3 164 211 psDrawLines + renderGC 0.687500 setForeground + renderGC + 174 132 + 166 209 + 169 211 + 4 174 132 psFillPoly + renderGC 0.000000 setForeground + renderGC + 166 209 + 169 211 + 3 174 132 psDrawLines + renderGC 0.687500 setForeground + renderGC + 166 209 + 174 132 + 175 24 + 4 166 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 174 132 + 175 24 + 3 166 209 psDrawLines + renderGC 0.562500 setForeground + renderGC + 159 202 + 164 211 + 162 209 + 4 159 202 psFillPoly + renderGC 0.000000 setForeground + renderGC + 164 211 + 162 209 + 3 159 202 psDrawLines + renderGC 0.500000 setForeground + renderGC + 166 209 + 162 209 + 164 211 + 4 166 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 209 + 164 211 + 3 166 209 psDrawLines + renderGC 0.687500 setForeground + renderGC + 162 209 + 157 187 + 159 202 + 4 162 209 psFillPoly + renderGC 0.000000 setForeground + renderGC + 157 187 + 159 202 + 3 162 209 psDrawLines + renderGC 0.562500 setForeground + renderGC + 157 187 + 162 209 + 160 184 + 4 157 187 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 209 + 160 184 + 3 157 187 psDrawLines + renderGC 0.500000 setForeground + renderGC + 167 213 + 173 178 + 169 211 + 4 167 213 psFillPoly + renderGC 0.000000 setForeground + renderGC + 173 178 + 169 211 + 3 167 213 psDrawLines + renderGC 0.562500 setForeground + renderGC + 177 139 + 169 211 + 173 178 + 4 177 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 169 211 + 173 178 + 3 177 139 psDrawLines + renderGC 0.375000 setForeground + renderGC + 169 211 + 177 139 + 174 132 + 4 169 211 psFillPoly + renderGC 0.000000 setForeground + renderGC + 177 139 + 174 132 + 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renderGC 0.750000 setForeground + renderGC + 73 149 + 65 142 + 69 145 + 4 73 149 psFillPoly + renderGC 0.000000 setForeground + renderGC + 65 142 + 69 145 + 3 73 149 psDrawLines + renderGC 0.812500 setForeground + renderGC + 80 158 + 72 149 + 75 155 + 4 80 158 psFillPoly + renderGC 0.000000 setForeground + renderGC + 72 149 + 75 155 + 3 80 158 psDrawLines + renderGC 0.812500 setForeground + renderGC + 72 149 + 80 158 + 76 153 + 4 72 149 psFillPoly + renderGC 0.000000 setForeground + renderGC + 80 158 + 76 153 + 3 72 149 psDrawLines + renderGC 0.750000 setForeground + renderGC + 76 153 + 69 145 + 72 149 + 4 76 153 psFillPoly + renderGC 0.000000 setForeground + renderGC + 69 145 + 72 149 + 3 76 153 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/compatan.ps b/books/ps/compatan.ps new file mode 100644 index 0000000..7cbee24 --- /dev/null +++ b/books/ps/compatan.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 237 96 7 156 psDrawLine + globGC (X) 242 91 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 92 54 178 221 psDrawLine + globGC (Y) 87 49 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 129 21 129 199 psDrawLine + globGC (Z) 134 16 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.625000 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psDrawLines + renderGC 0.687500 setForeground + renderGC + 183 156 + 195 161 + 191 154 + 4 183 156 psFillPoly + renderGC 0.000000 setForeground + renderGC + 195 161 + 191 154 + 3 183 156 psDrawLines + renderGC 0.687500 setForeground + renderGC + 195 161 + 183 156 + 187 162 + 4 195 161 psFillPoly + renderGC 0.000000 setForeground + renderGC + 183 156 + 187 162 + 3 195 161 psDrawLines + renderGC 0.687500 setForeground + renderGC + 174 157 + 187 162 + 183 156 + 4 174 157 psFillPoly + renderGC 0.000000 setForeground + renderGC + 187 162 + 183 156 + 3 174 157 psDrawLines + renderGC 0.687500 setForeground + renderGC + 187 162 + 174 157 + 178 164 + 4 187 162 psFillPoly + renderGC 0.000000 setForeground + renderGC + 174 157 + 178 164 + 3 187 162 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/compexp.ps b/books/ps/compexp.ps new file mode 100644 index 0000000..fd152b4 --- /dev/null +++ b/books/ps/compexp.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 0 300 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 195 74 67 269 psDrawLine + globGC (X) 200 69 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 15 115 285 205 psDrawLine + globGC (Y) 10 110 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 141 31 141 247 psDrawLine + globGC (Z) 146 26 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + 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0.750000 setForeground + renderGC + 250 133 + 261 120 + 252 124 + 4 250 133 psFillPoly + renderGC 0.000000 setForeground + renderGC + 261 120 + 252 124 + 3 250 133 psDrawLines + renderGC 0.750000 setForeground + renderGC + 261 120 + 250 133 + 259 130 + 4 261 120 psFillPoly + renderGC 0.000000 setForeground + renderGC + 250 133 + 259 130 + 3 261 120 psDrawLines + renderGC 0.750000 setForeground + renderGC + 252 124 + 264 109 + 254 114 + 4 252 124 psFillPoly + renderGC 0.000000 setForeground + renderGC + 264 109 + 254 114 + 3 252 124 psDrawLines + renderGC 0.750000 setForeground + renderGC + 264 109 + 252 124 + 261 120 + 4 264 109 psFillPoly + renderGC 0.000000 setForeground + renderGC + 252 124 + 261 120 + 3 264 109 psDrawLines + renderGC 0.812500 setForeground + renderGC + 254 114 + 267 96 + 257 102 + 4 254 114 psFillPoly + renderGC 0.000000 setForeground + renderGC + 267 96 + 257 102 + 3 254 114 psDrawLines + renderGC 0.750000 setForeground + renderGC + 267 96 + 254 114 + 264 109 + 4 267 96 psFillPoly + renderGC 0.000000 setForeground + renderGC + 254 114 + 264 109 + 3 267 96 psDrawLines + renderGC 0.812500 setForeground + renderGC + 257 102 + 270 81 + 259 89 + 4 257 102 psFillPoly + renderGC 0.000000 setForeground + renderGC + 270 81 + 259 89 + 3 257 102 psDrawLines + renderGC 0.750000 setForeground + renderGC + 270 81 + 257 102 + 267 96 + 4 270 81 psFillPoly + renderGC 0.000000 setForeground + renderGC + 257 102 + 267 96 + 3 270 81 psDrawLines + renderGC 0.750000 setForeground + renderGC + 259 89 + 274 63 + 263 73 + 4 259 89 psFillPoly + renderGC 0.000000 setForeground + renderGC + 274 63 + 263 73 + 3 259 89 psDrawLines + renderGC 0.750000 setForeground + renderGC + 274 63 + 259 89 + 270 81 + 4 274 63 psFillPoly + renderGC 0.000000 setForeground + renderGC + 259 89 + 270 81 + 3 274 63 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/compgamm.ps b/books/ps/compgamm.ps new file mode 100644 index 0000000..da2b450 --- /dev/null +++ b/books/ps/compgamm.ps @@ -0,0 +1,18791 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 37 224 239 81 psDrawLine + globGC (X) 32 229 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 262 224 60 81 psDrawLine + globGC (Y) 267 229 psDrawStr + globalGC1 109.000000 setForeground + globGC 109.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 24 149 229 psDrawLine + globGC (Z) 154 19 psDrawStr + globalGC1 109.000000 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168 98 psDrawLines + renderGC 0.812500 setForeground + renderGC + 162 109 + 168 98 + 165 103 + 4 162 109 psFillPoly + renderGC 0.000000 setForeground + renderGC + 168 98 + 165 103 + 3 162 109 psDrawLines + renderGC 0.625000 setForeground + renderGC + 168 99 + 173 93 + 170 97 + 4 168 99 psFillPoly + renderGC 0.000000 setForeground + renderGC + 173 93 + 170 97 + 3 168 99 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/complexexp.ps b/books/ps/complexexp.ps new file mode 100644 index 0000000..b598e04 --- /dev/null +++ b/books/ps/complexexp.ps @@ -0,0 +1,14050 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 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+ renderGC + 125 12 + 137 12 + 131 17 + 4 125 12 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 12 + 131 17 + 3 125 12 psDrawLines + renderGC 0.687500 setForeground + renderGC + 137 12 + 150 12 + 144 17 + 4 137 12 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 12 + 144 17 + 3 137 12 psDrawLines + renderGC 0.687500 setForeground + renderGC + 137 21 + 124 21 + 131 17 + 4 137 21 psFillPoly + renderGC 0.000000 setForeground + renderGC + 124 21 + 131 17 + 3 137 21 psDrawLines + renderGC 0.687500 setForeground + renderGC + 150 21 + 137 21 + 144 17 + 4 150 21 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 21 + 144 17 + 3 150 21 psDrawLines + renderGC 0.750000 setForeground + renderGC + 131 17 + 144 17 + 137 21 + 4 131 17 psFillPoly + renderGC 0.000000 setForeground + renderGC + 144 17 + 137 21 + 3 131 17 psDrawLines + renderGC 0.812500 setForeground + renderGC + 144 25 + 131 25 + 137 21 + 4 144 25 psFillPoly + renderGC 0.000000 setForeground + renderGC + 131 25 + 137 21 + 3 144 25 psDrawLines + renderGC 0.750000 setForeground + renderGC + 124 21 + 137 21 + 131 25 + 4 124 21 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 21 + 131 25 + 3 124 21 psDrawLines + renderGC 0.687500 setForeground + renderGC + 137 21 + 150 21 + 144 25 + 4 137 21 psFillPoly + renderGC 0.000000 setForeground + renderGC + 150 21 + 144 25 + 3 137 21 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/complexroot.ps b/books/ps/complexroot.ps new file mode 100644 index 0000000..bf60408 --- /dev/null +++ b/books/ps/complexroot.ps @@ -0,0 +1,14050 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set 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cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/cylcoord.ps b/books/ps/cylcoord.ps new file mode 100644 index 0000000..171942c --- /dev/null +++ b/books/ps/cylcoord.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 6 271 259 82 psDrawLine + globGC (X) 1 276 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 285 267 34 85 psDrawLine + globGC (Y) 290 272 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 142 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+%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state 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105 + 168 107 + 3 169 116 psDrawLines + renderGC 0.562500 setForeground + renderGC + 171 109 + 175 120 + 172 118 + 4 171 109 psFillPoly + renderGC 0.000000 setForeground + renderGC + 175 120 + 172 118 + 3 171 109 psDrawLines + renderGC 0.625000 setForeground + renderGC + 172 118 + 168 107 + 171 109 + 4 172 118 psFillPoly + renderGC 0.000000 setForeground + renderGC + 168 107 + 171 109 + 3 172 118 psDrawLines + renderGC 0.625000 setForeground + renderGC + 175 120 + 171 109 + 174 110 + 4 175 120 psFillPoly + renderGC 0.000000 setForeground + renderGC + 171 109 + 174 110 + 3 175 120 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/exit.ps b/books/ps/exit.ps new file mode 100644 index 0000000..400b9c0 --- /dev/null +++ b/books/ps/exit.ps @@ -0,0 +1,386 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 284.120003 559.000000 538.120003 +%%Creator: /u/jonms/XtoPS/xwd2ps +%%CreationDate: Tue Oct 8 20:20:30 1991 +%%Title: standard input +%%EndComments +% /u/jonms/XtoPS/xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% /u/jonms/XtoPS/xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 60, height = 30, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 60, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 30 +% starting column = 1 +% ending column = 60 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 60 30 8 + [60 0 0 -30 0 30] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%`0.750000 inch 3.960000 inch translate +matrix currentmatrix +.5 inch .16 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +dbdb70 +2f4f4f +23238e +d3d3d3 +ff0000 +000000 +004900 +004955 +0049aa +0049ff +006d00 +006d55 +006daa +006dff +009200 +009255 +0092aa +0092ff +00b600 +00b655 +00b6aa +00b6ff +00db00 +00db55 +00dbaa +00dbff +00ff00 +00ff55 +00ffaa +00ffff +240000 +240055 +2400aa +2400ff +242400 +242455 +2424aa +2424ff +244900 +244955 +2449aa +2449ff +246d00 +246d55 +246daa +246dff +249200 +249255 +2492aa +2492ff +24b600 +24b655 +24b6aa +24b6ff +24db00 +24db55 +24dbaa +24dbff +24ff00 +24ff55 +24ffaa +24ffff +490000 +490055 +4900aa +4900ff +492400 +492455 +4924aa +4924ff +494900 +494955 +4949aa +4949ff +496d00 +496d55 +496daa +496dff +499200 +499255 +4992aa +4992ff +49b600 +49b655 +49b6aa +49b6ff +49db00 +49db55 +49dbaa +49dbff +49ff00 +49ff55 +49ffaa +49ffff +6d0000 +6d0055 +6d00aa +6d00ff +6d2400 +6d2455 +6d24aa +6d24ff +6d4900 +6d4955 +6d49aa +6d49ff +6d6d00 +6d6d55 +6d6daa +6d6dff +6d9200 +6d9255 +6d92aa +6d92ff +6db600 +6db655 +6db6aa +6db6ff +6ddb00 +6ddb55 +6ddbaa +6ddbff +6dff00 +6dff55 +6dffaa +6dffff +920000 +920055 +9200aa +9200ff +922400 +922455 +9224aa +9224ff +924900 +924955 +9249aa +9249ff +926d00 +926d55 +926daa +926dff +929200 +929255 +9292aa +9292ff +92b600 +92b655 +92b6aa +92b6ff +92db00 +92db55 +92dbaa +92dbff +92ff00 +92ff55 +92ffaa +92ffff +b60000 +b60055 +b600aa +b600ff +b62400 +b62455 +b624aa +b624ff +b64900 +b64955 +b649aa +b649ff +b66d00 +b66d55 +b66daa +b66dff +b69200 +b69255 +b692aa +b692ff +b6b600 +b6b655 +b6b6aa +b6b6ff +b6db00 +b6db55 +b6dbaa +b6dbff +b6ff00 +b6ff55 +b6ffaa +b6ffff +db0000 +db0055 +db00aa +db00ff +db2400 +db2455 +db24aa +db24ff +db4900 +db4955 +db49aa +db49ff +db6d00 +db6d55 +db6daa +db6dff +db9200 +db9255 +db92aa +db92ff +dbb600 +dbb655 +dbb6aa +dbb6ff +dbdb00 +dbdb55 +dbdbaa +dbdbff +dbff00 +dbff55 +dbffaa +dbffff +ff0000 +ff0055 +ff00aa +ff00ff +ff2400 +ff2455 +ff24aa +ff24ff +ff4900 +ff4955 +ff49aa +ff49ff +ff6d00 +ff6d55 +ff6daa +ff6dff +ff9200 +ff9255 +ff92aa +ff92ff +ffb600 +ffb655 +ffb6aa +ffb6ff +ffdb00 +ffdb55 +ffdbaa +ffdbff +ffff00 +ffff55 +ffffaa +ffffff + + +drawcolorimage +3b073b073b073b070507090103070101070701010307010103070b01050705070a0102070201060702010207020102070c01040705070a0103070201040702010307020103070b01 +0407050702010b0702010407020103070201070702010907050702010c0702010207020104070201070702010907050702010c070201020702010407020107070201090705070201 +0d0702010007020105070201070702010907050702010d0702010007020105070201070702010907050702010e070401060702010707020109070507080108070401060702010707 +02010907050709010807020107070201070702010907050709010807020107070201070702010907050702010e07040106070201070702010907050702010e070401060702010707 +02010907050702010d0702010007020105070201070702010907050702010d0702010007020105070201070702010907050702010c07020102070201040702010707020109070507 +02010c0702010207020104070201070702010907050702010b07020104070201030702010707020109070507090104070201040702010307020107070201090705070a0102070201 +0607020102070201070702010907060709010307010107070101030701010807010109073b073b073b073b07 +pop pop setmatrix + +/#copies 1 def +grestore +%%Trailer diff --git a/books/ps/h-alldoms.ps b/books/ps/h-alldoms.ps new file mode 100644 index 0000000..0fd08ef --- /dev/null +++ b/books/ps/h-alldoms.ps @@ -0,0 +1,1367 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:13:14 1991 +%%Title: h-alldoms.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-alldoms.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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b/books/ps/h-allrank.ps new file mode 100644 index 0000000..51ebf3e --- /dev/null +++ b/books/ps/h-allrank.ps @@ -0,0 +1,1565 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:13:35 1991 +%%Title: h-allrank.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-allrank.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-atsearch.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:14:14 1991 +%%Title: h-brfront.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-brfront.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 745, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 745 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 745 8 + [622 0 0 -745 0 745] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +% 0.750000 inch 1.517878 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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100644 index 0000000..62f65d5 --- /dev/null +++ b/books/ps/h-comsearch.ps @@ -0,0 +1,1480 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:14:32 1991 +%%Title: h-comsearch.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-comsearch.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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24 07:52:57 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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b/books/ps/h-consearch.ps new file mode 100644 index 0000000..c9325ad --- /dev/null +++ b/books/ps/h-consearch.ps @@ -0,0 +1,1145 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:14:53 1991 +%%Title: h-consearch.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-consearch.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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100644 index 0000000..0bec001 --- /dev/null +++ b/books/ps/h-consearch2.ps @@ -0,0 +1,1392 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:15:09 1991 +%%Title: h-consearch2.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-consearch2.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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/dev/null +++ b/books/ps/h-crossref.ps @@ -0,0 +1,819 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 209.573372 559.000000 612.666634 +%%Creator: xwd2ps +%%CreationDate: Tue Sep 24 07:53:43 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-docsearch.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:15:44 1991 +%%Title: h-gensearch.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-gensearch.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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+%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:16:00 1991 +%%Title: h-matargs.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matargs.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+%%Title: h-matats.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matats.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+grestore +%%Trailer diff --git a/books/ps/h-matdesc.ps b/books/ps/h-matdesc.ps new file mode 100644 index 0000000..fe1fa02 --- /dev/null +++ b/books/ps/h-matdesc.ps @@ -0,0 +1,1041 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:16:31 1991 +%%Title: h-matdesc.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matdesc.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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/dev/null +++ b/books/ps/h-matexamp.ps @@ -0,0 +1,1074 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:16:46 1991 +%%Title: h-matexamp.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matexamp.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matexports.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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index 0000000..05d3e20 --- /dev/null +++ b/books/ps/h-matimp.ps @@ -0,0 +1,1404 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:17:22 1991 +%%Title: h-matimp.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matimp.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:17:36 1991 +%%Title: h-matinv.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matinv.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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b/books/ps/h-matmap.ps @@ -0,0 +1,1430 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:17:51 1991 +%%Title: h-matmap.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matmap.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:18:06 1991 +%%Title: h-matops.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matops.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matpage.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matrelops.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+grestore +%%Trailer diff --git a/books/ps/h-matrix.ps b/books/ps/h-matrix.ps new file mode 100644 index 0000000..d79ca53 --- /dev/null +++ b/books/ps/h-matrix.ps @@ -0,0 +1,871 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 209.573372 559.000000 612.666634 +%%Creator: xwd2ps +%%CreationDate: Tue Sep 24 07:49:42 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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b/books/ps/h-matrixops.ps @@ -0,0 +1,1114 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:18:55 1991 +%%Title: h-matrixops.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matrixops.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 2.7 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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b/books/ps/h-matrixops2.ps @@ -0,0 +1,871 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 209.573372 559.000000 612.666634 +%%Creator: xwd2ps +%%CreationDate: Tue Sep 24 07:49:42 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 2.4 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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b/books/ps/h-matsource.ps @@ -0,0 +1,1390 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 326.600014 469.000000 495.640009 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:20:55 1991 +%%Title: h-matsource.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.32 h-matsource.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 766, height = 389, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 768, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 389 +% starting column = 1 +% ending column = 766 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 766 389 8 + [766 0 0 -389 0 389] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 4.550000 inch translate +matrix currentmatrix +4.500000 inch 2.320000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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+%%CreationDate: Fri Sep 27 18:19:45 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w6.0 -h2.5 +% by matthew.watson.ibm.com:jenks () +% Information from XWD rasterfile header: +% width = 667, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 668, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 667 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 667 495 8 + [667 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%1.250000 inch 4.460000 inch translate +matrix currentmatrix +6 inch 2.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +0000ff +ff00ff +00ff00 +ffff00 +00ffff +ff0000 +000000 +000055 +0000ab +0000ff 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b/books/ps/h-matxref.ps @@ -0,0 +1,1248 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 280.520006 469.000000 541.719999 +%%Creator: xwd2ps +%%CreationDate: Fri Nov 22 15:19:16 1991 +%%Title: h-matxref.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 h-matxref.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.910000 inch translate +matrix currentmatrix +4.500000 inch 2.500000 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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EPSF-1.2 +%%BoundingBox: 143.000000 320.120003 469.000000 502.120003 +%%Creator: xwd2ps +%%CreationDate: Fri Sep 27 17:55:20 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 -h2.5 +% by matthew.watson.ibm.com:jenks () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 4.460000 inch translate +matrix currentmatrix +4.5 inch 2.5 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +0000ff +ff00ff +00ff00 +ffff00 +00ffff +ff0000 +000000 +000055 +0000ab +0000ff 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53.000000 209.573372 559.000000 612.666634 +%%Creator: xwd2ps +%%CreationDate: Tue Sep 24 07:51:58 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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grestore +%%Trailer diff --git a/books/ps/h-parameter.ps b/books/ps/h-parameter.ps new file mode 100644 index 0000000..579b6f3 --- /dev/null +++ b/books/ps/h-parameter.ps @@ -0,0 +1,926 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 209.573372 559.000000 612.666634 +%%Creator: xwd2ps +%%CreationDate: Tue Sep 24 07:51:58 1991 +%%Title: standard input +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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grestore +%%Trailer diff --git a/books/ps/h-root.ps b/books/ps/h-root.ps new file mode 100644 index 0000000..862e881 --- /dev/null +++ b/books/ps/h-root.ps @@ -0,0 +1,1335 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 143.000000 281.197176 469.000000 541.042830 +%%Creator: xwd2ps +%%CreationDate: Sat Nov 23 21:42:29 1991 +%%Title: h-root.xwd +%%EndComments +% xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps -w4.5 h-root.xwd +% by loan7:themos (Themos Tsikas, NAG Ltd) +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%2.000000 inch 3.919405 inch translate +matrix currentmatrix +4.500000 inch 3.581190 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +b3b3b3 +d8d8bf +dbdb70 +ffff00 +3299cc +236b8e +238e23 +cc7f32 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Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 622, height = 495, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 624, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 495 +% starting column = 1 +% ending column = 622 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 622 495 8 + [622 0 0 -495 0 495] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 2.924630 inch translate +matrix currentmatrix +4.5 inch 3 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +23238e +d3d3d3 +ff0000 +7c8498 +fce089 +aeb9d5 +444954 +292c32 +9db5cd +ffff00 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284.120003 559.000000 538.120003 +%%Creator: /u/jonms/XtoPS/xwd2ps +%%CreationDate: Tue Oct 8 20:20:34 1991 +%%Title: standard input +%%EndComments +% /u/jonms/XtoPS/xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% /u/jonms/XtoPS/xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 60, height = 30, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 60, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 30 +% starting column = 1 +% ending column = 60 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 60 30 8 + [60 0 0 -30 0 30] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 3.960000 inch translate +matrix currentmatrix +.5 inch .16 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +dbdb70 +2f4f4f +23238e +d3d3d3 +ff0000 +000000 +004900 +004955 +0049aa +0049ff +006d00 +006d55 +006daa +006dff +009200 +009255 +0092aa +0092ff +00b600 +00b655 +00b6aa +00b6ff +00db00 +00db55 +00dbaa +00dbff +00ff00 +00ff55 +00ffaa +00ffff +240000 +240055 +2400aa +2400ff +242400 +242455 +2424aa +2424ff +244900 +244955 +2449aa +2449ff +246d00 +246d55 +246daa +246dff +249200 +249255 +2492aa +2492ff +24b600 +24b655 +24b6aa +24b6ff +24db00 +24db55 +24dbaa +24dbff +24ff00 +24ff55 +24ffaa +24ffff +490000 +490055 +4900aa +4900ff +492400 +492455 +4924aa +4924ff +494900 +494955 +4949aa +4949ff +496d00 +496d55 +496daa +496dff +499200 +499255 +4992aa +4992ff +49b600 +49b655 +49b6aa +49b6ff +49db00 +49db55 +49dbaa +49dbff +49ff00 +49ff55 +49ffaa +49ffff +6d0000 +6d0055 +6d00aa +6d00ff +6d2400 +6d2455 +6d24aa +6d24ff +6d4900 +6d4955 +6d49aa +6d49ff +6d6d00 +6d6d55 +6d6daa +6d6dff +6d9200 +6d9255 +6d92aa +6d92ff +6db600 +6db655 +6db6aa +6db6ff +6ddb00 +6ddb55 +6ddbaa +6ddbff +6dff00 +6dff55 +6dffaa +6dffff +920000 +920055 +9200aa +9200ff +922400 +922455 +9224aa +9224ff +924900 +924955 +9249aa +9249ff +926d00 +926d55 +926daa +926dff +929200 +929255 +9292aa +9292ff +92b600 +92b655 +92b6aa +92b6ff +92db00 +92db55 +92dbaa +92dbff +92ff00 +92ff55 +92ffaa +92ffff +b60000 +b60055 +b600aa +b600ff +b62400 +b62455 +b624aa +b624ff +b64900 +b64955 +b649aa +b649ff +b66d00 +b66d55 +b66daa +b66dff +b69200 +b69255 +b692aa +b692ff +b6b600 +b6b655 +b6b6aa +b6b6ff +b6db00 +b6db55 +b6dbaa +b6dbff +b6ff00 +b6ff55 +b6ffaa +b6ffff +db0000 +db0055 +db00aa +db00ff +db2400 +db2455 +db24aa +db24ff +db4900 +db4955 +db49aa +db49ff +db6d00 +db6d55 +db6daa +db6dff +db9200 +db9255 +db92aa +db92ff +dbb600 +dbb655 +dbb6aa +dbb6ff +dbdb00 +dbdb55 +dbdbaa +dbdbff +dbff00 +dbff55 +dbffaa +dbffff +ff0000 +ff0055 +ff00aa +ff00ff +ff2400 +ff2455 +ff24aa +ff24ff +ff4900 +ff4955 +ff49aa +ff49ff +ff6d00 +ff6d55 +ff6daa +ff6dff +ff9200 +ff9255 +ff92aa +ff92ff +ffb600 +ffb655 +ffb6aa +ffb6ff +ffdb00 +ffdb55 +ffdbaa +ffdbff +ffff00 +ffff55 +ffffaa +ffffff + + +drawcolorimage +3b003b003b003b0004000101050001010300080103000101090007010700040002010400020102000901020002010800080106000400020104000201020009010200020108000901 +050004000201040002010200020109000201080002010300030104000400020104000201020002010900020108000201040002010400040002010400020102000201090002010800 +020104000201040004000201040002010200020109000201080002010400020104000400020104000201020002010900020108000201040002010400040002010400020102000201 +090002010800020104000201040004000a0102000701040002010800020103000301040004000a01020008010300020108000901050004000a010200080103000201080008010600 +0400020104000201020002010900020108000701070004000201040002010200020109000201080002010c0004000201040002010200020109000201080002010c00040002010400 +02010200020109000201080002010c0004000201040002010200020109000201080002010c0004000201040002010200020109000201080002010c00040002010400020102000201 +09000201080002010c0004000201040002010200080103000701030002010c0004000201040002010200090102000801020002010c00050001010500010103000801030008010200 +01010c003b003b003b003b00 +pop pop setmatrix + +/#copies 1 def +grestore +%%Trailer diff --git a/books/ps/home.ps b/books/ps/home.ps new file mode 100644 index 0000000..c3ad0b0 --- /dev/null +++ b/books/ps/home.ps @@ -0,0 +1,389 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%BoundingBox: 53.000000 284.120003 559.000000 538.120003 +%%Creator: /u/jonms/XtoPS/xwd2ps +%%CreationDate: Tue Oct 8 20:20:32 1991 +%%Title: standard input +%%EndComments +% /u/jonms/XtoPS/xwd2ps -- program written by Robert C. Tatar and Craig A. McGowan. +% The command used to create this file (missing quotes on strings): +% /u/jonms/XtoPS/xwd2ps +% by pi.watson.ibm.com:jonms () +% Information from XWD rasterfile header: +% width = 60, height = 30, depth = 8 +% file_version = 7, pixmap_format = 2, byte_order = 1 +% bitmap_unit = 32, bitmap_bit_order = 1, bitmap_pad = 32 +% bits_per_pixel = 8, bytes_per_line = 60, visual_class = 3 +% bits/rgb = 8, colormap entries = 256, ncolors = 256 +% Portion of raster image in this file: +% starting line = 1 +% ending line = 30 +% starting column = 1 +% ending column = 60 +gsave +/inch {72 mul} def +/buffer 2 string def +/rgbmap 768 string def +/rgb (000) def +/pixels 768 string def +%%Title: colorimage.ps +% Written 11-4-88 by Bob Tatar +% U.S. Mail: GE-CRD, PO Box 8, KW-C214, Schenectady, NY 12301 +% E-Mail: tatar@crd.ge.com +% colorimage procedure to be used on monochrome printers +% or when the colorimage procedure is not available +% NOTE: Only 1 color mode is supported: single proc. & RGB + +systemdict /colorimage known not { % only create if not in systemdict + % Utility procedure for colorimage operator. This procedure takes a + % string of rgb encoded values and creates a string 1/3 as long with + % monochrome values. This procedure assumes 8 bits/color (i.e. + % 1 character/color) + % storage format for input string: (r1 g1 b1 r2 g2 b2 r3 g3 b3 ... ) + % storage format for output string: (g1 g2 g3 ... ) + + /colortograyscale { %def % (string) + dup /rgbdata exch store % (string) + length 3 idiv % Ns/3 + /npixls exch store % ; npixls => Ns/3 + /indx 0 store % ; indx => 0 + /pixls npixls string store % ; pixls => (....) + 0 1 npixls -1 add { % counter + pixls exch % pixls counter + rgbdata indx get .3 mul % pixls counter .3*rgbdata(ind) + rgbdata indx 1 add get .59 mul add % pixls counter .3*rgbdata(ind) + + % .59*rgbdata(ind+1) + rgbdata indx 2 add get .11 mul add % pixls counter .3*rgbdata(ind) + .59 + % *rgbdata(ind+1)+.11*rgbdata(ind+2) + cvi % pixls counter + put % + /indx indx 3 add store % ; /ind => ind+3 + } for % repeat for each rgb value + pixls % (pixls) + } bind def % ; /colortograyscale -> dictionary + + % Utility procedure for colorimage operator. This procedure takes two + % procedures off the stack and merges them into a single procedure. + + /mergeprocs { %def % {proc1} {proc2} + dup length % {proc1} {proc2} N2 + 3 -1 roll % {proc2} N2 {proc1} + dup % {proc2} N2 {proc1} {proc1} + length % {proc2} N2 {proc1} N1 + dup % {proc2} N2 {proc1} N1 N1 + 5 1 roll % N1 {proc2} N2 {proc1} N1 + 3 -1 roll % N1 {proc2} {proc1} N1 N2 + add % N1 {proc2} {proc1} N1+N2 + array cvx % N1 {proc2} {proc1} { ... } + dup % N1 {proc2} {proc1} { ... } { ... } + 3 -1 roll % N1 {proc2} { ... } { ... } {proc1} + 0 exch % N1 {proc2} { ... } { ... } 0 {proc1} + putinterval % N1 {proc2} { <<{proc1}>> ... } + dup % N1 {proc2} { <<{proc1}>> ... } { <<{proc1}>> ... } + 4 2 roll % { <<{proc1}>> ... } { <<{proc1}>> ... } N1 {proc2} + putinterval % { <<{proc1}>> <<{proc2}>> } + } bind def % ; /mergeprocs => dictionary + + /colorimage { %def % {imageproc} multiproc ncolors + pop % {imageproc} multiproc ; assume 3 colors + pop % {imageproc} ; assume false + {colortograyscale} % {imageproc} {colortograyscale} + mergeprocs % {imageproc colortograyscale} + image % construct monochrome image + } bind def % ; /colorimage => dictionary +} if % only create if it doesn't already exist +/drawcolorimage { + 60 30 8 + [60 0 0 -30 0 30] + {currentfile buffer readhexstring pop pop % get run length & color info + /npixels buffer 0 get 1 add 3 mul store % number of pixels (run length) + /color buffer 1 get 3 mul store % color of pixels + % /pixels npixels string store % create string to hold colors + /rgb rgbmap color 3 getinterval store % get rgb value + 0 3 npixels -1 add { + pixels exch rgb putinterval + } for + pixels 0 npixels getinterval % Return color values + } + false 3 + colorimage +} bind def +%0.750000 inch 3.960000 inch translate +matrix currentmatrix +.5 inch .16 inch scale + +% get rgb color table +currentfile rgbmap readhexstring pop pop +000000 +ffffff +dbdb70 +2f4f4f +23238e +d3d3d3 +ff0000 +000000 +004900 +004955 +0049aa +0049ff 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+ +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 376 400 setDim + maxX maxY 0 0 rectangle clip % set clip path + + renderGC 0.375000 setForeground + renderGC + 195 125 + 187 129 + 194 124 + 4 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+ renderGC 0.500000 setForeground + renderGC + 161 262 + 173 268 + 164 259 + 4 161 262 psFillPoly + renderGC 0.500000 setForeground + renderGC + 193 284 + 171 271 + 191 287 + 4 193 284 psFillPoly + renderGC 0.562500 setForeground + renderGC + 171 271 + 193 284 + 173 268 + 4 171 271 psFillPoly + renderGC 0.562500 setForeground + renderGC + 212 261 + 225 262 + 214 257 + 4 212 261 psFillPoly + renderGC 0.562500 setForeground + renderGC + 225 262 + 212 261 + 223 266 + 4 225 262 psFillPoly + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/modbess.ps b/books/ps/modbess.ps new file mode 100644 index 0000000..04f8c66 --- /dev/null +++ b/books/ps/modbess.ps @@ -0,0 +1,534 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawIStr: +% psDrawIStr +% window type: title or window +% string +% y +% x +% graphics-context dictionary +% it draws a text string in foreground color on top of bounding box of +% string, which is in background color. + +/psDrawIStr + { gsave + newpath %% for rectangle + loadFont + + /window exch def %% get window type + + %% draw bounding box with background color + /str exch def %% get text string + str stringwidth pop 1 sub %% width + FontHeight 1 sub %% height + currentfont begin %% get font height + FontBBox + end + /ypos exch def pop %% define ypos + neg ypos add /offset exch def pop + /offset ypos offset div FontHeight mul def %% define offset + /h exch def /w exch def %% define h + /y0 exch def %% define y0 + /x0 exch def %% define x0 + w h x0 y0 offset sub + window (title) eq + {hVal moveto drawRect} %% draws in title window + {rectangle} ifelse %% draws in view window + begin + BGcolor setgray fill %% set background box color + + x0 y0 + window (title) eq + {hVal} %% print title text + {yVal} ifelse %% print window text + moveto str + FGcolor setgray show %% set text color + end + grestore } + def + +% operand stack configuration in order to use psFillArc: +% psFillArc +% y center of rectangle +% x center of rectangle +% angle2 +% angle1 +% width +% height +% y +% x +% graphics-context dictionary +% this draws and fills an arc whose origin is at x, y, and whose width +% and height specifies the rectangle which encases the arc. +% Origin is at upper left corner of rectangle. +% This function uses "scale" to make cricles and ellipses. +/psFillArc + { gsave + newpath + yVal moveto + /sfactor 4 index 4 index div def + 1 sfactor scale + 6 5 roll %% x on top of stack + 3 index 2 div add %% define x origin + 6 5 roll %% y on top of stack + 6 5 roll %% h on top of stack + 2 div add yVal sfactor div %% define y origin + 5 4 roll %% w on top of stack + 2 div %% define radius + 5 3 roll %% a1 a2 now on top + 1 index add + arcn %% draw clockwise arc + begin installGC fill end %% fills with foreground color + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 0 0 1 + 1072693248 0 /graphGC makeDict + 0 0 1 + 1072693248 0 /unitGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 300 85 0 85 psDrawLine + globalGC1 150 276 150 0 psDrawLine + unitGC 150 87 150 83 psDrawLine + unitGC 138 100 (0.00) (window) psDrawIStr + unitGC 203 87 203 83 psDrawLine + unitGC 191 100 (5.00) (window) psDrawIStr + unitGC 256 87 256 83 psDrawLine + unitGC 241 100 (10.00) (window) psDrawIStr + unitGC 96 87 96 83 psDrawLine + unitGC 81 100 (-5.00) (window) psDrawIStr + unitGC 42 87 42 83 psDrawLine + unitGC 24 100 (-10.00) (window) psDrawIStr + unitGC 152 27 148 27 psDrawLine + unitGC 105 32 (20.00) (window) psDrawIStr + unitGC 152 142 148 142 psDrawLine + unitGC 96 147 (-20.00) (window) psDrawIStr + unitGC 152 199 148 199 psDrawLine + unitGC 96 204 (-40.00) (window) psDrawIStr + unitGC 152 256 148 256 psDrawLine + unitGC 96 261 (-60.00) (window) psDrawIStr + globalGC1 19 84 3 3 0 360 20 85 psFillArc + globalGC1 20 85 20 85 psDrawLine + globalGC1 19 84 3 3 0 360 20 85 psFillArc + globalGC1 21 118 20 85 psDrawLine + globalGC1 20 117 3 3 0 360 21 118 psFillArc + globalGC1 21 147 21 118 psDrawLine + globalGC1 20 146 3 3 0 360 21 147 psFillArc + globalGC1 21 173 21 147 psDrawLine + globalGC1 20 172 3 3 0 360 21 173 psFillArc + globalGC1 22 195 21 173 psDrawLine + globalGC1 21 194 3 3 0 360 22 195 psFillArc + globalGC1 22 230 22 195 psDrawLine + globalGC1 21 229 3 3 0 360 22 230 psFillArc + globalGC1 23 252 22 230 psDrawLine + globalGC1 22 251 3 3 0 360 23 252 psFillArc + globalGC1 23 259 23 252 psDrawLine + globalGC1 22 258 3 3 0 360 23 259 psFillArc + globalGC1 24 264 23 259 psDrawLine + globalGC1 23 263 3 3 0 360 24 264 psFillArc + globalGC1 24 265 24 264 psDrawLine + globalGC1 23 264 3 3 0 360 24 265 psFillArc + globalGC1 24 266 24 265 psDrawLine + globalGC1 23 265 3 3 0 360 24 266 psFillArc + globalGC1 24 266 24 266 psDrawLine + globalGC1 23 265 3 3 0 360 24 266 psFillArc + globalGC1 24 266 24 266 psDrawLine + globalGC1 23 265 3 3 0 360 24 266 psFillArc + globalGC1 24 265 24 266 psDrawLine + globalGC1 23 264 3 3 0 360 24 265 psFillArc + globalGC1 25 263 24 265 psDrawLine + globalGC1 24 262 3 3 0 360 25 263 psFillArc + globalGC1 25 259 25 263 psDrawLine + globalGC1 24 258 3 3 0 360 25 259 psFillArc + globalGC1 26 246 25 259 psDrawLine + globalGC1 25 245 3 3 0 360 26 246 psFillArc + globalGC1 26 228 26 246 psDrawLine + globalGC1 25 227 3 3 0 360 26 228 psFillArc + globalGC1 27 208 26 228 psDrawLine + globalGC1 26 207 3 3 0 360 27 208 psFillArc + globalGC1 28 186 27 208 psDrawLine + globalGC1 27 185 3 3 0 360 28 186 psFillArc + globalGC1 28 163 28 186 psDrawLine + globalGC1 27 162 3 3 0 360 28 163 psFillArc + globalGC1 29 141 28 163 psDrawLine + globalGC1 28 140 3 3 0 360 29 141 psFillArc + globalGC1 30 120 29 141 psDrawLine + globalGC1 29 119 3 3 0 360 30 120 psFillArc + globalGC1 30 101 30 120 psDrawLine + globalGC1 29 100 3 3 0 360 30 101 psFillArc + globalGC1 31 85 30 101 psDrawLine + globalGC1 30 84 3 3 0 360 31 85 psFillArc + globalGC1 32 61 31 85 psDrawLine + globalGC1 31 60 3 3 0 360 32 61 psFillArc + globalGC1 33 53 32 61 psDrawLine + globalGC1 32 52 3 3 0 360 33 53 psFillArc + globalGC1 34 48 33 53 psDrawLine + globalGC1 33 47 3 3 0 360 34 48 psFillArc + globalGC1 34 47 34 48 psDrawLine + globalGC1 33 46 3 3 0 360 34 47 psFillArc + globalGC1 34 45 34 47 psDrawLine + globalGC1 33 44 3 3 0 360 34 45 psFillArc + globalGC1 35 45 34 45 psDrawLine + globalGC1 34 44 3 3 0 360 35 45 psFillArc + globalGC1 35 45 35 45 psDrawLine + globalGC1 34 44 3 3 0 360 35 45 psFillArc + globalGC1 35 45 35 45 psDrawLine + globalGC1 34 44 3 3 0 360 35 45 psFillArc + globalGC1 36 46 35 45 psDrawLine + globalGC1 35 45 3 3 0 360 36 46 psFillArc + globalGC1 36 49 36 46 psDrawLine + globalGC1 35 48 3 3 0 360 36 49 psFillArc + globalGC1 39 67 36 49 psDrawLine + globalGC1 38 66 3 3 0 360 39 67 psFillArc + globalGC1 42 85 39 67 psDrawLine + globalGC1 41 84 3 3 0 360 42 85 psFillArc + globalGC1 43 91 42 85 psDrawLine + globalGC1 42 90 3 3 0 360 43 91 psFillArc + globalGC1 44 93 43 91 psDrawLine + globalGC1 43 92 3 3 0 360 44 93 psFillArc + globalGC1 44 94 44 93 psDrawLine + globalGC1 43 93 3 3 0 360 44 94 psFillArc + globalGC1 45 95 44 94 psDrawLine + globalGC1 44 94 3 3 0 360 45 95 psFillArc + globalGC1 46 95 45 95 psDrawLine + globalGC1 45 94 3 3 0 360 46 95 psFillArc + globalGC1 47 94 46 95 psDrawLine + globalGC1 46 93 3 3 0 360 47 94 psFillArc + globalGC1 47 94 47 94 psDrawLine + globalGC1 46 93 3 3 0 360 47 94 psFillArc + globalGC1 49 92 47 94 psDrawLine + globalGC1 48 91 3 3 0 360 49 92 psFillArc + globalGC1 50 89 49 92 psDrawLine + globalGC1 49 88 3 3 0 360 50 89 psFillArc + globalGC1 51 87 50 89 psDrawLine + globalGC1 50 86 3 3 0 360 51 87 psFillArc + globalGC1 53 85 51 87 psDrawLine + globalGC1 52 84 3 3 0 360 53 85 psFillArc + globalGC1 53 84 53 85 psDrawLine + globalGC1 52 83 3 3 0 360 53 84 psFillArc + globalGC1 54 83 53 84 psDrawLine + globalGC1 53 82 3 3 0 360 54 83 psFillArc + globalGC1 55 83 54 83 psDrawLine + globalGC1 54 82 3 3 0 360 55 83 psFillArc + globalGC1 55 83 55 83 psDrawLine + globalGC1 54 82 3 3 0 360 55 83 psFillArc + globalGC1 57 82 55 83 psDrawLine + globalGC1 56 81 3 3 0 360 57 82 psFillArc + globalGC1 58 82 57 82 psDrawLine + globalGC1 57 81 3 3 0 360 58 82 psFillArc + globalGC1 59 83 58 82 psDrawLine + globalGC1 58 82 3 3 0 360 59 83 psFillArc + globalGC1 61 84 59 83 psDrawLine + globalGC1 60 83 3 3 0 360 61 84 psFillArc + globalGC1 63 85 61 84 psDrawLine + globalGC1 62 84 3 3 0 360 63 85 psFillArc + globalGC1 65 85 63 85 psDrawLine + globalGC1 64 84 3 3 0 360 65 85 psFillArc + globalGC1 66 85 65 85 psDrawLine + globalGC1 65 84 3 3 0 360 66 85 psFillArc + globalGC1 67 85 66 85 psDrawLine + globalGC1 66 84 3 3 0 360 67 85 psFillArc + globalGC1 69 85 67 85 psDrawLine + globalGC1 68 84 3 3 0 360 69 85 psFillArc + globalGC1 71 85 69 85 psDrawLine + globalGC1 70 84 3 3 0 360 71 85 psFillArc + globalGC1 74 84 71 85 psDrawLine + globalGC1 73 83 3 3 0 360 74 84 psFillArc + globalGC1 79 83 74 84 psDrawLine + globalGC1 78 82 3 3 0 360 79 83 psFillArc + globalGC1 85 83 79 83 psDrawLine + globalGC1 84 82 3 3 0 360 85 83 psFillArc + globalGC1 90 81 85 83 psDrawLine + globalGC1 89 80 3 3 0 360 90 81 psFillArc + globalGC1 96 79 90 81 psDrawLine + globalGC1 95 78 3 3 0 360 96 79 psFillArc + globalGC1 101 75 96 79 psDrawLine + globalGC1 100 74 3 3 0 360 101 75 psFillArc + globalGC1 106 70 101 75 psDrawLine + globalGC1 105 69 3 3 0 360 106 70 psFillArc + globalGC1 112 64 106 70 psDrawLine + globalGC1 111 63 3 3 0 360 112 64 psFillArc + globalGC1 117 56 112 64 psDrawLine + globalGC1 116 55 3 3 0 360 117 56 psFillArc + globalGC1 123 46 117 56 psDrawLine + globalGC1 122 45 3 3 0 360 123 46 psFillArc + globalGC1 128 35 123 46 psDrawLine + globalGC1 127 34 3 3 0 360 128 35 psFillArc + globalGC1 133 25 128 35 psDrawLine + globalGC1 132 24 3 3 0 360 133 25 psFillArc + globalGC1 139 16 133 25 psDrawLine + globalGC1 138 15 3 3 0 360 139 16 psFillArc + globalGC1 141 12 139 16 psDrawLine + globalGC1 140 11 3 3 0 360 141 12 psFillArc + globalGC1 144 9 141 12 psDrawLine + globalGC1 143 8 3 3 0 360 144 9 psFillArc + globalGC1 145 8 144 9 psDrawLine + globalGC1 144 7 3 3 0 360 145 8 psFillArc + globalGC1 147 8 145 8 psDrawLine + globalGC1 146 7 3 3 0 360 147 8 psFillArc + globalGC1 148 7 147 8 psDrawLine + globalGC1 147 6 3 3 0 360 148 7 psFillArc + globalGC1 150 7 148 7 psDrawLine + globalGC1 149 6 3 3 0 360 150 7 psFillArc + globalGC1 151 7 150 7 psDrawLine + globalGC1 150 6 3 3 0 360 151 7 psFillArc + globalGC1 152 8 151 7 psDrawLine + globalGC1 151 7 3 3 0 360 152 8 psFillArc + globalGC1 154 8 152 8 psDrawLine + globalGC1 153 7 3 3 0 360 154 8 psFillArc + globalGC1 155 9 154 8 psDrawLine + globalGC1 154 8 3 3 0 360 155 9 psFillArc + globalGC1 158 12 155 9 psDrawLine + globalGC1 157 11 3 3 0 360 158 12 psFillArc + globalGC1 160 16 158 12 psDrawLine + globalGC1 159 15 3 3 0 360 160 16 psFillArc + globalGC1 166 25 160 16 psDrawLine + globalGC1 165 24 3 3 0 360 166 25 psFillArc + globalGC1 171 35 166 25 psDrawLine + globalGC1 170 34 3 3 0 360 171 35 psFillArc + globalGC1 176 46 171 35 psDrawLine + globalGC1 175 45 3 3 0 360 176 46 psFillArc + globalGC1 182 56 176 46 psDrawLine + globalGC1 181 55 3 3 0 360 182 56 psFillArc + globalGC1 187 64 182 56 psDrawLine + globalGC1 186 63 3 3 0 360 187 64 psFillArc + globalGC1 193 70 187 64 psDrawLine + globalGC1 192 69 3 3 0 360 193 70 psFillArc + globalGC1 198 75 193 70 psDrawLine + globalGC1 197 74 3 3 0 360 198 75 psFillArc + globalGC1 203 79 198 75 psDrawLine + globalGC1 202 78 3 3 0 360 203 79 psFillArc + globalGC1 209 81 203 79 psDrawLine + globalGC1 208 80 3 3 0 360 209 81 psFillArc + globalGC1 214 83 209 81 psDrawLine + globalGC1 213 82 3 3 0 360 214 83 psFillArc + globalGC1 220 84 214 83 psDrawLine + globalGC1 219 83 3 3 0 360 220 84 psFillArc + globalGC1 225 84 220 84 psDrawLine + globalGC1 224 83 3 3 0 360 225 84 psFillArc + globalGC1 230 85 225 84 psDrawLine + globalGC1 229 84 3 3 0 360 230 85 psFillArc + globalGC1 236 85 230 85 psDrawLine + globalGC1 235 84 3 3 0 360 236 85 psFillArc + globalGC1 241 85 236 85 psDrawLine + globalGC1 240 84 3 3 0 360 241 85 psFillArc + globalGC1 246 85 241 85 psDrawLine + globalGC1 245 84 3 3 0 360 246 85 psFillArc + globalGC1 252 85 246 85 psDrawLine + globalGC1 251 84 3 3 0 360 252 85 psFillArc + globalGC1 257 85 252 85 psDrawLine + globalGC1 256 84 3 3 0 360 257 85 psFillArc + globalGC1 263 85 257 85 psDrawLine + globalGC1 262 84 3 3 0 360 263 85 psFillArc + globalGC1 268 85 263 85 psDrawLine + globalGC1 267 84 3 3 0 360 268 85 psFillArc + globalGC1 273 85 268 85 psDrawLine + globalGC1 272 84 3 3 0 360 273 85 psFillArc + globalGC1 279 85 273 85 psDrawLine + globalGC1 278 84 3 3 0 360 279 85 psFillArc + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/modbessc.ps b/books/ps/modbessc.ps new file mode 100644 index 0000000..8a59e1d --- /dev/null +++ b/books/ps/modbessc.ps @@ -0,0 +1,16338 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 276 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 67 207 216 93 psDrawLine + globGC (X) 62 212 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 232 207 83 93 psDrawLine + globGC (Y) 237 212 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 149 64 149 203 psDrawLine + globGC (Z) 154 59 psDrawStr + globalGC1 99.000000 setForeground + globGC 99.000000 setForeground + 1 0 1 renderGC setLineAttributes + renderGC 0.000000 setForeground + renderGC 0.437500 setForeground + renderGC + 138 74 + 145 68 + 142 69 + 4 138 74 psFillPoly + renderGC 0.000000 setForeground + renderGC + 145 68 + 142 69 + 3 138 74 psDrawLines + renderGC 0.375000 setForeground + renderGC + 149 65 + 142 69 + 145 68 + 4 149 65 psFillPoly + renderGC 0.000000 setForeground + renderGC + 142 69 + 145 68 + 3 149 65 psDrawLines + renderGC 0.375000 setForeground + renderGC + 142 69 + 149 65 + 146 62 + 4 142 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 65 + 146 62 + 3 142 69 psDrawLines + renderGC 0.375000 setForeground + renderGC + 134 77 + 141 71 + 138 74 + 4 134 77 psFillPoly + renderGC 0.000000 setForeground + renderGC + 141 71 + 138 74 + 3 134 77 psDrawLines + renderGC 0.375000 setForeground + renderGC + 145 68 + 138 74 + 141 71 + 4 145 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 138 74 + 141 71 + 3 145 68 psDrawLines + renderGC 0.562500 setForeground + renderGC + 146 62 + 153 62 + 149 56 + 4 146 62 psFillPoly + renderGC 0.000000 setForeground + renderGC + 153 62 + 149 56 + 3 146 62 psDrawLines + renderGC 0.625000 setForeground + renderGC + 153 62 + 146 62 + 149 65 + 4 153 62 psFillPoly + renderGC 0.000000 setForeground + renderGC + 146 62 + 149 65 + 3 153 62 psDrawLines + renderGC 0.625000 setForeground + renderGC + 149 65 + 157 65 + 153 62 + 4 149 65 psFillPoly + renderGC 0.000000 setForeground + renderGC + 157 65 + 153 62 + 3 149 65 psDrawLines + renderGC 0.375000 setForeground + renderGC + 129 75 + 137 72 + 134 77 + 4 129 75 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 72 + 134 77 + 3 129 75 psDrawLines + renderGC 0.375000 setForeground + renderGC + 141 71 + 134 77 + 137 72 + 4 141 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 134 77 + 137 72 + 3 141 71 psDrawLines + renderGC 0.687500 setForeground + renderGC + 154 67 + 162 67 + 157 65 + 4 154 67 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 67 + 157 65 + 3 154 67 psDrawLines + renderGC 0.625000 setForeground + renderGC + 157 65 + 149 65 + 154 67 + 4 157 65 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 65 + 154 67 + 3 157 65 psDrawLines + renderGC 0.500000 setForeground + renderGC + 145 68 + 154 67 + 149 65 + 4 145 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 154 67 + 149 65 + 3 145 68 psDrawLines + renderGC 0.500000 setForeground + renderGC + 141 71 + 149 69 + 145 68 + 4 141 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 69 + 145 68 + 3 141 71 psDrawLines + renderGC 0.562500 setForeground + renderGC + 154 67 + 145 68 + 149 69 + 4 154 67 psFillPoly + renderGC 0.000000 setForeground + renderGC + 145 68 + 149 69 + 3 154 67 psDrawLines + renderGC 0.625000 setForeground + renderGC + 158 68 + 166 69 + 162 67 + 4 158 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 166 69 + 162 67 + 3 158 68 psDrawLines + renderGC 0.625000 setForeground + renderGC + 162 67 + 154 67 + 158 68 + 4 162 67 psFillPoly + renderGC 0.000000 setForeground + renderGC + 154 67 + 158 68 + 3 162 67 psDrawLines + renderGC 0.625000 setForeground + renderGC + 149 69 + 158 68 + 154 67 + 4 149 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 158 68 + 154 67 + 3 149 69 psDrawLines + renderGC 0.500000 setForeground + renderGC + 137 72 + 145 70 + 141 71 + 4 137 72 psFillPoly + renderGC 0.000000 setForeground + renderGC + 145 70 + 141 71 + 3 137 72 psDrawLines + renderGC 0.500000 setForeground + renderGC + 149 69 + 141 71 + 145 70 + 4 149 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 141 71 + 145 70 + 3 149 69 psDrawLines + renderGC 0.687500 setForeground + renderGC + 162 70 + 170 71 + 166 69 + 4 162 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 170 71 + 166 69 + 3 162 70 psDrawLines + renderGC 0.625000 setForeground + renderGC + 166 69 + 158 68 + 162 70 + 4 166 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 158 68 + 162 70 + 3 166 69 psDrawLines + renderGC 0.500000 setForeground + renderGC + 125 68 + 133 70 + 129 75 + 4 125 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 133 70 + 129 75 + 3 125 68 psDrawLines + renderGC 0.437500 setForeground + renderGC + 137 72 + 129 75 + 133 70 + 4 137 72 psFillPoly + renderGC 0.000000 setForeground + renderGC + 129 75 + 133 70 + 3 137 72 psDrawLines + renderGC 0.562500 setForeground + renderGC + 145 70 + 154 69 + 149 69 + 4 145 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 154 69 + 149 69 + 3 145 70 psDrawLines + renderGC 0.625000 setForeground + renderGC + 158 68 + 149 69 + 154 69 + 4 158 68 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 69 + 154 69 + 3 158 68 psDrawLines + renderGC 0.625000 setForeground + renderGC + 154 69 + 162 70 + 158 68 + 4 154 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 70 + 158 68 + 3 154 69 psDrawLines + renderGC 0.562500 setForeground + renderGC + 133 70 + 141 70 + 137 72 + 4 133 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 141 70 + 137 72 + 3 133 70 psDrawLines + renderGC 0.500000 setForeground + renderGC + 145 70 + 137 72 + 141 70 + 4 145 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 137 72 + 141 70 + 3 145 70 psDrawLines + renderGC 0.687500 setForeground + renderGC + 166 71 + 174 72 + 170 71 + 4 166 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 174 72 + 170 71 + 3 166 71 psDrawLines + renderGC 0.687500 setForeground + renderGC + 170 71 + 162 70 + 166 71 + 4 170 71 psFillPoly + renderGC 0.000000 setForeground + renderGC + 162 70 + 166 71 + 3 170 71 psDrawLines + renderGC 0.562500 setForeground + renderGC + 141 70 + 149 70 + 145 70 + 4 141 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 149 70 + 145 70 + 3 141 70 psDrawLines + renderGC 0.562500 setForeground + renderGC + 154 69 + 145 70 + 149 70 + 4 154 69 psFillPoly + renderGC 0.000000 setForeground + renderGC + 145 70 + 149 70 + 3 154 69 psDrawLines + renderGC 0.687500 setForeground + renderGC + 158 70 + 166 71 + 162 70 + 4 158 70 psFillPoly + renderGC 0.000000 setForeground + renderGC + 166 71 + 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134 + 222 118 + 214 126 + 4 208 134 psFillPoly + renderGC 0.000000 setForeground + renderGC + 222 118 + 214 126 + 3 208 134 psDrawLines + renderGC 0.562500 setForeground + renderGC + 228 113 + 214 126 + 222 118 + 4 228 113 psFillPoly + renderGC 0.000000 setForeground + renderGC + 214 126 + 222 118 + 3 228 113 psDrawLines + renderGC 0.687500 setForeground + renderGC + 222 118 + 208 134 + 216 127 + 4 222 118 psFillPoly + renderGC 0.000000 setForeground + renderGC + 208 134 + 216 127 + 3 222 118 psDrawLines + renderGC 0.750000 setForeground + renderGC + 211 150 + 197 167 + 203 167 + 4 211 150 psFillPoly + renderGC 0.000000 setForeground + renderGC + 197 167 + 203 167 + 3 211 150 psDrawLines + renderGC 0.812500 setForeground + renderGC + 197 167 + 211 150 + 203 153 + 4 197 167 psFillPoly + renderGC 0.000000 setForeground + renderGC + 211 150 + 203 153 + 3 197 167 psDrawLines + renderGC 0.625000 setForeground + renderGC + 228 113 + 243 98 + 233 111 + 4 228 113 psFillPoly + renderGC 0.000000 setForeground + renderGC + 243 98 + 233 111 + 3 228 113 psDrawLines + renderGC 0.562500 setForeground + renderGC + 247 102 + 233 111 + 243 98 + 4 247 102 psFillPoly + renderGC 0.000000 setForeground + renderGC + 233 111 + 243 98 + 3 247 102 psDrawLines + renderGC 0.812500 setForeground + renderGC + 218 133 + 203 153 + 211 150 + 4 218 133 psFillPoly + renderGC 0.000000 setForeground + renderGC + 203 153 + 211 150 + 3 218 133 psDrawLines + renderGC 0.812500 setForeground + renderGC + 203 153 + 218 133 + 210 139 + 4 203 153 psFillPoly + renderGC 0.000000 setForeground + renderGC + 218 133 + 210 139 + 3 203 153 psDrawLines + renderGC 0.812500 setForeground + renderGC + 225 118 + 210 139 + 218 133 + 4 225 118 psFillPoly + renderGC 0.000000 setForeground + renderGC + 210 139 + 218 133 + 3 225 118 psDrawLines + renderGC 0.750000 setForeground + renderGC + 210 139 + 225 118 + 216 127 + 4 210 139 psFillPoly + renderGC 0.000000 setForeground + renderGC + 225 118 + 216 127 + 3 210 139 psDrawLines + renderGC 0.625000 setForeground + renderGC + 222 118 + 238 99 + 228 113 + 4 222 118 psFillPoly + renderGC 0.000000 setForeground + renderGC + 238 99 + 228 113 + 3 222 118 psDrawLines + renderGC 0.625000 setForeground + renderGC + 243 98 + 228 113 + 238 99 + 4 243 98 psFillPoly + renderGC 0.000000 setForeground + renderGC + 228 113 + 238 99 + 3 243 98 psDrawLines + renderGC 0.750000 setForeground + renderGC + 232 106 + 216 127 + 225 118 + 4 232 106 psFillPoly + renderGC 0.000000 setForeground + renderGC + 216 127 + 225 118 + 3 232 106 psDrawLines + renderGC 0.687500 setForeground + renderGC + 216 127 + 232 106 + 222 118 + 4 216 127 psFillPoly + renderGC 0.000000 setForeground + renderGC + 232 106 + 222 118 + 3 216 127 psDrawLines + renderGC 0.625000 setForeground + renderGC + 238 99 + 222 118 + 232 106 + 4 238 99 psFillPoly + renderGC 0.000000 setForeground + renderGC + 222 118 + 232 106 + 3 238 99 psDrawLines + + grestore % restore graphics state + + + cleartomark %% clearing operand stack + +end %% pops mainDict from dictionary stack + +showpage + +%-------------------------- end --------------------------% diff --git a/books/ps/newmap.ps b/books/ps/newmap.ps new file mode 100644 index 0000000..c499d38 --- /dev/null +++ b/books/ps/newmap.ps @@ -0,0 +1,16041 @@ +%!PS-Adobe-2.0 +%%DocumentFonts: Times-Roman +%%Creator: Axiom +%%CreationDate: today +%%Pages: 1 +%%processing (hard) limit: 250 pts or 500 values for the operand stack. +%%EndComments + +%------------------------------- prologue -------------------------------% +%-------------------------- support procedures --------------------------% + +%--------- first create user dictionary with 100 entries max ------------% +% (number can be changed to accomodate definitions) % + +100 dict begin %% using 100 entries in top level dictionary + +/FontHeight 12 def + +/inch + { 72 mul } + def + +% yVal and hVal are necessary because the Xwindow display origin +% is at the upper left corner, while the postscript display +% origin is at the lower left hand corner. + +/yVal %% get Y value -- make upper left corner origin + { maxY sub abs } %% maxY is viewWindow height + def + +/hVal %% get H value -- used for displaying title text + { maxH sub abs } %% maxH is viewWindow height+titleWindow height + def + +% loads in the font + +/loadFont + { /Times-Roman findfont FontHeight scalefont setfont } + def + +% draws a rectangle with input operand: +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/drawRect + { 1 index 1 add 0 rlineto %% draw first side + 0 exch 1 add neg rlineto %% draw second side + 1 add neg 0 rlineto %% draw third side + closepath } %% draw fourth side + def + +% create a rectangle with input operand in the view window: +% y +% x +% height +% width +% notice that this function does not "draw" or ink the rectangle. +/rectangle + { yVal moveto %% set currentpoint for line + drawRect } %% draws the rectangle + def + +% These are global variables that every draw procedure uses +% THe operand should be as follows: +% viewWindow width +% viewWindow height +% title height +/setDim + { /maxX exch def %% width of display + /maxY exch def %% height of display + /titleH exch def %% height of title + /maxH maxY titleH add def %% height of display + title + } def + +%-------------------------- major procedures --------------------------% + +/title %% draws a rectangle around the title of picture + { gsave + newpath + moveto %% lower left of title + titleH 1 add 0 exch rlineto %% draw first side + 1 add 0 rlineto %% draw second side + 1 add neg 0 exch rlineto + begin installGC stroke end %% draw third side + grestore } + def + +/drawFrame %% draw display frame + { gsave + newpath + maxX maxY 0 0 rectangle + begin installGC stroke end + grestore } + def + +% updates the foreground color of existing graphics-context dictionary: +% foreground color +% dictionary name +/setForeground + { /FGcolor exch put } + def + +% updates the background color of existing graphics-context dictionary: +% background color +% dictionary name +/setBackground + { /BGcolor exch put } + def + +% updates the line width, line style, cap style, join style of +% existing graphics-context dictionary: +% dictionary name +% join style +% cap style +% line width +/setLineAttributes + { begin + /JoinStyle exch def + /CapStyle exch def + /LineWidth exch def + end } + def + +% creates a graphics context dictionary with the following information: +% /dictionary name +% foreground color +% background color +% line width +% cap style +% join style +% this creates different graphical contexts for different drawing functions. +/makeDict + { 5 dict 2 copy def begin pop %% with dict name on top of stack + /FGcolor exch def %% define drawing attributes + /BGcolor exch def %% not heavily used + /LineWidth exch def + /CapStyle exch def + /JoinStyle exch def + end } + def + +% makes the current dictionary attributes effective +% this function takes the values in the current dictionary to set the context +% these are the values currently being used: foreground, cap, join, and width +/installGC + { + FGcolor currentgray ne + {FGcolor setgray} if %% foreground color + CapStyle currentlinecap ne + {CapStyle setlinecap} if %% cap style + JoinStyle currentlinejoin ne + {JoinStyle setlinejoin} if %% join style + LineWidth currentlinewidth ne + {LineWidth setlinewidth} if } %% line width + def + +% operand stack configuration in order to use psDrawFilled: +% psFillPoly +% XPoint[0].y +% XPoint[0].x +% n +% ... +% XPoint[n].y +% XPoint[n].x +% graphics-context dictionary +% this draws a polygon by connecting all the points and fills the +% region with foreground color + +/psFillPoly + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + closepath + begin installGC fill end %% fills with foreground color + grestore } + def + +% operand stack configuration in order to use psDrawLine: +% psDrawLine +% y0 +% x0 +% y1 +% x1 +% graphics-context dictionary +% this draws a line from (x0, y0) to (x1, y1). + +/psDrawLine + { gsave + newpath + yVal moveto + yVal lineto + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawLines: +% psDrawLines +% points[0].y +% points[0].x +% n +% ... +% points[n].y +% points[n].x +% graphics-context dictionary +% this draws lines connecting all the points. + +/psDrawLines + { gsave + newpath + yVal moveto + 1 sub { + yVal lineto + } repeat + begin installGC stroke end + grestore } + def + +% operand stack configuration in order to use psDrawStr: +% psDrawStr +% y +% x +% string +% graphics-context dictionary +% this function draws a text string at (x,y). + +/psDrawStr + { gsave + newpath + loadFont + yVal moveto + exch begin installGC show end + grestore } + def + +%-------------------------- script --------------------------% + +% 1 inch 1 inch translate + + mark %% mark bottom of our stack + + 0 0 1 + 1072693248 0 /globalGC1 makeDict + 0 0 1 + 1072693248 0 /globalGC2 makeDict + 0 0 1 + 1072693248 0 /trashGC makeDict + 0 0 1 + 1072693248 0 /componentGC makeDict + 0 0 1 + 1072693248 0 /opaqueGC makeDict + 0 0 1 + 1072693248 0 /renderGC makeDict + 0 0 1 + 1072693248 0 /globGC makeDict + 0 0 1 + 1072693248 0 /anotherGC makeDict + 1 0 1 + 1072693248 0 /renderGC makeDict + + gsave % save graphics state for clipping path + + 1.000000 1.000000 scale + + 24 303 300 setDim + maxX maxY 0 0 rectangle clip % set clip path + + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 33 287 245 118 psDrawLine + globGC (X) 28 292 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 267 287 54 118 psDrawLine + globGC (Y) 272 292 psDrawStr + globalGC1 107.000000 setForeground + globGC 107.000000 setForeground + globalGC1 0.000000 setForeground + globGC 0.000000 setForeground + globalGC1 150 87 150 275 psDrawLine + 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