.. role:: raw-html(raw)
   :format: html

.. _c-spsolvers:


***********************
Sparse Linear Equations
***********************

In this section we describe routines for solving sparse sets of linear 
equations.

A real symmetric or complex Hermitian sparse matrix is stored as an 
:class:`spmatrix <cvxopt.spmatrix>` object ``X``  of size 
(:math:`n`, :math:`n`) and an 
additional character argument ``uplo`` with possible values :const:`'L'` 
and :const:`'U'`.  If ``uplo`` is :const:`'L'`, the lower triangular part
of ``X`` contains the lower triangular part of the symmetric or Hermitian 
matrix, and the upper triangular matrix of ``X`` is ignored.  If ``uplo`` 
is :const:`'U'`, the upper triangular part of ``X`` contains the upper 
triangular part of the matrix, and the lower triangular matrix of ``X`` is 
ignored.

A general sparse square matrix of order :math:`n` is represented by an
:class:`spmatrix` object of size (:math:`n`, :math:`n`).

Dense matrices, which appear as right-hand sides of equations, are 
stored using the same conventions as in the BLAS and LAPACK modules.


.. _s-orderings:

Matrix Orderings
****************

CVXOPT includes an interface to the AMD library for computing approximate 
minimum degree orderings of sparse matrices.

.. seealso::

    * `AMD code, documentation, copyright, and license
      <http://www.cise.ufl.edu/research/sparse/amd>`_

    * P. R. Amestoy, T. A. Davis, I. S. Duff,  Algorithm 837: AMD, An 
      Approximate Minimum Degree Ordering Algorithm, ACM Transactions on 
      Mathematical Software, 30(3), 381-388, 2004.


.. function:: cvxopt.amd.order(A[, uplo = 'L'])

    Computes the approximate mimimum degree ordering of a symmetric  sparse
    matrix :math:`A`.  The ordering is returned as an integer dense matrix 
    with length equal to the order of :math:`A`.  Its entries specify a 
    permutation that reduces fill-in during the Cholesky factorization.
    More precisely, if ``p = order(A)`` , then ``A[p, p]`` has 
    sparser Cholesky factors than ``A``.   


As an example we consider the matrix 

.. math::

    \left[ \begin{array}{rrrr}
     10 &  0 & 3 &  0 \\
      0 &  5 & 0 & -2 \\
      3 &  0 & 5 &  0 \\
      0 & -2 & 0 &  2 
    \end{array}\right].


>>> from cvxopt import spmatrix, amd 
>>> A = spmatrix([10,3,5,-2,5,2], [0,2,1,2,2,3], [0,0,1,1,2,3])
>>> P = amd.order(A)
>>> print(P)
[ 1]
[ 0]
[ 2]
[ 3]


.. _s-umfpack:

General Linear Equations
************************

The module :mod:`cvxopt.umfpack` includes four functions for solving 
sparse non-symmetric sets of linear equations.  They call routines from 
the UMFPACK library, with all control options set to the default values 
described in the UMFPACK user guide.  

.. seealso::

    * `UMFPACK code, documentation, copyright, and license
      <http://www.cise.ufl.edu/research/sparse/umfpack>`_

    * T. A. Davis, Algorithm 832: UMFPACK -- an unsymmetric-pattern 
      multifrontal method with a column pre-ordering strategy, ACM 
      Transactions on Mathematical Software, 30(2), 196-199, 2004. 


.. function:: cvxopt.umfpack.linsolve(A, B[, trans = 'N'])

    Solves a sparse set of linear equations 
    
    .. math::

         AX & = B \quad (\mathrm{trans} = \mathrm{'N'}), \\
         A^TX & = B \quad (\mathrm{trans} = \mathrm{'T'}), \\
         A^HX & = B \quad (\mathrm{trans} = \mathrm{'C'}),
    
    where :math:`A` is a sparse matrix and :math:`B` is a dense matrix.
    The arguments ``A`` and ``B`` must have the same type 
    (:const:`'d'` or :const:`'z'`) as ``A``.  On exit ``B`` contains 
    the solution.  Raises an :exc:`ArithmeticError` if the coefficient 
    matrix is singular.

In the following example we solve an equation with coefficient matrix 

.. math:: 
    :label: e-sp-Adef

    A = \left[\begin{array}{rrrrr}
        2 & 3 & 0 & 0 & 0 \\
        3 & 0 & 4 & 0 & 6 \\
        0 &-1 &-3 & 2 & 0 \\
        0 & 0 & 1 & 0 & 0 \\
        0 & 4 & 2 & 0 & 1 
        \end{array}\right].


>>> from cvxopt import spmatrix, matrix, umfpack 
>>> V = [2, 3, 3, -1, 4, 4, -3, 1, 2, 2, 6, 1]
>>> I = [0, 1, 0,  2, 4, 1,  2, 3, 4, 2, 1, 4]
>>> J = [0, 0, 1,  1, 1, 2,  2, 2, 2, 3, 4, 4]
>>> A = spmatrix(V,I,J)
>>> B = matrix(1.0, (5,1))
>>> umfpack.linsolve(A,B)
>>> print(B)
[ 5.79e-01]
[-5.26e-02]
[ 1.00e+00]
[ 1.97e+00]
[-7.89e-01]

The function :func:`linsolve <cvxopt.umfpack.linsolve>`  is 
equivalent to the following three functions called in sequence.  

.. function:: cvxopt.umfpack.symbolic(A)

    Reorders the columns of ``A`` to reduce fill-in and performs a symbolic 
    LU factorization.  ``A`` is a sparse, possibly rectangular, matrix.
    Returns the symbolic factorization as an opaque C object that can be 
    passed on to :func:`numeric <cvxopt.umfpack.numeric>`.


.. function:: cvxopt.umfpack.numeric(A, F)

    Performs a numeric LU factorization of a sparse, possibly rectangular,
    matrix ``A``.   The argument ``F`` is the symbolic factorization
    computed by :func:`symbolic <cvxopt.umfpack.symbolic>` 
    applied to the matrix ``A``,
    or another sparse matrix with the same sparsity pattern, dimensions,
    and type.  The numeric factorization is returned as an opaque C object 
    that that can be passed on to 
    :func:`solve <cvxopt.umfpack.solve>`.  Raises an
    :exc:`ArithmeticError` if the matrix is singular.


.. function:: cvxopt.umfpack.solve(A, F, B[, trans = 'N'])

    Solves a set of linear equations

    .. math:: 

        AX & = B \quad (\mathrm{trans} = \mathrm{'N'}), \\
        A^TX & = B \quad (\mathrm{trans} = \mathrm{'T'}), \\
        A^HX & = B \quad (\mathrm{trans} = \mathrm{'C'}),

    where :math:`A` is a sparse matrix and :math:`B` is a dense matrix.
    The arguments ``A`` and ``B`` must have the same type.  The argument  
    ``F`` is a numeric factorization computed 
    by :func:`numeric <cvxopt.umfpack.numeric>`.  
    On exit ``B`` is overwritten by the 
    solution.


These separate functions are useful for solving several sets of linear 
equations with the same coefficient matrix and different right-hand sides, 
or with coefficient matrices that share the same sparsity pattern.
The symbolic factorization depends only on the sparsity pattern of
the matrix, and not on the numerical values of the nonzero coefficients. 
The numerical factorization on the other hand depends on the sparsity 
pattern of the matrix and on its the numerical values.

As an example, suppose :math:`A` is the matrix :eq:`e-sp-Adef` and 

.. math::

    B = \left[\begin{array}{rrrrr}
        4 & 3 & 0 & 0 & 0 \\
        3 & 0 & 4 & 0 & 6 \\
        0 &-1 &-3 & 2 & 0 \\
        0 & 0 & 1 & 0 & 0 \\
        0 & 4 & 2 & 0 & 2 
        \end{array}\right],

which differs from :math:`A` in its first and last entries.  The following 
code computes

.. math::

    \newcommand{\ones}{\mathbf 1}
    x = A^{-T}B^{-1}A^{-1}\ones.


>>> from cvxopt import spmatrix, matrix, umfpack
>>> VA = [2, 3, 3, -1, 4, 4, -3, 1, 2, 2, 6, 1]
>>> VB = [4, 3, 3, -1, 4, 4, -3, 1, 2, 2, 6, 2]
>>> I =  [0, 1, 0,  2, 4, 1,  2, 3, 4, 2, 1, 4]
>>> J =  [0, 0, 1,  1, 1, 2,  2, 2, 2, 3, 4, 4]
>>> A = spmatrix(VA, I, J)
>>> B = spmatrix(VB, I, J)
>>> x = matrix(1.0, (5,1))
>>> Fs = umfpack.symbolic(A)
>>> FA = umfpack.numeric(A, Fs)
>>> FB = umfpack.numeric(B, Fs)
>>> umfpack.solve(A, FA, x)
>>> umfpack.solve(B, FB, x)
>>> umfpack.solve(A, FA, x, trans='T')
>>> print(x)
[ 5.81e-01]
[-2.37e-01]
[ 1.63e+00]
[ 8.07e+00]
[-1.31e-01]


.. _s-cholmod:

Positive Definite Linear Equations
**********************************

:mod:`cvxopt.cholmod` is an interface to the Cholesky factorization routines
of the CHOLMOD package.  It includes functions for Cholesky factorization 
of sparse positive definite matrices, and for solving sparse sets of linear
equations with positive definite matrices. 
The routines can also be used for computing 
:raw-html:`LDL<sup><small>T</small></sup>`
(or 
:raw-html:`LDL<sup><small>H</small></sup>`
factorizations
of symmetric indefinite matrices (with :math:`L` unit lower-triangular and 
:math:`D` diagonal and nonsingular) if such a factorization exists.  

.. seealso::
   
     `CHOLMOD code, documentation, copyright, and license
     <http://www.cise.ufl.edu/research/sparse/cholmod>`_

.. function:: cvxopt.cholmod.linsolve(A, B[, p = None, uplo = 'L'])

    Solves

    .. math::

        AX = B 

    with :math:`A` sparse and real symmetric or complex Hermitian.  

    ``B`` is a dense matrix of the same type as ``A``.  On exit it 
    is overwritten with the solution.  The argument ``p`` is an integer 
    matrix with length equal to the order of :math:`A`, and specifies an 
    optional reordering.  If ``p`` is not specified, CHOLMOD uses a 
    reordering from the AMD library.

    Raises an :exc:`ArithmeticError` if the factorization does not exist.


As an  example, we solve 

.. math:: 
    :label: e-A-pd

        \left[ \begin{array}{rrrr}
            10 &  0 & 3 &  0 \\
             0 &  5 & 0 & -2 \\
             3 &  0 & 5 &  0 \\
             0 & -2 & 0 &  2 
        \end{array}\right] X = 
        \left[ \begin{array}{cc} 
             0 & 4 \\ 1 & 5 \\ 2 & 6 \\ 3 & 7
        \end{array} \right].


>>> from cvxopt import matrix, spmatrix, cholmod
>>> A = spmatrix([10, 3, 5, -2, 5, 2], [0, 2, 1, 3, 2, 3], [0, 0, 1, 1, 2, 3])
>>> X = matrix(range(8), (4,2), 'd')
>>> cholmod.linsolve(A,X)
>>> print(X)
[-1.46e-01  4.88e-02]
[ 1.33e+00  4.00e+00]
[ 4.88e-01  1.17e+00]
[ 2.83e+00  7.50e+00]


.. function:: cvxopt.cholmod.splinsolve(A, B[, p = None, uplo = 'L'])

    Similar to :func:`linsolve <cvxopt.cholmod.linsolve>` except that 
    ``B`` is an :class:`spmatrix <cvxopt.spmatrix>` and 
    that the solution is returned as an output argument (as a new 
    :class:`spmatrix`).  ``B`` is not modified.


The following code computes the inverse of the coefficient matrix 
in :eq:`e-A-pd` as a sparse matrix.

>>> X = cholmod.splinsolve(A, spmatrix(1.0,range(4),range(4)))
>>> print(X)
[ 1.22e-01     0     -7.32e-02     0    ]
[    0      3.33e-01     0      3.33e-01]
[-7.32e-02     0      2.44e-01     0    ]
[    0      3.33e-01     0      8.33e-01]


The functions :func:`linsolve <cvxopt.cholmod.linsolve>` and 
:func:`splinsolve <cvxopt.cholmod.splinsolve>` are equivalent to 
:func:`symbolic <cvxopt.cholmod.symbolic>` and 
:func:`numeric <cvxopt.cholmod.numeric>` called in sequence, followed by 
:func:`solve <cvxopt.cholmod.solve>`, respectively, 
:func:`spsolve <cvxopt.cholmod.spsolve>`.

.. function:: cvxopt.cholmod.symbolic(A[, p = None, uplo = 'L'])

    Performs a symbolic analysis of a sparse real symmetric or
    complex Hermitian matrix :math:`A` for one of the two factorizations:

    .. math:: 
        :label: e-chol-ll 

        PAP^T = LL^T, \qquad PAP^T = LL^H, 
    
    and 

    .. math:: 
        :label: e-chol-ldl

        PAP^T = LDL^T, \qquad PAP^T = LDL^H,

    where :math:`P` is a permutation matrix, :math:`L` is lower triangular 
    (unit lower triangular in the second factorization), and :math:`D` is 
    nonsingular diagonal.  The type of factorization depends on the value 
    of :attr:`options['supernodal']` (see below).

    If ``uplo`` is :const:`'L'`, only the lower triangular part of ``A`` 
    is accessed and the upper triangular part is ignored.
    If ``uplo`` is :const:`'U'`, only the upper triangular part of ``A`` 
    is accessed and the lower triangular part is ignored.

    If ``p`` is not provided, a reordering from the AMD library is used.

    The symbolic factorization is returned as an opaque C object that 
    can be passed to :func:`numeric <cvxopt.cholmod.numeric>`.


.. function:: cvxopt.cholmod.numeric(A, F)

    Performs a numeric factorization of a sparse symmetric matrix 
    as :eq:`e-chol-ll` or :eq:`e-chol-ldl`.  The argument ``F`` is the 
    symbolic factorization computed by 
    :func:`symbolic <cvxopt.cholmod.symbolic>` applied to 
    the matrix ``A``, or to another sparse  matrix with the same sparsity 
    pattern and typecode, or by 
    :func:`numeric <cvxopt.cholmod.numeric>` applied to a matrix
    with the same sparsity pattern and typecode as ``A``.

    If ``F`` was created by a 
    :func:`symbolic <cvxopt.cholmod.symbolic>` with ``uplo`` 
    equal 
    to :const:`'L'`, then only the lower triangular part of ``A`` is 
    accessed and the upper triangular part is ignored.  If it was created 
    with ``uplo`` equal to :const:`'U'`, then only the upper triangular 
    part of ``A`` is accessed and the lower triangular part is ignored.

    On successful exit, the factorization is stored in ``F``.
    Raises an :exc:`ArithmeticError` if the factorization does not exist.


.. function:: cvxopt.cholmod.solve(F, B[, sys = 0])

    Solves one of the following linear equations where ``B`` is a dense 
    matrix and ``F`` is the numeric factorization :eq:`e-chol-ll` 
    or :eq:`e-chol-ldl` computed by 
    :func:`numeric <cvxopt.cholmod.numeric>`.  
    ``sys`` is an integer with values between 0 and 8. 

    +---------+--------------------+ 
    | ``sys`` | equation           | 
    +---------+--------------------+
    | 0       | :math:`AX = B`     |
    +---------+--------------------+
    | 1       | :math:`LDL^TX = B` |
    +---------+--------------------+
    | 2       | :math:`LDX = B`    |
    +---------+--------------------+
    | 3       | :math:`DL^TX=B`    | 
    +---------+--------------------+
    | 4       | :math:`LX=B`       |
    +---------+--------------------+
    | 5       | :math:`L^TX=B`     |
    +---------+--------------------+
    | 6       | :math:`DX=B`       |
    +---------+--------------------+
    | 7       | :math:`P^TX=B`     |
    +---------+--------------------+
    | 8       | :math:`PX=B`       |
    +---------+--------------------+

    (If ``F`` is a Cholesky factorization of the form :eq:`e-chol-ll`, 
    :math:`D` is an identity matrix in this table.  If ``A`` is complex, 
    :math:`L^T` should be replaced by :math:`L^H`.)

    The matrix ``B`` is a dense :const:`'d'` or :const:`'z'` matrix, with 
    the same type as ``A``.  On exit it is overwritten by the solution.


.. function:: cvxopt.cholmod.spsolve(F, B[, sys = 0])

    Similar to :func:`solve <cvxopt.cholmod.solve>`, except that ``B`` is 
    a class:`spmatrix`, and the solution is returned as an output argument 
    (as an :class:`spmatrix`).  ``B`` must have the same typecode as ``A``.


For the same example as above:

>>> X = matrix(range(8), (4,2), 'd')
>>> F = cholmod.symbolic(A)
>>> cholmod.numeric(A, F)
>>> cholmod.solve(F, X)
>>> print(X)
[-1.46e-01  4.88e-02]
[ 1.33e+00  4.00e+00]
[ 4.88e-01  1.17e+00]
[ 2.83e+00  7.50e+00]


.. function:: cvxopt.cholmod.diag(F)

    Returns the diagonal elements of the Cholesky factor :math:`L` 
    in :eq:`e-chol-ll`, as a dense matrix of the same type as ``A``.
    Note that this only applies to Cholesky factorizations.  The matrix 
    :math:`D` in an :raw-html:`LDL<sup><small>T</small></sup>`
    factorization can be retrieved via :func:`solve <cvxopt.cholmod.solve>`
    with ``sys`` equal to 6.


In the functions listed above, the default values of the control 
parameters described in the CHOLMOD user guide are used, except for 
:c:data:`Common->print` which is set to 0 instead of 3 and 
:c:data:`Common->supernodal` which is set to 2 instead of 1.
These parameters (and a few others) can be modified by making an 
entry in the dictionary :attr:`cholmod.options`. 
The meaning of these parameters is as follows.

:attr:`options['supernodal']` 
    If equal to 0, a factorization :eq:`e-chol-ldl` is computed using a 
    simplicial algorithm.  If equal to 2, a factorization :eq:`e-chol-ll`
    is computed using a supernodal algorithm.  If equal to 1, the most 
    efficient of the two factorizations is selected, based on the sparsity 
    pattern.  Default: 2.

:attr:`options['print']` 
    A nonnegative integer that controls the amount of output printed to 
    the screen.  Default: 0 (no output).


As an example that illustrates :func:`diag  <cvxopt.cholmod.diag>` and the 
use of :attr:`cholmod.options`, we compute the logarithm of the determinant 
of the coefficient matrix in :eq:`e-A-pd` by two methods.


>>> import math
>>> from cvxopt.cholmod import options
>>> from cvxopt import log
>>> F = cholmod.symbolic(A)
>>> cholmod.numeric(A, F)
>>> print(2.0 * sum(log(cholmod.diag(F))))
5.50533153593
>>> options['supernodal'] = 0
>>> F = cholmod.symbolic(A)
>>> cholmod.numeric(A, F)
>>> Di = matrix(1.0, (4,1))
>>> cholmod.solve(F, Di, sys=6)
>>> print(-sum(log(Di)))
5.50533153593


Example: Covariance Selection
*****************************

This example illustrates the use of the routines for sparse Cholesky 
factorization.  We consider the problem 

.. math::
    :label: e-covsel

    \newcommand{\Tr}{\mathop{\bf tr}}
    \begin{array}{ll}
        \mbox{minimize} & -\log\det K + \Tr(KY) \\
        \mbox{subject to} & K_{ij}=0,\quad (i,j) \not \in S.
    \end{array}

The optimization variable is a symmetric matrix :math:`K` of order 
:math:`n` and the domain of the problem is the set of positive definite 
matrices.  The matrix :math:`Y` and the index set :math:`S` are given.  
We assume that all the diagonal positions are included in :math:`S`.
This problem arises in maximum likelihood estimation of the covariance
matrix of a zero-mean normal distribution, with constraints 
that specify that pairs of variables are conditionally independent.

We can express :math:`K` as

.. math::

    \newcommand{\diag}{\mathop{\bf diag}}
    K(x) = E_1\diag(x)E_2^T+E_2\diag(x)E_1^T

where :math:`x` are the nonzero elements in the lower triangular part of 
:math:`K`, with the diagonal elements scaled by 1/2, and

.. math::

    E_1 = \left[ \begin{array}{cccc}
        e_{i_1} & e_{i_2} & \cdots & e_{i_q} \end{array}\right], \qquad
    E_2 = \left[ \begin{array}{cccc}
        e_{j_1} & e_{j_2} & \cdots & e_{j_q} \end{array}\right], 

where (:math:`i_k`, :math:`j_k`) are the positions of the nonzero entries 
in the lower-triangular part of :math:`K`.  With this notation, we can 
solve problem :eq:`e-covsel` by solving the unconstrained problem

.. math::

    \newcommand{\Tr}{\mathop{\bf tr}}
    \begin{array}{ll}
    \mbox{minimize} & f(x) = -\log\det K(x) + \Tr(K(x)Y).
    \end{array}

The code below implements Newton's method with a backtracking line search.  
The gradient and Hessian of the objective function are given by

.. math:: 

    \newcommand{\diag}{\mathop{\bf diag}}
    \begin{split}
    \nabla f(x) 
        & = 2 \diag( E_1^T (Y - K(x)^{-1}) E_2)) \\
        & = 2\diag(Y_{IJ} - \left(K(x)^{-1}\right)_{IJ}) \\
    \nabla^2 f(x) 
        & = 2 (E_1^T K(x)^{-1} E_1) \circ (E_2^T K(x)^{-1} E_2) 
            + 2 (E_1^T K(x)^{-1} E_2) \circ (E_2^T K(x)^{-1} E_1) \\
        & = 2 \left(K(x)^{-1}\right)_{II} \circ \left(K(x)^{-1}\right)_{JJ}
            + 2 \left(K(x)^{-1}\right)_{IJ} \circ 
            \left(K(x)^{-1}\right)_{JI},
    \end{split}

where :math:`\circ` denotes Hadamard product.


::

    from cvxopt import matrix, spmatrix, log, mul, blas, lapack, amd, cholmod

    def covsel(Y):
        """
        Returns the solution of

             minimize    -log det K + Tr(KY)
             subject to  K_{ij}=0,  (i,j) not in indices listed in I,J.

        Y is a symmetric sparse matrix with nonzero diagonal elements.
        I = Y.I,  J = Y.J.
        """

        I, J = Y.I, Y.J
        n, m = Y.size[0], len(I) 
        N = I + J*n         # non-zero positions for one-argument indexing 
        D = [k for k in range(m) if I[k]==J[k]]  # position of diagonal elements

        # starting point: symmetric identity with nonzero pattern I,J
        K = spmatrix(0.0, I, J) 
        K[::n+1] = 1.0

        # Kn is used in the line search
        Kn = spmatrix(0.0, I, J)

        # symbolic factorization of K 
        F = cholmod.symbolic(K)

        # Kinv will be the inverse of K
        Kinv = matrix(0.0, (n,n))
        
        for iters in range(100):

            # numeric factorization of K
            cholmod.numeric(K, F)
            d = cholmod.diag(F)

            # compute Kinv by solving K*X = I 
            Kinv[:] = 0.0
            Kinv[::n+1] = 1.0
            cholmod.solve(F, Kinv)

            # solve Newton system
            grad = 2*(Y.V - Kinv[N])
            hess = 2*(mul(Kinv[I,J],Kinv[J,I]) + mul(Kinv[I,I],Kinv[J,J]))
            v = -grad
            lapack.posv(hess,v) 
            
            # stopping criterion
            sqntdecr = -blas.dot(grad,v) 
            print("Newton decrement squared:%- 7.5e" %sqntdecr)
            if (sqntdecr < 1e-12):
                print("number of iterations: ", iters+1)
                break

            # line search
            dx = +v
            dx[D] *= 2      # scale the diagonal elems        
            f = -2.0 * sum(log(d))    # f = -log det K
            s = 1
            for lsiter in range(50):
                Kn.V = K.V + s*dx
                try: 
                    cholmod.numeric(Kn, F)
                except ArithmeticError:
                    s *= 0.5
                else:
                    d = cholmod.diag(F)
                    fn = -2.0 * sum(log(d)) + 2*s*blas.dot(v,Y.V)
                    if (fn < f - 0.01*s*sqntdecr): 
                         break
                    s *= 0.5
                
            K.V = Kn.V

        return K
