GAP (HAP) - Chapter 6: Sparse Chain complexes

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6 Sparse Chain complexes

### 6 Sparse Chain complexes

`SparseMat(A)` Inputs a matrix A and returns the matrix in sparse format.
`TransposeOfSparseMat(A)` Inputs a sparse matrix A and returns its transpose sparse format.
`ReverseSparseMat(A)` Inputs a sparse matrix A and modifies it by reversing the order of the columns. This function modifies A and returns no value.
`SparseRowMult(A,i,k)` Multiplies the i-th row of a sparse matrix A by k. The sparse matrix A is modified but nothing is returned.
`SparseRowInterchange(A,i,k)` Interchanges the i-th and j-th rows of a sparse matrix A by k. The sparse matrix A is modified but nothing is returned.
`SparseRowAdd(A,i,j,k)` Adds k times the j-th row to the i-th row of a sparse matrix A. The sparse matrix A is modified but nothing is returned.
`SparseSemiEchelon(A)` Converts a sparse matrix A to semi-echelon form (which means echelon form up to a permutation of rows). The sparse matrix A is modified but nothing is returned.
`RankMatDestructive(A)` Returns the rank of a sparse matrix A. The sparse matrix A is modified during the calculation.
`RankMat(A)` Returns the rank of a sparse matrix A.
`SparseChainComplex(Y)` Inputs a regular CW-complex Y and returns a sparse chain complex which is chain homotopy equivalent to the cellular chain complex of Y. The function uses discrete vector fields to calculate a smallish chain complex.
`SparseChainComplexOfRegularCWComplex(Y)` Inputs a regular CW-complex Y and returns its cellular chain complex as a sparse chain complex. The function SparseChainComplex(Y) will usually return a smaller chain complex.
`SparseBoundaryMatrix(C,n)` Inputs a sparse chain complex C and integer n. Returns the n-th boundary matrix of the chain complex in sparse format.
`Bettinumbers(C,n)` Inputs a sparse chain complex C and integer n. Returns the n-th Netti number of the chain complex.

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