### 10 Sparse Chain complexes

#### 10.1

##### 10.1-1 SparseMat
 ‣ SparseMat( A ) ( function )

Inputs a matrix A and returns the matrix in sparse format.

##### 10.1-2 TransposeOfSparseMat
 ‣ TransposeOfSparseMat( A ) ( function )

Inputs a sparse matrix A and returns its transpose sparse format.

##### 10.1-3 ReverseSparseMat
 ‣ ReverseSparseMat( A ) ( function )

Inputs a sparse matrix A and modifies it by reversing the order of the columns. This function modifies A and returns no value.

##### 10.1-4 SparseRowMult
 ‣ SparseRowMult( A, i, k ) ( function )

Multiplies the i-th row of a sparse matrix A by k. The sparse matrix A is modified but nothing is returned.

##### 10.1-5 SparseRowInterchange
 ‣ SparseRowInterchange( A, i, k ) ( function )

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.

##### 10.1-6 SparseRowAdd
 ‣ SparseRowAdd( A, i, j, k ) ( function )

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.

##### 10.1-7 SparseSemiEchelon
 ‣ SparseSemiEchelon( A ) ( function )

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.

##### 10.1-8 RankMatDestructive
 ‣ RankMatDestructive( A ) ( function )

Returns the rank of a sparse matrix A. The sparse matrix A is modified during the calculation.

##### 10.1-9 RankMat
 ‣ RankMat( A ) ( function )

Returns the rank of a sparse matrix A.

##### 10.1-10 SparseChainComplex
 ‣ SparseChainComplex( Y ) ( function )

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.

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##### 10.1-11 SparseChainComplexOfRegularCWComplex
 ‣ SparseChainComplexOfRegularCWComplex( Y ) ( function )

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.

##### 10.1-12 SparseBoundaryMatrix
 ‣ SparseBoundaryMatrix( C, n ) ( function )

Inputs a sparse chain complex C and integer n. Returns the n-th boundary matrix of the chain complex in sparse format.

##### 10.1-13 Bettinumbers
 ‣ Bettinumbers( C, n ) ( function )

Inputs a sparse chain complex C and integer n. Returns the n-th Netti number of the chain complex.

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