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# taucs_chfact

cholesky factorisation of a sparse s.p.d. matrix

### Calling Sequence

C_ptr = taucs_chfact(A)

### Arguments

- A
a sparse real symmetric positive definite (s.p.d.) matrix

- C_ptr
a pointer to the Cholesky factors (C,p : A(p,p)=CC')

### Description

This function computes a Cholesky factorization of the sparse symmetric positive definite (s.p.d.) matrix A and retrieves at the scilab level, a pointer (C_ptr) to an handle of the Cholesky factors (C,p) (the memory used for them is "outside" scilab space).

If your matrix is s.p.d. this function must be used in place of umf_lufact or in place of the scilab function chfact for a gain in speed (also as chfact uses the scilab memory for the factors the user must set the stacksize with a large value because of the fill-in occuring in computing the factor C which then may take more memory than the initial matrix A).

When such a factorisation have been computed, a linear system must
be solved with taucs_chsolve. **To free the memory used by the Cholesky factors, use
taucs_chdel(C_ptr)**; to
retrieve the Cholesky factors at the scilab level (for example to display
their sparse patterns), use taucs_chget;
to get some information (number of non zeros in C), use
taucs_chinfo. To compute an
approximation of the condition number in norm 2 use
cond2sp.

### Remarks

taucs_chfact works only with the upper triangle of A, and the matrix A must be provided either in its complete form (that is with the lower triangle also) or only with its upper triangle;

currently taucs_chfact uses the genmmd (generalized minimum degree) algorithm of Liu to find in a first step the permutation p (so as to minimize the fill-in in the factorization); future versions will let the user choose his/her own reordering by providing a supplementary argument p.

### Examples

// Example #1 : a small linear test system // whom solution must be [1;2;3;4;5] A = sparse( [ 2 -1 0 0 0; -1 2 -1 0 0; 0 -1 2 -1 0; 0 0 -1 2 -1; 0 0 0 -1 2] ); b = [0 ; 0; 0; 0; 6]; Cp = taucs_chfact(A); x = taucs_chsolve(Cp,b) // do not forget to clear memory with taucs_chdel(Cp) // Example #2 a real example // first load a sparse matrix [A] = ReadHBSparse(SCI+"/modules/umfpack/examples/bcsstk24.rsa"); // compute the factorisation Cp = taucs_chfact(A); b = rand(size(A,1),1); // a random rhs // use taucs_chsolve for solving Ax=b x = taucs_chsolve(Cp,b); norm(A*x - b) // the same with one iterative refinement step x = taucs_chsolve(Cp,b,A); norm(A*x - b) // do not forget to clear memory taucs_chdel(Cp)

### Authors

- taucs by Sivan Toledo (see taucs_license)
- scilab interface by Bruno Pincon

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