# taucs_chfact

cholesky factorization 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 a 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 factorization 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 factorization
    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)

## See Also

- taucs_chsolve
- taucs_chdel
- taucs_chinfo
- taucs_chget
- cond2sp