condestsp

estimate the condition number of a sparse matrix

Calling Sequence

[K1] = condestsp(A, LUp, t)
[K1] = condestsp(A, LUp)
[K1] = condestsp(A, t)
[K1] = condestsp(A)

Arguments

:A a real or complex square sparse matrix : :LUp (optional) a pointer to (umf) LU factors of A obtained by a

call to umf_lufact ; if you have already computed the LU (= PAQ) factors it is recommended to give this optional parameter (as the factorization may be time consuming)

: :t (optional) a positive integer (default value 2) by increasing

this one you may hope to get a better (even exact) estimate

: :K1 estimated 1-norm condition number of A :

Description

Give an estimate of the 1-norm condition number of the sparse matrix A by Algorithm 2.4 appearing in :

"A block algorithm for matrix 1-norm estimation
 with an application to 1-`norm`_ pseudospectra"
 Nicholas J. Higham `and`_ Francoise Tisseur
 Siam J. Matrix Anal. Appl., vol 21, No 4, pp 1185-1201

Noting the exact condition number K1e = ||A||_1 ||A^(-1)||_1, we have always K1 <= K1e and this estimate gives in most case something superior to 1/2 K1e

Examples

A = `sparse`_( [ 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] );
K1 = condestsp(A)
// verif by direct computation
K1e = `norm`_(A,1)*`norm`_(`inv`_(`full`_(A)),1)

// another example
[A] = `ReadHBSparse`_(SCI+"/modules/umfpack/examples/arc130.rua");
K1 = condestsp(A)
// this example is not so big so that we can do the verif
K1e = `norm`_(A,1)*`norm`_(`inv`_(`full`_(A)),1)

// if you have already the lu factors condestsp(A,Lup) is faster
// because lu factors are then not computed inside condestsp
Lup = `umf_lufact`_(A);
K1 = condestsp(A,Lup)
`umf_ludel`_(Lup)         // clear memory

See Also

• umf_lufact lu factorisation of a sparse matrix
• rcond inverse condition number