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# cond2sp

computes an approximation of the 2-norm condition number of a s.p.d. sparse matrix

### Syntax

[K2, lm, vm, lM, vM] = cond2sp(A, C_ptr [, rtol, itermax, verb])

### Arguments

- A
a real symmetric positive definite sparse matrix

- C_ptr
a pointer to a Cholesky factorization (got with taucs_chfact)

- rtol
(optional) relative tolerance (default 1.e-3) (see details in DESCRIPTION)

- itermax
(optional) maximum number of iterations in the underlying algorithms (default 30)

- verb
(optional) boolean, must be %t for displaying the intermediary results, and %f (default) if you do not want.

- K2
estimated 2-norm condition number

`K2 = ||A||_2 ||A^(-1)||_2 = lM/lm`

- lm
(real positive scalar) minimum eigenvalue

- vm
associated eigenvector

- lM
(real positive scalar) maximum eigenvalue

- vM
associated eigenvector

### Description

This quick and dirty function computes `(lM,vM)`

using the iterative
power method and `(lm,vm)`

with the inverse iterative power method, then
`K2 = lM/lm`

. For each method the iterations are stopped until the following
condition is met :

| (l_new - l_old)/l_new | < rtol

but 4 iterations are nevertheless required and also the iterations are stopped
if itermax is reached (and a warning message is issued). As the matrix is symmetric
this is the rayleigh quotient which gives the estimated eigenvalue at each step
(`lambda = v'*A*v`

). You may called this function with named parameter, for
instance if you want to see the intermediary result without setting yourself the
rtol and itermax parameters you may called this function with the syntax :

[K2, lm, vm, lM, vM] = cond2sp(A , C_ptr, verb=%t )

### Caution

Currently there is no verification for the input parameters !

### Remark

This function is intended to get an approximation of the 2-norm condition number (K2) and
with the methods used, the precision on the obtained eigenvectors (vM and vm) are generally
not very good. If you look for a smaller residual `||Av - l*v||`

, you may apply some inverse
power iterations from v0 with the matrix :

B = A - l0*speye(A)

For instance, applied 5 such iterations for `(lm,vm)`

is done with :

[A] = ReadHBSparse(SCI+"/modules/umfpack/demos/bcsstk24.rsa"); C_ptr = taucs_chfact(A); [K2, lm, vm, lM, vM] = cond2sp(A , C_ptr, 1.e-5, 50, %t ); taucs_chdel(C_ptr) l0 = lm ; v0 = vm; // or l0 = lM ; v0 = vM; // to polish (lM,vM) B = A - l0*speye(A); LUp = umf_lufact(B); vr = v0; nstep = 5; for i=1:nstep, vr = umf_lusolve(LUp, vr, "Ax=b", B); vr = vr/norm(vr) ; end umf_ludel(LUp); // if you do not use anymore this factorization lr = vr'*A*vr; norm_r0 = norm(A*v0 - l0*v0); norm_rr = norm(A*vr - lr*vr); // Bauer-Fike error bound... mprintf(" first estimated eigenvalue : l0 = %e \n\t", l0) mprintf(" |l-l0| <= ||Av0-l0v0|| = %e , |l-l0|/l0 <= %e \n\r", norm_r0, norm_r0/l0) mprintf(" raffined estimated eigenvalue : lr = %e \n\t", lr) mprintf(" |l-lr| <= ||Avr-lrvr|| = %e , |l-lr|/lr <= %e \n\r", norm_rr, norm_rr/lr)

### Examples

[A] = ReadHBSparse(SCI+"/modules/umfpack/demos/bcsstk24.rsa"); C_ptr = taucs_chfact(A); [K2, lm, vm, lM, vM] = cond2sp(A , C_ptr, 1.e-5, 50, %t ); taucs_chdel(C_ptr)

### See also

- condestsp — estimate the condition number of a sparse matrix
- taucs_chfact — cholesky factorization of a sparse s.p.d. matrix
- rcond — inverse condition number

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