# cond2sp

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

## Calling Sequence

    [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/examples/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/examples/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
- taucs_chfact
- rcond