Preconditioned Iterative Methods in a Subspace
for Nonsymmetric Systems with
Large Jumps in the Coefficients
Andrew Knyazev
Department of Mathematics
University of Colorado at Denver
P.O. Box 173364, Campus Box 170
Denver, CO 80217-3364
Email aknyazev@tiger.cudenver.edu
We consider a family of nonsymmetric matrices $A_\omega = A_0 + \omega B,$
with a noninvertible matrix $A_0,$ an invertible matrix $B,$ and a nonnegative
parameter $\omega \leq 1.$ The matrix $A_{\omega}$ is expected to be poor
conditioned for $\omega = 1.$ Small $\omega$ leads to jumps in the
coefficients and makes the condition number even larger. Using a special
preconditioning and a symmetrization we convert the original system with the
matrix $A_{\omega}$ into a system with a symmetric matrix. A standard
iterative method, e.g. the conjugate gradient method, can be used for the new
system. We show, with a proper choice of the initial guess, the uniform in
${\omega}$ convergence of the method, even though the new matrix is still poor
conditioned for small $\omega.$