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 $\omega.$