On convergence bounds of GMRES algorithm

Abstract

We first make a brief review of GMRES convergence results. Then we derive new bounds for the GMRES residual norm by making use of a unitary matrix U and a Hermitian positive definite matrix P, which are GMRES-equivalent to the coefficient matrix A with respect to the initial residual r/sub 0/. The existence of such U and P was proved by Leonid (2000). As a GMRES residual norm bound for linear systems with Hermitian positive definite coefficient matrices is known and a GMRES residual norm bound for linear systems with unitary coefficient matrices can be readily derived from Liesen's (2000) work, our new bounds follow from the fact that two GMRES-equivalent matrices make the same residual.