THE SUPERLINEAR CONVERGENCE BEHAVIOR OF GMRES

GMRES is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems. It is well known that GMRES often has a so-called superlinear convergence behaviour, i.e., the rate of convergence seems to improve as the iteration proceeds. For the conjugate gradients method this phenomenon has been related to a (modest) degree of convergence of the Ritz values. It has been observed in experiments that for GMRES too, changes in the convergence behaviour seem to be related to the convergence of Ritz values. In this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by Ritz values, GMRES from then on converges at least as fast as for a related system in which these eigenvalues (and their eigenvector components) are missing.