Ritz and harmonic Ritz values and the convergence of FOM and GMRES

The Ritz and harmonic Ritz values are approximate eigenvalues, which can be computed cheaply within the FOM and GMRES Krylov subspace iterative method s for solving non-symmetric linear systems. They are also the zeros of the residual polynomials of FOM and GMRES, respectively. In this paper we show that the Walker-Zhou interpretation of GMRES enables us to formulate the re lation between the harmonic Ritz values and GMRES in the same way as the re lation between the Ritz values and FOM. We present an upper bound for the n orm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The differences between the Ritz and harmonic Ri tz values enable us to describe the breakdown of FOM and stagnation of GMRE S. Copyright (C) 1999 John Wiley & Sons, Ltd.