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FOM and GMRES are Krylov subspace iterative methods for solving nonsymmetric linear systems. The Ritz values are approximate eigenvalues, which can be computed cheaply within these algorithms. In this paper, we generalise the concept of Harmonic Ritz values, introduced by Paige et al. for symmetric matrices, to nonsymmetric matrices. We show that the zeroes of the residual polynomials of FOM and GMRES are the Ritz and Harmonic Ritz values respectively. We present an upper bound for the difference between the matrices from which the Ritz and Harmonic Ritz values are computed. The differences between these values allows us to describe breakdown of FOM and stagnation of GMRES.

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