Ritz and harmonic Ritz values and the convergence of FOM and GMRES (Q2760358)

The Ritz values are the eigenvalues of the well-known Hessenberg matrix \(H_m\) which is computed in the Arnoldi process within iterations by means of the full orthogonal method (FOM) or the generalized minimal residual (GMRES) method. On the basis of the Walker-Zhou formulation of GMRES, the authors construct the Hessenberg matrix so that the harmonic Ritz values are the eigenvalues of this one. On the basis of this result it is proved that the harmonic Ritz values are the zeros of the GMRES residual polynomial. This result is analogous to the proposition mentioned at the beginning of this paper that the Ritz values are the zeros of the FOM residual polynomials. Moreover, the authors present an upper bound for the norm of the difference between the matrices from which the Ritz and harmonic Ritz values are computed. The difference between the Ritz and harmonic Ritz values enables us to describe the breakdown of FOM and stagnation of GMRES.