Volume 47, pp. 37-56, 2017.

Any admissible harmonic Ritz value set is possible for GMRES

Kui Du, Jurjen Duintjer Tebbens, and Gérard Meurant


It is known that the harmonic Ritz values are the zeros of the GMRES residual polynomials. In this paper we show that any decreasing GMRES residual norm history is possible with any prescribed finite harmonic Ritz values in every iteration of the GMRES process. In addition, we characterize the admissible harmonic Ritz values when GMRES stagnates during some iterations, and show that with any prescribed, in this sense admissible harmonic Ritz values, any non-increasing GMRES residual norm history is possible. We also present a parameterization of the class of matrices and right-hand sides yielding prescribed GMRES residual norms and prescribed admissible harmonic Ritz values.

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Key words

Ritz values, harmonic Ritz values, GMRES convergence, prescribed residual norms, FOM convergence

AMS subject classifications

65F15, 65F10

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