Minimal Residual Rational Krylov Subspace Method for Sequences of Shifted Linear Systems

Abstract

Abstract. The solution of sequences of shifted linear systems is a classic problem in numerical linear algebra, and a variety of efficient methods have been proposed over the years. Nevertheless, there exist challenging scenarios for which there is still a lack of performant solvers. For instance, state-of-the-art procedures based on restarting may struggle to handle nonsymmetric problems where the shifts are possibly complex numbers that do not come as conjugate pairs. We design a novel projection strategy based on the rational Krylov subspace equipped with a minimal residual condition. We also devise a novel pole selection procedure, tailored to our problem, providing poles for the rational Krylov basis construction that yield faster convergence than those computed by available general-purpose schemes. A panel of diverse numerical experiments shows that our novel approach performs better than state-of-the-art techniques, especially on the very challenging problems mentioned above. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/palittaUniBO/MR_RKSM and in the supplementary materials ( MR_RKSM-main.zip [2.19MB]). [Formula: see text]