A comparison of Krylov methods for Shifted Skew-Symmetric Systems

TL;DR

This paper introduces the MRS3 solver, a short-recurrence Krylov method tailored for shifted skew-symmetric systems, demonstrating its superior speed and robustness through theoretical analysis and numerical experiments.

Contribution

The paper presents the MRS3 solver, a novel short-recurrence Krylov method specifically designed for shifted skew-symmetric systems, and compares it favorably to existing methods.

Findings

MRS3 is the fastest Krylov solver for shifted skew-symmetric systems.

MRS3 demonstrates higher robustness compared to other Krylov methods.

Numerical experiments confirm the theoretical advantages of MRS3.

Abstract

It is well known that for general linear systems, only optimal Krylov methods with long recurrences exist. For special classes of linear systems it is possible to find optimal Krylov methods with short recurrences. In this paper we consider the important class of linear systems with a shifted skew-symmetric coefficient matrix. We present the MRS3 solver, a minimal residual method that solves these problems using short vector recurrences. We give an overview of existing Krylov solvers that can be used to solve these problems, and compare them with the MRS3 method, both theoretically and by numerical experiments. From this comparison we argue that the MRS3 solver is the fastest and most robust of these Krylov method for systems with a shifted skew-symmetric coefficient matrix.