On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems

TL;DR

This paper extends Krylov subspace methods to singular skew-symmetric systems and systematically studies three methods (S$^3$CG, S$^3$MR, S$^3$LQ) based on Lanczos triangularization, providing new theoretical insights and numerical validation.

Contribution

It introduces the first study of S$^3$LQ and extends Krylov methods to singular skew-symmetric systems, establishing relationships among these methods and existing algorithms.

Findings

Theoretical results on S$^3$CG, S$^3$MR, and S$^3$LQ.

Relationships among the three methods and existing algorithms.

Numerical examples validating the theoretical findings.

Abstract

Krylov subspace methods for solving linear systems of equations involving skew-symmetric matrices have gained recent attention. Numerical equivalences among Krylov subspace methods for nonsingular skew-symmetric linear systems have been given in Greif et al. [SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1071--1087]. In this work, we extend the results of Greif et al. to singular skew-symmetric linear systems. In addition, we systematically study three Krylov subspace methods (called S$^3$CG, S$^3$MR, and S$^3$LQ) for solving shifted skew-symmetric linear systems. They all are based on Lanczos triangularization for skew-symmetric matrices, and correspond to CG, MINRES, and SYMMLQ for solving symmetric linear systems, respectively. To the best of our knowledge, this is the first work that studies S$^3$LQ. We give some new theoretical results on S$^3$CG, S$^3$MR, and S$^3$LQ. We also provide the relationship among the three methods and those based on Golub--Kahan bidiagonalization and Saunders--Simon--Yip tridiagonalization. Numerical examples are given to illustrate our theoretical findings.