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arXiv · 2307.16460

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

Kui Du·Jia-Jun Fan·Xiao-Hui Sun·Fang Wang·Ya-Lan Zhang
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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.

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Kui Du, Jia-Jun Fan, Xiao-Hui Sun, Fang Wang, Ya-Lan Zhang. 2023-07-31. On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems. https://arxiv.org/abs/2307.16460

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