On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems
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 SCG, SMR, and SLQ) 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 SLQ. We give some new theoretical results on SCG, SMR, and SLQ. 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.
Keywords
Cite
@article{arxiv.2307.16460,
title = {On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems},
author = {Kui Du and Jia-Jun Fan and Xiao-Hui Sun and Fang Wang and Ya-Lan Zhang},
journal= {arXiv preprint arXiv:2307.16460},
year = {2023}
}
Comments
23 pages, 3 figures