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

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 \(^3\) ${}^3$ CG, S \(^3\) ${}^3$ MR, and S \(^3\) ${}^3$ 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 \(^3\) ${}^3$ LQ. We give some new theoretical results on S \(^3\) ${}^3$ CG, S \(^3\) ${}^3$ MR, and S \(^3\) ${}^3$ LQ. We also provide relations 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.