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.

翻译：求解涉及斜对称矩阵的线性方程组的Krylov子空间方法近来受到关注。Greif等人[SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1071–1087]给出了非奇异斜对称线性系统Krylov子空间方法之间的数值等价关系。本文将其结果推广至奇异斜对称线性系统。此外，我们系统研究了三种求解移位斜对称线性系统的Krylov子空间方法（称为S$^3$CG、S$^3$MR和S$^3$LQ）。这三种方法均基于斜对称矩阵的Lanczos三角化，分别对应于求解对称线性系统的CG、MINRES和SYMMLQ方法。据我们所知，这是首次研究S$^3$LQ方法的工作。我们对S$^3$CG、S$^3$MR和S$^3$LQ提出了若干新的理论结果，并给出了这三种方法与基于Golub–Kahan双对角化和Saunders–Simon–Yip三对角化方法之间的关系。数值算例验证了我们的理论发现。