In this work, we introduce a generalization of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, named as generalized PMHSS (GPMHSS) iteration method, to solve a class of singular block two-by-two system of linear equations. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution for any initial guess no matter the system is consistent or inconsistent. Besides, with the preconditioner derived from the GPMHSS iteration method, the preconditioned generalized minimal residual (GMRES) method also determines the minimum norm least squares solution of the consistent singular block two-by-two linear systems at breakdown. Numerical experiments are presented to show the effectiveness and the robustness of the GPMHSS iteration method and the corresponding preconditioner.

在这项工作中，我们介绍了预处理的修正的Hermitian和Skew-Hermitian分裂（PMHSS）迭代方法的一般化，称为广义PMHSS（GPMHSS）迭代方法，以解决一类奇异的块二乘二线性方程组。理论分析表明，无论系统是一致还是不一致，对于任何初始猜测，GPMHSS迭代方法都会无条件收敛到最小范数最小二乘解。此外，利用从GPMHSS迭代方法派生的预处理器，预处理的广义最小残差（GMRES）方法还确定了击穿时一致奇异块二乘二线性系统的最小范数最小二乘解。