We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.

中文翻译：

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正确的MINRES预处理系统，适用于单个系统†

我们考虑解决大型稀疏对称奇异线性系统。我们首先介绍了一种右预条件最小残差算法（MINRES），并证明了该算法迭代收敛到预条件加权最小二乘解，即使线性系统是奇异且不一致。对于系统一致的特殊情况，我们证明，如果初始向量在正确的预处理系数矩阵的范围空间内，则迭代会收敛到针对预处理器的最小范数解。此外，我们提出了使用Eisenstat技巧的对称连续过度松弛（SSOR）进行正确预处理的MINRES。在电磁分析中的半定系统上的一些数值实验表明，该方法是有效且鲁棒的。最后，我们表明可以通过重新启动迭代来进一步减少残差范数。