The celebrated minimum residual method (MINRES), proposed in the seminal paper of Paige and Saunders, has seen great success and wide-spread use in solving linear least-squared problems involving Hermitian matrices, with further extensions to complex symmetric settings. Unless the system is consistent whereby the right-hand side vector lies in the range of the matrix, MINRES is not guaranteed to obtain the pseudo-inverse solution. Variants of MINRES, such as MINRES-QLP, which can achieve such minimum norm solutions, are known to be both computationally expensive and challenging to implement. We propose a novel and remarkably simple lifting strategy that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum norm solution with negligible additional computational costs. We study our lifting strategy in a diverse range of settings encompassing Hermitian and complex symmetric systems as well as those with semi-definite preconditioners. We also provide numerical experiments to support our analysis and showcase the effects of our lifting strategy.

翻译：经典的极小残量法（MINRES）由Paige和Saunders的奠基性论文提出，在求解涉及Hermitian矩阵的线性最小二乘问题中取得了巨大成功并被广泛应用，且已进一步推广至复对称情形。除非系统是相容的（即右端向量位于矩阵的值域内），否则MINRES无法保证获得伪逆解。已知能够实现此类最小范数解的MINRES变体（如MINRES-QLP）在计算上既昂贵又难以实现。我们提出了一种新颖且极其简单的提升策略，该策略能与MINRES的最终迭代无缝集成，从而以可忽略的额外计算代价获得最小范数解。我们在涵盖Hermitian系统、复对称系统以及带有半定预条件子的系统等多种情形下研究了该提升策略，并提供了数值实验以支持我们的分析并展示该策略的效果。