Obtaining Pseudoinverse Solutions with MINRES

.The celebrated minimum residual method (MINRES), proposed in the seminal paper of Paige and Saunders [SIAM J. Numer. Anal., 12 (1975), pp. 617–62951], has seen great success and widespread use in solving Hermitian (and complex-symmetric) systems \(A x = b\). Unless the system is consistent, MINRES is not guaranteed to obtain the pseudoinverse solution. We propose a novel and remarkably simple minimum-norm (MN) refinement that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum-norm solution with negligible additional computational cost. We extend our MN refinement to complex-symmetric systems, building on S.-C. Choi's extension of MINRES for solving these systems. Given the flexibility of MINRES to accommodate singular preconditioners, we further investigate the MN refinement in preconditioned settings that involve singular preconditioners. We also provide numerical experiments to support our analysis and showcase the effects of our MN refinement.