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Matrix Anal. Appl."],"published-print":{"date-parts":[[2025,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The celebrated minimum residual method (MINRES), proposed in the seminal paper of Paige and Saunders [ SIAM J. Numer. Anal., 12 (1975), pp. 617\u201362951], has seen great success and widespread use in solving Hermitian (and complex-symmetric) systems [Formula: see text]. 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\u2019s 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.<\/jats:p>","DOI":"10.1137\/24m1638422","type":"journal-article","created":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T08:18:28Z","timestamp":1754554708000},"page":"1887-1916","source":"Crossref","is-referenced-by-count":2,"title":["Obtaining Pseudoinverse Solutions with MINRES"],"prefix":"10.1137","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2381-2111","authenticated-orcid":true,"given":"Yang","family":"Liu","sequence":"first","affiliation":[{"name":"Mathematical Institute, University of Oxford, Oxford, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6784-5417","authenticated-orcid":true,"given":"Andre","family":"Milzarek","sequence":"additional","affiliation":[{"name":"School of Data Science, The Chinese University of Hong Kong, Shenzhen, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6920-7072","authenticated-orcid":true,"given":"Fred","family":"Roosta","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia, and International Computer Science Institute, Berkeley, CA 94704 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,8,7]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1016\/B978-012387582-2\/50038-1"},{"key":"ref2","first-page":"26","volume":"9","author":"Ammar G.","year":"1999","journal-title":"Electron. 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