{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T08:57:25Z","timestamp":1768726645441,"version":"3.49.0"},"reference-count":33,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T00:00:00Z","timestamp":1579824000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/http\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"name":"Grant-in-Aid for Scientific Research (C), Japan","award":["15K04768"],"award-info":[{"award-number":["15K04768"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2020,5]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>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\u2010hand\u2010side 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\u2010norm 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\u2010relaxation (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.<\/jats:p>","DOI":"10.1002\/nla.2277","type":"journal-article","created":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T22:46:41Z","timestamp":1579906001000},"update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Right preconditioned MINRES for singular systems<sup>\u2020<\/sup>"],"prefix":"10.1002","volume":"27","author":[{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0003-4939-1893","authenticated-orcid":false,"given":"Kota","family":"Sugihara","sequence":"first","affiliation":[{"name":"National Institute of Informatics Tokyo Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/orcid.org\/0000-0001-9640-586X","authenticated-orcid":false,"given":"Ken","family":"Hayami","sequence":"additional","affiliation":[{"name":"National Institute of Informatics Tokyo Japan"},{"name":"SOKENDAI, The Graduate University for Advanced Studies Tokyo Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ning","family":"Zheng","sequence":"additional","affiliation":[{"name":"National Institute of Informatics Tokyo Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2020,1,24]]},"reference":[{"key":"e_1_2_10_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"e_1_2_10_4_1","unstructured":"HayamiK YinJ\u2010F.On the convergence of Krylov subspace methods for rank\u2010deficient least squares problems. 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In: Proceedings of the Third International Conference on Scientific Computing and Partial Differential Equations (SCPDE08);2008;Hong Kong."},{"key":"e_1_2_10_5_1","unstructured":"HayamiK.Convergence of the conjugate gradient method on singular systems. arXiv preprint arXiv:1809.00793.2018."},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(88)90358-5"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2015.09.056"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02142495"},{"key":"e_1_2_10_9_1","unstructured":"MorikuniK.Symmetric inner\u2010iteration preconditioning for rank\u2010deficient least squares problems. arXiv preprint arXiv:1504.00889.2015."},{"issue":"1","key":"e_1_2_10_10_1","first-page":"62","article-title":"Inner\u2010iteration preconditioning with symmetric splitting matrices for symmetric singular linear systems","volume":"29","author":"Morikuni K","year":"2019","journal-title":"Trans Jpn Soc Ind Appl Math"},{"key":"e_1_2_10_11_1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198528678.001.0001","volume-title":"Finite elements and fast iterative solvers with applications in incompressible fluid dynamics","author":"Elman H","year":"2005"},{"key":"e_1_2_10_12_1","volume-title":"Iterative methods for solving linear systems","author":"Greenbaum A","year":"1987"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511615115"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"e_1_2_10_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894262339"},{"key":"e_1_2_10_17_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.737"},{"key":"e_1_2_10_18_1","doi-asserted-by":"publisher","DOI":"10.1137\/100787921"},{"key":"e_1_2_10_19_1","unstructured":"SugiharaK HayamiK ZhengN.Right preconditioned MINRES using Eisenstat\u2010SSOR for positive semidefinite systems. 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