{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:29:57Z","timestamp":1787336997562,"version":"build-2736575974"},"reference-count":51,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>\n                    Iterative methods are developed and studied for near-singular linear systems Cx = b. Our approach, called the transformed minimal residual algorithm (TMRES), is derived from any convergent iterative scheme Sx\n                    <jats:sub>k+1<\/jats:sub>\n                    = Tx\n                    <jats:sub>k<\/jats:sub>\n                    + b associated with a splitting C = S - T. In each step of TMRES, the transformed residual S\n                    <jats:sup>-1<\/jats:sup>\n                    (b- Cx) is minimized over a Krylov space generated by S\n                    <jats:sup>-1<\/jats:sup>\n                    T . The original iterative scheme typically converges slowly when C is nearly singular, while a Krylov space generated by S\n                    <jats:sup>-1<\/jats:sup>\n                    T often contains a much better approximation to a solution. TMRES is algebraically equivalent to the generalized minimal residual algorithm (GMRES) preconditioned by S\n                    <jats:sup>-1<\/jats:sup>\n                    , although there are numerical differences since a different matrix S\n                    <jats:sup>-1<\/jats:sup>\n                    C is used to generate the Krylov space in preconditioned GMRES. Special attention is given to sparsity and convergence issues related to linear systems of the form $({\\bf AA}\\tr + \\sigma{\\bf I}){\\bf x} =$ ${\\bf b}$, where $\\sigma \\ge 0$.\n                  <\/jats:p>","DOI":"10.1137\/s106482759834634x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"747-766","source":"Crossref","is-referenced-by-count":25,"title":["Iterative Methods for Nearly Singular Linear Systems"],"prefix":"10.1137","volume":"22","author":[{"given":"William W.","family":"Hager","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":", Linear and nonlinear conjugate gradient\u2010related methods, Proceedings of the AMS\u2010IMS\u2010SIAM Summer Research Conference held at the University of Washington, Seattle, WA, July 9\u201313, 1995, Society for Industrial and Applied Mathematics (SIAM), 1996, 0\u20130, xvi+16497k:90004"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/1022003"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42792"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596305258"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01930845"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894262339"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0724076"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01733786"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"M. 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