{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:12:02Z","timestamp":1787382722950,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000057","name":"National Institute of General Medical Sciences","doi-asserted-by":"publisher","award":["U01GM102098"],"award-info":[{"award-number":["U01GM102098"]}],"id":[{"id":"10.13039\/100000057","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We propose an iterative method named LSLQ for solving linear least-squares problems of any shape. The method is based on the Golub and Kahan (1965) process, where the dominant cost consists in products with the linear operator and its transpose. In the rank-deficient case, LSLQ identifies the minimum-length least-squares solution. LSLQ is formally equivalent to SYMMLQ applied to the normal equations, so that the current estimate's Euclidean norm increases monotonically, while the associated error norm decreases monotonically. We provide lower and upper bounds on the error in the Euclidean norm along the LSLQ iterations. The upper bound translates to an upper bound on the error norm along the LSQR iterations, which was previously unavailable, and provides an error-based stopping criterion involving a transition to the LSQR point. We report numerical experiments on standard test problems and on a full-wave inversion problem arising from geophysics in which an approximate least-squares solution corresponds to an approximate gradient of a relevant penalty function that is to be minimized.<\/jats:p>","DOI":"10.1137\/17m1113552","type":"journal-article","created":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T13:07:56Z","timestamp":1550149676000},"page":"254-275","source":"Crossref","is-referenced-by-count":11,"title":["LSLQ: An Iterative Method for Linear Least-Squares with an Error Minimization Property"],"prefix":"10.1137","volume":"40","author":[{"given":"Ron","family":"Estrin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dominique","family":"Orban","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3800-4982","authenticated-orcid":true,"given":"Michael A.","family":"Saunders","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,2,14]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1088570"},{"key":"atypb2","first-page":"9780898719857","volume":"1137","author":"Conn A. 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Saunders (2016),\n                      Euclidean-norm error bounds for CG via SYMMLQ\n                      , Cahier du GERAD G-2016-70, GERAD, Montr\u00e9al, QC, Canada."},{"key":"atypb7","unstructured":"D. C.L. Fong (2011),\n                      Minimum-Residual Methods for Sparse Least-Squares Using Golub-Kahan Bidiagonalization\n                      , Ph.D. thesis, Stanford University, Stanford, CA."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.24200\/squjs.vol17iss1pp44-62"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/0702016"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510247"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"M. Hegland (1990),\n                      On the computation of breeding values\n                      , in CONPAR 90-VAPP IV, Joint International Conference on Vector and Parallel Processing, Lecture Notes in Comput. Sci. 457, Springer, Berlin, Heidelberg, pp. 232-242,https:\/\/doi.org\/10.1007\/3-540-53065-7_103.","DOI":"10.1007\/3-540-53065-7_103"},{"key":"atypb13","unstructured":"M. Hegland (1993),\n                      Description and Use of Animal Breeding Data for Large Least Squares Problems\n                      , Technical Report TR\/PA\/93\/50, CERFACS, Toulouse, France."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-005-1528-0"},{"key":"atypb17","unstructured":"D. Orban (2016),\n                      Optimizers\/Animal: Initial Release\n                      ,https:\/\/github.com\/optimizers\/animal."},{"key":"atypb18","unstructured":"D. Orban (2017),\n                      Krylov.jl: A Julia Basket of Hand-Picked Krylov Methods\n                      ,https:\/\/github.com\/JuliaSmoothOptimizers\/Krylov.jl."},{"key":"atypb19","first-page":"9781611974737","volume":"1137","author":"Orban D.","year":"2017","journal-title":"SIAM Spotlights 3, SIAM, Philadelphia, https:\/\/doi.org\/10."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1145\/355993.356000"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01739829"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/0725052"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597319519"},{"key":"atypb26","first-page":"56","volume":"13","author":"Strako\u0161 Z.","year":"2002","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"T. van Leeuwen and F. J. Herrmann (2016),\n                      A penalty method for PDE-constrained optimization in inverse problems\n                      , Inverse Problems, 32, 015007,https:\/\/doi.org\/10.1088\/0266-5611\/32\/1\/015007.","DOI":"10.1088\/0266-5611\/32\/1\/015007"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/0805005"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1113552","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:42Z","timestamp":1787327262000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1113552"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1113552"],"URL":"https:\/\/doi.org\/10.1137\/17m1113552","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}