{"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":1787382722759,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","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 describe LNLQ for solving the least-norm problem min $\\|x\\|$ subject to $Ax=b$, using the Golub--Kahan bidiagonalization of $[b\\ A]$. Craig's method is known to be equivalent to applying the conjugate gradient method to the normal equations of the second kind ($AA^T y = b$, $x = A^T y$); LNLQ is equivalent to applying SYMMLQ. If an underestimate of the smallest singular value is available, error upper bounds for both $x$ and $y$ are available cheaply at each iteration. LNLQ is a companion method to the least-squares solver LSLQ [R. Estrin, D. Orban, and M. A. Saunders, SIAM J. Matrix Anal. Appl., 40 (2019b), pp. 235--253] which is equivalent to SYMMLQ on the conventional normal equations. We show that the error bounds are tight and comparable to the bounds suggested by Arioli [ SIAM J. Matrix Anal. Appl., 34 (2013), pp. 571--592] for CRAIG. A sliding window technique allows us to tighten the error bound for $y$ at the expense of a few additional scalar operations per iteration. We illustrate the tightness of the error bounds on two standard test problems and on the computation of an inexact gradient in the context of a penalty method for PDE-constrained optimization.<\/jats:p>","DOI":"10.1137\/18m1194948","type":"journal-article","created":{"date-parts":[[2019,9,12]],"date-time":"2019-09-12T14:16:51Z","timestamp":1568297811000},"page":"1102-1124","source":"Crossref","is-referenced-by-count":9,"title":["LNLQ: An Iterative Method for Least-Norm Problems 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":[[2019,9,12]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/120866543"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/sapm195534164"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1145\/2049662.2049663"},{"key":"atypb4","unstructured":"R. Estrin, M. P. Friedlander, D. Orban, and M. A. Saunders (2018),\n                      Implementing a smooth exact penalty function for nonlinear optimization\n                      , Cahier du GERAD, in preparation."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/16M1094816"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/17M1113552"},{"key":"atypb7","unstructured":"R. Fletcher,\n                      A class of methods for nonlinear programming: III. Rates of convergence\n                      (1973), in Numerical Methods for Nonlinear Optimization, F. A. Lootsma, ed., Academic Press, New York."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.24200\/squjs.vol17iss1pp44-62"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0702016"},{"key":"atypb10","unstructured":"G. H. Golub and G. Meurant (1994),\n                      Matrices, moments and quadrature\n                      , in Numerical Analysis 1993 (Dundee, 1993), Pitman Res. Notes Math. Ser. 303, Longman, Harlow, UK, pp. 105-156."},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"G. H. Golub and G. Meurant (2010),\n                      Matrices, Moments and Quadrature with Applications\n                      , Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ.","DOI":"10.1515\/9781400833887"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02142693"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/0709016"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"atypb16","unstructured":"G. Meurant and P. Tich\u00fd (2014),\n                      A New Algorithm for Computing Quadrature-Based Bounds in Conjugate Gradients\n                      ,http:\/\/www.cs.cas.cz\/tichy\/download\/present\/2014Spa.pdf."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"G. Meurant and P. Tich\u00fd (2018),\n                      Approximating the extreme Ritz values and upper bounds for the A-norm of the error in CG\n                      , Numer. Algorithms,http:\/\/doi.org\/10.1007\/s11075-018-0634-8.","DOI":"10.1007\/s11075-018-0634-8"},{"key":"atypb18","first-page":"9781611974737","volume":"1137","author":"Orban D.","year":"2017","journal-title":"SIAM Spotlights 3, SIAM, Philadelphia, https:\/\/doi.org\/10."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/0711019"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/355993.356000"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"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","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\/18M1194948","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:36:22Z","timestamp":1787326582000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1194948"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1194948"],"URL":"https:\/\/doi.org\/10.1137\/18m1194948","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}