{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T05:21:17Z","timestamp":1787376077598,"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>For positive definite and semidefinite consistent $Ax_\\star=b$, we use the Gauss--Radau approach of Golub and Meurant (1997) to obtain an upper bound on the error $\\|x_\\star-x_k^L\\|_2$ for SYMMLQ iterates, assuming exact arithmetic. Such a bound, computable in constant time per iteration, was not previously available. We show that the CG error $\\|x_\\star-x_k^C\\|_2$ is always smaller and can also be bounded in constant time per iteration. Our approach is computationally cheaper than other bounds or estimates of the CG error in the literature. As with other approaches using Gauss--Radau quadrature, we require a positive lower bound on the smallest nonzero eigenvalue of $A$. For indefinite $A$, we obtain an estimate of $\\|x_\\star-x_k^L\\|_2$. Numerical experiments demonstrate that our bounds are remarkably tight for SYMMLQ on positive definite systems and therefore provide reliable bounds for CG.<\/jats:p>","DOI":"10.1137\/16m1094816","type":"journal-article","created":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T13:07:56Z","timestamp":1550149676000},"page":"235-253","source":"Crossref","is-referenced-by-count":7,"title":["Euclidean-Norm Error Bounds for SYMMLQ and CG"],"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","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\/120866543"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385494"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597328510"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(72)90264-8"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"G. Dahlquist, G. H. Golub, and S. G. Nash (1979),\n                      Bounds for the error in linear systems\n                      , in Semi-Infinite Programming, Lecture Notes in Control and Inform. Sci. 15, Springer, Berlin, New York, pp. 154-172.","DOI":"10.1007\/BFb0003890"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/2049662.2049663"},{"key":"atypb7","unstructured":"R. Estrin, D. Orban, and M. A. Saunders (2016),\n                      LSLQ: An Iterative Method for Linear Least-Squares Problems with a Forward Error Minimization Property\n                      , Cahier du GERAD G-2017-05, GERAD, Montr\u00e9al, QC, Canada."},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"B. Fischer (1996),\n                      Polynomial Based Iteration Methods for Symmetric Linear Systems\n                      , Wiley-Teubner Ser. Adv. Numer. Math., John Wiley & Sons, Ltd., Chichester, UK, B. G. Teubner, Stuttgart, Germany,https:\/\/doi.org\/10.1007\/978-3-663-11108-5.","DOI":"10.1007\/978-3-663-11108-5"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.24200\/squjs.vol17iss1pp44-62"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/110859749"},{"key":"atypb11","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 Sci. Tech., Harlow, UK, pp. 105-156."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510247"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"G. H. Golub and G. Meurant (2009),\n                      Matrices, Moments and Quadrature with Applications\n                      , Princeton University Press, Princeton, NJ.","DOI":"10.1515\/9781400833887"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02142693"},{"key":"atypb15","unstructured":"G. H. Golub and C. F. Van Loan (2013),\n                      Matrix Computations\n                      , 4th ed., Johns Hopkins University Press, Baltimore, MD."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019178811767"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-005-1528-0"},{"key":"atypb20","first-page":"9780898718140","volume":"1137","author":"Meurant G.","year":"2006","journal-title":"Software Environ. Tools 19, SIAM, Philadelphia, https:\/\/doi.org\/10."},{"key":"atypb21","unstructured":"G. Meurant and P. Tich\u00fd (2015),\n                      On the Numerical Behavior of Quadrature Based Bounds for the A-Norm of the Error in CG\n                      ,http:\/\/www.cs.cas.cz\/~tichy\/download\/present\/2015ALA.pdf."},{"key":"atypb22","first-page":"9781611974737","volume":"1137","author":"Orban D.","year":"2017","journal-title":"SIAM Spotlights 3, SIAM, Philadelphia, https:\/\/doi.org\/10."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01739829"},{"key":"atypb25","unstructured":"M. A. Saunders (2016),\n                      CME 338 class notes 4: Iterative methods for symmmetric $Ax=b$\n                      ,http:\/\/stanford.edu\/class\/msande318\/notes.html."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/0720042"},{"key":"atypb27","first-page":"56","volume":"13","author":"Strako\u0161 Z.","year":"2002","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb28","first-page":"51","volume":"1","author":"Szyld D. B.","year":"1993","journal-title":"East-West J. Numer. Math."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1094816","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:40Z","timestamp":1787327260000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1094816"}},"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\/16M1094816"],"URL":"https:\/\/doi.org\/10.1137\/16m1094816","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}