{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:42:57Z","timestamp":1787330577133,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \"Francesco Severi\"","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009112","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>The Generalized Minimal RESidual (gmres) method is a well-established strategy for iteratively solving a large linear system $A x = b$, where $A\\in\\mathbb{R}^{n\\times n}$ is a nonsymmetric and nonsingular coefficient matrix, and $b\\in\\mathbb{R}^n$. In the analysis of its convergence for $A$ diagonalizable, a much used upper bound for the relative residual norm involves a min-max polynomial problem over the set of eigenvalues of $A$, magnified by the condition number of the eigenvector matrix of $A$. This latter factor may cause a huge overestimation of the residual norm, making the bound nondescriptive in practice. We show that when a large condition number is caused by the almost linear dependence of a few of the eigenvectors, a more descriptive analysis of the method's behavior can be performed, irrespective of the location of the corresponding eigenvalues. The new analysis aims at capturing how the gmres polynomial deals with the ill-conditioning; as a by-product, a new upper bound for the gmres residual norm is obtained. A variety of numerical experiments illustrate our findings.<\/jats:p>","DOI":"10.1137\/17m1141291","type":"journal-article","created":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T13:24:35Z","timestamp":1557408275000},"page":"542-563","source":"Crossref","is-referenced-by-count":4,"title":["A GMRES Convergence Analysis for Localized Invariant Subspace Ill-Conditioning"],"prefix":"10.1137","volume":"40","author":[{"given":"Giulia","family":"Sacchi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0795-5865","authenticated-orcid":true,"given":"Valeria","family":"Simoncini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,5,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510405"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144503433077"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2005.04.027"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01733786"},{"key":"atypb5","unstructured":"F. Chatelin,\n                      Valeurs Propres de Matrices\n                      , Masson, Paris, 1988."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/110832793"},{"key":"atypb7","unstructured":"M. Embree,\n                      How Descriptive are GMRES Convergence Bounds?\n                      , Tech. report 08, Oxford University Computer Laboratory, 1999."},{"key":"atypb8","first-page":"77","author":"Freund R. W.","year":"1992","journal-title":"Basel"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1506(199906)6:4<281::AID-NLA158>3.0.CO;2-B"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894275030"},{"key":"atypb11","first-page":"22","author":"Greenbaum A.","year":"1994","journal-title":"New York"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/0915025"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479898341669"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803424967"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503430746"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"J. Liesen and Z. Strako\u0161,\n                      Krylov Subspace Methods. Principles and Analysis\n                      , Oxford University Press, 2013.","DOI":"10.1093\/acprof:oso\/9780199655410.001.0001"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201490008"},{"key":"atypb18","first-page":"7","author":"MathWorks The","year":"2017","journal-title":"MATLAB"},{"key":"atypb19","unstructured":"Matrix Market,\n                      A Visual Repository of Test Data for Use in Comparative Studies of Algorithms for Numerical Linear Algebra\n                      , Mathematical and Computational Sciences Division, National Institute of Standards and Technology,http:\/\/math.nist.gov\/MatrixMarket, 2007."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/0613049"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"J. Nocedal and S. Wright,\n                      Numerical Optimization\n                      , Springer, New York, 1999.","DOI":"10.1007\/b98874"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680020205"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"atypb24","unstructured":"G. Sacchi,\n                      A New Convergence Model for the GMRES Method\n                      , Master's thesis, Alma Mater Studiorum, Universit\u00e0 di Bologna, 2017,https:\/\/amslaurea.unibo.it\/13501\/1\/sacchi_giulia_tesi_mag.pdf."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/120884328"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022302215539"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1002\/nla.499"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-008-0145-y"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"L. Trefethen and M. Embree,\n                      Spectra and Pseudospectra. The Behavior of Non-Normal Matrices and Operators\n                      , Princeton University Press, Princeton, NJ, 2005.","DOI":"10.1515\/9780691213101"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1141291","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:13:32Z","timestamp":1787328812000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1141291"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1141291"],"URL":"https:\/\/doi.org\/10.1137\/17m1141291","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}