{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T14:35:04Z","timestamp":1777300504928,"version":"3.51.4"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2019,12,12]],"date-time":"2019-12-12T00:00:00Z","timestamp":1576108800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,12,12]],"date-time":"2019-12-12T00:00:00Z","timestamp":1576108800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004240","name":"Akademie V\u011bd \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["RVO: 67985807"],"award-info":[{"award-number":["RVO: 67985807"]}],"id":[{"id":"10.13039\/501100004240","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,8]]},"DOI":"10.1007\/s11075-019-00846-z","type":"journal-article","created":{"date-parts":[[2019,12,12]],"date-time":"2019-12-12T17:04:57Z","timestamp":1576170297000},"page":"1329-1352","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the residual norms, the Ritz values and the harmonic Ritz values that can be generated by restarted GMRES"],"prefix":"10.1007","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0553-854X","authenticated-orcid":false,"given":"Jurjen","family":"Duintjer Tebbens","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e9rard","family":"Meurant","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,12,12]]},"reference":[{"issue":"4","key":"846_CR1","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1007\/BF02510405","volume":"38","author":"M Arioli","year":"1998","unstructured":"Arioli, M., Pt\u00e1k, V., Strako\u0161, Z.: Krylov sequences of maximal length and convergence of GMRES. BIT 38(4), 636\u2013643 (1998)","journal-title":"BIT"},{"issue":"1","key":"846_CR2","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1137\/0912003","volume":"12","author":"PN Brown","year":"1991","unstructured":"Brown, P.N.: A theoretical comparison of the Arnoldi and GMRES algorithms. SIAM J. Sci. Statist. Comput. 12(1), 58\u201378 (1991)","journal-title":"SIAM J. Sci. Statist. Comput."},{"issue":"1","key":"846_CR3","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1002\/(SICI)1099-1506(199701\/02)4:1<43::AID-NLA99>3.0.CO;2-Z","volume":"4","author":"A Chapman","year":"1997","unstructured":"Chapman, A., Saad, Y.: Deflated and augmented Krylov subspace techniques. Numer. Linear Algebra Appl. 4(1), 43\u201366 (1997)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"846_CR4","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1137\/S0036142997315950","volume":"36","author":"E De Sturler","year":"1999","unstructured":"De Sturler, E.: Truncation strategies for optimal Krylov subspace methods. SIAM J. Numer. Anal. 36(3), 864\u2013889 (1999)","journal-title":"SIAM J. Numer. Anal."},{"issue":"SI","key":"846_CR5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1553\/etna_vol47s37","volume":"47","author":"K Du","year":"2017","unstructured":"Du, K., Duintjer Tebbens, J., Meurant, G.: Any admissible harmonic Ritz value set is possible for GMRES. Electron. Trans. Numer. Anal. 47(SI), 37\u201356 (2017)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"3","key":"846_CR6","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1137\/110843666","volume":"33","author":"J Duintjer Tebbens","year":"2012","unstructured":"Duintjer Tebbens, J., Meurant, G.: Any Ritz value behavior is possible for Arnoldi and for GMRES. SIAM J. Matrix Anal. Appl. 33(3), 958\u2013978 (2012)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"846_CR7","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s11075-013-9695-x","volume":"65","author":"J Duintjer Tebbens","year":"2014","unstructured":"Duintjer Tebbens, J., Meurant, G.: Prescribing the behavior of early terminating GMRES and Arnoldi iterations. Num. Algor. 65(1), 69\u201390 (2014)","journal-title":"Num. Algor."},{"issue":"1","key":"846_CR8","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10543-015-0564-y","volume":"56","author":"J Duintjer Tebbens","year":"2016","unstructured":"Duintjer Tebbens, J., Meurant, G.: On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems. BIT 56(1), 77\u201397 (2016)","journal-title":"BIT"},{"key":"846_CR9","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.laa.2014.02.044","volume":"450","author":"J Duintjer Tebbens","year":"2014","unstructured":"Duintjer Tebbens, J., Meurant, G., Sadok, H., Strako\u0161, Z.: On investigating GMRES convergence using unitary matrices. Lin. Alg. Appl. 450, 83\u2013107 (2014)","journal-title":"Lin. Alg. Appl."},{"key":"846_CR10","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1017\/S0962492901000046","volume":"10","author":"M Eiermann","year":"2001","unstructured":"Eiermann, M., Ernst, O.G.: Geometric aspects of the theory of Krylov subspace methods. Acta Numer. 10, 251\u2013312 (2001)","journal-title":"Acta Numer."},{"issue":"1\u20132","key":"846_CR11","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/S0377-0427(00)00398-8","volume":"123","author":"M Eiermann","year":"2000","unstructured":"Eiermann, M., Ernst, O.G., Schneider, O.: Analysis of acceleration strategies for restarted minimal residual methods. J. Comput. Appl. Math. 123(1\u20132), 261\u2013292 (2000)","journal-title":"J. Comput. Appl. Math."},{"key":"846_CR12","volume-title":"Iterative Methods for Large Sparse Nonsymmetric Systems of Linear Equations. PhD thesis. Department of Computer Science","author":"H Elman","year":"1982","unstructured":"Elman, H.: Iterative Methods for Large Sparse Nonsymmetric Systems of Linear Equations. PhD thesis. Department of Computer Science. Yale University, New Haven (1982)"},{"issue":"2","key":"846_CR13","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1137\/S003614450139961","volume":"45","author":"M Embree","year":"2003","unstructured":"Embree, M.: The tortoise and the hare restart GMRES. SIAM Rev. 45(2), 259\u2013266 (2003)","journal-title":"SIAM Rev."},{"issue":"3","key":"846_CR14","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1137\/060678087","volume":"46","author":"V Faber","year":"2008","unstructured":"Faber, V., Liesen, J., Tich\u00fd, P.: The Faber-Manteuffel theorem for linear operators. SIAM J. Numer. Anal. 46(3), 1323\u20131337 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"846_CR15","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1137\/0721026","volume":"21","author":"V Faber","year":"1984","unstructured":"Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21(2), 352\u2013362 (1984)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"846_CR16","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF01385726","volume":"60","author":"RW Freund","year":"1991","unstructured":"Freund, R.W., Nachtigal, N.M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60(3), 315\u2013339 (1991)","journal-title":"Numer. Math."},{"issue":"2","key":"846_CR17","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1137\/110820713","volume":"34","author":"A Gaul","year":"2013","unstructured":"Gaul, A., Gutknecht, M.H., Liesen, J., Nabben, R.: A framework for deflated and augmented Krylov subspace methods. SIAM J. Matrix Anal. Appl. 34 (2), 495\u2013518 (2013)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"846_CR18","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1137\/S0895479894275030","volume":"17","author":"A Greenbaum","year":"1996","unstructured":"Greenbaum, A., Pt\u00e1k, V., Strako\u0161, Z.: Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl. 17(3), 465\u2013469 (1996)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"846_CR19","doi-asserted-by":"crossref","unstructured":"Greenbaum, A., Strako\u0161, Z.: Matrices that generate the same Krylov residual spaces. In: Recent Advances in Iterative Methods, IMA Vol. Math. Appl., vol. 60, pp 95\u2013118. Springer, New York (1994)","DOI":"10.1007\/978-1-4613-9353-5_7"},{"issue":"6","key":"846_CR20","doi-asserted-by":"crossref","first-page":"1989","DOI":"10.1137\/S1064827503430746","volume":"26","author":"J Liesen","year":"2005","unstructured":"Liesen, J., Strako\u0161, Z.: GMRES convergence analysis for a convection-diffusion model problem. SIAM J. Sci. Comput. 26(6), 1989\u20132009 (2005). (electronic)","journal-title":"SIAM J. Sci. Comput."},{"key":"846_CR21","volume-title":"Krylov Subspace Methods, Principles and Analysis. Numerical Mathematics and Scientific Computation","author":"J Liesen","year":"2013","unstructured":"Liesen, J., Strako\u0161, Z.: Krylov Subspace Methods, Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)"},{"key":"846_CR22","unstructured":"Meurant, G.: Necessary and sufficient conditions for GMRES complete and partial stagnation. Submitted"},{"key":"846_CR23","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s11075-014-9891-3","volume":"68","author":"G Meurant","year":"2015","unstructured":"Meurant, G., Duintjer Tebbens, J.: The role eigenvalues play in forming GMRES residual norms with non-normal matrices. Numer. Algor. 68, 143\u2013165 (2015)","journal-title":"Numer. Algor."},{"issue":"4","key":"846_CR24","doi-asserted-by":"crossref","first-page":"1154","DOI":"10.1137\/S0895479893253975","volume":"16","author":"RB Morgan","year":"1995","unstructured":"Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16(4), 1154\u20131171 (1995)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"4","key":"846_CR25","doi-asserted-by":"crossref","first-page":"1112","DOI":"10.1137\/S0895479897321362","volume":"21","author":"RB Morgan","year":"2000","unstructured":"Morgan, R.B.: Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations. SIAM J. Matrix Anal. Appl. 21(4), 1112\u20131135 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"846_CR26","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1137\/S1064827599364659","volume":"24","author":"RB Morgan","year":"2002","unstructured":"Morgan, R.B.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24(1), 20\u201337 (2002)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"846_CR27","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1137\/0613050","volume":"13","author":"NM Nachtigal","year":"1992","unstructured":"Nachtigal, N.M., Reichel, L., Trefethen, L.N.: A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM J. Matrix Anal. Appl. 13(3), 796\u2013825 (1992)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"7","key":"846_CR28","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1016\/j.laa.2007.10.040","volume":"428","author":"B Parlett","year":"2008","unstructured":"Parlett, B., Strang, G.: Matrices with prescribed Ritz values. Linear Algebra Appl. 428(7), 1725\u20131739 (2008)","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"846_CR29","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1137\/S0895479895294289","volume":"18","author":"Y Saad","year":"1997","unstructured":"Saad, Y.: Analysis of augmented Krylov subspace methods. SIAM J. Matrix Anal. Appl. 18(2), 435\u2013449 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"846_CR30","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","volume":"7","author":"Y Saad","year":"1986","unstructured":"Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856\u2013869 (1986)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"846_CR31","first-page":"133","volume":"45","author":"M Schweitzer","year":"2016","unstructured":"Schweitzer, M.: Any finite convergence curve is possible in the initial iterations of restarted FOM. Electron. Trans. Numer. Anal. 45, 133\u2013145 (2016)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"2","key":"846_CR32","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1137\/S0895479898348507","volume":"22","author":"V Simoncini","year":"2000","unstructured":"Simoncini, V.: On the convergence of restarted Krylov subspace methods. SIAM J. Matrix Anal. Appl. 22(2), 430\u2013452 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"846_CR33","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s00211-008-0145-y","volume":"109","author":"V Simoncini","year":"2008","unstructured":"Simoncini, V., Szyld, D.B.: New conditions for non-stagnation of minimal residual methods. Numer. Math. 109(3), 477\u2013487 (2008)","journal-title":"Numer. Math."},{"key":"846_CR34","unstructured":"de Surler, E.: Personal communication (2013)"},{"issue":"1","key":"846_CR35","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1137\/080727403","volume":"32","author":"E Vecharynski","year":"2010","unstructured":"Vecharynski, E., Langou, J.: The cycle-convergence of restarted GMRES for normal matrices is sublinear. SIAM J. Sci. Comput. 32(1), 186\u2013196 (2010)","journal-title":"SIAM J. Sci. Comput."},{"key":"846_CR36","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1002\/nla.739","volume":"18","author":"E Vecharynski","year":"2011","unstructured":"Vecharynski, E., Langou, J.: Any admissible cycle-convergence behavior is possible for restarted GMRES at its initial cycles. Num. Lin. Algebr. Appl. 18, 499\u2013511 (2011)","journal-title":"Num. Lin. Algebr. Appl."},{"key":"846_CR37","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/S0024-3795(02)00612-2","volume":"367","author":"I Zavorin","year":"2003","unstructured":"Zavorin, I., O\u2019Leary, D.P., Elman, H.: Complete stagnation of GMRES. Linear Algebra Appl. 367, 165\u2013183 (2003)","journal-title":"Linear Algebra Appl."},{"issue":"6","key":"846_CR38","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1002\/nla.589","volume":"15","author":"B Zhong","year":"2008","unstructured":"Zhong, B., Morgan, R.B.: Complementary cycles of restarted GMRES. Numer. Linear Algebra Appl. 15(6), 559\u2013571 (2008)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"846_CR39","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1002\/(SICI)1099-1506(200004\/05)7:3<117::AID-NLA189>3.0.CO;2-Z","volume":"7","author":"J Z\u00edtko","year":"2000","unstructured":"Z\u00edtko, J.: Generalization of convergence conditions for a restarted GMRES. Numer. Linear Algebra Appl. 7(3), 117\u2013131 (2000)","journal-title":"Numer. Linear Algebra Appl."},{"key":"846_CR40","first-page":"221","volume":"31","author":"J Z\u00edtko","year":"2008","unstructured":"Z\u00edtko, J.: Some remarks on the restarted and augmented GMRES method. Electron. Trans. Numer. Anal. 31, 221\u2013227 (2008)","journal-title":"Electron. Trans. Numer. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00846-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00846-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00846-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,11]],"date-time":"2020-12-11T01:06:22Z","timestamp":1607648782000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00846-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,12]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["846"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00846-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,12]]},"assertion":[{"value":"24 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}