{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T17:36:01Z","timestamp":1776533761842,"version":"3.51.2"},"reference-count":18,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T00:00:00Z","timestamp":1678838400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["463329614"],"award-info":[{"award-number":["463329614"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008982","name":"National Science Foundation of Sri Lanka","doi-asserted-by":"publisher","award":["DMS\u20101913364"],"award-info":[{"award-number":["DMS\u20101913364"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Gradient\u2010type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD\u2010id) is one of such methods. The convergence behavior of the PSD\u2010id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non\u2010asymptotic estimates indicate a superlinear convergence of the PSD\u2010id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD\u2010id using a restricted formulation of the PSD\u2010id. More importantly, we extend the new convergence analysis of the PSD\u2010id to a practically preferred block version of the PSD\u2010id, or BPSD\u2010id, and show the cluster robustness of the BPSD\u2010id. Numerical examples are provided to validate the theoretical estimates.<\/jats:p>","DOI":"10.1002\/nla.2498","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T09:51:02Z","timestamp":1678873862000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation"],"prefix":"10.1002","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7096-1649","authenticated-orcid":false,"given":"Ming","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Mathematics University of Rostock  Rostock Mecklenburg\u2010Vorpommern Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaojun","family":"Bai","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Department of Mathematics University of California, Davis  Davis California USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunfeng","family":"Cai","sequence":"additional","affiliation":[{"name":"Cognitive Computing Lab Baidu Research  Beijing China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Klaus","family":"Neymeyr","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of Rostock  Rostock Mecklenburg\u2010Vorpommern Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2023,3,15]]},"reference":[{"key":"e_1_2_10_2_1","first-page":"105","article-title":"The steepest descent method for an eigenvalue problem with semi\u2010bounded operators","volume":"5","author":"Samokish BA","year":"1958","journal-title":"Izv Vyssh Uchebn Zaved Mat"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02127702"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2005.06.039"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.1703-m2016-0580"},{"key":"e_1_2_10_6_1","first-page":"38","article-title":"Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method","volume":"15","author":"Knyazev AV","year":"2003","journal-title":"Electron Trans Numer Anal."},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90379-B"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/100784928"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/11084488X"},{"key":"e_1_2_10_10_1","volume-title":"Matrix perturbation theory","author":"Stewart GW","year":"1990"},{"key":"e_1_2_10_11_1","first-page":"93","article-title":"The block preconditioned steepest descent iteration for elliptic operator eigenvalue problems","volume":"41","author":"Neymeyr K","year":"2014","journal-title":"Electron Trans Numer Anal."},{"key":"e_1_2_10_12_1","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3446"},{"key":"e_1_2_10_13_1","volume-title":"Numerical methods for large eigenvalue problems","author":"Saad Y","year":"1992"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801399596"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/040620643"},{"key":"e_1_2_10_16_1","volume-title":"The symmetric eigenvalue problem","author":"Parlett BN","year":"1980"},{"key":"e_1_2_10_17_1","doi-asserted-by":"publisher","DOI":"10.1515\/rnam.1987.2.5.371"},{"key":"e_1_2_10_18_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.246"},{"key":"e_1_2_10_19_1","unstructured":"ZhouM NeymeyrK.Convergence rates of individual Ritz values in block preconditioned gradient\u2010type eigensolvers. Technical report. 2022. Available from.https:\/\/arxiv.org\/abs\/2206.00585"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.2498","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/nla.2498","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.2498","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T03:01:17Z","timestamp":1694401277000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.2498"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,15]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["10.1002\/nla.2498"],"URL":"https:\/\/doi.org\/10.1002\/nla.2498","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,15]]},"assertion":[{"value":"2022-09-08","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-02-26","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2023-03-15","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2498"}}