Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Ovtchinnikov Evgueni E and Xanthis Leonidas S 2001Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimatesProc. R. Soc. Lond. A.457441–451http://doi.org/10.1098/rspa.2000.0674SectionRestricted accessResearch articleSuccessive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates Evgueni E Ovtchinnikov Evgueni E Ovtchinnikov Centre for Techno-Mathematics and Scientific Computing Laboratory, University of Westminster, London HA1 3TP, UK (; ) Google Scholar Find this author on PubMed Search for more papers by this author and Leonidas S Xanthis Leonidas S Xanthis Centre for Techno-Mathematics and Scientific Computing Laboratory, University of Westminster, London HA1 3TP, UK (; ) Google Scholar Find this author on PubMed Search for more papers by this author Evgueni E Ovtchinnikov Evgueni E Ovtchinnikov Centre for Techno-Mathematics and Scientific Computing Laboratory, University of Westminster, London HA1 3TP, UK (; ) Google Scholar Find this author on PubMed Search for more papers by this author and Leonidas S Xanthis Leonidas S Xanthis Centre for Techno-Mathematics and Scientific Computing Laboratory, University of Westminster, London HA1 3TP, UK (; ) Google Scholar Find this author on PubMed Search for more papers by this author Published:08 February 2001https://doi.org/10.1098/rspa.2000.0674AbstractWe present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem Au =λBu for symmetric positive definite operators A and B. We call this method successive eigenvalue relaxation, or the SER method (homoechon of the classical successive over‐relaxation, or SOR method for linear systems). In particular, there are two significant features of SER which render it computationally attractive: (i) it can effectively deal with preconditioned large‐scale eigenvalue problems, and (ii) its practical implementation does not require any information about the preconditioner used: it can routinely accommodate sophisticated preconditioners designed to meet more exacting requirements (e.g. three‐dimensional elasticity problems with small thickness parameters). We endow SER with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a discretized partial diwfferential equation exhibiting clusters of eigenvalues. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Knyazev A and Neymeyr K (2003) A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra and its Applications, 10.1016/S0024-3795(01)00461-X, 358:1-3, (95-114), Online publication date: 1-Jan-2003. Hiptmair R and Neymeyr K (2002) Multilevel Method for Mixed Eigenproblems, SIAM Journal on Scientific Computing, 10.1137/S1064827501385001, 23:6, (2141-2164), Online publication date: 1-Jan-2002. Knyazev A (2001) Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method, SIAM Journal on Scientific Computing, 10.1137/S1064827500366124, 23:2, (517-541), Online publication date: 1-Jan-2001. This Issue08 February 2001Volume 457Issue 2006 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0674Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/02/2001Published in print08/02/2001 License: Citations and impact Keywordsmultiple and clustered eigenvalueslarge-scale eigenvalue problemsconvergence ratesubspace iterationeigensolvers with preconditioning