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

By Evgueni Ovtchinnikov and Leonidas Xanthis

Abstract

We present a new subspace iteration method for the efficient computation of several smallest eigenvalues of the generalized eigenvalue problem Au = lambda Bu 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,\ud 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\ud preconditioned large-scale eigenvalue problems, and (ii) its practical implementation does not require any information about the preconditioner used: it can routinely\ud accommodate sophisticated preconditioners designed to meet more exacting requirements (e.g. three-dimensional elasticity problems with small thickness parameters).\ud We endow SER with theoretical convergence estimates allowing for multiple and clusters of eigenvalues and illustrate their usefulness in a numerical example for a\ud discretized partial differential equation exhibiting clusters of eigenvalues.\u

Topics: UOW3
OAI identifier: oai:westminsterresearch.wmin.ac.uk:529
Provided by: WestminsterResearch

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