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

Ovtchinnikov, E. and Xanthis, L. 2001. Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 457 (2006), pp. 441-451.

TitleSuccessive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates
AuthorsOvtchinnikov, E. and Xanthis, L.
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,

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 differential equation exhibiting clusters of eigenvalues.

Keywordslarge-scale eigenvalue problems, eigensolvers with preconditioning, subspace iteration, convergence rate, multiple and clustered eigenvalues
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Journal citation457 (2006), pp. 441-451
ISSN1364-5021
Year08 Feb 2001
FileOvtchinnikov_Xanthis_2001_final.pdf
Digital Object Identifier (DOI)doi:10.1098/rspa.2000.0674
Publication dates
Published08 Feb 2001

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