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

Abstract

We 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.