|Title||Successive eigenvalue relaxation: a new method for the generalized eigenvalue problem and convergence estimates|
|Authors||Ovtchinnikov, E. and Xanthis, L.|
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.
|Keywords||large-scale eigenvalue problems, eigensolvers with preconditioning, subspace iteration, convergence rate, multiple and clustered eigenvalues|
|Journal||Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|
|Journal citation||457 (2006), pp. 441-451|
|Year||08 Feb 2001|
|Digital Object Identifier (DOI)||https://doi.org/10.1098/rspa.2000.0674|
|Published||08 Feb 2001|