Shift for nonsymmetric generalised eigenvalue problems

J Berns-Muller, A Spence

OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/28160072-0087-4be1-a555-3f79b9f6b1c4 • DOI: https://doi.org/10.1137/050623255

In this paper we analyze inexact inverse iteration for the nonsymmetric generalized eigenvalue problem Ax = λMx, where M is symmetric positive definite and the problem is diagonalizable. Our analysis is designed to apply to the case when A and M are large and sparse and preconditioned iterative methods are used to solve shifted linear systems with coefficient matrix A − σM. We prove a convergence result for the variable shift case (for example, where the shift is the Rayleigh quotient) which extends current results for the case of a fixed shift. Additionally, we consider the approach from [V. Simoncini and L. Elden, BIT, 42 (2002), pp. 159–182] to modify the right-hand side when using preconditioned solves. Several numerical experiments are presented thatillustrate the theory and provide a basis for the discussion of practical issues.

Eigenvalue approximation Inverse iteration iterative methods