Shift for nonsymmetric generalised eigenvalue problems

J Berns-Muller, A Spence

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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.
Original languageEnglish
Pages (from-to)1069-1082
Number of pages14
JournalSIAM Journal On Matrix Analysis and Applications (SIMAX)
Issue number4
Publication statusPublished - Dec 2006


  • Eigenvalue approximation
  • Inverse iteration
  • Iterative methods


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