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.
|Number of pages||14|
|Journal||SIAM Journal On Matrix Analysis and Applications (SIMAX)|
|Publication status||Published - Dec 2006|
- Eigenvalue approximation
- Inverse iteration
- Iterative methods
Berns-Muller, J., & Spence, A. (2006). Shift for nonsymmetric generalised eigenvalue problems. SIAM Journal On Matrix Analysis and Applications (SIMAX), 28(4), 1069-1082. https://doi.org/10.1137/050623255