Inexact Inverse Iteration with Variable Shift for Nonsymmetric Generalized Eigenvalue Problems
Jörg Berns-Müller, Alastair A. Spence · SIAM Journal on Matrix Analysis and Applications · 2006
In this paper we analyze inexact inverse iteration for the nonsymmetric generalized eigenvalue problem $\bf{A}\bf{x} = \lambda \bf{M}\bf{x}$, where $\bf{M}$ is symmetric positive definite and the problem is diagonalizable. Our analysis is designed to apply to the case when $\bf{A}$ and $\bf{M}$ are large and sparse and preconditioned iterative methods are used to solve shifted linear systems with coefficient matrix $\bf{A}-\sigma \bf{M}$. 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. Eldén, BIT, 42 (2002), pp. 159–182] to modify the right‐hand side when using preconditioned solves. Several numerical experiments are presented that illustrate the theory and provide a basis for the discussion of practical issues.