Journal of Computational Physics
Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves
SIAM Journal on Matrix Analysis and Applications
On convergence of the inexact Rayleigh quotient iteration with MINRES
Journal of Computational and Applied Mathematics
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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. Elde´n, 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.