Inexact inverse iteration for symmetric matrices

In this paper we analyse inexact inverse iteration for the real symmetric eigenvalue problem Av = λv. Our analysis is designed to apply to the case when A is large and sparse and where iterative methods are used to solve the shifted linear systems (A − σI)y = x which arise. We present a convergence theory that is independent of the nature of the inexact solver used, and, though the use of the Rayleigh quotient is emphasised, our analysis also extends to quite general choices for shift and inexact solver strategies. Additionally, the convergence framework allows us to treat both standard preconditioning and to present a new analysis of the variation introduced by Simoncini and Eldén (BIT, vol. 42, pp.159–182, 2002). Also, we provide an analysis of the performance of inner iteration solves when preconditioned MINRES is used as the inexact solver. This analysis provides descriptive bounds which are shown to predict well the actual behaviour observed in practice. Also, it explains the improvement in performance of the modification introduced by Simoncini and Eldén over the standard preconditioned form. Importantly, our analysis shows that letting the shift tend to the eigenvalue, as is the case if the Rayleigh quotient is used, does not harm significantly the performance of the iterative method for the shifted systems. Throughout the paper numerical results are given to illustrate the theory.