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 rst present a general convergence theory that is independent of the nature of the inexact solver used. Next we consider in detail the performance of preconditioned MINRES as the solver for (A − σI)y = x. In particular we analyse the cost of one solve and then the overall cost of the whole algorithm. Also we present a new treatment of the approach discussed by [18] to set up an appropriate right hand side for the preconditioned system. Our analysis shows that, if MINRES is the inexact solver then the optimal strategy is to let the shift tend to the desired eigenvalue as quickly as possible. In most practical situations that will mean that the shift should be taken as the Rayleigh quotient, just as if direct solves were used. Numerical results are given to illustrate the theory in the paper. AMS subject classification: Primary 65F15, Secondary 65F10