Rayleigh Quotient Iteration and Simplified Jacobi-davidson with Preconditioned Iterative Solves for Generalised Eigenvalue Problems

The computation of a right eigenvector and corresponding finite eigenvalue of a large sparse generalised eigenproblem Ax = λMx using preconditioned Rayleigh quotient iteration and the simplified JacobiDavidson method is considered. Both methods are inner-outer iterative methods and we consider GMRES and FOM as iterative algorithms for the (inexact) solution of the inner systems that arise. The performance of the methods is measured in terms of the number of inner iterations needed at each outer solve. For inexact Rayleigh quotient iteration we present a detailed analysis of both unpreconditioned and preconditioned GMRES and it is shown that the number of inner iterations increases as the outer iteration proceeds. We discuss a tuning strategy for the generalised eigenproblem, and show how a rank one change to the preconditioner produces significant savings in overall costs for the Rayleigh quotient iteration. Furthermore for a specially tuned preconditioner we prove an equivalence result between inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method. The theory is illustrated by several examples, including the calculation of a complex eigenvalue arising from a stability analysis of the linearised Navier-Stokes equations.