Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi-Davidson Method

We study inexact Rayleigh quotient iteration (IRQI) for computing a simple interior eigenpair of the generalized eigenvalue problem Av = λBv, providing new insights into a special type of preconditioners with “tuning” for the efficient iterative solution of the shifted linear systems that arise in this algorithm. We first give a new convergence analysis of IRQI, showing that locally cubic and quadratic convergence can be achieved for Hermitian and non-Hermitian problems, respectively, if the shifted linear systems are solved by a generic Krylov subspace method with a tuned preconditioner to a reasonably small fixed tolerance. We then refine the study by Freitag and Spence [Linear Algebra Appl., Vol. 428 (2008), pp. 2049–2060] on the equivalence of the inner solves of IRQI and inexact single-vector Jacobi-Davidson method where a preconditioned full orthogonalization method with a tuned preconditioner is used as the inner solver. We also provide some new perspectives on the tuning strategy, showing that tuning is essentially needed only in the first inner iteration in the non-Hermitian case. Based on this observation, we propose a flexible GMRES algorithm with a special configuration in the first inner step, and show that this method is as efficient as GMRES with the tuned preconditioner.