Numerical Analysis and Scientific Computing Seminar

Efficient Inexact Rayleigh Quotient Iteration and its Connections to the Jacobi-Davidson Method

Speaker: Fei Xu, Temple University

Location: Warren Weaver Hall 1302

Date: March 25, 2011, 10 a.m.


We study inexact Rayleigh quotient iteration (IRQI) for computing a simple interior eigenpair of the generalized eigenvalue problem Av = \lambda Bv, providing new insights into three aspects of a special type of preconditioners with “tuning” for the efficient solution of the shifted linear systems arising in this algorithm. We ?rst show that full asymptotic convergence rates of IRQI can be achieved, if the shifted linear systems are solved by a Krylov subspace method with a tuned preconditioner to a moderately small ?xed tolerance. We also discuss the equivalence of the inner solves of IRQI and the single-vector Jacobi-Davidson method. A ?exible GMRES (FGMRES) algorithm with a special con?guration in the ?rst inner step is proposed to simplify the use of tuning, and is shown to be as efficient as GMRES with the tuned preconditioner. The success of this FGMRES is also explained by its connection to the Jacobi-Davidson method.