The convergence of Jacobi-Davidson for Hermitian eigenproblems

Abstract

Rayleigh Quotient iteration is an iterative method with some attractive convergence properties for nding (interior) eigenvalues of large sparse Hermitian matrices. However, the method requires the accurate (and, hence, often expensive) solution of a linear system in every iteration step. Unfortunately, replacing the exact solution with a cheaper approximation may destroy the convergence. The (Jacobi-)Davidson correction equation can be seen as a solution for this problem. In this paper we deduce quantitative results to support this viewpoint and we relate it to other methods. This should make some of the experimental observations in practice more quantitative in the Hermitian case. Asymptotic convergence bounds are given for xed preconditioners and for the special case if the correction equation is solved with some xed relative residual precision. A new dynamic tolerance is proposed and some numerical illustration is presented.