Inexact Rayleigh Quotient-Type Methods for Eigenvalue Computations

We consider the computation of an eigenvalue and corresponding eigenvector of a Hermitian positive definite matrix A , assuming that good approximations of the wanted eigenpair are already available, as may be the case in applications such as structural mechanics. We analyze efficient implementations of inexact Rayleigh quotient-type methods, which involve the approximate solution of a linear system at each iteration by means of the Conjugate Residuals method. We show that the inexact version of the classical Rayleigh quotient iteration is mathematically equivalent to a Newton approach. New insightful bounds relating the inner and outer recurrences are derived. In particular, we show that even if in the inner iterations the norm of the residual for the linear system decreases very slowly, the eigenvalue residual is reduced substantially. Based on the theoretical results, we examine stopping criteria for the inner iteration. We also discuss and motivate a preconditioning strategy for the inner iteration in order to further accelerate the convergence. Numerical experiments illustrate the analysis.