Computing interior eigenvalues of large matrices

A method is given that does a better job of computing eigenvalues from the interior of the spectrum of a large matrix and a priori bounds can be given for the accuracy of interior eigenvalue and eigenvector approximations.

Abstract

Computing eigenvalues from the interior of the spectrum of a large matrix is a difficult problem. The Rayleigh-Ritz procedure is a standard way of reducing it to a smaller problem, but it is not optimal for interior eigenvalues. Here a method is given that does a better job. In contrast with standard Rayleigh-Ritz, a priori bounds can be given for the accuracy of interior eigenvalue and eigenvector approximations. When applied to the Lanczos algorithm, this method yields better approximations at early stages. Applied to preconditioning methods, the convergence rate is improved.