A note on harmonic Ritz values and their reciprocals

This note summarizes an investigation of harmonic Ritz values to approximate the interior eigenvalues of a real symmetric matrix A while avoiding the explicit use of the inverse A −1 . We consider a bounded functional ψ that yields the reciprocals of the harmonic Ritz values of a symmetric matrix A . The crucial observation is that with an appropriate residual s , many results from Rayleigh quotient and Rayleigh–Ritz theory naturally extend. The same is true for the generalization to matrix pencils ( A, B ) when B is symmetric positive definite. These observations have an application in the computation of eigenvalues in the interior of the spectrum of a large sparse matrix. The minimum and maximum of ψ correspond to the eigenpairs just to the left and right of zero (or a chosen shift). As a spectral transformation, this distinguishes ψ from the original harmonic approach where an interior eigenvalue remains at the interior of the transformed spectrum. As a consequence, ψ is a very attractive vehicle for a matrix‐free, optimization‐based eigensolver. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ . Copyright © 2009 John Wiley & Sons, Ltd.