Harmonic Ritz Values and Their Reciprocals

One application of harmonic Ritz values is to approximate, with a projection method, the interior eigenvalues of a matrix A while avoiding the explicit use of the inverse A. In this context, harmonic Ritz values are commonly derived from a Petrov-Galerkin condition for the residual of a vector from the test space. In this paper, we investigate harmonic Ritz values from a slightly different perspective. 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 that differs from the commonly used one, 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 such as the steepest descent, the nonlinear Preconditioned Conjugate Gradient (PCG), or the Locally Optimal Block PCG (LOBPCG) method. Instead of computing the smallest/largest eigenvalues by minimizing/maximizing the Rayleigh quotient, one can compute interior eigenvalues as the minimum/maximum of ψ. AMS subject classifications. 65F15, 65Y15.