A method for computing a few eigenpairs of large generalized eigenvalue problems

Abstract

Many methods for the computation of selected eigenpairs of generalized eigenproblems for matrix pairs use a shift-and-invert technique. When applied to large-scale problems, this requires the solution of large linear systems of equations. This paper proposes an application of an Arnoldi method described in Voss (2004) [21] to the computation of a few extreme eigenpairs of a matrix pair. An advantage of this approach, when compared to methods that use the shift-and-invert technique, is that no large systems of equations have to be solved. We compare this approach to using a technique for simultaneously reducing a pair of large matrices to a pair of small matrices by a generalized Arnoldi process described in Li and Ye (2003) [12] and Hoffnung et al. (2006) [6]. The latter technique does not require the solution of large linear systems of equations either. Computed examples show the proposed method to yield approximations of the desired eigenpairs of higher accuracy when using about the same amount of computer storage space.