Shift-invert and Cayley transforms for detection of rightmost eigenvalues of nonsymmetric matrices

Abstract

This manuscript is concerned with the determination of the rightmost eigenvalues of large sparse real nonsymmetric matrices. Specifically, the use of subspace iteration preconditioned by the Cayley transform and/or shift-invert is discussed. The convergence properties of subspace iteration are used to construct a strategy to validate the rightmost eigenvalue, which is computed by an iterative method. The motivation behind this paper is that rational preconditioners are very reliable in general but they can miss rightmost eigenvalues with large imaginary part. Numerical examples are given to illustrate the theory.