Eigenvalue perturbation and generalized Krylov subspace method

Abstract

In this paper, we study the computational aspect of eigenvalue perturbation theory. In previous research, high order perturbation terms were often derived from Taylor series expansion. Computations based on such an approach can be both unstable and highly complicated. We present here an approach based on the differential formulation of perturbation theory where the high order perturbation can be naturally obtained. The high order perturbation can be interpreted as a generalized Krylov subspace approximation and its convergence rate can be analyzed accordingly. This approach provides a simple and stable method to compute a few eigenvalues of a slightly modified system.