Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm

Abstract

It is well known that a Ritz vector obtained by the Lanczos method and the Arnoldi method can be characterized by an elegant explicit polynomial, whose roots are just the other m − 1 Ritz values of the matrix A in question from the m-dimensional Krylov subspace involved. Analogous results hold for the recently developed harmonic Lanczos and Arnoldi methods, and the roots of the corresponding polynomial are now the other m − 1 harmonic Ritz values of A from the same subspace. In this paper, we investigate a polynomial characterization of the refined Arnoldi method proposed by the author in recent years. We derive a polynomial characterization of the refined Arnoldi method and apply the implicitly restarting scheme proposed by Sorensen to the refined Arnoldi method. The roots of this polynomial are used as shifts, called refined shifts, within an implicitly restarted refined Arnoldi algorithm. The shifts are interpreted to have the same nature as the exact shifts used within the implicitly restarted Arnoldi algorithm proposed by Sorensen. Numerical experiments compare the use of refined shifts with exact shifts within an implicitly restarted Arnoldi algorithm. The results also show that implicitly restarting the refined Arnoldi method is far superior to the Saad's explicitly restarted scheme.