Inexact Shift-and-Invert Arnoldi for Toeplitz Matrix Exponential

TL;DR

This paper develops an inexact shift-and-invert Arnoldi method for Toeplitz matrix exponentials, introducing new stability analysis, a relation between system errors and residuals, and a practical stopping criterion to improve computational efficiency.

Contribution

It provides a new stability analysis of the Gohberg-Semencul formula, establishes a relation between Toeplitz system errors and residuals, and proposes an inexact algorithm with a practical stopping criterion.

Findings

The new stability analysis offers sharper bounds for large Toeplitz matrices.

Low-accuracy solutions suffice when the GSF condition number is moderate.

Numerical experiments confirm the effectiveness of the proposed inexact method.

Abstract

We revisit the shift-and-invert Arnoldi method proposed in [S. Lee, H. Pang, and H. Sun. {\it Shift-invert Arnoldi approximation to the Toeplitz matrix exponential}, SIAM J. Sci. Comput., 32: 774--792, 2010] for numerical approximation to the product of Toeplitz matrix exponential with a vector. In this approach, one has to solve two large scale Toeplitz linear systems in advance. However, if the desired accuracy is high, the cost will be prohibitive. Therefore, it is interesting to investigate how to solve the Toeplitz systems inexactly in this method. The contribution of this paper is in three regards. First, we give a new stability analysis on the Gohberg-Semencul formula (GSF) and define the GSF condition number of a Toeplitz matrix. It is shown that, when the size of the Toeplitz matrix is large, our result is sharper than the one given in [M. Gutknecht and M. Hochbruck. {\it The stability of inversion formulas for Toeplitz matrices}, Linear Algebra Appl., 223/224: 307--324, 1995]. Second, we establish a relation between the error of Toeplitz systems and the residual of Toeplitz matrix exponential. We show that if the GSF condition number of the Toeplitz matrix is medium sized, then the Toeplitz systems can be solved in a low accuracy. Third, based on this relationship, we present a practical stopping criterion for relaxing the accuracy of the Toeplitz systems, and propose an inexact shift-and-invert Arnoldi algorithm for the Toeplitz matrix exponential problem. Numerical experiments illustrate the numerical behavior of the new algorithm, and show the effectiveness of our theoretical results.