Inexact Shift-and-Invert Arnoldi for Toeplitz Matrix Exponential

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.