  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.
