不精确位移-反转Arnoldi方法用于Toeplitz矩阵指数
Inexact Shift-and-Invert Arnoldi for Toeplitz Matrix Exponential
AI总结：
本文提出不精确位移-反转Arnoldi方法用于计算Toeplitz矩阵指数与向量的乘积，通过改进GSF稳定性分析和建立误差与残差关系，提出新的停止准则和算法，通过数值实验验证有效性。
摘要：
我们回顾了Lee等人在2010年提出的位移-反转Arnoldi方法，用于数值近似Toeplitz矩阵指数与向量的乘积。该方法需要预先求解两个大规模Toeplitz线性系统，但若所需精度高则成本过高。因此，研究如何在该方法中不精确求解Toeplitz系统是有意义的。本文的贡献在于三个方面：首先，我们对Gohberg-Semencul公式（GSF）进行了新的稳定性分析，并定义了Toeplitz矩阵的GSF条件数。其次，我们建立了Toeplitz系统误差与Toeplitz矩阵指数残差之间的关系。第三，基于此关系，我们提出了放松Toeplitz系统精度的实用停止准则，并提出了一种不精确位移-反转Arnoldi算法用于Toeplitz矩阵指数问题。数值实验展示了新算法的数值行为，并证明了理论结果的有效性。
英文摘要：
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.
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