动力学问题中矩阵函数的高效算法

Efficient computational method for matrix function in dynamic problems

本文基于Paterson-Stockmeyer(PS)算法和过滤技术提出了一种能精确和高效计算动力学问题中矩阵函数的算法.借助对截断误差和过滤引误差的分析,探究了计算中的误差增长规律,并基于此给出了自适应过滤阈值,使得提出的算法能更加高效地达到与原PS算法相似的精度.数值实验采用了包括30个不同带宽的随机矩阵,10个复杂网络动力学问题中的邻接矩阵来验证提出算法的效率与精度.数值实验的结果表明,提出的算法在计算所考虑的动力学问题的矩阵函数时能获得良好的精度和效率.

An algorithm based on the Paterson-Stockmeyer(PS)scheme and filtering technology is developed to compute the large matrix functions in dynamic problems accurately and efficiently.With the assistance of analysis on truncation error and error caused by filtering,the error growth law during the computation is studied,based on which an adaptive filtering threshold is proposed to help the proposed algorithm more efficiently achieve similar accuracy as the original PS scheme.Numerical examples,including 30 random matrices with different bandwidths,10 adjacency matrices in the complex network dynamic problems,and a trampoline vibration problem,are given to verify the efficiency and accuracy of the proposed algorithm.Numerical results suggest that the proposed algorithm can achieve good accuracy and efficiency in computing the matrix function in the considered dynamic problems.