Research And Application Of High Performance Algorithm Of Matrix Function

This topic is mainly aimed at two types of special matrix functions and design an efficient and stable numerical method,where the ?-function is called an exponential-type matrix function,and the ?-function is called an oscillating matrix function.These two types of problems appear respectively in the exponential integration scheme for solving semi-linear rigid problems and the multidimensional and multi-frequency ERKN method.In actual problems,the?-function often involves the product of a large sparse matrix and a vector,so designing the product of the ?-function and the vector is of great significance.The efficient calculation of the exponential matrix function determines the calculation efficiency of the exponential integral,and the efficient calculation of the oscillating matrix function determines the calculation efficiency of the multi-dimensional and multi-frequency ERKN method.Therefore,it is necessary to design efficient and stable algorithms to calculate these two special matrix functions.Firstly,for the small and medium matrix A,a scaling method based on Taylor series is designed to calculate ?l(A),and a quasi-backward error analysis is proposed.Scaling parameter s and Taylor series expansion term m is determined based on the quasi-backward error,and two optimal selection methods for m and s are designed.Numerical experiments show that the algorithm in this paper is stable.And the calculation results are relatively more accurate and takes less time than the[13/13]padéapproximation.Secondly,for the large sparse matrix A,a scaling method based on Taylor series is designed to calculate ?l(A)b.The error analysis is consistent with that in Chapter 3.Scaling parameter s and Taylor series expansion term m is determined based on the quasi-backward error,and the amount of calculation is mainly generated by the number of times the matrix and the vector are multiplied.The experimental results show that the algorithm in this paper is stable,and the calculation time and calculation accuracy are still comparable to the most popular Krylov subspaces methods at this stage.Thirdly,for the small and medium matrix A,a scaled quartic method based on Taylor series is designed to calculate ?2(A).A quasi-backward error analysis is also given.Scaling parameter s and Taylor series expansion term m is determined based on the quasi-backward error,and the optimal selection method of m and s is given.Numerical experiments show that the algorithm in this paper is stable,has high computational accuracy,and has a shorter computational time than the ODE45 method.