李群方法里的矩阵指数计算

矩阵指数计算与力学计算中的动力学问题、最优控制的计算问题等密切相关,是数值代数里研究得最为广泛的课题之一。目前虽有以PSSA和PIM为代表的经典算法以及其最佳运算量的估计,但远未令人满意。近年来在国外流行的李群方法,由于具有重大的科学价值,在李群算法发展的需求下,李群、李代数里的指数计算,也成为了研究的热点。由于它要求逼近计算在李群与李代数里进行,一般不能直接使用经典的方法。因此,它比经典的指数逼近计算更为困难。本文系统地阐述了目前流行的几种主要逼近算法,对这些算法进行了详细的评估,并提出了一些有待深入研究的问题。

Computation of Matrix Exponential in Lie Group Methods

The computation of matrix exponential which ties up structural mechanics and optimum control is one of the extensive topics in numerical mathematics.Though there are many algorithms about the matrix exponential such as Pade-scaling and squaring-approximation(PSSA),precise integration method(PIM) and some results about those optimal computational cost,it is far from satisfying to us.Currently Lie group methods are of huge value and they are popular abroad,and lead to the computation of matrix exponential to be a hot topic.Since numerical approximation must run in Lie groups or Lie algebras,some classic methods are no effect. It is more difficult than classic method.In this paper a few popular computation methods of matrix exponential are reviewed systematically and evaluated in detail,and some problems which need our thorough research are put forward.