MatrixPolynomials and the Matrix Exponential

Mark Sofroniou

Wolfram Research, Inc.

Abstract

This talk begins by reviewing an algorithm of Paterson and Stockmeyer for the efficient evaluation of polynomials of matrices and presents some new results that minimize work and storage. The results are then applied to the computation of the matrix exponential for which many methods have been proposed—nineteen approaches are described in a recent survey article of Moler and Van Loan, where it is pointed out that many of them are of dubious numerical quality. Particular attention will be paid to the algorithm used in the function expm of MATLAB and some recent enhancements will be described. Numerical experiments are presented to illustrate some improvements obtained in a new implementation in Mathematica.