Application of Weeks Method for the Numerical Inversion of the Laplace Transform to the Matrix Exponential

Abstract

Weeks method is a well established algorithm for the numerical inversion of scalar Laplace space functions. In this paper, we extend the method to the inversion of matrix functions of a single time variable and assess the qualities of this approach. To illustrate and quantify our discussion, we compute the matrix exponential by means of an FFT based algorithm. Particular attention is paid to a comparison of algorithms for the automated selection of two tuning parameters. In addition to selection algorithms from the literature, we introduce a pseudospectra based approach for the particular case of the matrix exponential. Finally, applications involving both pathological matrices and the numerical solution of differential equations highlight the utility of the method.