An improved algorithm to compute the exponential of a matrix

Abstract: In this work, we present a new way to compute the Taylor polynomial of the matrix exponential which reduces the number of matrix multiplications in comparison with the de-facto standard Patterson-Stockmeyer method. This reduction is sufficient to make the method superior in performance to Pad\'e approximants by 10-30% over a range of values for the matrix norms and thus we propose its replacement in standard software kits. Numerical experiments show the performance of the method and illustrate its stability.