An improved algorithm to compute the exponential of a matrix

TL;DR

This paper introduces an improved algorithm for computing the matrix exponential that reduces matrix multiplications, outperforming existing methods like Padé approximants by 10-30% in speed, with demonstrated stability and efficiency.

Contribution

The paper presents a novel Taylor polynomial-based algorithm that decreases matrix multiplications, offering a more efficient alternative to the Patterson-Stockmeyer method and Padé approximants.

Findings

The new method reduces matrix multiplications compared to standard approaches.

It achieves 10-30% performance improvement over Padé approximants.

Numerical experiments confirm the method's stability and efficiency.

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.