Polynomial Form of the Matrix Exponential

TL;DR

This paper introduces a polynomial expansion method for computing the matrix exponential, offering a more reliable alternative to Pade approximations with comparable efficiency and convergence.

Contribution

The paper derives a polynomial form of the matrix exponential using symbolic computation, providing an alternative to Pade approximations with improved reliability.

Findings

Polynomial form is comparable in operation count to Pade methods.

The polynomial approach is more reliable due to simpler matrix evaluations.

Both rational and polynomial forms of the exponential are derived.

Abstract

An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to be comparable in operation count and convergence with the state--of--the--art method which is based on a Pade approximation of the exponential matrix function. The present polynomial form, however, is more reliable because the evaluation requires only linear combinations of the input matrix. We also show that the technique used to solve the differential equation, when implemented symbolically, leads to a rational as well as a polynomial form of the solution function. The rational form is the well-known diagonal Pade approximation of $e^x$. The polynomial form, after some rearranging to minimize operation count, will be used to evaluate the exponential of a matrix so as to illustrate its advantages as compared with the Pade form.