Computing the Reciprocal of a $\phi$-function by Rational Approximation

TL;DR

This paper introduces a new family of rational approximations for the reciprocal of a phi-function used in solving linear differential equations and manifold integration, demonstrating improved properties and numerical performance.

Contribution

The paper presents a novel family of rational approximations for phi-function reciprocals, with proven decay properties and advantages over classical methods.

Findings

Approximations exhibit decay properties comparable to theoretical bounds.

Numerical tests show improved accuracy over Taylor polynomials.

Method benefits manifold integration and differential equation solutions.

Abstract

In this paper we introduce a family of rational approximations of the reciprocal of a $\phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.