Computing the Reciprocal of a $ϕ$-function by Rational Approximation

Abstract

In this paper we introduce a family of rational approximations of the inverse of a φ function involved in the explicit solutions of certain linear differential equations as well as in integration schemes evolving on manifolds. For symmetric banded matrices these novel approximations provide a computable reconstruction of the associated matrix function which exhibits decaying properties comparable to the best existing theoretical bounds. Numerical examples show the benefits of the proposed rational approximations w.r.t. the classical Taylor polynomials.Keywords rational approximation • φ-functions • band matrices • matrix functions Mathematics Subject Classification (2000) 65F60 z with φ 0 (z) = e z [12]. Here we focus on the related issue of approximating the inverse φ 1 matrix function ψ 1 (A) where ψ 1 (z) is a meromorphic function defined by