Computing the Inverse of a φ-Function by Rational Approximation

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.