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

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.