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In this paper we introduce a family of rational approximations of thereciprocal of a $\phi$-function involved in the explicit solutions of certainlinear differential equations, as well as in integration schemes evolving onmanifolds. The derivation and properties of this family of approximationsapplied to scalar and matrix arguments are presented. Moreover, we show thatthe matrix functions computed by these approximations exhibit decayingproperties comparable to the best existing theoretical bounds. Numericalexamples highlight the benefits of the proposed rational approximationsw.r.t.~the classical Taylor polynomials and other rational functions.

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