On the backward and forward error of approximations of analytic functions and applications to the computation of matrix functions

In this paper we give a new formula to write the forward error of Taylor approximations of analytical functions in terms of the backward error of those approximations, considering exact arithmetic in both errors. Using this formula, a method to compute a backward error given by the power series centered in the same expansion point as the Taylor approximation is provided. The application of this method for Padé approximations is also shown. Based on the previous method, a MATLAB implementation for computing the first power series terms of the backward error for Taylor and Padé approximations of a given analytical function is provided, and examples of its use are given. Applications to the computation of matrix functions are given that overcome limitations of other backward error analyses use inverse compositional functions in the literature.