An Algorithm for the Computation of Hermite–Pade Approximations to the Exponential Function: Divided Differences and Hermite–Pade Forms

摘要

We present the first of two different algorithms for the explicit computation of Hermite–Pade forms (HPF) associated with the exponential function. Some roots of the algebraic equation associated with a given HPF are good approximants to the exponential in some subsets of the complex plane: they are called Hermite–Pade approximants (HPA) to this function. Our algorithm is recursive and based upon the expression of HPF as divided differences of the function t → exp(xt) at multiple integer nodes. Using this algorithm, we find again the results obtained by Borwein and Driver for quadratic HPF. As an example, we give an interesting family of quadratic HPA to the exponential.