Abstract
We present the first of two different algorithms for the explicit computation of Hermite–Padé 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–Padé 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.
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Sablonnière, P. An Algorithm for the Computation of Hermite–Padé Approximations to the Exponential Function: Divided Differences and Hermite–Padé Forms. Numerical Algorithms 33, 443–452 (2003). https://doi.org/10.1023/A:1025557231329
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DOI: https://doi.org/10.1023/A:1025557231329