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Manuscript Title: Programs for the approximation of real and imaginary single- and multi-valued functions by means of Hermite-Pade-Approximants.
Authors: T.M. Feil, H.H.H. Homeier
Program title: hp.sr
Catalogue identifier: ADSO
Journal reference: Comput. Phys. Commun. 158(2004)124
Programming language: Maple V.5.
Computer: Sun Ultra 10.
Operating system: Sun Solaris 7.0.
RAM: 8M words
Word size: 32
Keywords: Summation, Hermite-Padé, Padé, Divergent series, Perturbation, multi-valued functions, General purpose, Other numerical methods, Computer algebra.
Classification: 4.12, 5.

Nature of problem:
Many physical and chemical quantum systems lead to the problem of evaluating a function for which only a limited series expansion is known. These functions can be numerically approximated by summation methods even if the corresponding series is only asymptotic. With the help of Hermite-Padé approximants many different approximation schemes can be realised. Padé and algebraic approximants are just well known examples. Hermite-Padé approximants combine the advantages of highly accurate numerical results with the additional advantage of being able to sum complex multi-valued functions.

Solution method:
Special type Hermite-Padé polynomials are calculated for a set of divergent series. These polynomials are then used to implicitly define approximants for one of the functions of this set. This approximant can be numerically evaluated at any point of the Riemann surface of this function. For an approximation order smaller than 3 the approximants can alternatively be expressed in closed form and then be used to approximate the desired function on its complete Riemann surface.

In principle, the algorithm is only limited by the available memory and speed of the underlying computer system. Furthermore the achievable accuracy of the approximation only depends on the number of known series coefficients of the function to be approximated assumimg of course that these coefficients are known with enough accuracy.

Running time:
10 minutes with parameters comparable to the test runs.