Methods for the computation of slowly convergent series and finite sums based on Gauss-Christoffel quadratures
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[1] Serbian Academy of Sciences and Arts
Serbian Academy of Sciences and Arts
Serbia
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- Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 6, Nº. 1, 2014, págs. 37-68
- Idioma: inglés
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Resumen
In this paper we give an account on summation/integration methods for the computation of slowly convergent series and finite sums. The methods are based on Gauss-Christoffel quadrature rules with respect to some nonclassical weight functions over R or R+. For constructing such rules we use the recent progress in symbolic computation and variable-precision arithmetic, as well as our Mathematica package OrthogonalPolynomials (4, 26). Together with our own results, we are also taking into consideration the use of other known results, especially the classical summation formulae of Euler-Maclaurin and Abel-Plana, in order to apply them afterwards in the computational techniques that we have developed recently. We present the Laplace transform method [15] and the contour integration method (21, 23), and give several numerical examples in order to illustrate the efficiency of different summation/integration methods.
