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A fast numerical algorithm for the integration of rational functions

A fast numerical algorithm for the integration of rational functions
Dante V. Manna, Luis A. Medina, Victor H. Moll, Armin Straub — Numerische Mathematik — Volume 115, Number 2, 2010, Pages 289-307

Implementation in Mathematica

This paper is accompanied by an implementation of the described algorithm in Mathematica. The package (Landen.m) containing the algorithm as well as some examples (examples.nb) are available.

Abstract

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral with \(m\)-th order convergence. Examples illustrating the automatic generation of these formulas and the numerical behaviour of this method are given.

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203.18 KB Preprint (PDF, 16 pages) 1461
10.76 KB Mathematica package 903
30.35 KB Mathematica notebook with examples 966

BibTeX

@article{numlanden-2010,
    author = {Dante V. Manna and Luis A. Medina and Victor H. Moll and Armin Straub},
    title = {A fast numerical algorithm for the integration of rational functions},
    journal = {Numerische Mathematik},
    year = {2010},
    volume = {115},
    number = {2},
    pages = {289--307},
    doi = {10.1007/s00211-009-0284-9},
}
Date: 2009/02/17
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