A fast numerical algorithm for the integration of rational functions

This article by Dante Manna, Luis Medina, Victor H. Moll and Armin Straub has been published in Numerische Mathematik (Volume 115, Number 2, April 2010, Pages 289-307) and is available at doi:10.1007/s00211-009-0284-9.

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.

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 attached.

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