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29-05-2019 | Original Paper | Issue 4/2020

Numerical Algorithms 4/2020

Symbolic-numeric integration of rational functions

Journal:
Numerical Algorithms > Issue 4/2020
Authors:
Robert H. C. Moir, Robert M. Corless, Marc Moreno Maza, Ning Xie
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Abstract

We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the reliable yet efficient computation of symbolic antiderivatives while avoiding issues of ill-conditioning to which numerical methods are susceptible. We propose two alternative methods for exact input that compute the rational part of the integral using Hermite reduction and then compute the transcendental part two different ways using a combination of exact integration and efficient numerical computation of roots. The symbolic computation is done within bpas, or Basic Polynomial Algebra Subprograms, which is a highly optimized environment for polynomial computation on parallel architectures, while the numerical computation is done using the highly optimized multiprecision rootfinding package MPSolve. We provide for both algorithms computable expressions for the first-order term of a structured forward and backward error and show how, away from singularities, tolerance proportionality is achieved by adjusting the precision of the rootfinding tasks.

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