Symbolic-Numeric Integration of Rational Functions

TL;DR

This paper introduces a hybrid symbolic-numeric method for integrating rational functions that combines symbolic algebra with numerical rootfinding to achieve stable and efficient antiderivative computation.

Contribution

It presents two novel hybrid algorithms that integrate symbolic and numerical techniques for rational function integration, ensuring stability and efficiency.

Findings

Methods are forward and backward stable

Achieve numerical stability by adjusting rootfinding precision

Effective integration of rational functions using hybrid approach

Abstract

We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the stable 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 show that both methods are forward and backward stable in a structured sense and away from singularities tolerance proportionality is achieved by adjusting the precision of the rootfinding tasks.