Symbolic-Numeric Integration of Rational Functions
This work addresses the need for stable and efficient integration of rational functions in computer algebra systems, offering a practical hybrid approach for exact input.
The paper presents hybrid symbolic-numeric methods for integrating rational functions, combining Hermite reduction with numerical rootfinding to achieve stable and efficient computation of antiderivatives. The methods are shown to be forward and backward stable, with tolerance proportionality achieved by adjusting rootfinding precision.
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.