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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Notes
Note that strict preservation of the form of the integrand is not quite achieved for the PFD method described below, since the derivative cannot be simplified into this form without using approximate gcd. Thus, with exact computation, the degree of the numerator and denominator of the nearby integrand is larger in general than the exact integrand.
Note that throughout this section we assume that the error for the numerical rootfinding for a polynomial P(x) satisfies the relation |Δr|≤ ε|r|, where r is the value of the computed root and Δr is the distance in the complex plane to the exact root. This can be accomplished using MPSolve by specifying an error tolerance of ε. Given the way that MPSolve isolates and then approximates roots, the bound is generally satisfied by several orders of magnitude.
Note that we assume that ε is sufficiently small to avoid spurious identification of residues in this analysis. Even with spurious identification, however, the backward error analysis would only change slightly, viz., to use the maximum error among the nearby residues, rather than the error of the selected representative residue. A different but related issue occurs when several distinct residues are clustered within an interval of width of some small multiple of ε. This presents a more challenging situation that we reserve for future work, but discuss briefly in Section 7 below.
This means that when several computed residues are coalesced, the chosen representative root \(\hat {\gamma }_{\tilde {k}}\) must be used to evaluate \(\hat {a}_{k}\) and \(\hat {b}_{k}\), and \(c^{\prime }(\hat {\gamma }_{k})\) must be used to evaluate all of the error terms corresponding to the roots \(\hat {\gamma }_{k}\) that have the same residue.
Note that this may account for the differences in stability between the LRT-based and PFD-based methods observed in Section 6 below. Because roots are isolated using multiprecision arithmetic, however, both methods nonetheless produce stable results.
The data for this section was collected using an internal December 2017 version of the BPAS library.
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Moir, R.H.C., Corless, R.M., Maza, M.M. et al. Symbolic-numeric integration of rational functions. Numer Algor 83, 1295–1320 (2020). https://doi.org/10.1007/s11075-019-00726-6
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DOI: https://doi.org/10.1007/s11075-019-00726-6