Reliability Conditions in Quadrature Algorithms

TL;DR

This work tackles the reliability of automatic adaptive quadrature for parametric integrals by analyzing the integrand profile at quadrature knots to validate local error estimates. It introduces a framework of well-conditioned profiles and six consistency criteria tied to monotonicity, basis-polynomial properties, and derivative behavior to detect insufficient resolution and isolated difficult points. Numerical experiments with Gauss-Kronrod 10–21 rules show that the proposed diagnostics expand the range of reliable outputs and reduce spurious error assessments compared to standard QUADPACK and previous self-validating approaches, particularly for oscillatory and nonmonotonic integrands. The method enhances robustness of quadrature in complex physical models and can be integrated into automatic control routines, with companion documentation in AA02; error estimates can be sharpened toward near-100% reliability under suitable conditions.

Abstract

The detection of insufficiently resolved or ill-conditioned integrand structures is critical for the reliability assessment of the quadrature rule outputs. We discuss a method of analysis of the profile of the integrand at the quadrature knots which allows inferences approaching the theoretical 100% rate of success, under error estimate sharpening. The proposed procedure is of the highest interest for the solution of parametric integrals arising in complex physical models.

Figures (8)

• Figure 1: Relative errors $\rho\sb{qdp}$ and $\rho\sb{2n}$, Eq. (\ref{['eq:cerel']}), $\varepsilon\sb{Q}$, Eq. (\ref{['eq:erel']}), of the GK 10-21 outputs for the family of integrals (\ref{['eq:pow']}) at exponents $n \leq 200$. The upper leftmost solid line arrow points to the accuracy basin extension established by the QUADPACK code using the error estimate (\ref{['eq:eqdp']}). The solid line arrow on the same vertical points to the upper accuracy of the quadrature sum $q\sb{2n}$ retained as reliable by the QUADPACK code. The next pair of solid arrows show the result of the analysis done in ref. AA01. The left interrupted line arrow represents the extension of the reliability basin established by the present analysis, while the right one shows the exponent threshold above which the criterion (\ref{['eq:0.5']}) supersedes the need of reliability analysis.
• Figure 2: Same as fig. \ref{['fig:fr21pow']} for the family of integrals (\ref{['eq:atg']}), at upper integration ranges $b \leq 260$. The arrows bear the same significance.
• Figure 3: Outputs of the GK 10-21 quadrature rule for the family of integrals (\ref{['eq:oc10']}) at $p = 1$. The significances of the solid line arrows are the same as in fig. \ref{['fig:fr21pow']}. The left interrupted line arrow represents the extension of the reliability basin up to which the present analysis validates all the outputs $q\sb{2n}$. Inbetween the two interrupted line arrows, the diagnostics of the present analysis is too conservative in about one third of the solved cases.
• Figure 4: Same as fig. \ref{['fig:oc10p1']} for the family of integrals (\ref{['eq:os10']}) at $p = 1$.
• Figure 5: Same as fig. \ref{['fig:oc10p1']} for the family of integrals (\ref{['eq:oc11']}) at $p = 1$.