Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants

TL;DR

This paper introduces two new adaptive quadrature algorithms that improve reliability by novel integrand representation and error estimation methods, tested in Matlab and outperforming existing methods in reliability.

Contribution

The paper presents two innovative adaptive quadrature routines with enhanced reliability, differing in integrand representation, handling of non-numerical values, divergence, and error estimation.

Findings

More reliable than existing adaptive integrators

Effective in handling improper divergent integrals

Implemented successfully in Matlab

Abstract

We present two new adaptive quadrature routines. Both routines differ from previously published algorithms in many aspects, most significantly in how they represent the integrand, how they treat non-numerical values of the integrand, how they deal with improper divergent integrals and how they estimate the integration error. The main focus of these improvements is to increase the reliability of the algorithms without significantly impacting their efficiency. Both algorithms are implemented in Matlab and tested using both the "families" suggested by Lyness and Kaganove and the battery test used by Gander and Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases less efficient, than other commonly-used adaptive integrators.