AMGKQ: An Efficient Implementation of Adaptive Multivariate Gauss-Kronrod Quadrature for Simultaneous Integrands in Octave/MATLAB

TL;DR

AMGKQ introduces an efficient adaptive multivariate Gauss-Kronrod quadrature algorithm in Octave/MATLAB capable of simultaneously approximating multiple integrals over hyper-rectangular regions, with improved speed and handling of singularities.

Contribution

It presents a novel implementation of adaptive multivariate Gauss-Kronrod quadrature that efficiently handles multiple integrals and singularities in high-dimensional spaces.

Findings

Significantly faster than recursive routines in 2D and 3D.

Effectively manages improper integrals with singularity transformations.

Performance limited mainly by memory constraints.

Abstract

The algorithm AMGKQ for adaptive multivariate Gauss-Kronrod quadrature over hyper-rectangular regions of arbitrary dimensionality is proposed and implemented in Octave/MATLAB. It can approximate numerically any number of integrals over a common domain simultaneously. Improper integrals are addressed through singularity weakening coordinate transformations. Internal singularities are addressed through the use of breakpoints. Its accuracy performance is investigated thoroughly, and its running time is compared to other commonly available routines in two and three dimensions. Its running time can be several orders of magnitude faster than recursively called quadrature routines. Its performance is limited only by the memory structure of its operating environment. Included with the software are numerous examples of its invocation.