Custom-made Gauss quadrature for statisticians

TL;DR

This paper introduces an R package that computes custom Gauss quadrature rules using high-precision arithmetic, enabling statisticians to create tailored numerical integration schemes with up to 33 nodes.

Contribution

The authors developed a new R package implementing a moment-based method for custom Gauss quadrature, expanding computational tools for statisticians.

Findings

Supports up to 33 nodes with high-precision moments

Provides free R package with implementation details

Encourages statisticians to adopt custom quadrature methods

Abstract

The theory and computational methods for custom-made Gauss quadrature have been described in Gautschi's 2004 monograph. Gautschi has also provided Fortran and MATLAB code for the implementation and illustration of these methods. We have written an R package, implemented in the high-precision arithmetic provided by the R package Rmpfr, that uses a moment-based method via moment determinants to compute a Gauss quadrature rule, with up to 33 nodes, provided that the moments can be computed to arbitrary precision using the standard mathematical functions provided by the Rmpfr package. Our hope is that the provision of our free R package and the numerical results that we present will encourage other statisticians to also consider the custom-made construction of Gauss quadrature rules.