A unified framework for the computation of polynomial quadrature weights and errors

TL;DR

This paper introduces a unified framework that simplifies the computation of polynomial quadrature weights and errors using convenient bases and the undetermined coefficients method, applicable to various classical rules.

Contribution

It presents a novel, unified approach for calculating quadrature weights and errors for multiple classical rules using a specific basis and the undetermined coefficients method.

Findings

Framework effectively computes weights and errors for Newton-Cotes, Adams-Bashforth, Adams-Moulton, and Gaussian rules.

Simplifies the process of analyzing polynomial quadrature rules.

Demonstrates versatility across different classical quadrature methods.

Abstract

For the class of polynomial quadrature rules we show that conveniently chosen bases allow to compute both the weights and the theoretical error expression of a $n$-point rule via the undetermined coefficients method. As an illustration, the framework is applied to some classical rules such as Newton-Cotes, Adams-Bashforth, Adams-Moulton and Gaussian rules.