Quadrature as a least-squares and minimax problem

TL;DR

This paper explores the weights of interpolatory quadrature rules as solutions to least-squares and minimax problems, establishing their relationship and introducing parameters to evaluate and compare different quadrature rules.

Contribution

It introduces a novel framework linking quadrature weights to least-squares and minimax solutions, and proposes parameters for assessing and comparing quadrature rules.

Findings

The weights are the least-squares solutions of the fundamental system.

The minimax solution has equal residual norm to the least-squares solution.

Parameters like the rule's angle effectively compare different quadrature rules.

Abstract

The vector of weights of an interpolatory quadrature rule with $n$ preassigned nodes is shown to be the least-squares solution $\omega$ of an overdetermined linear system here called {\em the fundamental system} of the rule. It is established the relation between $\omega$ and the minimax solution $\stackrel{\ast}{z}$ of the fundamental system, and shown the constancy of the $\infty$-norms of the respective residual vectors which are equal to the {\em principal moment} of the rule. Associated to $\omega$ and $\stackrel{\ast}{z}$ we define several parameters, such as the angle of a rule, in order to assess the main properties of a rule or to compare distinct rules. These parameters are tested for some Newton-Cotes, Fej\'er, Clenshaw-Curtis and Gauss-Legendre rules.