On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions

TL;DR

This paper presents a robust, step-by-step numerical scheme for computing Gaussian quadrature rules for Chebyshev sets, applicable to integrals with non-smooth functions, overcoming limitations of previous methods.

Contribution

It introduces a new, generally applicable method for computing Gaussian quadrature rules for complete Chebyshev sets, ensuring success and robustness.

Findings

The scheme guarantees convergence for complete Chebyshev sets.

Quadrature rules are computed incrementally, increasing exactness at each step.

Applicable to integrals with non-smooth, non-polynomial functions.

Abstract

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points and weights. Moreover, all weights are positive and the points lie inside the interval $[a,b]$. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first $k$ elements also form a Chebyshev set for each $k$. The points of the quadrature rule are computed one by one, increasing exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with non-smooth integrands that are not well approximated by polynomials.