SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1168-1192, June 2022.
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 the 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 nonsmooth integrands that are not well approximated by polynomials.

中文翻译：

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线性无关函数切比雪夫集高斯求积规则的计算

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1168-1192 页，2022 年 6 月。
我们考虑计算对于区间 $[a,b]$ 上的一组线性独立函数 Chebyshev 精确的求积规则。切比雪夫集的一般理论保证了具有高斯性质的规则的存在，在某种意义上，$2l$ 基函数可以精确地与 $l$ 点和权重积分。此外，所有权重都是正数，并且点位于区间 $[a,b]$ 内。然而，这些点不是正交多项式或任何其他已知特殊函数的根，如在常规高斯求积的情况下。这些规则的特点是非线性方程组，早期的数值方法主要集中在为牛顿迭代寻找合适的起始值来求解这个系统。在本文中，我们描述了一种替代方案，该方案稳健且普遍适用于所谓的完整切比雪夫集。这些是有序的 Chebyshev 集，其中第一个 $k$ 元素也为每个 $k$ 形成一个 Chebyshev 集。正交规则的点是逐个计算的，增加了每一步规则的准确性。每一步都简化为找到单变量和单调函数的唯一根。因此，本文的方案保证成功。求积规则对于具有非光滑被积函数的积分很有意义，这些积分不能很好地被多项式逼近。每一步都简化为找到单变量和单调函数的唯一根。因此，本文的方案保证成功。求积规则对于具有非光滑被积函数的积分很有意义，这些积分不能很好地被多项式逼近。每一步都简化为找到单变量和单调函数的唯一根。因此，本文的方案保证成功。求积规则对于具有非光滑被积函数的积分很有意义，这些积分不能很好地被多项式逼近。