SIAM Review, Volume 64, Issue 1, Page 132-150, February 2022.
The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We review the subject from this point of view and show that this principle fails to predict the actual behavior in four of the best-known cases: Newton--Cotes, Clenshaw--Curtis, Gauss--Legendre, and Gauss--Hermite quadrature. New results include (i) the observation that $x^k$ is integrated accurately by the Newton--Cotes formula even though the Chebyshev polynomial $T_k(x)$ is not; (ii) the introduction of a parameter-free variant of band-limited quadrature for arbitrary integrands, which is demonstrated to have a factor $\pi/2$ advantage over Gauss quadrature in integrating complex exponentials; (iii) a theorem establishing that chopping the real line to a finite interval achieves $O(\exp(-Cn^{2/3}))$ convergence for $n$-point quadrature of Gauss--Hermite integrands, whereas for the Gauss--Hermite formula it is just $O(\exp(-Cn^{1/2}))$; and (iv) an explanation of how this result is consistent with the “optimality” of the Gauss--Hermite formula.

中文翻译：

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求积公式的精确性

SIAM 评论，第 64 卷，第 1 期，第 132-150 页，2022 年 2 月。
求积公式的标准设计原则是它们对于给定类的被积函数应该是精确的，例如固定次数的多项式。我们从这个角度回顾了这个主题，并表明这个原理无法预测四个最著名案例中的实际行为：Newton--Cotes、Clenshaw--Curtis、Gauss--Legendre 和 Gauss--Hermite 求积. 新结果包括 (i) 观察到 $x^k$ 被牛顿-科茨公式准确积分，即使切比雪夫多项式 $T_k(x)$ 不是；(ii) 为任意被积函数引入了带限正交的无参数变体，证明在积分复指数方面比高斯正交具有$\pi/2$ 的优势；(iii) 一个定理确定将实线切割成有限区间对于高斯--Hermite 被积函数的$n$-点求积实现$O(\exp(-Cn^{2/3}))$ 收敛，而对于Gauss--Hermite 公式就是 $O(\exp(-Cn^{1/2}))$; (iv) 解释这个结果如何与 Gauss-Hermite 公式的“最优性”相一致。