SIAM Review, Volume 63, Issue 1, Page 167-180, January 2021.
The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in nonperiodic cases. Commonly used improvements, such as Simpson\textquoteright s rule and the Newton--Cotes formulas, are not much (if at all) better than the even more classical quadrature formulas described by James Gregory in 1670. For increasing orders of accuracy, these methods all suffer from the Runge phenomenon (the fact that polynomial interpolants on equispaced grids become violently oscillatory as their degree increases). In the context of quadrature methods on equispaced nodes, and for orders of accuracy around 10 or higher, this leads to weights of oscillating signs and large magnitudes. This article develops further a recently discovered approach for avoiding these adverse effects.

中文翻译：

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提高梯形规则的精度

SIAM 评论，第 63 卷，第 1 期，第 167-180 页，2021 年 1 月。
梯形规则在等距节点使用函数值。它对于周期间隔上的积分非常准确，但在非周期情况下通常非常不准确。常用的改进，例如 Simpson\textquoterright 规则和 Newton--Cotes 公式，并不比 James Gregory 在 1670 年描述的更经典的正交公式好多少（如果有的话）。为了提高准确性，这些方法都受到龙格现象的影响（等距网格上的多项式插值随着次数的增加而剧烈振荡）。在等距节点上的正交方法的上下文中，对于大约 10 或更高数量级的精度，这会导致振荡符号的权重和大的幅度。