The Euler–Maclaurin (EM) formulae relate sums and integrals. Discovered nearly 300 years ago, they have lost none of their importance over the years, and are nowadays routinely taught in scientific computing and numerical analysis courses. The two common versions can be viewed as providing error expansions for the trapezoidal rule and for the midpoint rule, respectively. More importantly, they provide a means for evaluating many infinite sums to high levels of accuracy. However, in all but the simplest cases, calculating very high-order derivatives analytically becomes prohibitively complicated. When approximating such derivatives with finite differences (FD), the choice of step size typically requires a severe trade-off between errors due to truncation and to rounding. We show here that, in the special case of EM expansions, FD approximations can provide excellent accuracy without the step size having to go to zero. While FD approximations of low-order derivatives to high orders of accuracy have many applications for solving ODEs and PDEs, the present context is unusual in that it relies on FD approximations to derivatives of very high orders. The application to infinite sums ensures that one can use centred FD formulae (which are not subject to the Runge phenomenon).

中文翻译：

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没有解析导数的欧拉-麦克劳林展开式

欧拉-麦克劳林 (EM) 公式涉及和和积分。它们在近 300 年前被发现，多年来一直没有失去其重要性，现在经常在科学计算和数值分析课程中教授。可以将这两个常见版本视为分别为梯形规则和中点规则提供误差扩展。更重要的是，它们提供了一种以高准确度评估许多无限和的方法。然而，除了最简单的情况外，分析计算非常高阶的导数变得非常复杂。当用有限差分 (FD) 逼近此类导数时，步长的选择通常需要在截断和舍入导致的误差之间进行严格的权衡。我们在这里表明，在 EM 扩展的特殊情况下，FD 近似可以提供出色的精度，而步长不必为零。虽然低阶导数到高阶精度的 FD 逼近在求解 ODE 和 PDE 方面有很多应用，但目前的上下文是不寻常的，因为它依赖于对非常高阶导数的 FD 逼近。无穷和的应用确保可以使用居中的 FD 公式（不受龙格现象的影响）。