The Euler–Maclaurin (EM) formulas relate sums to integrals, and provide both insights and computational opportunities. Discovered nearly 300 years ago, these have lost none of their importance over the years. However, using many terms requires derivatives of high orders. For most non-trivial functions, the algebraic complexity grows prohibitively fast with repeated differentiations. We will here see that, with only little loss of accuracy, these analytic differentiations can be replaced by equispaced finite difference (FD) approximations. The three cases we consider are:

Case 1: Numerical contour integration in the complex plane. Using only function values at grid points spaced h apart, the accuracy of the trapezoidal rule (TR) can readily be improved from $O(h^2)$ to the $O(h^{30})$-$O(h^{50})$ range.

Case 2: High-accuracy approximations of infinite sums.

Case 3: Improving the Trapezoidal Rule for integration with equispaced nodes along an interval.
The TR weights are $h\{\frac{1}{2},1, 1, 1,\dots,1,\frac{1}{2}\}$. By using a larger number of ‘non-trivial’ weights near each interval end, already James Gregory (1638-1675) showed that the accuracy could be greatly improved. However, like with the Newton-Cotes formulas, the magnitudes of the weights grow quickly with increasing accuracy orders. We describe how this growth can be greatly reduced, giving schemes with all weights small and positive for accuracy orders up to around $O(h^{20})$. This includes an $O(h^{10})$ scheme for which all the weights are simple rational numbers.