Euler–Maclaurin expansions without analytic derivatives

Abstract

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).

Generalizing the trapezoidal rule in the complex plane

“…Exact values for these coefficients (as well as for the 5-line hexagonal counterpart discussed next) are given in Table 1. 5 Table 1 Weights to use in the 5-line periodic TR generalizations…”

“…These may be carried out using analytic derivatives, or (requiring no analytic differentiations) with finite difference (FD) approximations. Recent articles analyze the cases when these (equispaced) FD stencils use data points only within the interval [6], extend only in the direction of the line of integration [5], or extend over a region of the complex plane surrounding each end point [4].…”

“…If not, it would be more cost effective to double and triple the number of nodes in the 1-line TR than to apply the 3-line and 5-line TR versions, respectively 8. Appendix A in[5] sketches three derivations with references to still further ones.…”