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).