Introduction

The question of numerical accuracy has a long history and is a difficult one. We are mainly concerned with the practical matter of employing convergent Taylor series in order to obtain statistically viable results. At the simplest level, the basic integration schemes can be tested for the two-body problem, whereas trajectories in larger systems exhibit error growth on short time-scales. However, it is possible to achieve solutions of high accuracy for certain small systems when using regularization methods. There are no generally agreed test problems at present but we suggest some desirable objectives, including comparison with Monte Carlo methods. Since large simulations inevitably require the maximum available resources, due attention must be paid to the formulation of efficient procedures. The availability of different types of hardware adds another dimension to programming design, which therefore becomes very specialized. Aspects of optimization and alternative hardware are also discussed, together with some performance comparisons.

Error analysis

It is a fact of computer applications that an error is made every time two arbitrary real numbers are added. Hence the task is to control the propagation of numerical errors and if possible keep them below an acceptable level. Since the N-body problem constitutes a system of non-linear differential equations the error growth tends to be exponential, as was demonstrated right at the outset of such investigations [Miller, 1964] and emphasized in a subsequent study [Miller, 1974].