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A second disadvantage is that the coefficient in the principal part of the local truncation error is an extremely complicated function of y and its derivatives. I n trajectory calculations it is important to use the largest integrating step possible for a required accuracy in order to reduce both machine time and round-off error. The difficulty of evaluating the error term makes the choice of the proper integrating step difficult. Various methods have been proposed for eliminating this defect. One suggestion is that the integration for yt+l be performed first with a step of length h and then with two steps of length h / 2 .

The assumption is then made that similar accuracies are attainable on nonidealized orbits. This approach is of limited value for the following reasons : (1) The two-body problem is esseiitially two-dimensional in nature. The effects and the accuracy on true three-dimensional orbits may be significantly different. (2) This method provides no check a t all on the Encke and Variation of Parameter methods which use two-body motion as a reference orbit. The only special perturbation method which does not become degenerate for two-body motion is the Cowell method.

Round-off error will be discussed in more detail later. To summarize, Runge-Kutta methods are capable of good accuracy, follow the solution curves very well, do not suffer from excessive growth of round-off error, and can change the stepsize at will. On the other hand the usual method is of fourth order so that relatively small step sizes must be taken: it is slow because four derivative evaluations are required at each step; and there is no simple criterion for deciding on how to change the step size.

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