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This is not recommended as a general procedure since it involves the storing of all dependent variables at each of the desired times, but it is useful when good accuracy using a single precision routine is required. 3 Double Precision Operations For an idealized problem in which all constants are assumed known exactly, a completely double precision program will allow the use of integration steps so small that truncation errors will be negligible without incurring the danger of round-off error accumulation.

They are closely related to the concept of periodic integrals. An approximation theory is then developed based on this assumption. The theory is characterized by the fact that it splits the description of the solutions into two distinct parts; one element is the description of the periodic surface, the other is a description of the position on the surface. 10) T , = Xz(&, 82, J ) . These functioiis are obtained to first order in J by an espansiori procedure for the corresponding periodic integrals.

Use of numerical methods which estimate the propagated truncat,ion error and the accumulated round-off error. The relative efficiency of these tests will be discussed in the following sections. 36 SAMUEL D. 1 Comparison with Analytic Formulas To determine the accuracy of a particular method, one can compare the results for a n idealized two-body motion where the analytic solution is known. 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.

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