Advances in Computers, Vol. 3

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This process is a t once much simpler than that described by Kopal, much less time consuming, and more convenient for machine computation. Moreover, it is based on a firm mathematical foundation. 4). For the n-body problem the matrix (uJ is not difficult to derive. 4) are also required to obtain the partial derivatives of the vehicle's position and velocity with respect to initial parameters. This use of the variational system for this purpose will be described in Section 6. 1). 1) can also be extended to a consideration of round-off errors following the method outlined in Section 3 but this will not be pursued here.

Thus for the two-body problem the energy integral is constant, and for the motion of a satellite about an oblate earth the vertical component of the angular momentum is invariant. These integrals are usually stable in the presence of errors so that errors in the position and velocity cancel when evaluated for the integral. A physical explanation of this lies in the fact that the major portion of the error is along the line of motion rather than perpendicular to it. Thus, the major error is accounted for by a time discrepancy so that the integral is actually a t a point on the orbit but a t a different time.

He also assumed symmetric rounding and fixed point arithmetic, but he was concerned primarily with hand computations. His results indicate that, for a fixed step size, the error growth is of order N3I2where N is the number of integration steps. 1 is used. Henrici [IS] has recently SAMUEL D. CONTE 26 made a study of round-off error propagation and it may be worthwhile to summarize some of his results here. , fixed point or floating point; (3) how the rounding operations are performed. For simplicity we consider second order equations of the form y“ = f(x, y) which are typical for the free flight problem.

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