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Extra resources for Advances in Computers, Vol. 3
CONTE On the other hand they suffer from a number of disadvantages, both theoretical and practical. From the theoretical aspect, although mathematicians and astronomers have investigated this problem for hundreds of years, no theoretical proof of convergence of the series expansions has yet been given nor have any estimates of the truncation error in these expansions been obtained, except for the special case of equatorial orbits. In addition the general perturbation methods do not naturally lend themselves to the inclusion of nonconservative forces such as drag and this imposes a severe limitation on the type of orbit to which they are applicable.
This will serve to increase the speed and minimize the accumulation of round-off error. ( 2 ) It should be capable of automatically changing the step size based on an adequate estimate of the local truncation error. (3) The procedure for changing the step size or for restarting should be as inexpensive as possible. (4) Adequate safeguards should be provided against excessive accumulation of round-off error. (5) Provision should be made for the output of information at any desired time based on the use of interpolation formulas.
Normally b(t) will have to be obtained numerically since the solution y(t) will not be known analytically. 6) leads t o circular motion for which an explicit solution is known. The solution is given by y ( t ) = (cos t, -sin 2, sin t, cos t ) and the vector b(t) by (sin t, cos t , -cos t , sin t). 7) is then h(t) = (t sin t , t cos t, - t cos t, t sin t). The accumulated truncation error e(t) a t any time t is thus seen to oscillate with a period of 27r and t o grow linearly with time. Its numerical value can be obtained from the formula e(t) = -ch5h(t), where h is the iiitegration step size.
Advances in Computers, Vol. 3