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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. It may be worthwhile to illustrate this approach to determining the discretization error with a simple example. 6) 0 + yZ2,and y = (g1, y2, y3, y4) is a vcctor of position aiitl 41 SATELLITE ORBIT TRAJECTORIES velocity of the vehicle in a rectangular coordinate system. [$-y] 0 O1 0 P m 0 r5 0 0 where h is the vector of errors corresponding t o y.
I n the absence of an absolute standard it is SATELLITE ORBIT TRAJECTORIES 37 difficult to detcrmine this optimum step sizc. Moreover, thcrc arc ccrtain systematic errors, such as thosc caused by cancellation, which are not uncovered by simply reducing tho step size. Thus it is possible for two successive integrations to agree to N significant digits over a broad range of time values and yet the results may both be correct to somewhat fewer significant digits. Thus the consistency check does not provide a positive test of accuracy, although it does give useful information especially when it can be ascertained that round-off and cancellation errors are not major factors.
2) where = (af,/dy,). 2) is solved n times under n sets of initial conditions. If s,(T) represents the error in yz a t time T , the total truncation error will be given by z,(T) = J' k5 b k ( t ) X % L ( t ) dt =l 0 (i = 1, . . 3) where b k is the local truncation error a t time t, h , k ( t ) is the solution of ( 5 . 2 ) beginning a t the time t = T with initial conditions h&(T)= 6ak = 1 =0 (i = k), (i # k), the functions b k ( t ) are estimates of the local truncation error which for a multistep integration method will have the form Chp+'y(p+')(t).