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**Additional resources for Advances in Computers, Vol. 3**

**Example text**

11) POfn1. Let g, denote the numerically calculated values of y, and letf, and define the local round-off errors E, by ffkgn+k + . + aOgn = h2[bkfn+k * + .. + * PCLfn] + = f(z,, jj,), En. We now adopt a statistical model of round-off propagation. 12) where p , q are functions of x but independent of h, and p , u are functions of the step-size h but independent of x. 11) E(r,) = V(rn>= L5 h2 $ . 13) + O(h)) where T , = 5, - yn is the accumulated round-off error after n integration steps and m(z), v(x) are known functions which satisfy certain differential equations.

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. 6) is typical. If infinitely many places were carried by computing machines the results would be the same for either method. There is a considerable difference, however, if finitely many places are carried.

To put it differently, the error criterion for each of the programs is optimum and the results obtained are the best possible with a single precision program. The Cowell method required almost twice as many integrating steps as the Encke and Herrick methods but only about 15% more computing time than the Encke method.