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Extra info for Advances in Computers, Vol. 3
A very simple, effective, and inexpensive method for reducing accumulated round-off error in any numerical integration method is that of double precision accumulations. I n this procedure the dependent variable is always carried in double precision form, the derivative calculations are carried out in single precision, and the increment Ayi is added to the dependent variable yi in double precision. This process is effective in either fixed point or floating point operations. Round-off error will be discussed in more detail later.
See, for example, the comparison of special perturbation methods on a typical earth satellite orbit given in Section 5. In the latter case the use of reference orbits more complicated than conic sections holds promise of yielding comparable computational efficiency. The major disadvantages of the Encke method are its complexity, and the need for special care in handling the analytic expressions for the reference orbit in order to obtain maximum efficiency. 3 Variation of Parameter Methods In the Encke method the elements of a reference orbit are determined a t a fixed instant of time and the deviations from this reference orbit computed.
AOYtr = h2[Pkfn+k + .. 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.