When the numerical solution of differential equations (DE's) for dynamical simulation is performed with short-word-length computing devices, it is essential that truncation (round-off) error be understood and controlled where possible. This paper presents a simple but enlightening analysis of truncation errors in the computer solution of DE's, an analysis which develops the rationale for the residue register, the distinguishing feature of the digital differential analyzer (DDA), for the suppression of (second-order) truncation error. The analysis is extended formally to complete the theory of nth order truncation error suppression in linear DE's. After discussing the application of residue retention techniques to higher order integration formulas, the utility of third-order truncation error suppression strategy is illustrated in the suppression of "limit cycle" oscillations that occur in marginally damped oscillatory systems. The use of residue retention is then shown to be more critical in the case of DE's with multiplicative nonlinearities. Finally, design strategies are discussed briefly.

In a recent paper, Hannington and Whitehead1describe the design of a floating point digital differential analyzer (DDA). There are, however, shortcomings in the design which we will describe in this correspondence.

A combinational two-dimensional array for the rapid generation of sines and cosines is described. The array is a spatial realization of a continued product algorithm which uses multiplication by (1 + jEk2-k). Using 10-ns logic, it is estimated that 16-bit sines and cosines may be generated in less than 2 μs.

de Lugish [1] has defined efficient algorithms in radix 2 for certain elementary functions such as Y[X,Y/X1/2, Y + lnX, Y.exp (X), etc. His technique requires a systematic 1-bit left shift of a partially converged result, together with two 4-bit comparisons to select a ternary digit for the next iteration. This selection of digits reduces the average number of full precision additions to about 1/3 of those required in conventional schemes [3]. This paper develops modified algorithms in radix 2 which are more efficient when the time for a full precision addition is comparable to the time for a shift and comparison. The modified procedure is developed for Y/X in detail where more than a 40 percent decrease in execution time is achieved for only a marginal increase in cost.

This correspondence presents generalized higher radix algorithms for some elementary functions which use fast parallel m-bit multipliers where radix = 2m. These algorithms are extensions of those iterative schemes which are based on multiplications by (1 + 2-i) and the use of prestored values of ln (1 + 2-i) and tan-1(2-i). The particular functions under consideration are y/x, y/x1/2, y. exp (x), y + 1n (x), sin (x) and cos (x) [and hence tan (x)]. The extended algorithms rely on multiplication by (1 + dir-k) where di, 0 ≤ dir, is an m-bit integer. Using a simple selection procedure for di, simulations show that p(radix r) digits of a function may be generated, on the average, in less than p + 1 iterations.

This correspondence comments on the method of circle generation described by Armstrong [1], and discusses the methods by which satisfactory circles for computer graphics may be obtained with digital differential analyzers (DDA's) using a single-phase Adam's procedure.