90 RANDOM NUMBERS 3.3.4 Fig. 8. (a) The (Free web design)
90 RANDOM NUMBERS 3.3.4 Fig. 8. (a) The two-dimensional grid formed by all pairs of successive points (Xn, X,+1), when Xn+l = (137X, + 187) mod 256. (b) The three-dimensional grid of triplets (Xn, X=+1, Xn+z). [Illustrations courtesy of Bruce G. Baumgart.] Perhaps the most striking thing about the pattern of boxes in Fig. 8 is that we can cover them all by a fairly small number of parallel lines; indeed, there are many different families of parallel lines that will hit all the points. For example, a set of 20 nearly vertical lines will do the job, as will a set of 21 lines that tilt upward at roughly a 300 angle. We commonly observe similar patterns when driving past farmlands that have been planted in a systematic manner. If the same generator is considered in three dimensions, we obtain 256 points in a cube, obtained by appending a height component s(s(z)) to each of the 256 points (5, S(X)) in the plane of Fig. 8(a), as shown in Fig. 8(b). Let s imagine that this 3-D crystal structure has been made into a physical model, a cube that we can turn in our hands; as we rotate it, we will notice various families of parallel planes that encompass all of the points. In the words of Wallace Givens, the random numbers stay mainly in the planes. At first glance we might think that such systematic behavior is so nonrandom as to make congruential generators quite worthless; but more careful reflection, remembering that m is quite large in practice, provides a better insight. The regular structure in Fig. 8 is essentially the grain we see when examining our random numbers under a high-power microscope. If we take truly random numbers between 0 and 1, and round or truncate them to finite accuracy so that each is an integer multiple of l/v for some given number V, then the t- dimensional points (1) we obtain will have an extremely regular character when viewed through a microscope. Let l/v2 be the maximum distance between lines, taken over all families of parallel straight lines that cover the points {(z/m, s(z)/m)} in two dimen- sions. We shall call ~2 the two-dimensional accuracy of the random number generator, since the pairs of successive numbers have a fine structure that is
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