Free web hosting services - 3.3.4 THE SPECTRAL TEST 91 essentially good to
3.3.4 THE SPECTRAL TEST 91 essentially good to one part in v2. Similarly, let l/z+ be the maximum distance between planes, taken over all families of parallel planes that cover all points { (4m7 s(z)lm w4Ym)); we shall call ~3 the accuracy in three dimensions. The t-dimensional accuracy vt is the reciprocal of the maximum distance between hyperplanes, taken over all families of parallel (t -1)-dimensional hyperplanes that cover all points {(s/m, s(z)/m, . . . , s - (z)/m)}. The essential difference between periodic sequences and truly random se- quences that have been truncated to multiples of l/v is that the accuracy of truly random sequences is the same in all dimensions, while the accuracy of periodic sequences decreases as t increases. Indeed, since there are only m points in the t-dimensional cube when m is the period length, we can t achieve a t-dimensional accuracy of more than about milt. When the independence of t consecutive values is considered, computer- generated random numbers will behave essentially as if we took truly random numbers and truncated them to lgv, bits, where tit decreases with increasing t. In practice, such varying accuracy is usually all we need. We don t insist that the lo-dimensional accuracy be 235, in the sense that all (235)10 possible lo-tuples (Un, &x+1,. , ) should be equally likely on a 35-bit machine; for such large . . &x+9 values of t we want only a few of the leading bits of (Un, Un+i, . . . , Un+t-i) to behave as if they were independently random. On the other hand when an application demands high resolution of the random number sequence, simple linear congruential sequences will necessarily be inadequate; a generator with larger period should be used instead, even though only a small fraction of the period will actually be generated. Squaring the period will essentially square the accuracy in higher dimensions, i.e., it will double the effective number of bits of precision. The spectral test is based on the values of z,+ for small t, say 2 2 t 2 6. Dimensions 2, 3, and 4 seem to be adequate to detect important deficiencies in a sequence, but since we are considering the entire period it seems best to be somewhat cautious and go up into another dimension or two; on the other hand the values of Vt for t > 10 seem to be of no practical significance whatever. (This is fortunate, because it appears to be rather difficult to calculate Vt when t 2 10.) Note that there is a vague relation between the spectral test and the serial test; for example, a special case of the serial test, taken over the entire period as in exercise 3.3.3-19, counts the number of boxes in each of 64 subsquares of Fig. 8(a). The main difference is that the spectral test rotates the dots so as to discover the least favorable orientation. We shall return to a consideration of the serial test later in this section. It may appear at first that we should apply the spectral test only for one suitably high value of t; if a generator passes the test in three dimensions, it seems plausible that it should also pass the 2-D test, hence we might as well omit the latter. The fallacy in this reasoning occurs because we apply more stringent conditions in lower dimensions. A similar situation occurs with the serial test: Consider a generator that (quite properly) has almost the same number of points
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