3.3.4 THE SPECTRAL TEST 101 (Note that the (Web site directory)
3.3.4 THE SPECTRAL TEST 101 (Note that the first row Vi has actually gotten longer in this transformation, although eventually the rows of U should get shorter.) The next fourteen iterations of step S5 have (j, Q, qs, qs) = (2, -2, *, 0), (37% 3, *), (1, f, -10, -11, (2, -1, *, –6), (3, –1,O, $1, (1, *t 0,2), (2,0, *, -1): (3,3,4, $1, (1, *, O,O), (2, -5, *I 01, (3,1,0, *It (1, *, -3, -11, (2,0, *, 01, (3,&O, *I. Now the transformation process is stuck, but the rows of the matrices have become significantly shorter: -1479 -888874 601246 -2994234 u = -3022 -2809871 438109 (34) -227 -983 -854296 -9749816 -1707736 The search limits (zi, 22, zs) in step S8 turn out to be (O,O, l), so Us is the shortest solution to (33); we have v3 = J2272 + 9832 + 1302 zz 1017.21089. Note that only a few iterations were needed to find this value, although condition (33) looks quite difficult to deal with at first glance. All points (Un, Un+l, Un+2) produced by this random number generator lie on a family of parallel planes about 0.001 units apart. E. Ratings for various generators. So far we haven t really given a criterion that tells us whether or not a particular random number generator passes or flunks the spectral test. In fact, this depends on the application, since some applications demand higher resolution than others. It appears that vt 2 230/t for 2 2 t 5 6 will be quite adequate in most applications (although the author must admit choosing this criterion partly because 30 is conveniently divisible by 2, 3, 5, and 6). For some purposes we would like a criterion that is relatively independent of m, so we can say that a particular multiplier is good or bad with respect to the set of all other multipliers for the given m, without examining any others. A reasonable figure of merit for rating the goodness of a particular multiplier seems to be the volume of the ellipsoid in t-space defined by the relation (zim -22~ - * * . - xtut-1)2 + x; + . . . + xt < u;, since this volume tends to indicate how likely it is that nonzero integer points (zi, . . . , ~-corresponding to solutions of (15)-are in the ellipsoid. We therefore propose to calculate this volume, namely +I2 u; pt = (t/2)! m (35) as an indication of the effectiveness of the multiplier a for the given m. In this formula. (a)! = (a)(; -l)…(&G, for t odd. (36)