Mixed Congruential Random Number Generators for Decimal Machines

Abstract

Random number generators of the mixed eongruential type have recently been proposed. They appear to have some advantages over those of the multiplieative congruential type, but they have not been thoroughly tested. This paper summarizes the results of extensive testing of these generators which has been carried out on a decimal machine. Most results are for word length 10, and special attention is given to simple multipliers which give fast generators. But other word lengths and many other multipliers are considered. A variety of t~dditive constants is also used. It turns out that these mixed generators, in contrast to the multiplieative ones, are not consistently good from a statistical point of view. The cases which are bad seem to belong to a well-defined class which, unfortunately, includes most of the generators associated with the simple multipliers. However, a surprise result is that all generators associated with one of the simplest and fastest multipliers, namely 101, turn out to be consistently good for word lengths greater than seven digits. A final section of the paper suggests a simple theoretical explanation of these experimental results. Generators To Be TestedAlmost all random number generators that are used in practice can be obtained as special cases of the following procedure. One begins with the non-negative integers x0, a, c, and m, where m is the largest. Then one defines xl, x2, ... to be the non-negative integers less than m generated by xi+l-= ax~+c (modm) i = 1,2, -.-Finally, the sequence xo/m, x~/m, •. • is taken to be the sequence of random numbers. Tile hope is that the parameters x0, a, c, and m have been chosen so that the resulting sequence appears to be drawn at random from the uniform distribution on [0, 1]. Such a sequence is often called "pseudo-random." A general treatment of these generators is given in [6], along with an extensive bibliography. In practice it is especially convenient to choose m according to the particular computer being used, e.g. 101° or 2 ~5, and then the problem is to choose the remaining parameters x0, a, and c so that the period of the resulting sequence is as great as possible. Finally, one tests subsequences to see if they appear to be random.Until recently the only cases considered were those associated with choosing e = 0. Corresponding generators arc called "multiplieative" congruential generators. By choosing x, and a according to certain specifications [6], one can ensure that the resulting sequences are as long as is possible for this case, c = 0. For example, the maximal periods are 5 X 10 s when m = 10 ~°, and 23s when m = 2 sh. Using m = 10 r" on a decimal machine, the fastest generators yielding maximal •

Mixed Congruential Random Number Generators for Binary Machines

“…More complicated values of c tend to improve the statistical behavior. The improvement was slight and similar to what was reported earlier in [1]. So no further results will be given here.…”

“…However, the effect of changing either x0 or c was also tested. As expected both from the theory and on the basis of earlier results with a decimal machine [1], changing x0 had no effect. But changing c does have an effect.…”

“…In fact,, some mixed generators are completely unacceptable, including most of those associated with the "fast" multipliers of the form 10 ~ + 1 or 2 ~ -V 1. Details are given in [1] for the ease of a decimal machine. The purpose of this paper is to present the corresponding results for the ease of a binary machiac.…”

“…The same point should have been made more explicit in connection with the multiplier 101 for a decimal machine [1].…”

Some New Results in Pseudo-Random Number Generation

“…Proposed generators would be subjected to certain statistical tests, typically a ×~ test for uniformity, and tests of means, variances, and serial correlation. Muny generators have been reported in the literature as being satisfactory from the standpoint of these tests [1][2][3][4][5][6][7][8][9].…”