An exhaustive search for optimal multipliers

• Authors: Louis R. Moore;George S. Fishman
• Venue: WSC '84 Proceedings of the 16th conference on Winter simulation
• Year: 1984

This paper presents the results of an exhaustive search to find optimal multipliers A for the multiplicative congruential random number generator Zi @@@@ A Zi-1 (mod M) with prime modulus M &equil; 231-1. Since Marsaglia (1968) has shown that k-tuples from this and the more general class of linear congruential generators lie on sets of parallel hyperplanes, it has become common practice to evaluate multipliers in terms of their induced hyperplane structures. This study continues the practice and regards a multiplier as optimal if for k &equil; 2,...,6 and each set of parallel hyperplanes the Euclidean distance between adjacent hyperplanes does not exceed the minimal achievable distance by more than a prespecified amount. The concept of using this distance measure to evaluate multipliers orginated in the spectral test of Coveyou and MacPherson (1967) and has been used notably by Knuth (1981). However, the criterion of optimality defined here is considerably more stringent than the criteria that these writers proposed. First proposed by Lehmer (1951), the multiplicative congruential random number generator has come to be the most commonly employed mechanism for generating random numbers. Jannson (1966) collected the then known properties of these generators. Shortly there-after Marsaglia (1968) showed that all such generators share a common theoretical flaw and Coveyou and MacPherson (1967), Beyer, Roof and Williamson (1971), Marsaglia (1972) and Smith (1971) proposed alternative procedures for rating the seriousness of this flaw for individual multipliers. Later Niederreiter (1976, 1977, 1978a,b) proposed a rating system based on the concept of discrepancy, a measure of error used in numerical integration. With regard to empirical evaluation, Fishman and Moore (1982) described a comprehensive battery of statistical tests and illustrated how they could be used to detect local departures from randomness in samples of moderate size taken from these generators.