Multiplicative Congruential Random Number Generators with Modulus $2^\beta$: An Exhaustive Analysis for $\beta = 32$ and a Partial Analysis for $\beta = 48$
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Medientyp:
E-Artikel
Titel:
Multiplicative Congruential Random Number Generators with Modulus $2^\beta$: An Exhaustive Analysis for $\beta = 32$ and a Partial Analysis for $\beta = 48$
Beteiligte:
Fishman, George S.
Erschienen:
American Mathematical Society, 1990
Erschienen in:
Mathematics of Computation, 54 (1990) 189, Seite 331-344
Sprache:
Englisch
ISSN:
0025-5718;
1088-6842
Entstehung:
Anmerkungen:
Beschreibung:
This paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli $2^{32}$ and $2^{48}$. Here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which $k$-tuples lie does not exceed the minimal achievable distance by more than 25 percent for $k = 2,\ldots, 6$. This criterion is considerably more stringent than prevailing standards of acceptability and leads to a total of only 132 multipliers out of the more than 536 million candidate multipliers that exist for modulus $2^{32}$ and to only 42 multipliers in a sample of about 67.1 million tested among the more than $351 \times 10^{11}$ candidate multipliers for modulus $2^{48}$. Section 1 reviews the basic properties of multiplicative congruential generators and $\S2$ describes worst case performance measures. These include the maximal distance between adjacent parallel hyperplanes, the minimal number of parallel hyperplanes, the minimal distance between $k$-tuples and the discrepancy. For modulus $2^{32}, \S3$ presents the ten best multipliers and compares their performances with those of two multipliers that have been recommended in the literature. Comparisons using packing measures in the space of $k$-tuples and in the dual space are also made. For modulus $2^{48}, \S4$ also presents analogous results for the five best multipliers and for two multipliers suggested in the literature.