An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31}  - 1$

Fishman, George A; Moore, Louis R.

doi:10.1137/0907002

Cited by 175 publications

(69 citation statements)

References 18 publications

(4 reference statements)

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“…L'Ecuyer and Simard (2007) also reported that both MRG32k3a and some large-order MRGs (including DX-1597 and DX-47, proposed in Deng 2005) passed the stringent Crush battery of tests in the TestU01 library. In addition, they reported that all LCGs, including the "best" (under spectral tests) LCGs found in Fishman and Moore (1986), failed the Crush battery badly.…”

Section: Copyrightmentioning

confidence: 99%

“…It is common to perform a theoretical test to search for the best generator among a class of LCGs or smallorder MRGs based on their lattice structures in a specific dimension t; for example, the popular spectral test is to calculate the maximum distance d t k between adjacent parallel hyperplanes for a chosen dimension t. A measure suggested by Fishman and Moore (1986) for comparing generators with different values of modulus is the spectral value defined as S t k ≡ d * t k /d t k , where d * t k is the theoretical minimum of d t k ; however, the exact values of d * t k are known only for small t-say, t ≤ 8. For MRGs of any orders, the commonly used performance measures are the maximum distance d t k and the approximated spectral value p −k/t /d t k .…”

Section: Copyrightmentioning

confidence: 99%

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Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1

Deng

Shiau

2012

INFORMS Journal on Computing

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T he performance of a maximum-period multiple recursive generator (MRG) depends on the choices of the recurrence order k, the prime modulus p, and the multipliers used. For a maximum-period MRG, a largeorder k not only means a large period length (i.e., p k − 1) but, more importantly, also guarantees the equidistribution property in high dimensions (i.e., up to k dimensions), a desirable feature for a good random-number generator. As to generating efficiency, in addition to the multipliers, some special choices of the prime modulus p can significantly speed up the generation of pseudo-random numbers by replacing the expensive modulo operation with efficient logical operations. To construct efficient maximum-period MRGs of a large order, we consider the prime modulus p = 2 31 − 1 and, via extensive computer search, find two large values of k, 7 499 and 20 897, for which p k − 1 can be completely factorized. The successful search is achieved with the help of some results in number theory as well as some modern factorization methods. A general class of MRGs is introduced, which includes several existing classes of efficient generators. With the factorization results, we are able to identify via computer search within this class many portable and efficient maximum-period MRGs of order 7 499 or 20 897 with prime modulus 2 31 − 1 and multipliers of powers-of-two decomposition. These MRGs all pass the stringent TestU01 test suite empirically.

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Section: Copyrightmentioning

confidence: 99%

Section: Copyrightmentioning

confidence: 99%

Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1

Deng

Shiau

2012

INFORMS Journal on Computing

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“…This is done using the RMULT and RMOD commands. Given a prime modulus of 2 3 1 -1, Fishman and Moore (1985) identified the following five "best" multipliers out of more than 267 million that were tested:…”

Section: Standard Normal Deviate Generationmentioning

confidence: 99%

Statistical simulation on microcomputers

Bradley

Senko

Stewart

1990

Behavior Research Methods, Instruments, & Computers

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The statistical simulation program DATASIM is designed to conduct large-scale sampling experiments on microcomputers. Monte Carlo procedures are used to investigate the Type I and Type IT error rates for statistical tests when one or more assumptions are systematically violatedassumptions, for example, regarding normality, homogeneity of variance or covariance, minimum expected cell frequencies, and the like. In the present paper, we report several initial tests of the data-generating algorithms employed by DATASIM. The results indicate that the uniform and standard normal deviate generators perform satisfactorily. Furthermore, KolmogorovSmirnov tests show that the sampling distributions of z, t, F, x 2 , and r generated by DATASIM simulations follow the appropriate theoretical distributions. Finally, estimates of Type I error rates obtained by DATASIM under various patterns of violations of assumptions are in close agreement with the results of previous analytical and empirical studies. These converging lines of evidence suggest that DATASIM may well prove to be a reliable and productive tool for conducting statistical simulation research.Although statistical simulations have traditionally been conducted on mainframe or minicomputers, the increasing availability of powerful yet inexpensive personal computers provides another tool for researchers in this area. A laboratory equipped with 20 to 30 microcomputers has computing power roughly comparable to a minicomputer running in a timesharing environment. Although the individual microcomputer is unable to match the minicomputer in sheer number-crunching performance, the availability of multiple processors allows many simulations to be run concurrently. The microcomputer's disadvantage in processing speed is therefore largely offset by an advantage in distributed processing. An additional benefit is that microcomputers allow accurate real-time control of stimulus and response events for conducting on-line research, whereas timesharing systems are necessarily deficient in this respect. Furthermore, the linking of microcomputers to a network can provide the same access to centralized resources (software, databases, communications, traffic logging, etc.) that is available with timesharing systems.We have 25 PCs networked together that students use to conduct on-line research and statistical simulations. 1 A general purpose data simulator is used to perform the statistical simulations. The software, called DATASIM (Bradley, 1988), is designed to generate large numbers of randomized datasets from populations specified by the user, and to "capture" and analyze the sampling distributions of statistics computed on the individual datasets.'This provides a convenient way to conduct sampling experiments investigating the effects of violating assumptions on the Type I or Type II error rates of statistical tests (Bradley, 1989a(Bradley, , 1989b.Correspondence may be addressed 10 Drake R. Bradley. Department of Psychology, Bales College. Lewiston. ME 04240.Copyright 1990 Psycho...

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“…For small moduli it is possible to conduct an exhaustive search. Fishman and Moore [9], Fishman [11], Sezgin [25][26][27], Warford [28], L'Ecuyer et al [18] and Kao and Wong [13] give examples of this application. For very large modulus values most authors apply random searches.…”

Section: Introductionmentioning

confidence: 99%

“…3. Fishman [9] uses the multiplicative inverse for multipliers noting that for each multiplier there is an inverse a * such that aa * ≡ 1 (mod M ). The multipliers a and a * produce the same sequence but in the reverse order.…”

Section: Introductionmentioning

confidence: 99%

A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

Sezgin

2004

BIT Numerical Mathematics

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Abstract.This paper presents a method of systematic search for optimal multipliers for congruential random number generators. The word-size of computers is a limiting factor for development of random numbers. The generators for computers up to 32 bit wordsize are already investigated in detail by several authors. Some partial works are also carried out for moduli of 2 48 and higher sizes. Rapid advances in computer technology introduced recently 64 bit architecture in computers. There are considerable efforts to provide appropriate parameters for 64 and 128 bit moduli. Although combined generators are equivalent to huge modulus linear congruential generators, for computational efficiency, it is still advisable to choose the maximum moduli for the component generators. Due to enormous computational price of present algorithms, there is a great need for guidelines and rules for systematic search techniques. Here we propose a search method which provides 'fertile' areas of multipliers of perfect quality for spectral test in two dimensions. The method may be generalized to higher dimensions. Since figures of merit are extremely variable in dimensions higher than two, it is possible to find similar intervals if the modulus is very large. AMS subject classification (2000): 65C10, 65Y05, 68Q22, 11A55.

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An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$

Cited by 175 publications

References 18 publications

Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1

Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1

Statistical simulation on microcomputers

A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

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An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$

Cited by 175 publications

References 18 publications

Large-Order Multiple Recursive Generators with Modulus 231 − 1

Large-Order Multiple Recursive Generators with Modulus 231 − 1

Statistical simulation on microcomputers

A Method of Systematic Search for Optimal Multipliers in Congruential Random Number Generators

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Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1

Large-Order Multiple Recursive Generators with Modulus 2³¹ − 1