Efficient and portable combined random number generators

In this paper we present an efficient way to combine two or more Multiplicative Linear Congruential Generators (MLCGs) and propose several new generators. The individual MLCGs, making up the proposed combined generators, satisfy stringent theoretical criteria for the quality of the sequence they produce (based on the Spectral Test) and are easy to implement in a portable way. The proposed simple combination method is new and produces a generator whose period is the least common multiple of the individual periods. Each proposed generator has been submitted to a comprehensive battery of statistical tests. We also describe portable implementations, using 16-bit or 32-bit integer arithmetic. The proposed generators have most of the beneficial properties of MLCGs. For example, each generator can be split into many independent generators and it is easy to skip a long subsequence of numbers without doing the work of generating them all.

[1]  R. W. Driver,et al.  The Influence of Computer Graphics on the Recall of Information , 1983, MIS Q..

[2]  Harald Niederreiter,et al.  Optimal multipliers for pseudo-random number generation by the linear congruential method , 1983 .

[3]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[4]  W. Wilcox.  Numbers and the News: Graph, Table or Text? , 1964 .

[5]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[6]  Shu Tezuka,et al.  The k-distribution of generalized feedback shift register pseudorandom numbers , 1983, CACM.

[7]  I. D. Hill,et al.  An Efficient and Portable Pseudo‐Random Number Generator , 1982 .

[8]  Roy E. Welsch.  Graphics for data analysis , 1976, Comput. Graph..

[9]  U. Dieter,et al.  How to calculate shortest vectors in a lattice , 1975 .

[10]  G. S. Fishman,et al.  A Statistical Evaluation of Multiplicative Congruential Random Number Generators with Modulus 231 — 1 , 1982 .

[11]  W. H. Payne,et al.  Coding the Lehmer pseudo-random number generator , 1969, CACM.

[12]  Stephen D. Roberts,et al.  Implementing a portable FORTRAN Uniform (0,1) generator , 1983 .

[13]  Richard E. Nance,et al.  Some Experimental Observations on the Behavior of Composite Random Number Generators , 1978, Oper. Res..

[14]  G. Marsaglia.  Random numbers fall mainly in the planes. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Arne Thesen.  An efficient generator of uniformly distributed random variates between zero and one , 1985 .

[16]  Kenneth Shostack,et al.  Management by Computer Graphics. , 1971 .

[17]  H. Niederreiter.  Quasi-Monte Carlo methods and pseudo-random numbers , 1978 .

[18]  Paul Bratley,et al.  A guide to simulation , 1983 .

[19]  R. R. Coveyou,et al.  Fourier Analysis of Uniform Random Number Generators , 1967, JACM.

[20]  Linus Schrage,et al.  A More Portable Fortran Random Number Generator , 1979, TOMS.

[21]  Hirotaka Takeuchi,et al.  New Promise of Computer Graphics , 1980 .

[22]  Paul Bratley,et al.  A guide to simulation (2nd ed.) , 1986 .

[23]  Dileep R. Sule,et al.  New lagged product test for random number generators , 1985 .

[24]  Thomas S. Tullis,et al.  An Evaluation of Alphanumeric, Graphic, and Color Information Displays , 1981 .

[25]  Ruth Gertrude Strickland.  A Study of the Possibilities of Graphs As a Means of Instruction in the First Four Grades of the Elementary School , 1938 .

[26]  R. N. Clark.  A pseudorandom number generator , 1985 .

[27]  Roy E. Welsch,et al.  Graphics for data analysis , 1974, SIGGRAPH '74.

[28]  Linus Schrage,et al.  A guide to simulation , 1983 .

[29]  J. N. Washburne.  An experimental study of various graphic, tabular, and textual methods of presenting quantitative material , 1927 .

[30]  L. R. Moore,et al.  An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$ , 1986 .

[31]  Doan T. Modianos,et al.  Random Number Generation on Microcomputers , 1984 .

[32]  P. A. W. Lewis,et al.  A Pseudo-Random Number Generator for the System/360 , 1969, IBM Syst. J..

[33]  Herbert A. Simon,et al.  Information-processing models of cognition , 1981, J. Am. Soc. Inf. Sci..