Monte Carlo pp 587-682 | Cite as

Generating Pseudorandom Numbers

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)


Every Monte Carlo experiment relies on the availability of a procedure that supplies sequences of numbers from which arbitrarily selected nonoverlapping subsequences appear to behave like statistically independent sequences and where the variation in an arbitrarily chosen subsequence of length k (≥1) resembles that of a sample drawn from the uniform distribution on the k-dimensional unit hyper-cube \({\mathcal{I}^k}\). The words “appear to behave” and “resemble” alert the reader to yet another potential source of error that arises in Monte Carlo sampling. In practice, many procedures exist for generating these sequences. In addition to this error of approximation, the relative desirability of each depends on its computing time, on its ease of use, and on its portability By portability, we mean the ease of implementing a procedure or algorithm on a variety of computers, each with its own hardware peculiarities.


Random Number Generator Pseudorandom Number Primitive Root Pseudorandom Number Generator Multiplicative Inverse 
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  1. Afflerbach, L. and H. Grothe (1985). Calculation of Minkowski-reduced lattice bases, Comput, 35, 269–276.CrossRefGoogle Scholar
  2. Afflerbach, L. and R. Weilbächer (1989). The exact determination of rectangle discrepancy for linear congruential pseudorandom generators, Math. Comp, 53, 343–354.CrossRefGoogle Scholar
  3. Afflerbach, L. (1991). Private communication.Google Scholar
  4. Anderson, T.W. and D.A. Darling (1952). Asymptotic theory of goodness of fit criteria based on stochastic processes, Ann. Math. Statist, 23, 191–211.Google Scholar
  5. Anderson, T.W. and D.A. Darling (1954). A test of goodness of fit, J. Amer. Statist. Assoc, 49, 765–769.CrossRefGoogle Scholar
  6. André, D.A., G.L. Mullen, and H. Niederreiter (1990). Figures of merit for digital multistep pseudorandom numbers, Math. Comp, 54, 737–748.Google Scholar
  7. Beyer, W.A. (1972). Lattice structure and reduced bases of random vectors generated by linear recurrences, in Applications of Number Theory to Numerical Analysis, S.K. Zaremba ed., Academic Press, New York, pp. 361–370.Google Scholar
  8. Beyer, W.A. (1988). Private communication.Google Scholar
  9. Beyer, W.A., R.B. Roof, and D. Williamson (1971). The lattice structure of multiplicative congruential pseudorandom vectors, Math. Comp, 25, 345–363.CrossRefGoogle Scholar
  10. Borosh, I. and H. Niederreiter (1983). Optimal multipliers for pseudorandom number generation by the linear congruential method, BIT, 23, 65–74.CrossRefGoogle Scholar
  11. Bradley, G.H. (1993). Generating pseudorandom integers over an interval, Operations Research Department, Naval Postgraduate School, Monterey, CA.Google Scholar
  12. Brown, M. and H. Solomon (1979). On combining pseudorandom number generators, Ann. Statist, 7, 691–695.CrossRefGoogle Scholar
  13. Cassels, J.W.S. (1959). An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  14. Couture, R. and P. L’Ecuyer (1993). On the lattice structure of certain linear congruentialsequences related to AWC/SWB generators, University of Montreal, Canada.Google Scholar
  15. Couture, R., P. L’Ecuyer, and S. Tezuka (1991). On the distribution of k-dimension vectors for simple and combined Tauseworthe sequences, GERAD Tech. Rep. G-91–43, Groupe d’études et de Recherche en Analyse des Décisions, Montreal, Canada.Google Scholar
  16. Coveyou, R.R. (1970). Random number generation is too important to be left to chance, Stud. Appl. Math, 3, 70–111.Google Scholar
  17. Coveyou, R.R. and R.D. MacPherson (1967). Fourier analysis of uniform random number generators, J. ACM, 14, 100–119.CrossRefGoogle Scholar
  18. Devaney, R.L. (1986). An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings, Menlo Park, CA.Google Scholar
  19. Dieter, U. (1971). Pseudo-random numbers: the exact distribution of pairs, Math. Comp, 29, 827–833.CrossRefGoogle Scholar
  20. Dieter, U. (1975). How to calculate shortest vectors in a lattice, Math. Comp, 29, 827–833.CrossRefGoogle Scholar
  21. Dieter, U. and J.H. Ahrens (1977). Pseudorandom Numbers, University of Graz, Austria. Durst, M. (1989). Private communication.Google Scholar
  22. Dwass, M. (1958). On several statistics related to empirical distribution functions, Ann. Math Statist, 29, 188–191.CrossRefGoogle Scholar
  23. Eichenauer, J., J.H. Grothe, and J. Lehn (1988). Marsaglia’s lattice test and nonlinear congruential pseudorandom number generators, Metrika, 35, 241–240.CrossRefGoogle Scholar
  24. Eichenauer, J. and J. Lehn (1986). A non-linear congruential pseudorandom number generator, Statist. Papers, 27, 315–326.Google Scholar
  25. Eichenauer, J. and J. Lehn (1987). On the structure of quadratic congruential sequences, Manuscripta Math, 58, 129–140.CrossRefGoogle Scholar
  26. Eichenauer, J., J. Lehn, and A. Topuzoglu (1988). A nonlinear congruential pseudorandom number generator with power of two modulus, Math. Comp, 51, 757–759.CrossRefGoogle Scholar
  27. Eichenauer-Herrmann, J. and H. Niederreiter (1991). On the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math, 34, 243–249.CrossRefGoogle Scholar
  28. Eichenauer-Herrmann, J. and H. Niederreiter (1992). Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus, Math. Comp, 58, 775–779.CrossRefGoogle Scholar
  29. Ferrenberg, A.M., D.P. Landau and Y.J. Wong (1992). Monte Carlo simulations: hidden errors from “good” random number generators, Phys. Rev. Letters, 69, 3382–3384.CrossRefGoogle Scholar
  30. Fishman, G.S. (1990). Multiplicative congruential random number generators with modulus 2fl: an exhaustive analysis for ß = 32 and a partial analysis for ß = 48, Math. Comp, 54, 331–334.Google Scholar
  31. Fishman, G.S. and L.R. Moore (1982). A statistical evaluation of multiplicative congruential random number generators with modulus 231 — 1, J. Amer. Statist. Assoc, 77, 129–136.Google Scholar
  32. Fishman, G.S. and L.R. Moore (1986). An exhaustive analysis of multiplicative congruential random number generators with modulus 231 — 1, SIAM J. Sci. and Statist. Comput, 7, 24–45.CrossRefGoogle Scholar
  33. Fushimi, M. (1983). A reciprocity theorem on the random number generation based on m-sequences and its applications (in Japanese), Trans. Inform. Process Soc. Japan, 24, 576–579.Google Scholar
  34. Fushimi, M. (1989). An equivalence relation between Tausworthe and GFSR sequences and applications, Appl. Math. Letters, 2, 135–137.CrossRefGoogle Scholar
  35. Fushimi, M. (1990). Random number generation with the recursion X, = X_31,$ Xf_3q, J. Comp. Appl. Math, 31, 105–118.CrossRefGoogle Scholar
  36. Fushimi, M. and S. Tezuka (1983). The k-distribution of the generalized feedback shift register pseudorandom numbers, Comm. ACM, 26, 516–523.CrossRefGoogle Scholar
  37. Hörmann, W. (1994). Personal communication.Google Scholar
  38. Hörmann, W. and G. Derflinger (1993). A portable random number generator well suited for the rejection method, ACM Trans. Math. Software, 19, 489–495.CrossRefGoogle Scholar
  39. Hull, T.T. and A.R. Dobell (1962). Random number generators, SIAM Rev, 4, 230–254. Jannson, B. (1966). Random Number Generators, Almqvist and Wiksell, Stockholm.Google Scholar
  40. Keifer, J. (1961). On large deviations of the empiric d.f. of vector chance variables and a law of the iterated logarithm, Pacific J. Math, 11, 649–660.CrossRefGoogle Scholar
  41. Knuth, D. (1981). The Art of Computer Programming: Semi-numerical Algorithms, Vol. 2, 2nd ed., Addison-Wesley, Reading, MA.Google Scholar
  42. L’Ecuyer, P. (1986). Efficient and portable combined pseudo-ramdom number generatorsGoogle Scholar
  43. Rapport de recherche. DIUL-RR-8612, Université Laval, Quebec, Canada.Google Scholar
  44. L’Ecuyer, P. (1988). Efficient and portable combined pseudo-random number generators,Comm. ACM, 31, 742–749, 774.Google Scholar
  45. L’Ecuyer, P. (1990). Random numbers for simulation, Comm. ACM, 33, 85–97. L’Ecuyer, P. (1991). Private communication.Google Scholar
  46. L’Ecuyer, P. and F. Blouin (1988). Linear congruential generators or order k > 1. 1988Winter Simulation Conference Proceedings, IEEE Press, New York, pp. 432–439.Google Scholar
  47. L’Ecuyer, P. and F. Blouin (1990). Multiple recursive and matrix linear congruential genera-tors, Départment d’Informatique, Université Laval.Google Scholar
  48. L’Ecuyer, P. and S. Tezuka (1991). Structural properties for two classes of combined random number generators, Math. Comp, 57, 735–746.Google Scholar
  49. Lehmer, D.H. (1951). Mathematical methods in large-scale computing methods Ann. Comp. Lab. 26 141–146 Google Scholar
  50. Levene, H. (1952). On the power function of tests of randomness based on runs up and down, Ann. Math. Statistics, 23, 34–56.CrossRefGoogle Scholar
  51. Lewis, P.A., A.S. Goodman, and J.M. Miller (1969). A pseudorandom number generator for the System/360, IBM Syst. J, 8, 136–146.CrossRefGoogle Scholar
  52. Lewis, T.G. and W.H. Payne (1973). Generalized feedback shift register pseudorandom number algorithms, J. ACM, 20, 456–468.CrossRefGoogle Scholar
  53. Lidl, R. and H. Niederreiter (1986). Introduction to Finite Fields and Their Applications, Cambridge University Press, New York.Google Scholar
  54. Lipson, J.D. (1981). Elements of Algebra and Algebraic Computing, Addison-Wesley, Reading, MA.Google Scholar
  55. MacLaren, M.D. and G. Marsaglia (1965). Uniform random number generators J. Assoc. Comp. Mach. 12 83–89 Google Scholar
  56. McLeod, A.I. (1985). A remark on Algorithm AS183, an efficient and portable pseudorandom number generator, Appl. Statist, 34, 198–202.CrossRefGoogle Scholar
  57. Marsaglia, G. (1968). Random numbers fall mainly in the planes Proc. National Academy of Sciences U.S.A. 61 25–28 Google Scholar
  58. Marsaglia, G. (1972). The structure of linear congruential sequences, in Applications of Number Theory to Numerical Analysis, S.K. Zaremba ed., Academic Press, New York, pp. 249–285.Google Scholar
  59. Marsaglia, G., B. Narsimhan and A. Zaman (1990). A random number generator for PC’s Comput. Phys. Commun. 60 345–349 Google Scholar
  60. Marshall, A.W. and I.Olkin (1977). Majorization in multivariate distributions, Ann. Statist, 2, 1189–1200.CrossRefGoogle Scholar
  61. Mullen, G.L. and H. Niederreiter (1987). Optimal characteristic polynomials for digital multistep pseudorandom numbers, Comput, 39, 155–163.CrossRefGoogle Scholar
  62. Niederreiter, H. (1977). Pseudo-random numbers and optimal coefficients, Adv. Math, 26, 99–181.CrossRefGoogle Scholar
  63. Niederreiter, H. (1978a). The serial test for linear congruential pseudo-random numbers, Bull. Amer. Math. Soc., 84, 273–274.CrossRefGoogle Scholar
  64. Niederreiter, H. (1978b). Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84, 957–1041.CrossRefGoogle Scholar
  65. Niederreiter, H. (1982). Statistical tests for Tausworthe pseudo-random numbers, in Probability and Statistical Inference, W. Grossmann et al. eds. Reidel, Dordrecht, pp. 265274.Google Scholar
  66. Niederreiter, H. (1985). The serial test for pseudo-random numbers generated by the linear congruential method, Numer. Math., 46, 51–68.Google Scholar
  67. Niederreiter, H. (1987). A statistical analysis of generalized feedback shift register pseudorandom number generators, SIAM J. Sci. Stat. Comput., 8, 1035–1051.CrossRefGoogle Scholar
  68. Niederreiter, H. (1988a). The serial test for digital k-step pseudorandom numbers, Mathematical J. Okayama University, 30, 93–119.Google Scholar
  69. Niederreiter, H. (1988b). Statistical independence of nonlinear congruential pseudorandom numbers, Monatshefte fir Mathematik, 106, 149–159.CrossRefGoogle Scholar
  70. Niederreiter, H. (1989). The serial test for congruential pseudorandom numbers generated by inversions, Math. Comp, 52, 135–144.CrossRefGoogle Scholar
  71. Niederreiter, H. (1992). Recent trends in random number and random vector generation, Ann. Oper. Res, 31, 323–346.CrossRefGoogle Scholar
  72. Park, S.K. and K.W. Miller (1988). Random number generators: good ones are hard to find, Comm. ACM, 31, 1192–1201.CrossRefGoogle Scholar
  73. Payne, W.H J.R. Rabung and T.P Bogyo (1969). Coding the Lehmer pseudorandom number generator, Comm, ACM,12 85–86.Google Scholar
  74. Schrage, L. (1979). A more portable fortran random number generator, ACM Trans. Math. Software, 5, 132–138.CrossRefGoogle Scholar
  75. Smith, C.S. (1971). Multiplicative pseudorandom number generators with primal modulus, J. ACM, 18, 586–593.CrossRefGoogle Scholar
  76. Tausworthe, R.C. (1965). Random numbers generated by linear recurrence modulo two, Math. Comp, 19, 201–209.CrossRefGoogle Scholar
  77. Tezuka, S. and P. L’Ecuyer (1991). Efficient and portable combined Tausworthe randomnumber generators, ACM Trans. Modeling and Comput. Simul, 2, 99–112.CrossRefGoogle Scholar
  78. Whittlesley, J.R.B (1968). A comparison of the correlational behavior of random numbergenerators for the IBM 360, Comm. ACM,11 641–644.Google Scholar
  79. Wichmann, B.A. and I.D. Hill (1982). An efficient and portable pseudo-random numbergenerator, Appl Statist, 31, 188–190; correction, (1984) Appl. Statist, 33, 123.CrossRefGoogle Scholar
  80. Wolfowitz, J. (1944). Asymptotic distribution of runs up and down, Ann. Math. Statistics, 7, 1052–1057.Google Scholar
  81. Zierler, N. and J. Brillhart (1968). On primitive trinomials (mod 2), I, Inform. Contr, 13, 541–554.CrossRefGoogle Scholar
  82. Zierler, N. and J. Brillhart (1969). On primitive trinomials (mod 2), II, Inform. Contr, 14, 566–569.Google Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

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