Advertisement

Nonlinear Methods for Pseudorandom Number and Vector Generation

  • Harald Niederreiter
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 374)

Abstract

The principal aim of pseudorandom number generation is to devise and analyze deterministic algorithms for generating sequences of numbers which simulate a sequence of i.i.d random variables with given distribution function. We shall deal here exclusively with pseudorandom numbers for the uniform distribution on the interval [0,1], i.e. with uniform pseudorandom numbers. We refer to Knuth [16], Niederreiter [18], Ripley [25], and to the recent survey by Niederreiter [24] for a general background on uniform pseudorandom number generation.

Keywords

Pseudorandom Number Primitive Polynomial Congruential Generator Congruential Method Linear Congruential Generator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Afflerbach and H. Grothe: The lattice structure of pseudo-random vectors generated by matrix generators, J. Comput. Appl Math. 23, 127–131 (1988).CrossRefGoogle Scholar
  2. 2.
    J. Eichenauer, H. Grothe, and J. Lehn: Marsaglia’s lattice test and non-linear congruential pseudo random number generators, Metrika 35, 241–250 (1988).CrossRefGoogle Scholar
  3. 3.
    J. Eichenauer and J. Lehn: A non-linear congruential pseudo random number generator, Statist. Papers 27, 315–326 (1986).Google Scholar
  4. 4.
    J. Eichenauer and J. Lehn: On the structure of quadratic congruential sequences, Manuscripta Math. 58, 129–140 (1987).CrossRefGoogle Scholar
  5. 5.
    J. Eichenauer, J. Lehn, and A. Topuzoğlu: A nonlinear congruential pseudorandom number generator with power of two modulus, Math. Comp. 51, 757–759 (1988).CrossRefGoogle Scholar
  6. 6.
    J. Eichenauer and H. Niederreiter: On Marsaglia’s lattice test for pseudorandom numbers, Manuscripta Math: 62, 245–248 (1988).CrossRefGoogle Scholar
  7. 7.
    J. Eichenauer-Herrmann: Inversive congruential pseudorandom numbers avoid the planes, Math. Comp., to appear.Google Scholar
  8. 8.
    J. Eichenauer-Herrmann: On the discrepancy of inversive congruential pseudorandom numbers with prime power modulus, preprint, Technische Hochschule Darmstadt, 1990.Google Scholar
  9. 9.
    J. Eichenauer-Herrmann, H. Grothe, H. Niederreiter, and A. Topuzoğlu: On the lattice structure of a nonlinear generator with modulus 2α, J. Comput. Appl. Math. 31, 81–85 (1990).CrossRefGoogle Scholar
  10. 10.
    J. Eichenauer-Herrmann and H. Niederreiter: On the discrepancy of quadratic congruential pseudorandom numbers, J. Comput. Appl. Math., to appear.Google Scholar
  11. 11.
    J. Eichenauer-Herrmann and H. Niederreiter: Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus, preprint, Technische Hochschule Darmstadt, 1990.Google Scholar
  12. 12.
    J. Eichenauer-Herrmann and A. Topuzoğlu: On the period length of congruential pseudorandom number sequences generated by inversions, J. Comput. Appl. Math. 31, 87–96 (1990).CrossRefGoogle Scholar
  13. 13.
    G.S. Fishman: Multiplicative congruential random number generators with modulus 2β: An exhaustive analysis for β = 32 and a partial analysis for β = 48, Math. Comp. 54, 331–344 (1990).Google Scholar
  14. 14.
    H. Grothe: Matrixgeneratoren zur Erzeugung gleichverteilter Pseudozufallsvektoren, Dissertation, Technische Hochschule Darmstadt, 1988.Google Scholar
  15. 15.
    J. Kiefer: 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 (1961).Google Scholar
  16. 16.
    D.E. Knuth: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed., Addison-Wesley, Reading, Mass., 1981.Google Scholar
  17. 17.
    R. Lidl and H. Niederreiter: Finite Fields, Addison-Wesley, Reading, Mass., 1983.Google Scholar
  18. 18.
    H. Niederreiter: Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84, 957–1041 (1978).CrossRefGoogle Scholar
  19. 19.
    H. Niederreiter: Remarks on nonlinear congruential pseudorandom numbers, Metrika 35, 321–328 (1988).CrossRefGoogle Scholar
  20. 20.
    H. Niederreiter: Statistical independence of nonlinear congruential pseudorandom numbers, Monatsh. Math. 106, 149–159 (1988).CrossRefGoogle Scholar
  21. 21.
    H. Niederreiter: The serial test for congruential pseudorandom numbers generated by inversions, Math. Comp. 52, 135–144 (1989).CrossRefGoogle Scholar
  22. 22.
    H. Niederreiter: Lower bounds for the discrepancy of inversive congruential pseudorandom numbers, Math. Comp. 55, 277–287 (1990).CrossRefGoogle Scholar
  23. 23.
    H. Niederreiter: Statistical independence properties of pseudorandom vectors produced by matrix generators, J. Comput. Appl. Math. 31, 139–151 (1990).CrossRefGoogle Scholar
  24. 24.
    H. Niederreiter: Recent trends in random number and random vector generation, Ann. Operations Research, to appear.Google Scholar
  25. 25.
    B.D. Ripley: Stochastic Simulation, Wiley, New York, 1987.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Harald Niederreiter
    • 1
  1. 1.Institute for Information ProcessingAustrian Academy of SciencesAustria

Personalised recommendations