Harald Niederreiter
Abstract
Pseudorandom vectors are of importance for parallelized simulation methods. In this article we carry out a detailed analysis of the inversive method for the generation of uniform pseudorandom vectors. This method can be viewed as an analog of the inversive congruential method for pseudorandom number generation. We study, in particular, the periodicity properties and the behavior under the serial test for sequences of pseudorandom vectors generated by the inversive method.
- AFFLERBACH, L. AND GROTHE, H. 1988. The lattice structure of pseudo-random vectors generated by matrix generators. J. Comput. Appl. Math. 23, 127-131. Google Scholar
- ANDERSON, S. L. 1990. Random number generators on vector supercomputers and other advanced architectures. SIAM Rev. 32, 221 251. Google Scholar
- BHAVSAR, V. C. AND ISAAC, J.R. 1987. Design and analysis of parallel Monte Carlo algorithms. SIAM J. Sci. Stat. Comput. 8, s73 s95. Google Scholar
- CHOU, W.-S. 1993. On inversive maximal period polynomials over finite fields. Preprint, Austrian Academy of Sciences, Vienna.Google Scholar
- EDDY, W. F. 1990. Random number generators for parallel processors. J. Comput. Appl. Math. 31, 63 71. Google Scholar
- EICHENAUER, J. AND LEHN, J. 1986. A non-linear congruential pseudo random number generator. Stat. Papers 27, 315-326.Google Scholar
- EICHENAUER-HERRMANN, J. 1994. Improved lower bounds for the discrepancy of inversive congruential pseudorandom numbers. Math. Comput. to be published. Google Scholar
- EICHENAUER-HERRMANN, J. 1992. Inversive congruential pseudorandom numbers: A tutorial. Int. Stat. Rev. 60, 167 176.Google Scholar
- FLAHIVE, M. AND NIEDERREITER, H. 1992. On inversive congruential generators for pseudorandom numbers. In Finite Fields, Coding Theory, and Advances tn Communications and Computing, G. L. Mullen and P. J.-S. Shine, Eds. Dekker, New York, 75 80.Google Scholar
- KNUTH, D.E. 1981. The Art of Computer ProgrammLng. Vol. 2, Seminumer~cal Algorithms, 2nd ed. Addison-Wesley, Reading, Mass. Google Scholar
- KuPE~S, L. AND N~EDERREITER, H. 1974. Uniform Dtstributzon of Sequences. Wiley, New York.Google Scholar
- L'Ec~ER, P. 1990. Random numbers for simulation. Commun. ACM 33, 10, 85 97. Google Scholar
- LmL, R. AND NIEDERREITER, H. 1986. Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge, England. Google Scholar
- LIDL, R. AND NIEDERREITER, H. 1983. Finite Fields. Addison-Wesley, Reading, Mass (Now distributed by Cambridge University Press.)Google Scholar
- MORENO, C. J. AND MORENO, O. 1991. Exponential sums and Goppa codes: I. Prec. Am. Math. Soc. 111,523-531.Google Scholar
- NmDERREITE~, H. 1993. Factorization of polynomials and some linear-algebra problems over finite fields. Lin. Alg. Appl. 192, 301 328.Google Scholar
- NIEDERREITER, H. 1992a. Nonlinear methods for pseudorandom number and vector generation. In Simulation and Optimization, G. Pfiug and U. Dieter, Eds. Lecture Notes in Economics and Mathematical Systems, vol. 374. Springer, Berhn, 145 153.Google Scholar
- NIEDERREITER, H. 1992b. Random Number Generatlon and Quasi-Monte Carlo Methods. SIAM, Philadelphia, Pa. Google Scholar
- NIEDERREITER, H. 1991. Finite fields and their applications. In Contributions to General Algebra 7 (Vienna, 1990). Teubner, Stuttgart, 251 264.Google Scholar
- NIEDERREITER, H. 1990a. Statistical independence properties of pseudorandom vectors produced by matrix generators. J Comput. Appl Math. 31, 139-151. Google Scholar
- NIEDERREITER, H. 1990b. Lower bounds for the discrepancy ofinversive congruentml pseudorandom numbers. Math Comput. 55, 277 287.Google Scholar
- NIEDERREITER, H. 1978. Quasi-Monte Carlo methods and pseudo-random numbers Bull. Am. Math. Soc. 84, 957-1041.Google Scholar
- NIEDERREITER, H. 1977. Pseudo-random numbers and optimal coefficients. Adv. Math. 26, 99-181.Google Scholar
Index Terms
- Pseudorandom vector generation by the inversive method
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Reviews
William J. J. Rey
The inversive method provides an algorithm for pseudo-random vector generation with several attractive properties. A criterion for the maximal period length can be given, the behavior under the serial test is described and no additional complicated calculation of figures of merit as in the matrix method is necessary to guarantee a good behavior. The fact…shows that there is a considerable amount of irregularity in the sequence of pseudo-random vectors. Since this feature of irregularity will also be present in a sequence of truly random vectors, this property can be advantageous for various simulation purposes. (From the conclusion). The generation of pseudorandom numbers by the inversive method is now well known, and Niederreiter, a world expert in this domain, investigates an extension of the principle to the generation of vectors. The extension is straightforward, consisting of taking as vector entries the sequence of generated pseudorandom numbers. The treatment runs from lemmas to theorems and corollaries; it makes a nice exercise in number theory. The main tool is the evaluation of discrepancies as well as their lower and upper bounds. No numerical experimentation is reported. The possibility of implementing the method is touched upon briefly. This paper certainly meets the needs of many readers concerned with number theory. It is a nice piece of work, but does it help the statistician I wonder. It is troubling to see that the vectors are less uniformly distributed than with the matrix congruential method (inasmuch as this can be indicated by a “supremum” in the definition of the discrepancy). Can this irregularity be a favorable feature We will certainly read much more on the inversive method before finding it accepted by the computer community.
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