Abstract
The inversive congruential method for generating uniform pseudorandom numbers has been introduced recently as an alternative to linear congruential generators with their coarse lattice structure. In the present paper the statistical independence properties of pairs of consecutive pseudorandom numbers obtained from an inversive congruential generator with prime power modulus are analysed by means of the serial test. Upper bounds for the discrepancy of these pairs are established which are essentially best possible. The results show that the inversive congruential method with prime power modulus performs uniformly satisfactorily under the serial test. The methods of proof rely heavily on the evaluation of certain exponential sums which resemble Kloosterman sums.
This is a preview of subscription content, access via your institution.
References
Eichenauer, J., Grothe, H. and Lehn, J.: Marsaglia's lattice test and non-linear congruential pseudo random number generators, Metrika 35, 241–250 (1988)
Eichenauer, J. and Lehn, J.: A non-linear congruential pseudo random number generator, Statist. Papers 27, 315–326 (1986)
Eichenauer, J. and Lehn, J.: On the structure of quadratic congruential sequences, manuscripta math. 58, 129–140 (1987)
Eichenauer, J., Lehn, J. and Topuzoĝlu, A.: A nonlinear congruential pseudorandom number generator with power of two modulus, Math. Comp. 51, 757–759 (1988)
Eichenauer-Herrmann, J.: Inversive congruential pseudorandom numbers avoid the planes, Math. Comp. (to appear)
Eichenauer-Herrmann, J., Grothe, H., Niederreiter, H. and Topuzoĝlu, A.: On the lattice structure of a nonlinear generator with modulus 2α, J. Comp. Appl. Math. 31, 81–85 (1990)
Eichenauer-Herrmann, J. and Niederreiter, H.: Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus (submitted to Math. Comp.)
Eichenauer-Herrmann, J. and Niederreiter, H.: On the discrepancy of quadratic congruential pseudorandom numbers, J. Comp. Appl. Math. (to appear)
Eichenauer-Herrmann, J. and Topuzoĝlu, A.: On the period length of congruential pseudorandom number sequences generated by inversions, J. Comp. Appl. Math. 31, 87–96 (1990)
Lidl, R. and Niederreiter, H.: Finite Fields, Addison-Wesley, Reading, Mass., 1983
Niederreiter, H.: Pseudo-random numbers and optimal coefficients, Adv. in Math. 26, 99–181 (1977)
Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84, 957–1041 (1978)
Niederreiter, H.: The serial test for pseudo-random numbers generated by the linear congruential method, Number. Math. 46, 51–68 (1985)
Niederreiter, H.: Remarks on nonlinear congruential pseudorandom numbers, Metrika 35, 321–328 (1988)
Niederreiter, H.: Statistical independence of nonlinear congruential pseudorandom numbers, Monatsh. Math. 106, 149–159 (1988)
Niederreiter, H.: The serial test for congruential pseudorandom numbers generated by inversions, Math. Comp. 52, 135–144 (1989)
Niederreiter, H.: Lower bounds for the discrepancy of inversive congruential pseudorandom numbers, Math. Comp. 55, 277–287 (1990)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eichenauer-Herrmann, J. On the discrepancy of inversive congruential pseudorandom numbers with prime power modulus. Manuscripta Math 71, 153–161 (1991). https://doi.org/10.1007/BF02568399
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02568399
Keywords
- Serial Test
- Pseudorandom Number
- Congruential Generator
- Congruential Method
- Linear Congruential Generator