Jürgen Eichenauer-Herrmann

Statistical independence of a new class of inversive congruential pseudorandom numbers

Linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations. This was the motiviation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers. It is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus. In the present paper a new class of such inversive congruential generators is introduced and analyzed. It is shown that they have excellent statistical independence properties and model true random numbers very closely. The methods of proof rely heavily on Weil-Stepanov bounds for rational exponential sums. 39 refs.

A survey of quadratic and inversive congruential pseudorandom numbers

This review paper deals with nonlinear methods for the generation of uniform pseudorandom numbers in the unit interval. The emphasis is on results of the theoretical analysis of quadratic congruential and (recursive) inversive congruential generators, which are scattered over a fairly large number of articles. Additionally, empirical results of some sample generators in a two—level overlapping serial test are given.

Inversive congruential pseudorandom numbers avoid the planes

Nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed. These generators do not show the lattice structure of the widely used linear congruential method. In the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property: Any hyperplane in (/-space contains at most d of these points, that is to say, the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points. This feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided.

On the discrepancy of quadratic congruential pseudorandom numbers

Abstract One of the alternatives to linear congruential pseudorandom number generators with their undesirable lattice structure is the quadratic congruential method which is due to Knuth. In the present paper the statistical independence properties of pairs of consecutive pseudorandom numbers generated according to this method 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 quadratic congruential method performs uniformly satisfactorily if a reasonable choice of one of the parameters is made. The method of proof relies heavily on the evaluation of certain exponential sums.

On the period length of congruential pseudorandom number sequences generated by inversions

Abstract Congruential pseudorandom number sequences generated by inversions have been studied recently. These sequences do not show the undesirable lattice structure of the linear congruential method. The necessary and sufficient condition for the generated sequence to have the maximal period length was given by Eichenauer (1988) for the case of 2e modulus. Generalization of this result to the case of an arbitrary prime power modulus is obtained.

On the lattice structure of a nonlinear generator with modulus 2 a

Abstract Nonlinear congruential pseudorandom number generators based on inversions have been introduced and analysed recently. These generators do not show the simple lattice structure of the widely used linear congruential generators which are too regular for certain simulation purposes. In the present paper a nonlinear congruential generator based on inversions with respect to a power of two modulus is considered. It is shown that the set of points formed by consecutive pseudorandom numbers has a more complicated lattice structure: it forms a superposition of shifted lattices. The corresponding lattice bases are explicitly determined and analysed.

Digital inversive pseudorandom numbers

A new algorithm, the digital inversive method, for generating uniform pseudorandom numbers is introduced. This algorithm starts from an inversive recursion in a large finite field and derives pseudorandom numbers from it by the digital method. If the underlying finite field has q elements, then the sequences of digital inversive pseudorandom numbers with maximum possible period length q can be characterized. Sequences of multiprecision pseudorandom numbers with very large period lengths are easily obtained by this new method. Digital inversive pseudorandom numbers satisfy statistical independence properties that are close to those of truly random numbers in the sense of asymptotic discrepancy. If q is a power of 2, then the digital inversive method can be implemented in a very fast manner.

A new inversive congruential pseudorandom number generator with power of two modulus

Pseudorandom numbers are important ingredients of stochastic simulations. Their quality is fundamental for the strength of the simulation outcome. The inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators, which show many undesirable regularities. In the present paper a new inversive congruential generator with power of two modulus is introduced. Periodicity and statistical independence properties of the generated sequences are analyzed. The results show that these inversive congruential generators perform very satisfactorily.

Computation of critical distances within multiplicative congruential pseudorandom number sequences

Abstract Multiplicative congruential pseudorandom number generators with prime moduli are considered. Recently, it has been shown by two of the authors (Eichenauer-Herrmann and Grothe (1989)) that terms located far apart in the generated sequences are strongly correlated. However, the critical distances between the terms were not obtained explicitly. Since a straight approach for computing them through tabulations cannot be used for large values of the modulus, an efficient method has been developed and is described in the present note. The resulting algorithm has been applied to several generators and tables of the most critical distances for two generators with large moduli are included.

On generalized inversive congruential pseudorandom numbers

The inversive congruential method with prime modulus for generating uniform pseudorandom numbers has several very promising properties. Very recently, a generalization for composite moduli has been introduced. In the present paper it is shown that the generated sequences have very attractive statistical independence properties.