Abstract not found

this paper is to provide good CMRGs of different sizes, selected via the spectral test up to 32 (or 24) dimensions, and a faster implementation than in L'Ecuyer (1996) using floating-point arithmetic. Why do we need different parameter sets? Firstly, different types of implementations require different constraints on the modulus and multipliers. For example, a floating-point implementation with 53 bits of precision allows moduli of more than 31 bits and this can be exploited to increase the period length for free. Secondly, as 64-bit computers get more widespread, there is demand for generators implemented in 64-bit integer arithmetic. Tables of good parameters for such generators must be made available. Thirdly, RNGs are somewhat like cars: a single model and single size for the entire world is not the most satisfactory solution. Some people want a fast and relatively small RNG, while others prefer a bigger and more robust one, with longer period and good equidistribution properties in larger dimensions. Naively, one could think that an RNG with period length near 2

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

We survey some of the recent developments on quasi-Monte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a high-dimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration. 1 2 1.

This paper deals with the inversive congruential method with power of two modulus m for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that, on the average over the parameters in the inversive congruential method, the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between m \Gamma1=2 and m \Gamma1=2 (log m)³. The method of proof relies on a detailed discussion of the properties of certain exponential sums.

this paper is to explain the implementation of a software package for analyzing the lattice structure of linear congruential or multiple recursive random number generators. Such generators are based on linear recurrences of the form

: A portable package for uniform random number generation is proposed, based on a backbone generator with period length near 2 121 , which is a combination of four linear congruential generators. The package provides for multiple (virtual) generators evolving in parallel. Each generator also has many disjoint subsequences, and software tools are provided to reset the state of any generator to the beginning of its first, previous, or current subsequence. Such facilities are helpful to maintain synchronization for implementing variance reduction methods in simulation. Computer implementations are available in the C and Modula-2 languages. Keywords: Random number generation, jump-ahead, software package. Authors' Addresses: P. L'Ecuyer, D'epartement d'Informatique et de Recherche Op'erationnelle (IRO), Universit'e de Montr'eal, C.P. 6128, Succ. Centre-Ville, Montr'eal, H3C 3J7, Canada; e-mail: lecuyer@iro.umontreal.ca www: http://www.iro.umontreal.ca/¸lecuyer T. Andres, AECL Whiteshel...

In this article we present background, rationale, and a description of the Scalable Parallel Random Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not consider splitting methods such as the leap-frog or blocking methods. We describe in detail parameterized versions of the following pseudorandom number generators: (i) linear congruential generators, (ii) shift-register generators, and (iii) lagged-Fibonacci generators. We briey describe the methods, detail some advantages and disadvantages of each method, and recount results from number theory that impact our understanding of their quality in parallel applications. SPRNG was designed around the uniform implementation of dierent families of parameterized random number generators. We then present a short description of SPRNG. The description contained within this document is meant only to outline the rationale behind and the capabilities of SPRNG. Much more information, including examples and detailed documentation aimed at helping users with putting and using SPRNG on scalable systems is available at the URL: http://sprng.cs.fsu.edu/RNG. In this description of SPRNG we discuss the random number generator library as well as the suite of tests of randomness that is an integral part of SPRNG. Random number tools for parallel Monte Carlo applications must be subjected to classical as well as new types of empirical tests of ran- domness to eliminate generators that show defects when used in scalable environments.

Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in that respect. In this paper, we propose new generators of that form, with better equidistribution and “bit-mixing ” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.

Stochastic simulation requires a reliable source of randomness. Inversive methods are an interesting and very promising new approach to produce uniform pseudorandom numbers. In this paper, we present evidence that these methods are an important contribution to our toolbox. We survey the outstanding performance of inversive pseudorandom number generators in theoretical and empirical tests, in comparison to linear generators. In addition, this paper contains tables of parameters to implement inversive congruential generators. More empirical results as well as an implementation of inversive generators in C are available in the Internet from our Web-site http:// random.mat.sbg.ac.at. 1 INTRODUCTION Pseudorandom number generators are essential elements in the toolbox of stochastic simulation. Their task is to simulate realizations of independent, identically U([0; 1[)-distributed random variables. Other distributions will be obtained by transformation methods, see Devroye (1986), and the...