Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

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

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.

. We provide sets of parameters for multiplicative linear congruential generators (MLCGs) of different sizes and good performance with respect to the spectral test. For ` = 8; 9; : : : ; 64; 127; 128, we take as a modulus m the largest prime smaller than 2 ` , and provide a list of multipliers a such that the MLCG with modulus m and multiplier a has a good lattice structure in dimensions 2 to 32. We provide similar lists for power-of-two moduli m = 2 ` , for multiplicative and non-multiplicative LCGs. 1. Introduction A multiplicative linear congruential generator (MLCG) is defined by a recurrence of the form xn = axn\Gamma1 mod m (1) where m and a are integers called the modulus and the multiplier , respectively, and xn 2 Zm = f0; : : : ; m \Gamma 1g is the state at step n. To obtain a sequence of "random numbers" in the interval [0; 1), one can define the output at step n as un = xn=m: (2) We use the expression "the MLCG (m; a)" to denote a sequence that obeys (1) and (2). Th...

: 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...

: Usually, the t-dimensional spectral test for linear congruential generators examines the lattice structure of all the points formed by taking t successive values in the sequence. In this paper, we consider the case where the t values taken are not successive, but separated by lags that are chosen a priori. For certain classes of linear congruential and multiple recursive generators, and for certain choices of the lags, we give lower bounds on the distance between hyperplanes. In some cases, those lower bounds are quite large, even in dimensions as small as t = 3. We give illustrations with specific classes of generators that have been proposed in the literature, and discuss the possible implications. Additional Key Words and Phrases: Random Number Generation; Linear Congruential Generators; Lattice Structure; Spectral Test Author's Address: 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'e...

Abstract. We study the multiply-with-carry family of generators proposed by Marsaglia as a generalization of previous add-with-carry families. We define for them an infinite state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the d-dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test. Our results are stated in a general context and are applied to a related complementary multiply-with-carry family of generators. 1.

. In this paper we present a new approach to finding good lattice points. We employ the spectral test, a well-known figure of merit for uniform random number generators. This concept leads to an assessment of lattice points g that is closely related to the classical Babenko-Zaremba quantity ae(g; N ). The associated lattice rules are good uniformly over a whole range of dimensions. Our numerical examples suggest that this simple approach leads to quasi-Monte Carlo node sets that perform very well in comparison to the best available (t; m; s)-nets. 1 Introduction There is no contradiction in the title of this paper. We show how to employ concepts that belong to the field of random number generation to obtain excellent node sets for quasi-Monte Carlo integration in high dimensions. Our approach uses linear congruential generators ("LCGs") and the spectral test to find good lattice points ("GLPs") of the Korobov type. LCGs are a classical method for generating uniform random numbers, see...

Most random number generators used in practice are based on linear recurrences, with linear output transformations. This gives long periods, fast implementations, and structures that are easy to analyze. But the points produced by these generators have very regular structures. Nonlinear generators can have less regular structures, but they are generally slower and much harder to analyze when their period is long. In this paper, combined generators with one large linear component, and a second component of a different type (nonlinear or linear), are proposed and studied. The structure of vectors of successive and non-successive output values produced by the combined generators is analyzed. Under mild conditions, these vector sets are proved to have at least as much uniformity than the corresponding sets for the linear component alone. In empirical statistical tests, these combined generators perform better than simple linear generator of comparable period lengths, because of their less regular structure. Efficient implementation methods are suggested.

We study a class of combined multiple recursive random number generators constructed in a way that each component runs fast and is easy to implement, while the combination enjoys excellent structural properties as measured by the spectral test. Each component is a linear recurrence of order k>1, modulo a large prime number, and the coefficients are either 0 or are of the form a or a . This allows a simple and very fast implementation, because each modular multiplication by a power of 2 can be implemented via a shift, plus a few additional operations for the modular reduction. We select the parameters in terms of the performance of the combined generator in the spectral test. We provide a specific implementation.