Inversive congruential pseudorandom numbers : a tutorial
TL;DR: In this paper, the authors dealt exclusively with the problem of pseudorandom number generation in the interval [0, 1] and showed that for a large positive integer m and integers a, b, yo a linear congruential sequence (y, n),n-o of nonnegative integers less than m is defined by
Abstract: Uniform pseudorandom numbers in the interval [0, 1) are basic ingredients of any stochastic simulation. Their quality is of fundamental importance for the simulation outcome. The present paper deals exclusively with this problem. General background material on pseudorandom number generation can be found in the book of Knuth [21] and in the survey article of Niederreiter [26]. The classical standard method of generating uniform pseudorandom numbers in the interval [0, 1) is the linear congruential method. For a large positive integer m and integers a, b, yo a linear congruential sequence (y,),n-o of nonnegative integers less than m is defined by
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TL;DR: TestU01 as discussed by the authors is a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs).
Abstract: We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator's period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.
972 citations
Book•
27 Feb 2009
TL;DR: The Monte Carlo method has been used in many applications, e.g., for algebra, beyond numerical integration, this article, and for error and variance analysis for Halton sequences.
Abstract: The Monte Carlo method.- Sampling from known distributions.- Pseudorandom number generators.- Variance reduction techniques.- Quasi-Monte Carlo constructions.- Using quasi-Monte Carlo constructions.- Using quasi-Monte Carlo in practice.- Financial applications.- Beyond numerical integration.- Review of algebra.- Error and variance analysis for Halton sequences.- References.- Index.
517 citations
Cites background from "Inversive congruential pseudorandom..."
...An explicit inversive congruential generator (EICG) [94] is described by a transition function...
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TL;DR: In this article, the authors examine practical ways of generating (deterministic approximations to) such uniform variates on a computer and compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness.
Abstract: In typical stochastic simulations, randomness is produced by generating a sequence of independent uniform variates (usually real-valued between 0 and 1, or integer-valued in some interval) and transforming them in an appropriate way. In this paper, we examine practical ways of generating (deterministic approximations to) such uniform variates on a computer. We compare them in terms of ease of implementation, efficiency, theoretical support, and statistical robustness. We look in particular at several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear congruential over fields of formal series, and combined generators, and show how all of them can be analyzed in terms of their lattice structure. We also mention other classes of generators, like non-linear generators, discuss other kinds of theoretical and empirical statistical tests, and give a bibliographic survey of recent papers on the subject.
235 citations
01 Jan 1995
TL;DR: A survey of recent and new developments in the areas of uniform pseudorandom number and uniform Pseudo-Pseudorandom vector generation is presented.
Abstract: A survey of recent and new developments in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on generators for which a detailed theoretical analysis is available.
99 citations
TL;DR: A new class of inversive congruential generators is introduced and it is shown that they have excellent statistical independence properties and model true random numbers very closely.
Abstract: 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.
79 citations
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Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
TL;DR: In this paper, the authors propose a method to solve the problem of the problem: this paper...,.. ].. ).. ]... )...
Abstract: CONTENTS
820 citations
"Inversive congruential pseudorandom..." refers background in this paper
...There are a lot of theoretical results on the structural and statistical properties of linear congruential pseudorandom numbers which are covered by the articles [25], [26], and [27]....
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...General background material on pseudorandom number generation can be found in the book of Knuth [21] and in the survey article of Niederreiter [26]....
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...[26], p....
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TL;DR: The paper gives details of the degree of regularity of congruential random number generators in terms of sets of relatively few parallel hyperplanes which contain all of the points produced by the generator.
Abstract: : Most of the world's computer centers use congruential random number generators. This note points out that such random number generators produce points in 2,3,4,... dimensions which are too regular for many Monte Carlo calculations. The trouble is that the points fall exactly on a lattice with quite a gross structure. The paper gives details of the degree of regularity of such generators in terms of sets of relatively few parallel hyperplanes which contain all of the points produced by the generator.
492 citations
"Inversive congruential pseudorandom..." refers background in this paper
...Marsaglia in his famous paper 'Random numbers fall mainly in the planes' [23] and was worked out in detail by Marsaglia [24] and...
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CERN1
TL;DR: The conclusion is that pseudorandom number generators with the required properties are now available, but the generators actually used are often not good enough.
411 citations
Additional excerpts
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