Uniform Pseudorandom Number Generator Research Articles

RNGAVXLIB: Program library for random number generation, AVX realization

Abstract The library RNGSSELIB for random number generators (RNGs) based upon the SSE2 command set is presented. The library contains realization of a number of modern and most reliable generators. Usage of SSE2 command set allows to substantially improve performance of the generators. Three new RNG realizations are also constructed. We present detailed analysis of the speed depending on compiler usage and associated optimization level, as well as results of extensive statistical testing for all generators using available test packages. Fast SSE implementations produce exactly the same output sequence as the original algorithms. Program summary Program title: RNGSSELIB Catalogue identifier: AEIT_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIT_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4177 No. of bytes in distributed program, including test data, etc.: 21 228 Distribution format: tar.gz Programming language: C. Computer: PC. Operating system: UNIX, Windows. RAM: 1 Mbytes Classification: 4.13. Nature of problem: Any calculation requiring uniform pseudorandom number generator, in particular, Monte Carlo calculations. Solution method: The library contains realization of a number of modern and reliable generators: mt19937 , mrg32k3a and lfsr113 . Also new realizations for the method based on parallel evolution of an ensemble of dynamical systems are constructed: GM19 , GM31 and GM61 . The library contains both usual realizations and realizations based on SSE command set. Usage of SSE commands allows the performance of all generators to be substantially improved. Restrictions: For SSE realizations of the generators, Intel or AMD CPU supporting SSE2 command set is required. In order to use the realization lfsr113sse , CPU must support SSE4 command set. Running time: Running time is of the order of 20 sec for generating 109 pseudorandom numbers with a PC based on Intel Core i7-940 CPU. Running time is analysed in detail in Section 5 of the paper.

PRAND: GPU accelerated parallel random number generation library: Using most reliable algorithms and applying parallelism of modern GPUs and CPUs

The library PRAND for pseudorandom number generation for modern CPUs and GPUs is presented. It contains both single-threaded and multi-threaded realizations of a number of modern and most reliable generators recently proposed and studied in Barash (2011), Matsumoto and Tishimura (1998), L’Ecuyer (1999,1999), Barash and Shchur (2006) and the efficient SIMD realizations proposed in Barash and Shchur (2011). One of the useful features for using PRAND in parallel simulations is the ability to initialize up to 1019 independent streams. Using massive parallelism of modern GPUs and SIMD parallelism of modern CPUs substantially improves performance of the generators. Program summaryProgram title: PRANDCatalogue identifier: AESB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 45979No. of bytes in distributed program, including test data, etc.: 23953564Distribution format: tar.gzProgramming language: Cuda C, Fortran.Computer: PC, workstation, laptop, or server with NVIDIA GPU (tested on Tesla X2070, Fermi C2050, GeForce GT540M) and with Intel or AMD processor.Operating system: Linux with CUDA version 5.0 or later. Should also run on MacOs, Windows, or UNIX.RAM: 4 MbytesClassification: 4.13.Nature of problem:Any calculation requiring uniform pseudorandom number generator, in particular, Monte Carlo calculations. Any calculation or simulation requiring uncorrelated parallel streams of uniform pseudorandom numbers.Solution method:The library contains realization of a number of modern and reliable generators: MT19937, MRG32K3A and LFSR113. Also new realizations of the method based on parallel evolution of an ensemble of transformations of two-dimensional torus are included in the library: GM19, GM29, GM31, GM61, GM55, GQ58.1, GQ58.3 and GQ58.4. The library contains: single-threaded and multi-threaded realizations for GPU, single-threaded realizations for CPU, realizations for CPU based on SSE command set. Also, the library contains the abilities to jump ahead inside RNG sequence and to initialize independent random number streams with block splitting method for each of the RNGs.Restrictions:Nvidia Cuda Toolkit version 5.0 or later should be installed. To use GPU realizations, Nvidia GPU supporting CUDA and the corresponding Nvidia driver should be installed. For SSE realizations of the generators, Intel or AMD CPU supporting SSE2 command set is required. In order to use the SSE realization of LFSR113, CPU must support SSE4 command set.Additional comments:A version of this program, which only contains the realizations for CPUs, is held in the Library as Catalog Id., AEIT_v2_0 (RNGSSELIB). It does not require a GPU device or CUDA compiler.Running time:The tests and the examples included in the package take less or about one minute to run. Running time is analyzed in detail in Section 8 of the paper.

RNGSSELIB: Program library for random number generation. More generators, parallel streams of random numbers and Fortran compatibility

In this update, we present the new version of the random number generator (RNG) library RNGSSELIB, which, in particular, contains fast SSE realizations of a number of modern and most reliable generators \cite{RNGSSELIB1}. The new features are: i) Fortran compatibility and examples of using the library in Fortran; ii) new modern and reliable generators; iii) the abilities to jump ahead inside RNG sequence and to initialize up to $10^{19}$ independent random number streams with block splitting method.

RNGSSELIB: Program library for random number generation, SSE2 realization

The library RNGSSELIB for random number generators (RNGs) based upon the SSE2 command set is presented. The library contains realization of a number of modern and most reliable generators. Usage of SSE2 command set allows to substantially improve performance of the generators. Three new RNG realizations are also constructed. We present detailed analysis of the speed depending on compiler usage and associated optimization level, as well as results of extensive statistical testing for all generators using available test packages. Fast SSE implementations produce exactly the same output sequence as the original algorithms. Program summaryProgram title: RNGSSELIBCatalogue identifier: AEIT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIT_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4177No. of bytes in distributed program, including test data, etc.: 21 228Distribution format: tar.gzProgramming language: C.Computer: PC.Operating system: UNIX, Windows.RAM: 1 MbytesClassification: 4.13.Nature of problem: Any calculation requiring uniform pseudorandom number generator, in particular, Monte Carlo calculations.Solution method: The library contains realization of a number of modern and reliable generators: mt19937, mrg32k3a and lfsr113. Also new realizations for the method based on parallel evolution of an ensemble of dynamical systems are constructed: GM19, GM31 and GM61. The library contains both usual realizations and realizations based on SSE command set. Usage of SSE commands allows the performance of all generators to be substantially improved.Restrictions: For SSE realizations of the generators, Intel or AMD CPU supporting SSE2 command set is required. In order to use the realization lfsr113sse, CPU must support SSE4 command set.Running time: Running time is of the order of 20 sec for generating 109 pseudorandom numbers with a PC based on Intel Core i7-940 CPU. Running time is analysed in detail in Section 5 of the paper.

Pseudo-random number generators for Monte Carlo simulations on ATI Graphics Processing Units

Basic uniform pseudo-random number generators are implemented on ATI Graphics Processing Units (GPU). The performance results of the realized generators (multiplicative linear congruential (GGL), XOR-shift (XOR128), RANECU, RANMAR, RANLUX and Mersenne Twister (MT19937)) on CPU and GPU are discussed. The obtained speed up factor is hundreds of times in comparison with CPU. RANLUX generator is found to be the most appropriate for using on GPU in Monte Carlo simulations. The brief review of the pseudo-random number generators used in modern software packages for Monte Carlo simulations in high-energy physics is presented.

Effects of chaotic exploration on reinforcement learning in target capturing task

A process of trial and error plays an important role in not only the human learning but also the machine learning. Such a process is called exploration in the reinforcement learning which has originated from experimental studies on learning in psychology. A uniform pseudorandom number generator appears to be suitable for exploration. However, it is known that a chaotic source also provides a random-like sequence as like as a stochastic source. By applying this random-like feature of a deterministic chaotic generator for exploration in a nonstationary shortcut maze problem, we have observed that a deterministic chaotic generator provides a better performance than a stochastic random exploration generator when used for exploration based on a logistic map. In this study, in order to confirm this difference in the performances of the two generators, we examine another nonstationary task - target capturing. The simulation result of this task agrees with the result of our previous study. From the view of multi-agent system, it is an inhomogeneous or heterogeneous system composed of some kinds of agents in many cases. In such situations, the exploration of them is not uniform. Chaotic exploration may suit well this heterogeneity in such a multi-agent system.

A Repetition Test for Pseudo-Random Number Generators

A new statistical test for uniform pseudo-random number generators (PRNGs) is presented. The idea is that a sequence of pseudo-random numbers should have numbers reappear with a certain probability. The expectation time that a repetition occurs provides the metric for the test. For linear congruential generators (LCGs) failure can be shown theoretically. Empirical test results for a number of commonly used PRNGs are reported, showing that some PRNGs considered to have good statistical properties fail. A sample implementation of the test is provided over the Internet.

CHAOTIC EXPLORATION EFFECTS ON REINFORCEMENT LEARNING IN SHORTCUT MAZE TASK

Reinforcement learning is usually required in the process of trial and error called exploration, and the uniform pseudorandom number generator is considered effective in that process. As a generator for the exploration, chaotic sources are also useful in creating a random-like sequence such as in the case of stochastic sources. In this research, we investigate the efficiency of the deterministic chaotic generator for the exploration in learning a nonstationary shortcut maze problem. As a result, it is found that the deterministic chaotic generator based on the logistic map is better in the performance of the exploration than in the stochastic random generator. This has been made clear by analyzing the difference of the performances between the two generators in terms of the patterns of exploration occurrence. We also examine the tent map, which is homeomorphic to the logistic map, compared with other generators.

A Repetition Test for Pseudo-Random Number Generators

A new statistical test for uniform pseudo-random number generators (PRNGs) is presented. The idea is that a sequence of pseudo-random numbers should have numbers reappear with a certain probability. The expectation time that a repetition occurs provides the metric for the test. For linear congruential generators (LCGs) failure can be shown theoretically. Empirical test results for a number of commonly used PRNGs are reported, showing that some PRNGs considered to have good statistical properties fail. A sample implementation of the test is provided over the Internet.

JMASM6: An Algorithm For Generating Exact Critical Values For the Kruskal-Wallis One-Way ANOVA

A Fortran 77 subroutine is provided for computing exact critical values for the Kruskal-Wallis test on k independent groups with equal or unequal samples sizes. The subroutine requires the user to provide sorting and ranking routines and a uniform pseudo-random number generator. The program is available from the author on request.

JMASM1: RANGEN 2.0 (Fortran 90/95)

Rangen 2.0 is Fortran 90 module of subroutines used to generate uniform and nonuniform pseudo-random deviates. It includes uni1, an uniform pseudo-random number generator, and non-uniform generators based on unil. The subroutines in Rangen 2.0 were written using Essential Lahey Fortran 90, a proper subset of Fortran 90. It includes both source code for the subroutines and a short description of each subroutine, its purpose, and the arguments including data type and usage.

Random Number Generation for the New Century

Use of empirical studies based on computer-generated random numbers has become a common practice in the development of statistical methods, particularly when the analytical study of a statistical procedure becomes intractable. The quality of any simulation study depends heavily on the quality of the random number generators. Classical uniform random number generators have some major defects—such as the (relatively) short period length and the lack of higher-dimension uniformity. Two recent uniform pseudo-random number generators (MRG and MCG) are reviewed. They are compared with the classical generator LCG. It is shown that MRG/MCG are much better random number generators than the popular LCG. Special forms of MRG/MCG are introduced and recommended as the random number generators for the new century. A step-by-step procedure for constructing such random number generators is also provided.

The Multiple-Recursive Matrix Method for Pseudorandom Number Generation

We carry out an in-depth analysis of the multiple-recursive matrix method for uniform pseudorandom number generation which was introduced in an earlier paper of the author. This method yields much larger period lengths than the GFSR method with the same order of the recursion and the same precision. Besides periodicity properties, we establish also uniformity properties of s-tuples of successive pseudorandom numbers generated by the multiple-recursive matrix method and we study the performance under the s-dimensional serial test. The uniformity properties and the behavior under the serial test depend on an appropriate figure of merit in the case where the dimension s exceeds the order of the recursion.

On a new class of pseudorandom numbers for simulation methods

The deficiencies of classical methods for uniform pseudorandom number generation have recently led to the development of alternative methods, such as nonlinear congruential methods. A particularly attractive nonlinear congruential method is the explicit inversive congruential method recently introduced by Eichenauer-Herrmann. We establish several new results on the structural and statistical properties of pseudorandom numbers generated by this method. These results demonstrate that this method is eminently suitable for the generation of parallel streams of pseudorandom numbers with desirable properties.

Pseudorandom number generator for massively parallel molecular-dynamics simulations.

A class of uniform pseudorandom number generators is proposed for modeling and simulations on massively parallel computers. The algorithm is simple, nonrecursive, and is easily transported to serial or vector computers. We have tested the procedure for uniformity, independence, and correlations by several methods. Related, less complex sequences passed some of these tests well enough; however, inadequacies were revealed by tests for correlations and in an interesting application, namely, annealing from an initial lattice that is mechanically unstable. In the latter case, initial velocities chosen by a random number generator that is not sufficiently random lead quickly to unphysical regularity in grain structure. The new class of generators passes this dynamical diagnostic for unwanted correlations.

Recent trends in random number and random vector generation

A survey of recent work in the areas of uniform pseudorandom number and uniform pseudorandom vector generation is presented. The emphasis is on methods for which a detailed theory is available. A progress report on the construction of quasirandom points for efficient multidimensional numerical integration is also given.

Lower bounds for the discrepancy of inversive congruential pseudorandom numbers

The inversive congruential method is a uniform pseudorandom number generator which was introduced recently. For a prime modulus p the discrepancy D p ( k ) D_p^{(k)} of k-tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers. It was shown earlier by the author that \[ D p ( k ) = O ( p − 1 / 2 ( log ⁡ p ) k ) for 2 ≤ k > p . D_p^{(k)} = O({p^{ - 1/2}}{(\log p)^k})\quad {\text {for}}\;2 \leq k > p. \] Here it is proved that this bound is essentially best possible. In fact, for a positive proportion of the admissible parameters in the inversive congruential method the discrepancy D p ( k ) D_p^{(k)} is at least of the order of magnitude p − 1 / 2 {p^{ - 1/2}} for all k ≥ 2 k \geq 2 .

Additive vs multiplicative uniform pseudo-random number generators in the generation of Erlang variates

This paper presents the comparative results of several test runs involving the use of several unit uniform random number generators with an inverse transform to generate Erland random variates with λ=1 and k = 3, 4, 5, and 6. The results show that two commonly used generators (a multiplicative and a mixed one) produce unacceptable results. Two others (an additive one and a multiplicative one) produce acceptable results. The computer system used is an IBM 370/3033.