Linear Congruential Random Number Generator Research Articles

Minimising the makespan in open shop scheduling problems using variants of discrete firefly algorithm

Minimising makespan in open shop scheduling problem is the aim of this research work. The scheduling problems consist of 'n' jobs and 'm' machines, in which each job has 'O' operations. The processing time for 50 open shop scheduling problems was generated using a linear congruential random number generator. A non-traditional algorithm called discrete firefly algorithm is proposed to minimise the makespan of open shop scheduling problems and this method is referred as A1. Discrete firefly algorithm is also hybridised along with other heuristic algorithms in two ways: 1) firefly algorithm is hybridised with longest total remaining processing on other machines and local search algorithms which is referred as A2; 2) firefly algorithm is hybridised with simulated annealing and referred as A3. Novelty of this work focuses on the application of open shop scheduling problems in quality control of a production industry.

Improved CUDA programs for GPU computing of Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models

Improved CUDA programs for GPU computing of Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models

Dimensionality of Spectral Test for a Linear Congruential Random Number Generator

This paper considers the successive dimensions of the spectral test for a linear congruential generator (LCG) based on three types of the upper bound on the center density. We conduct an exhaustive search for the maximum and difference normalized spectral value in dimensions 2, 3, …, 32 of an LCG. The experimental result indicates that the appropriate dimensions for implementing the spectral test are 2, 3, …, 8 with the objective of maximizing the normalized spectral value.

Finding the best portable congruential random number generators

Linear congruential random number generators must have large moduli to attain maximum periods, but this creates integer overflow during calculations. Several methods have been suggested to remedy this problem while obtaining portability. Approximate factoring is the most common method in portable implementations, but there is no systematic technique for finding appropriate multipliers and an exhaustive search is prohibitively expensive. We offer a very efficient method for finding all portable multipliers of any given modulus value. Letting M=AB+C, the multiplier A gives a portable result if B−C is positive. If it is negative, the portable multiplier can be defined as A=⌊M/B⌋. We also suggest a method for discovering the most fertile search region for spectral top-quality multipliers in a two-dimensional space. The method is extremely promising for best generator searches in very large moduli: 64-bit sizes and above. As an application to an important and challenging problem, we examined the prime modulus 263−25, suitable for 64-bit register size, and determined 12 high quality portable generators successfully passing stringent spectral and empirical tests.

An exhaustive search for good 64-bit linear congruential random number generators with restricted multiplier

This paper explores that the different prime moduli can affect both the number of primitive root and the spectral test performance for 64-bit linear congruential generators (LCGs). Three forms of prime modulus and two types restriction on multiplier are considered in this paper. We perform computerized experiments that indicate significant differences exist among the number of primitive root of three forms of prime modulus. These differences can affect the performance of the spectral test. Two good 64-bit LCGs with significantly better spectral values and excellent empirical performance are presented. They are suitable for the requirements of todayʼs computer simulation studies.

A 32-bit linear congruential random number generator with prime modulus

This paper explores that the different prime moduli can affect both the number of primitive root and the spectral test performance for a 32-bit linear congruential generator (LCG). We consider five forms of prime modulus: the Mersenne prime modulus, the largest prime modulus, the Sophie-Germain prime modulus, the twin prime modulus and the factorial prime modulus. We perform a computerized experiment that indicates significant differences exist among the number of primitive root of the five forms of prime modulus, and demonstrate that these differences can affect the performance of spectral test.

An Efficient Randomized Quasi-Monte Carlo Algorithm for the Pareto Distribution

This paper studies a new randomized quasi-Monte Carlo method for estimating the mean and variance of the Pareto distribution. In many Monte Carlo simulations, there are some stability problems for estimating the population Pareto variance by using the sample variance. In this paper, we propose a randomized quasi-random number generator [quasi- RNG] to generate Pareto random samples, such that the sample mean and sample variance estimators gain more efficiency. The efficiency of this generator relative to the popular linear congruential random number generator [LC-RNG] is studied by using the simulation mean square errors. We also compare the results of the Kolmogorov-Smirnov goodness-of-fit tests using these two sample generators.

ANOMALOUS DIFFUSION AT PERCOLATION THRESHOLD IN HIGH DIMENSIONS ON 1018 SITES

Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven dimensions, at the percolation threshold with L7 sites and L≤420, we confirm the expected time-dependence of the end-to-end distance (including the corrections to the asymptotic behavior).

Beware of linear congruential generators with multipliers of the form a = ±2 q ±2 r

Linear congruential random-number generators with Mersenne prime modulus and multipliers of the form a = ±2 q ± r have been proposed recently. Their main advantage is the availability of a simple and fast implementation algorithm for such multipliers. This note generalizes this algorithm, points out statistical weaknesses of these multipliers when used in a straightforward manner, and suggests in what context they could be used safely.

A statistical analysis of seven multipliers for linear congruential random number generators with modulus 231?1

Twelve statistical tests were performed on the sequences produced by seven linear congruential pseudorandom number generators with modulus 231−1. The seven generators had, respectively, the multipliers 742938285, 950706376, 1226874159, 62089911, 1343714438, 397204094 and 16807. The worst results were obtained from the sequences produced by the multipliers 1226874159, 397204094 and 16807, while the best ones corresponded to the sequences produced by 1343714438 and 742938285.

A hyperbolic tangent identity and the geometry of Padé sign function iterations

The rational iterations obtained from certain Pade approximations associated with computing the matrix sign function are shown to be equivalent to iterations involving the hyperbolic tangent and its inverse. Using this equivalent formulation many results about these Pade iterations, such as global convergence, the semi-group property under composition, and explicit partial fraction decompositions can be obtained easily. In the second part of the paper it is shown that the behavior of points under the Pade iterations can be expressed, via the Cayley transform, as the combined result of a completely regular iteration and a chaotic iteration. These two iterations are decoupled, with the chaotic iteration taking the form of a multiplicative linear congruential random number generator where the multiplier is equal to the order of the Pade approximation.

A portable random number generator well suited for the rejection method

Up to now, all known efficient portable implementations of linear congruential random number generators with modulus 2 31 – 1 have worked only with multipliers that are small compared with the modulus. We show that for nonuniform distributions, the rejection method may generate random numbers of bad qualify if combined with a linear congruential generator with small multiplier. A method is described that works for any multiplier smaller than 2 30 . It uses the decomposition of multiplier and seed in high-order and low-order bits to compute the upper and lower half of the product. The sum of the two halfs gives the product of multiplier and seed modulo 2 21 – 1. Coded in ANSI-C and FORTRAN77 the method results in a portable implementation of the linear congruential generator that is as fast or faster than other portable methods.

On the distribution of 𝑘-dimensional vectors for simple and combined Tausworthe sequences

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F 2 {\mathbb {F}_2} . The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into 2 k l {2^{kl}} identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.

An Expanded Set of Correlation Tests for Linear Congruential Random Number Generators

An analytic study is made of the correlation structure of Tausworthe and linear congruential random number generators. The former case is analyzed by the bit mask correlations recently introduced by Compagner. The latter is studied first by an extension to word masks, which include spectral test coefficients as special cases, and then by the bit mask procedure. Although low order bit mask coefficients vanish in both cases, the Tausworthe generator appears to produce a substantially smaller non-vanishing correlation set for large masks – but with larger correlation values – than does the linear congruential.

Random access to a random number sequence

We have expanded the method in a recent note by Koniges and Leith [ 1 ] for a direct jump to any desired k th element κ k of the standard linear congruential random number generator specified by Eqs. (1) and (2). The digits of k (base b) in Eq. (6) are indices for selecting elements from two precalculated arrays, P and S, Eqs. (6) and (7). These elements are combined in Eqs. (10) and (11) to give the addends of κ k = κ k + Ω k . The choice of the base b determines the number of digits in k (base b). For each nonzero digit, the evaluation of Eqs. (10) and (11) require two multiplications (mod m), and one add. The larger the choice for b, the fewer the digits in k (base b), and the less the arithmetic needed to evaluate Eqs. (10) and (11). For modulus m = 2 48, values of b = 2, 8, and 256 require at most 48, 16, and 6 multiplications, respectively, to evaluate Eq. (10). The expected number of multiplications is somewhat less. As noted above, only the nonzero digits of k lead to modular arithmetic calculations (zero digits imply the addition of zero and/or multiplication by unity). For a random k, base b, the probability that a random bit is nonzero is (b − 1) b . Thus we find that the expected number of multiplications to evaluate Eq. (10) for b = 2, 8, or 256 drops to 24, 14, or 5.98, respectively. The drawback for very large base b is the storage needed for the precalculated P and S arrays, Eqs. (8) and (9). Each has b( J+ 1) elements, where J+ 1 is the maximum number of digits in k (base b). For m = 2 48 , and b = 2, 8, 256, there are respectively 2 × 48 = 96, 8 × 15 = 120, and 256 × 6 = 1536 elements in P and in S. If b is further increased, then each array size b( J+ 1) grows rapidly. On the other hand, the J+ 1 number of multiplications (mod m) needed to evaluate Eq. (10) decreases slowly. We suggest that for a 48-bit machine word length b = 256 is a good compromise between table size and achievable speedup. For a standard generator with zero additive term using base 256 reduces the overall operations count from 48 if tests, 48 register shifts, 24 multiplies, and 72 logical ands to 6 register shifts, 5.98 multiplies, and 11.98 logical ands, yielding more than a fourfold increase in speed over the base 2 case.

Long range correlations in linear congruential generators

Primitive linear congruential random number generators introduce correlations among elements separated by powers of two which thereby introduce errors into Monte Carlo procedures. A generalization of this phenomenon is made in this paper to all linear congruential generators with fixed multiplier using a power of two as modulus. (AIP)