Multiple Recursive Random Number Generators Research Articles

Spectral test of DX and DL multiple recursive random number generators

This paper conducts some computational experiments to exhaustive searches for the worst-case spectral test of DX and DL multiple recursive generators. The results indicate differences of the spectral test exist among the order, the generator and the number of multipliers vectors. We demonstrate that these differences can affect the worst-case performance of spectral tests.

On the Lattice Structure of a Special Class of Multiple Recursive Random Number Generators

We examine some properties of the points produced by certain classes of long-period linear multiple recursive random number generators proposed by L.-Y. Deng and his co-authors in several papers. These generators have their parameters selected in special ways to make the implementation faster. We show that as a result, the points produced by these generators have a poor lattice structure, and a poor initialization of the state can have long-lasting impact, because of the limited diffusion capacity of the recurrence.

A partial exhaustive search for good two-term third-order multiple recursive random number generators

This paper considers the problem of searching for a vector of multipliers in a two-term third-order multiple recursive generator (3MRG) with the objective of maximizing the spectral test performance. On computers with double precision floating point, the maximum spectral value can serve as an initial threshold one. The spectral value found by this paper is superior to the best spectral value previously published for the full period 3MRG.

Searching for good two-term fifth-order multiple recursive random number generators

An implementation of the spectral test for a full period two-term fifth-order multiple recursive generator with unrestricted multipliers is performed. Generators of this type with significantly better spectral values are presented. They are suitable for extensive and sophisticated large-scale simulation studies.

An exhaustive analysis of two-term multiple recursive random number generators with double precision floating point restricted multipliers

This paper performs an exhaustive search for the maximum spectral value in full period two-term k th-order multiple recursive generators (kMRGs) with the double precision floating point (DF) restricted multipliers for orders k = 2, 3, …, 7. For a two-term kMRG, computational experiment is conducted to compare and evaluate the numbers of the number of possible multipliers and the spectral values with those of the approximate factoring (AF) restricted multipliers. According to the experiments we perform, the results indicate differences exist among the numbers of possible multipliers for the AF and DF multiplier restrictions. We demonstrate that these differences can affect the performance of spectral tests.

Equivalency in periodicity and spectral test for two associated multiple recursive random number generators

This paper considers the problem of reducing the number of possible vectors of multipliers in a multiple recursive generator (MRG). We construct an associated MRG that satisfies a linear relationship between original and associated ones. It is shown that these two generators are equal in both periodicity and spectral value, thereby reducing the number of possible vectors of multipliers.

A backward heuristic algorithm for two-term multiple recursive random number generators

This paper considers the problem of searching for the maximum spectral value in a full period two-term kth-order multiple recursive generator with the unrestricted multipliers. The maximum spectral value with the double precision floating-point restricted multipliers can serve as an initial threshold spectral value. Based on equivalence properties of full period and spectral test, a backward heuristic algorithm with the threshold spectral value for efficiently calculating spectral value and checking full period is presented and is suitable for the parallel computations.

Theoretical analyses of forward and backward heuristics of multiple recursive random number generators

This article focuses on heuristic algorithms that are capable of finding multiple recursive generators (MRGs) with maximum spectral value criterion in a short period of time. Two promising algorithms, provided by Kao and Tang, are the forward and backward heuristics. Here we provide a deep understanding of the forward and backward heuristics for an MRG. The forward heuristic finds a sequence of partial sets of multipliers that lead toward a better spectral value. For the second-order MRGs, the performance of the backward heuristic theoretically outperforms that of the forward heuristic. The effectiveness of the backward heuristic is numerically confirmed by the experiments we present.

A revised forward and backward heuristic for two-term multiple recursive random number generators

This paper is concerned with designing a full period two-term kth-order multiple recursive generator (MRG) with the objective of maximizing the spectral value. A new heuristic that combines a forward and backward method with random and best-improving is presented. We adopt a statistical analysis to compare the performance of the new heuristic to the random search method for the full period two-term MRGs of orders three and four. We also perform a single longer experiment with maximum allowed CPU 604 800 s to compare the spectral value found using the new heuristic with the best of the previously published one. The results presented in this paper show that our method is indeed very effective with respect to the spectral value. Therefore, the new heuristic is a good way of obtaining an ideal two-term MRG of higher orders.

Searching for good two-term multiple recursive random number generators using a backpropagation algorithm

We describe an approximation algorithm for the problem of finding the maximum spectral value in a full period two-term multiple recursive generator (MRG). This algorithm is based on the principle of the backpropagation algorithm and random search. A welldesigned backpropagation algorithm is developed by an extensive set of computational comparisons of various parameters of the algorithm. Using this algorithm, a random search is conducted. Computational experiences show that our algorithm can find larger spectral value and shorter running time than forward/backward method for the full period twoterm MRG of order three. An application to the full period two-term MRG of order four is also discussed.

An exhaustive analysis of two-term multiple recursive random number generators with efficient multipliers

This paper presents results of an exhaustive computer search for the maximum spectral value in full period two-term kth-order multiple recursive generators (MRGs) with efficient multipliers. For orders k = 2 , 3 , … , 7 , we provide a table of kth-order MRGs with largest spectral values. These MRGs not only have the best spectral values found so far but also can be implemented efficiently by applying the approximate factoring method.

Searching for Good Multiple Recursive Random Number Generators via a Genetic Algorithm

In designing ideal multiple recursive random number (RN) generators (MRGs), the best set of multipliers, in terms of the lattice structure of the RNs produced, is sought. As the order of the MRG increases, the number of possible sets of multipliers to be examined grows exponentially. This paper proposes a genetic algorithm for designing good MRGs. The set of multipliers associated with the MRG is encoded as a binary string. Via the operations of reproduction, crossover, and mutation, new sets of multipliers are generated. The spectral values of the MRGs are calculated to guide the search process. As an illustration, the proposed algorithm is employed to find good sets of multipliers for MRGs of orders three and four. The results are better than those derived from other studies. To conclude, this paper not only finds better MRGs of orders three and four, but also develops an algorithm for designing MRGs of higher orders.

Using an adaptive genetic algorithm with reversals to find good second-order multiple recursive random number generators

This paper considers the problem of searching for good second-order multiple recursive generators (MRGs) with long period and good lattice structure. An adaptive genetic algorithm with reversals is proposed. The proposed algorithm is compared with forward/backward and random methods, and its effectiveness and efficiency is numerically confirmed by the experiments. The extensively tested second-order MRG (1259791845, 1433587751) found from the proposed algorithm possesses the properties of long period and good lattice structure and is therefore recommended.

Symmetry properties of multiple recursive random number generators in full period and spectral test

We analyze the full period and spectral test of multiple recursive generators (MRGs). Every MRG has associated with it another MRG, which possesses the same full period and spectral test as the original MRG. Based on this property, the computational effort required in testing full period and spectral test is reduced by half.

A statistical analysis of the screening measure of multiple recursive random number generators of orders one and two

This paper presents the results of a comprehensive empirical analysis of the screening measure of multiple recursive generators (MRGs) of orders one and two. Two kinds of screening measures are distinguished: spectral test and lattice test. With regard to these screening measures, two exhaustive searches of the twenty best MRGs of orders one and two are conducted. Some empirical comparisons reveal that the screening procedure with maximum spectral value criterion is preferred in terms of efficiency and thus, is a good way of obtaining ideal MRGs of higher orders. Several extensively tested second-order MRGs are also presented and are therefore recommended.

Variance Reduction via Lattice Rules

This is a review article on lattice methods for multiple integration over the unit hypercube, with a variance-reduction viewpoint. It also contains some new results and ideas. The aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science. These models can usually be reformulated as integration problems over the unit hypercube with a large (sometimes infinite) number of dimensions. We examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low-dimensional subspaces. The criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators. Numerical examples illustrate the effectiveness of the approach.

Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators

Combining parallel multiple recursive sequences provides an efficient way of implementing random number generators with long periods and good structural properties. Such generators are statistically more robust than simple linear congruential generators that fit into a computer word. We made extensive computer searches for good parameter sets, with respect to the spectral test, for combined multiple recursive generators of different sizes. We also compare different implementations and give a specific code in C that is faster than previous implementations of similar generators.

Several extensively tested multiple recursive random number generators

The Multiple Recursive Generator (MRG) has been considered by many scholars as a very good Random Number (RN) generator. This paper applies a sequential search to identify the MRGs of orders one, two, and three which are able to produce RNs with good lattice structure in terms of the spectral value and Beyer quotient. To detect departures from local randomness and homogeneity, extensive statistical tests including runs, auto-correlation, chi-square, serial, and the sparse occupancy tests have been conducted. In approximately 19.3 billion candidates, only four MRGs, namely, (1280550, −45991), (0, 45991, 1758790), (885300443, 0, 1552858447), and (885300443, 1546795921, 598295599), have passed all the theoretical and empirical tests. Among which (0, 45991, 1758790) can be implemented efficiently by applying the approximate factoring method and is therefore most recommended.

An Implementation of the Lattice and Spectral Tests for Multiple Recursive Linear Random Number Generators

We discuss the implementation of theoretical tests to assess the structural properties of simple or combined linear congruential and multiple recursive random number generators. In particular, we describe a package implementing the so-called spectral and lattice tests for such generators. Our programs analyze the lattices generated by vectors of successive or nonsuccessive values produced by the generator, analyze the behavior of generators in high dimensions, and deal with moduli of practically unlimited sizes. We give numerical illustrations. We also explain how to build lattice bases in several different cases, e.g., for vectors of far-apart nonsuccessive values, or for sublattices generated by the set of periodic states or by a subcycle of a generator, and, for all these cases, how to increase the dimension of a (perhaps partially reduced) basis.

Upper bounds in spectral test for multiple recursive random number generators with missing terms

One method for generating random numbers (RNs) of long period recommended by many scholars is the multiple recursive generator (MRG), in that the current RN is essentially a linear combination of the k preceding ones. In this paper, the upper bounds for a figure of merit adopted in the spectral test are derived for the k th order MRG with p ≤ k terms. As p gets smaller, the bounds become smaller as well. The simplest form of the k th order MRG with two terms frequently discussed in literature is found to have the worst bound.