Multiple Recursive Generators Research Articles

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.

Reverse multiple recursive random number generators

Every multiple recursive generator (MRG) has an associated reverse MRG that generates the same sequence but in reverse order. An explicit formula for the recurrence of this reverse MRG is derived based on finite field arithmetic. These two MRGs are equal in both periodicity and spectral value. This property could be exploited to reduce the number of candidates when searching for good MRG parameters.

A new heuristic for second-order multiple recursive random number generator

This paper considers the problem of designing sets of multipliers in a full period multiple recursive generator (MRG) with the objective of maximizing the spectral value. We propose a new heuristic that is based on the principles of forward and backward, random and best-improving. An application to the second-order MRG is presented. It is shown that the spectral value found by proposed method is better than published ones. Such secondorder MRG also passes a comprehensive battery of empirical statistical tests and is therefore recommended.

Efficient and portable multiple recursive generators of large order

Deng and Xu [2003] proposed a system of multiple recursive generators of prime moduluspand orderk, where all nonzero coefficients of the recurrence are equal. This type of generator is efficient because only a single multiplication is required. It is common to choosep= 231−1 and some multipliers to further improve the speed of the generator. In this case, some fast implementations are available without using explicit division or multiplication. For such ap, Deng and Xu [2003] provided specific parameters, yielding the maximum period for recurrence of orderk, up to 120. One problem of extending it to a largerkis the difficulty of finding a complete factorization ofpk−1. In this article, we apply an efficient technique to findksuch that it is easy to factorpk−1, withp= 231−1. The largest one found isk= 1597. To find multiple recursive generators of large orderk, we introduce an efficient search algorithm with an early exit strategy in case of a failed search. Fork= 1597, we constructed several efficient and portable generators with the period length approximately 1014903.1.

Effective and efficient restriction on producing the multipliers for the multiple recursive random number generator

The most efficient and easiest way for implementing a multiple recursive random number generator is the approximate factoring (AF) method. In the literature, the AF method can be applied in the three types of restriction on multiplier. This paper considers the effectiveness and efficiency of these restrictions from the theoretical and empirical points of view. The most effective and efficient restriction is presented. The number of arithmetic operations required by the AF method is also discussed.

Simulated division with approximate factoring for the multiple recursive generator with both unrestricted multiplier and non-mersenne prime modulus

This paper focuses on devising a general and efficient way of generating random numbers for the multiple recursive generator with both unrestricted multiplier and non-Mersenne prime modulus. We propose a new algorithm that embeds the technique of approximate factoring into the simulated division method. The proposed new algorithm improves the decomposition method in terms of both the suitability for various word-sizes of the computers and the efficiency characteristics, such as the number of arithmetic operations required and the computational time. Empirical simulations are conducted to compare and evaluate the computational time of this algorithm with the decomposition method for various computers.

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators

We propose a system of multiple recursive generators of moduluspand orderkwhere all nonzero coefficients of the recurrence are equal. The advantage of this property is that a single multiplication is needed to compute the recurrence, so the generator would run faster than the general case. Forp= 231− 1, the most popular modulus used, we provide tables of specific parameter values yielding maximum period for recurrence of orderk= 102 and 120. Forp= 231− 55719 andk= 1511, we have found generators with a period length approximately 1014100.5.

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.

Sum-discrepancy test on pseudorandom number generators

We introduce a non-empirical test on pseudorandom number generators (prng), named sum-discrepancy test. We compute the distribution of the sum of consecutive m outputs of a prng to be tested, under the assumption that the initial state is uniformly randomly chosen. We measure its discrepancy from the ideal distribution, and then estimate the sample size which is necessary to reject the generator. These tests are effective to detect the structure of the outputs of multiple recursive generators with small coefficients, in particular that of lagged Fibonacci generators such as random( ) in BSD-C library, as well as add-with-carry and subtract-with-borrow generators like RCARRY. The tests show that these generators will be rejected if the sample size is of order 10 6. We tailor the test to generators with a discarding procedure, such as ran_array and RANLUX, and exhibit empirical results. It is shown that ran_array with half of the output discarded is rejected if the sample size is of the order of 4×10 10. RANLUX with luxury level 1 (i.e. half of the output discarded) is rejected if the sample size is of the order of 2×10 8, and RANLUX with luxury level 2 (i.e. roughly 3/4 is discarded) will be rejected for the sample size of the order of 2.4×10 18. In our previous work, we have dealt with the distribution of the Hamming weight function using discrete Fourier analysis. In this work, we replace the Hamming weight with the continuous sum, using a classical Fourier analysis, i.e. Poisson’s summation formula and Levy’s inversion formula.

Modified decomposition method for multiple recursive random number generator

This paper considers the problem of generally and efficiently generating random numbers (RNs) for the multiple recursive generators with unrestricted multipliers. A new algorithm based on the decomposition method is proposed. The new algorithm improves the decomposition method in terms of both generality and efficiency. It is shown to be suitable for the signed and unsigned magnitude number systems on a computer, and to require fewer numbers of arithmetic operations than the decomposition method. Several randomly constructed numerical examples illustrate the low variance and efficiency of the new algorithm compared with the decomposition method for various computers.

Combined random number generator via the generalized Chinese remainder theorem

The combined random number (RN) generator has been considered by many scholars as a good RN generator. One promising type of combined RN generator, recommended by L'Ecuyer (Oper. Res. 44 (1996) 816; 47 (1999) 159), is the combined multiple recursive generator (MRG). This paper analyzes the combined MRG via the Chinese remainder theorem. A new combined generator based on the generalized Chinese remainder theorem and on the Ore algorithm (Amer. Math. Monthly 59 (1952) 365) is presented. The proposed combined generator improves the combined MRG in terms of both the suitability for various types of RN generators and the restriction on the moduli of the individual generators. Therefore, the proposed combined generator is an ideal RN generator and is most recommended.

Lower bounds in spectral tests for vectors of nonsuccessive values produced by multiple recursive generator with some zero multipliers

L'Ecuyer develops lower bounds for the maximum distance of the parallel hyperplanes generated by the vectors of nonsuccessive random numbers (RNs) obtained from multiple recursive generators (MRGs). In this paper, a stronger bound, twice that obtained by L'Ecuyer, is derived. For k th-order MRGs with fewer than k terms in the recursive relationship, much stronger bounds are found. Large distance implies bad lattice structure. In designing RN generators, this factor should be taken into consideration.

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.

The revised recursive reduction for efficiently generating random numbers

One of the general methods for implementing a multiple recursive generator is recursive reduction method (RRM). This paper provides an analysis of its number of operations. We also propose a new method that modifies the RRM by considering the stop conditions and the order of multiplier and multiplicand. Some empirical comparisons reveal that the new algorithm is more efficient with a reduction of iterative numbers in the range of 13.1416%-20.5118% depending on the moduli and multipliers being used.

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.

Implementing a multiple recursive generator with mersenne prime modulus

This paper considers the problem of efficiently and correctly generating random numbers for the multiple recursive generators with Mersenne prime modulus. For the appropriate Mersenne prime modulus, simulated division is an efficient way to generate random numbers. The general algorithm for any Mersenne prime modulus is also presented.

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.