Pseudorandom Vectors Research Articles

Pseudorandom vector generation by the inversive method

Pseudorandom vectors are of importance for parallelized simulation methods. In this article we carry out a detailed analysis of the inversive method for the generation of uniform pseudorandom vectors. This method can be viewed as an analog of the inversive congruential method for pseudorandom number generation. We study, in particular, the periodicity properties and the behavior under the serial test for sequences of pseudorandom vectors generated by the inversive method.

Factorization of polynomials and some linear-algebra problems over finite fields

The recently developed algorithm of Niederreiter for the factorization of polynomials over finite fields provides a “linearization” of the factorization problem via differential equations in rational function fields. We show that the range of applicability of this algorithm can be extended by various techniques, such as the application of normal bases and the use of Hasse-Teichmüller derivatives. We also discuss topics from pseudorandom vector generation and combinatorial linear algebra that lead to interesting linear algebra problems over finite fields.

Random Number Generation and Quasi-Monte Carlo Methods.

Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.

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.

Statistical independence properties of pseudorandom vectors produced by matrix generators

The uniformity test and the serial test are standard statistical tests for equidistribution and statistical independence, respectively, in sequences of uniform pseudorandom numbers. We introduce analogues of these tests for sequences of uniform pseudorandom vectors and we apply these tests to matrix generators for uniform pseudorandom vector generation. The results show that these generators behave well under these tests provided that the matrix in the generation algorithm is chosen carefully. The essential condition is that a certain figure of merit attached to the matrix be large. We also prove a general theorem which guarantees the existence of matrices with large figure of merit.

Testing of differential cascade voltage switch (DCVS) circuits

The problem of testing differential cascode voltage switch (DCVS) circuits is analyzed. These circuits have several potential applications in fault-tolerant, highly available system design due to their inherent self-checking capability. It is shown how concurrent (online) testing of DCVS circuits, which is very effective under single transistor fault assumptions, can be performed. The impact of multiple faults of DCVS circuits is examined, and analytical results are derived. These results indicate that periodic offline tests on DCVS circuits are necessary in order to achieve high multiple-fault coverage. Single-fault test sets and/or pseudorandom vectors were successfully used in the offline tests to detect many of the multiple faults which reduce the efficiency of online tests. The results show the need for a comprehensive mixed-test strategy combining offline and online tests for DCVS circuits. >

Alternative specifications of the error process in the stochastic simulation of econometric models

SUMMARY This paper analyses the stochastic simulation of econometric models using three different methods for specifying the probability distribution of the structural error terms. The impact of these different assumptions on the simulation bias and model variance is explored empirically. Monte Carlo variance reduction techniques are used to distinguish the effects of the different specifications. A macroeconometric model is a set of behavioural equations which represent endogenous variables as a function of predetermined variables and error terms. The model may be written in the form f(yt, xt) = ut, where yt is a vector of endogenous variables, xt is a vector of predetermined variables, and ut is a vector of stochastic error terms. The parameters of the structural functions f are estimated using econometric methods. When econometric models are used to predict the future, a point prediction is usually obtained with deterministic solution. The deterministic solution is calculated by solving f(yt, xt) = 0 for yt. This is equivalent to assuming that the stochastic terms ut are equal to zero in the future period. It is well known that, if the model is nonlinear, then E(yt I xt) is not equal to the deterministic solution. For example, see, Fisher and Salmon (1986), where large discrepancies between E(yt I xt) arid the deterministic solution are shown in some cases. Alternatively, predictions may be made using stochastic simulation. In stochastic simulation a distribution is assumed for the error vector ut. Then a pseudo-random vector st is drawn from the specified distribution. The model is then solved; the value of Yt which solves f(yt, xt) = st is found. This process is repeated n times and the resulting vectors are averaged. Since this is a Monte Carlo technique, the economist can increase accuracy by performing more replications (increasing n). The results will depend upon the distribution assumed for the error terms ut, and adopted for generating the pseudo-random error terms St. 1 Estimation errors of the structural form parameters could also be considered in simulation studies. However, it was shown in Bianchi and Calzolari (1980) that the impact of errors in estimated coefficients can be treated with analytical or analytical-numerical methods even for nonlinear models, without need of stochastic simulation. Therefore, in this study we shall not deal with errors in coefficients; coefficients will be treated as given.

On the period length of pseudorandom vector sequences generated by matrix generators

In Tahmi [5], Niederreiter [4], Afflerbach and Grothe [1], and Grothe [2] linear recursive congruential matrix generators for generating r-dimensional pseudorandom vectors are analyzed. In particular, conditions are established which ensure that the period length equals p r − 1 {p^r} - 1 for any nonzero starting vector in case of a prime modulus p. For a modulus of the form p α {p^\alpha } , α ≥ 2 \alpha \geq 2 and p prime, this paper describes a simple method for constructing matrix generators having the maximal possible period length ( p r − 1 ) ⋅ p α − 1 ({p^r} - 1) \cdot {p^{\alpha - 1}} for any starting vector which is nonzero modulo p.

On the Period Length of Pseudorandom Vector Sequences Generated by Matrix Generators

On the Period Length of Pseudorandom Vector Sequences Generated by Matrix Generators

The lattice structure of pseudo-random vectors generated by matrix generators

In many areas of science the problems treated by Monte-Carlo simulations become more and more complex and more extensive. Because of that generators like linear congruential matrix generators are needed which produce enormously many pseudo-random numbers. In order to assess stochastical properties of the generated pseudo-random vectors the lattice structure of these matrix generators is studied here.

Matrix generators for pseudo-random vector generation

In this paper a method for the direct generation of pseudo-random vectors is considered. Thereby the n-th pseudo-random vector is recursively generated from the (n−1)-th pseudo-random vector by multiplication with a matrix. The period lengths of this generator type are examined and characterized. Furthermore it is shown that some popular pseudo-random number generators can be regarded as special cases of this method.

The simulation of spherical distributions in the Fisher-Bingham family

The problem of generating pseudo-random unit vectors from the Fisher-Bingham distribution is considered. Attention is focussed on a subfamily known as FB6' which includes many of the spherical distributions of practical interest, including those of Dimroth-Watson, FB4' FB5 and Bingham type. The rejection procedures suggested here for these distributions should prove adequate for most practical purposes. Possibilities for simulating from the general Fisher-Bingham distribution are also mentioned briefly.

Computer Generation of Distributions on the m-Sphere

A technique is developed for constructing algorithms to generate pseudo random vectors from families of distributions on the unit m‐sphere. The method is applied to the Fisher‐von Mises distribution to provide a generator which is easily programmed and very efficient. The Fisher–von Mises generator is then used in a brief Monte Carlo study on the validity of a test procedure developed by Stephens (1982).

The Lattice Structure of Multiplicative Congruential Pseudo-Random Vectors

The lattice structure of points in an n-dimensional space produced by an appropriate grouping of pseudo-random numbers obtained from multiplicative congruential generators is discussed. Examples are given for $2 \leqq n \leqq 6$. The work is based on the theory of the reduction of positive quadratic forms in n variables.

The lattice structure of multiplicative congruential pseudo-random vectors

The lattice structure of points in an n-dimensional space produced by an appropriate grouping of pseudo-random numbers obtained from multiplicative congruential generators is discussed. Examples are given for 2 ≦ n ≦ 6 2 \leqq n \leqq 6 . The work is based on the theory of the reduction of positive quadratic forms in n variables.

Difference methods for stochastic ordinary differential equations

Numerical methods for linear systems of the form (1.1) are given in [7] and [8]. These methods do not apply to nonlinear equations. The purpose of this paper is to provide an analysis of the random error which occurs when the stochastic, nonlinear equations (1.1) are approximated by finitedifference equations. For At > 0 we approximate the random vector x(t) defined by (1.1) by the vector x) defined by the difference equation (1.3) x(n+l) = xn) + Atf(x (n) + g(fn) (n = 0, 1, ... ) x() where t = nAt and where g(n) is a pseudo-random vector simulating a random