Pseudo-random number generators

Fig. 4. A schematic two-dimensional illustration of the idea for an information theory model of hydrophobic hydration. Direct insertion of a solute of substantial size (the larger circle) will be impractical. For smaller solutes (the smaller circles) the situation is tractable a successful insertion is found, for example, in the upper panel on the right. For either the small or the large solute, statistical information can be collected that leads to reasonable but approximate models of the hydration free energy, Eq. (7). An important issue is that the solvent configurations (here, the point sets) are supplied by simulation or X-ray or neutron scattering experiments. Therefore, solvent structural assumptions can be avoided to some degree. The point set for the upper panel is obtained by pseudo-random-number generation so the correct inference would be of a Poisson distribution of points and = kTpv where v is the van der Waals volume of the solute. Quasi-random series were used for the bottom panel so those inferences should be different. See Pratt et al. (1999).

Consider the case each DCT block where a bit is inserted. The key K is used as a seed of pseudo random number generator to get random index values from [Z, Z2], 1index values are matched with orders of the JPEG-like zigzag pattern in the DCT block. Now two index sets To and h, each containing 2n elements, are generated using the secret key Ki as follows ... 

The choice of other variables R, r, h, 0, and r appropriate for Monte-Carlo averaging is made by pseudo random numbers generated on computer. The reactive cross section can be found by averaging the reaction probability over the impact parameter and rotational state... 

Matsumoto M and Nishimura T 1998 Mersenne twister A 623-dimensionally equidistrihuted uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation 8(1), 3-30. 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

Figure 23. El and E2 plots (a) Lorenz attractor, (b) Time series produced by a (pseudo) random number generator.

The first big success were pseudo-random number generators, see, in particular, [BlMi84, Yao82a, ImLL89] and for an overview, [Gold95]. 

Octave and Matiab provide many convenient built-in functions. Some that we have used in this text include the matrix exponential, expm incomplete gamma function, gammai pseudo-random number generators, rand and randm and several others. Type help -i and select the menu item Functi on Index for a complete list of Octave s built-in functions,... 

We next mention the implications of correlation and cycle length on parallel pseudo-random number generators (PPRNGs). 

Simulation experiments with the Lotka-Volterra reaction are given (Fig. 5.2). It can be seen that even the qualitative results of the simulation can depend primarily on the distance from the equilibrium, and even on the operation of the pseudo-random number generator. The set of realisations can be classified ... 

The initial data should be chosen from quite extensional table or created via true or pseudo random number generator excluding wrong data. 

W. Janke, Pseudo Random Numbers Generation and Quality Checks, m Proceedings of the Winter School Quantum Simulations of Complex Many-Body Systems From Theory to Algorithms, John von Neumann Institute for Computing, Jiilich, NIC Series vol. 10, ed. J. Grotendorst, D. Marx, and A. Muramatsu (Jiilich NIC, 2002), p. 447. 

We want to stress that in [Caravenna et al. (2006)] particular care has been put into analyzing the role of the pseudo-random number generator, above all when applying the statistical test. Several random number generators have been compared and also true randomness has been used (see the paper for details). 

In the numerical experiments reported in this section we have used G5 CPUs clocked at 2 MHz. We have implemented the algorithms in Section 9.1.1 in C the codes are available from the webpage of the author. The pseudo-random number generator that we have used is a standard implementation of the Mersenne-Twister algorithm [Matsumoto and Nishimura (1998)]. The figures in this section have been done with R [R Development Core Team (2004)]. 

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