## Random Number Generators for Parallel Applications (1998)

Venue: | in Monte Carlo Methods in Chemical Physics |

Citations: | 18 - 7 self |

### BibTeX

@INPROCEEDINGS{Srinivasan98randomnumber,

author = {Ashok Srinivasan and David M. Ceperley and Michael Mascagni},

title = {Random Number Generators for Parallel Applications},

booktitle = {in Monte Carlo Methods in Chemical Physics},

year = {1998},

pages = {13--36},

publisher = {John Wiley and Sons}

}

### OpenURL

### Abstract

this article is devoted, because these com1 putations require the highest quality of random numbers. The ability to do a multidimensional integral relies on properties of uniformity of n-tuples of random numbers and/or the equivalent property that random numbers be uncorrelated. The quality aspect in the other uses is normally less important simply because the models are usually not all that precisely specified. The largest uncertainties are typically due more to approximations arising in the formulation of the model than those caused by lack of randomness in the random number generator. In contrast, the first class of applications can require very precise solutions. Increasingly, computers are being used to solve very well-defined but hard mathematical problems. For example, as Dirac [1] observed in 1929, the physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known and it is only necessary to find precise methods for solving the equations for complex systems. In the intervening years fast computers and new computational methods have come into existence. In quantum chemistry, physical properties must be calculated to "chemical accuracy" (say 0.001 Rydbergs) to be relevant to physical properties. This often requires a relative accuracy of 10

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Citation Context ...exponential sum bounds, as in the case of the prime modulus LCG. This similarity is no accident, and is based on the fact that both generators are maximal period linear recursions over a finite field =-=[25]-=-. 3.3 Lagged-Fibonacci Generators The Additive Lagged-Fibonacci Generator (ALFG) is: x n = x n\Gammaj + x n\Gammak (mod 2 m ); j ! k: (3) In recent years the ALFG has become a popular generator for se... |

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Citation Context ...s to the lattice is slow in high dimensions. The discrepancy of a lattice does not decrease smoothly with increasing N . A remarkable fact is that there is a lower (Roth) bound to the its discrepancy =-=[39]-=-: D s (x i )sC s N \Gamma1 (log N) (s\Gamma1)=2 ; s ? 3: (14) This gives us a target to aim at for the construction of low-discrepancy point sets (quasi-random numbers). For comparison, the estimated ... |

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Citation Context ...ly prime to m \Gamma 1, and (2) does the good inter-stream correlation also ensure good intra-stream independence via the spectral test? 3.2 Shift-Register Generators Shift Register Generators (SRGs) =-=[21, 22]-=- are of the form: x n+k = k\Gamma1 X i=0 a i x n+i (mod 2); (2) where the x n 's and the a i 's are either 0 or 1. The maximal period of 2 k \Gamma 1 and can be achieved using as few as two non-zero v... |

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Citation Context ...ly prime to m \Gamma 1, and (2) does the good inter-stream correlation also ensure good intra-stream independence via the spectral test? 3.2 Shift-Register Generators Shift Register Generators (SRGs) =-=[21, 22]-=- are of the form: x n+k = k\Gamma1 X i=0 a i x n+i (mod 2); (2) where the x n 's and the a i 's are either 0 or 1. The maximal period of 2 k \Gamma 1 and can be achieved using as few as two non-zero v... |

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Citation Context ... An important new type of PRNG that, as yet, has not found any widely distributed implementation is the Inversive Congruential Generator (ICG). This generator comes in two versions, the recursive ICG =-=[30, 31] x n = a-=-x n\Gamma1 + b (mod m); (5) and the explicit ICG [32] x n = an + b (mod m): (6) In both the above equations �� c denotes the multiplicative inverse modulo m in the sense that c��c j 1 (mod m) ... |

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Citation Context ... is cheap to compute and it does well on standard statistical tests [11], especially when the lag k is sufficiently high (such as k = 1279). The maximal period of the ALFG is (2 k \Gamma 1)2 m\Gamma1 =-=[26, 27]-=- and has 2 (k\Gamma1)\Theta(m\Gamma1) different full-period cycles [28]. Another advantage of the ALFG is that one can implement these generators directly in floating-point to avoid the conversion fro... |

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Citation Context ...s can probe other qualities of random number generators such as inter-process correlation. There is a recent review which covers parallel random number generation in somewhat more depth by Coddington =-=[12]-=-. The interested reader can also refer to [13, 14, 15, 16, 17] for work related to parallel random number generation and testing. This article is structured as follows. First we discuss the desired pr... |

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Citation Context ...enerators such as inter-process correlation. There is a recent review which covers parallel random number generation in somewhat more depth by Coddington [12]. The interested reader can also refer to =-=[13, 14, 15, 16, 17]-=- for work related to parallel random number generation and testing. This article is structured as follows. First we discuss the desired properties that random number generators should have. Next we di... |

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Citation Context ...�� c denotes the multiplicative inverse modulo m in the sense that c��c j 1 (mod m) when c 6= 0, and �� 0 = 0. An advantage of ICGs over LCGs are that tuples made from ICGs do not fall in =-=hyperplanes [33, 34]-=-. Unfortunately the cost of doing modular inversion is considerable: it is O(log 2 m) times the cost of multiplication. 3.5 Combination Generator Better quality sequences can often be obtained by comb... |

25 |
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Citation Context ...enerators such as inter-process correlation. There is a recent review which covers parallel random number generation in somewhat more depth by Coddington [12]. The interested reader can also refer to =-=[13, 14, 15, 16, 17]-=- for work related to parallel random number generation and testing. This article is structured as follows. First we discuss the desired properties that random number generators should have. Next we di... |

24 | Parallel linear congruential generators with Sophie-Germain moduli
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Citation Context ...enerators such as inter-process correlation. There is a recent review which covers parallel random number generation in somewhat more depth by Coddington [12]. The interested reader can also refer to =-=[13, 14, 15, 16, 17]-=- for work related to parallel random number generation and testing. This article is structured as follows. First we discuss the desired properties that random number generators should have. Next we di... |

23 |
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Citation Context ...ng, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect and it has been discovered that the random number generator can influence the results =-=[3, 4, 5, 6]-=-. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. Given ... |

18 |
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Citation Context ...dely distributed implementation is the Inversive Congruential Generator (ICG). This generator comes in two versions, the recursive ICG [30, 31] x n = ax n\Gamma1 + b (mod m); (5) and the explicit ICG =-=[32] x n = an + -=-b (mod m): (6) In both the above equations �� c denotes the multiplicative inverse modulo m in the sense that c��c j 1 (mod m) when c 6= 0, and �� 0 = 0. An advantage of ICGs over LCGs are... |

15 | Techniques for testing the quality of parallel pseudo-random number generators
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Citation Context |

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A Fast, High-Quality, and Reproducible Lagged-Fibonacci Pseudorandom Number Generator
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Citation Context ...specially when the lag k is sufficiently high (such as k = 1279). The maximal period of the ALFG is (2 k \Gamma 1)2 m\Gamma1 [26, 27] and has 2 (k\Gamma1)\Theta(m\Gamma1) different full-period cycles =-=[28]-=-. Another advantage of the ALFG is that one can implement these generators directly in floating-point to avoid the conversion from integer to floating-point that accompanies the use of other generator... |

9 |
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Citation Context ...ng, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect and it has been discovered that the random number generator can influence the results =-=[3, 4, 5, 6]-=-. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. Given ... |

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Citation Context ...umber generators and has often detected defects in generators. We can test parallel generators on the Ising model application by assigning distinct random number sequences to subsets of lattice sites =-=[37]-=-. How good are the popular generators on the Ising model? Ferrenberg, et al [3] found that certain generators such as the particular shift register generator they used (R250) failed with the Wolff alg... |

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Citation Context ...ng, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect and it has been discovered that the random number generator can influence the results =-=[3, 4, 5, 6]-=-. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. Given ... |

5 |
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Citation Context ...ponding ALFG [27], it has empirical properties considered to be superior to ALFGs [11]. Of interest for parallel computing is that a parameterization analogous to that of the ALFG exists for the MLFG =-=[29]-=-. 3.4 Inversive Congruential Generators An important new type of PRNG that, as yet, has not found any widely distributed implementation is the Inversive Congruential Generator (ICG). This generator co... |

4 |
Cluster-flipping Monte Carlo algorithm and correlations in "good" random number generators
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4 |
Statistical independence of nonlinear congruential pseudorandom numbers
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(Show Context)
Citation Context ... An important new type of PRNG that, as yet, has not found any widely distributed implementation is the Inversive Congruential Generator (ICG). This generator comes in two versions, the recursive ICG =-=[30, 31] x n = a-=-x n\Gamma1 + b (mod m); (5) and the explicit ICG [32] x n = an + b (mod m): (6) In both the above equations �� c denotes the multiplicative inverse modulo m in the sense that c��c j 1 (mod m) ... |

4 |
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(Show Context)
Citation Context ...rput sequence. There have been many others which are thought to be more general and have provably smaller discrepancy. Of particular note are the explicit constructions of Faure [41] and Niederreiter =-=[42]-=-. The use of quasi-random numbers in quadrature has not been widespread because the claims of superiority of quasi-random over pseudo-random for quadrature have not been shown empirically especially f... |

2 | Recent developments in parallel pseudorandom number generation
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(Show Context)
Citation Context |

2 |
Using Permutations to Reduce Discrepancy
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(Show Context)
Citation Context ...40] use the van der Corput sequence. There have been many others which are thought to be more general and have provably smaller discrepancy. Of particular note are the explicit constructions of Faure =-=[41]-=- and Niederreiter [42]. The use of quasi-random numbers in quadrature has not been widespread because the claims of superiority of quasi-random over pseudo-random for quadrature have not been shown em... |

1 |
Funktionen von beschrankter variation in der theorie der gleichverteiling
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Citation Context ...5]: jI \GammasIjsD s (x i )V (f); (12) where D s (x i ) is the discrepancy of the points x i as defined by Eq. (13), and V (f) is the total variation of f on [0; 1) s in the sense of Hardy and Krause =-=[38]-=-. The total variation is roughly the average absolute value of the s th derivative of f(x). Note that Eq. (12) gives a deterministic error bound for integration because V (f) depends only on the natur... |