## Uniform Random Number Generators for Supercomputers (1992)

### Cached

### Download Links

Venue: | Proc. Fifth Australian Supercomputer Conference |

Citations: | 26 - 11 self |

### BibTeX

@INPROCEEDINGS{Brent92uniformrandom,

author = {Richard P. Brent},

title = {Uniform Random Number Generators for Supercomputers},

booktitle = {Proc. Fifth Australian Supercomputer Conference},

year = {1992},

pages = {95--104}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the requirements for uniform pseudo-random number generators on modern vector and parallel supercomputers, consider the pros and cons of various classes of methods, and outline what is currently available. We propose a class of random number generators which have good statistical properties and can be implemented efficiently on vector processors and parallel machines. A good method for initialization of these generators is described, and an implementation on a Fujitsu VP 2200/10 vector processor is discussed. 1

### Citations

675 |
The Art of Computer Programming, volume 2: Seminumerical Algorithms
- Knuth
- 1988
(Show Context)
Citation Context ...s distributions (e.g. normal, exponential, Poisson, : : :) but the algorithms used to generate these random numbers almost invariably require a good uniform random number generator -- see for example =-=[2, 16, 29]-=-. In this paper we consider only the generation of uniformly distributed numbers. Usually we are concerned with real numbers un which are intended to be uniformly distributed on the interval [0; 1]. S... |

234 |
Shift register sequences
- Golomb
- 1967
(Show Context)
Citation Context ...od m) or "exclusive or" (mod m = 2 w ). We abbreviate these operators by +; \Gamma;sand \Phi respectively. Generators using \Phi are also called "shift register" generators or &quo=-=t;Tausworthe" generators [11, 18, 31]-=-. If ` is + or \Gamma (mod m) then a theory of generalized Fibonacci generators can be based on the generating function G(x) = 1 X n=0 Unx n which is given by G(x) = P (x)=Q(x) mod m; where Q(x) = 1 \... |

229 |
Random number generators: Good ones are hard to find
- Park, Miller
- 1988
(Show Context)
Citation Context ...Gamma 1, even if c = 0. It is sometimes convenient that an exact zero does not occur in the sequence. There is much theory regarding the best choice of multiplier a for linear congruential generators =-=[16, 27]-=-, and an exhaustive search has been performed for certain moduli [8]. Marsaglia [19] pointed out the fundamental weakness of the class of linear congruential generators. If d-tuples (u dn ; u dn+1 ; :... |

206 |
Various techniques used in connection with random digits
- Neumann
- 1951
(Show Context)
Citation Context ...a Fujitsu VP 2200/10 vector processor is discussed. 1 Introduction -- Requirements Pseudo-random numbers have been used in Monte Carlo calculations [1, 3, 15] since the pioneering days of Von Neumann =-=[26]-=-. With the increasing speed of vector processors and parallel computers, considerable attention must be paid to the quality of random number generators available in subroutine libraries. A program run... |

105 | E¢ cient and portable combined random number generators - L’Ecuyer - 1988 |

71 |
Random numbers fall mainly in the planes
- Marsaglia
- 1968
(Show Context)
Citation Context ...the sequence. There is much theory regarding the best choice of multiplier a for linear congruential generators [16, 27], and an exhaustive search has been performed for certain moduli [8]. Marsaglia =-=[19]-=- pointed out the fundamental weakness of the class of linear congruential generators. If d-tuples (u dn ; u dn+1 ; : : : ; u dn+d\Gamma1 ) of normalized numbers are considered as points in the d-dimen... |

60 |
A Review of Pseudorandom Number Generators
- James
- 1990
(Show Context)
Citation Context ...e generators is described, and an implementation on a Fujitsu VP 2200/10 vector processor is discussed. 1 Introduction -- Requirements Pseudo-random numbers have been used in Monte Carlo calculations =-=[1, 3, 15]-=- since the pioneering days of Von Neumann [26]. With the increasing speed of vector processors and parallel computers, considerable attention must be paid to the quality of random number generators av... |

59 |
Toward a Universal Random Number Generator, Florida State University FSU-SCRI-87-50
- Marsaglia, Zaman
- 1987
(Show Context)
Citation Context ...tor processor. Siemens Nixdorf in collaboration with the University of Karlsruhe have implemented a package of random number generators (RAND/VP). The generators are adapted from the generator UNI of =-=[23], whi-=-ch is based on the generalized Fibonacci generator F (97; 33; \Gamma). Marsaglia [22] now prefers his VLP generators (described below) . The algorithm used in RAND/VP "has been modified to genera... |

56 |
Random number generators on vector supercomputers and other advanced architectures
- Anderson
- 1990
(Show Context)
Citation Context ...e generators is described, and an implementation on a Fujitsu VP 2200/10 vector processor is discussed. 1 Introduction -- Requirements Pseudo-random numbers have been used in Monte Carlo calculations =-=[1, 3, 15]-=- since the pioneering days of Von Neumann [26]. With the increasing speed of vector processors and parallel computers, considerable attention must be paid to the quality of random number generators av... |

44 |
Generalized feedback shift register pseudorandom number algorithm
- Lewis, Payne
- 1973
(Show Context)
Citation Context ...od m) or "exclusive or" (mod m = 2 w ). We abbreviate these operators by +; \Gamma;sand \Phi respectively. Generators using \Phi are also called "shift register" generators or &quo=-=t;Tausworthe" generators [11, 18, 31]-=-. If ` is + or \Gamma (mod m) then a theory of generalized Fibonacci generators can be based on the generating function G(x) = 1 X n=0 Unx n which is given by G(x) = P (x)=Q(x) mod m; where Q(x) = 1 \... |

39 |
On primitive trinomials (mod 2
- Zierler, Brillhart
- 1968
(Show Context)
Citation Context ... 2 and the initial values are not all zero, then the sequence has maximal period 2 r \Gamma 1 if and only if Q(x) is a primitive polynomial (mod 2). Tables of such primitive polynomials are available =-=[17, 32]-=-. Verification is particularly simple if r is the exponent of a Mersenne prime (i.e. 2 r \Gamma 1 is prime) because then we only need to check that x = x 2 r mod (Q(x); 2) which can be done by r squar... |

37 | A new class of random number generators. The Annals of Applied Probability - Marsaglia, Zaman - 1991 |

31 | The k-distribution of generalized feedback shift register pseudorandom numbers - FUSHIMI, TEZUKA - 1983 |

30 |
Primitive trinomials whose degree is a Mersenne exponent, Inform, and Control 15(1969), 67-69. National Research Laboratory of Metrology, Umezono
- Zierler
(Show Context)
Citation Context ...rime (i.e. 2 r \Gamma 1 is prime) because then we only need to check that x = x 2 r mod (Q(x); 2) which can be done by r squarings of polynomials (mod 2), involving a total of only O(r 2 ) operations =-=[33]-=-. The more usual formulation in terms of r by r matrices [20, 21] instead of polynomials is less efficient computationally because matrix multiplication is more expensive than polynomial multiplicatio... |

30 | Matrices and the structure of random number sequences. Linear Algebra and its Applications 67 - Marsaglia, Tsay - 1985 |

28 | On the periods of generalized fibonacci recurrences
- Brent
- 1994
(Show Context)
Citation Context ...\Gamma 1) if ` = \Sigma mod m, 2 w\Gamma3 (2 r \Gamma 1) if ` =smod m. The initial values must be odd for ` = , not all even for ` = \Sigma, and not all zero for ` = \Phi. For precise conditions, see =-=[5, 21]-=-. We see one advantage of the generalized Fibonacci generators over linear congruential generators -- the period can be made very large by choosing r large. However, one should refrain from using more... |

26 | A current view of random number generators,” in Computer Science and Statistics: The Interface - Marsaglia - 1985 |

25 | Random number generators for MIMD parallel processors - Percus, Kalos - 1989 |

23 | Primitive t-nomials (t =3, 5) over GF(2) whose degree is a Mersenne exponent - Kurita, Matsumoto - 1991 |

23 | Tausworthe, “Random numbers generated by linear recurrence modulo two - C - 1965 |

21 | A class of parallel random number generators - Matteis, Pagnutti - 1990 |

20 |
A random number generator for PC’s
- Marsaglia, Narasimhan, et al.
- 1990
(Show Context)
Citation Context ...emented a package of random number generators (RAND/VP). The generators are adapted from the generator UNI of [23], which is based on the generalized Fibonacci generator F (97; 33; \Gamma). Marsaglia =-=[22] now prefe-=-rs his VLP generators (described below) . The algorithm used in RAND/VP "has been modified to generate several streams of random numbers in parallel" [13]. Presumably this means that the rec... |

13 | Algorithm 488: A Gaussian pseudo-random number generator - Brent |

11 |
Some Vectorized Random Number Generators for Uniform
- Petersen
- 1988
(Show Context)
Citation Context ...s distributions (e.g. normal, exponential, Poisson, : : :) but the algorithms used to generate these random numbers almost invariably require a good uniform random number generator -- see for example =-=[2, 16, 29]-=-. In this paper we consider only the generation of uniformly distributed numbers. Usually we are concerned with real numbers un which are intended to be uniformly distributed on the interval [0; 1]. S... |

9 |
Analysis of Additive Random Number Generators, Ph. D. thesis and
- Reiser
- 1977
(Show Context)
Citation Context ...ability 1=6 for a random sequence ([16], ex. 3.2.2.2). Attempts have been made to generalize the Fibonacci recurrence to obtain "generalized Fibonacci " or "lagged Fibonacci" rando=-=m number generators [12, 16, 30]. Marsagli-=-a [20] considers generators F (r; s; `) which satisfy Un = Un\Gammar `Un\Gammas for fixed "lags" r and s (r ? s ? 0) and nsr. Here ` is some binary operator, e.g. addition (mod m), subtracti... |

9 | Quasi-Random Number Sequences from a Long-Period TLP Generator with Remarks on Application to Cryptography - Bright, Enison - 1979 |

8 | Uniform Random Number Generators for Vector and Parallel Computers
- Brent
- 1992
(Show Context)
Citation Context ... ? s ? t ? 0 are suitably chosen lags. We call such generators "4-term generalized Fibonacci " generators in contrast to the usual 3term generators. For a discussion of 4-term generators, we=-= refer to [4]-=-. 4 Comments on some available generators We have discussed RANDU in the sub-section on linear congruential generators, and shown why similar generators are not to be recommended. Even with an improve... |

7 |
Empirical tests of an additive random number generator
- Green, Smith, et al.
- 1959
(Show Context)
Citation Context ...ability 1=6 for a random sequence ([16], ex. 3.2.2.2). Attempts have been made to generalize the Fibonacci recurrence to obtain "generalized Fibonacci " or "lagged Fibonacci" rando=-=m number generators [12, 16, 30]. Marsagli-=-a [20] considers generators F (r; s; `) which satisfy Un = Un\Gammar `Un\Gammas for fixed "lags" r and s (r ? s ? 0) and nsr. Here ` is some binary operator, e.g. addition (mod m), subtracti... |

5 |
A new class of random number generators", The Annals of Applied Probability 1
- Marsaglia, Zaman
- 1991
(Show Context)
Citation Context ...orsF (r; s; \Gamma) with lags 0 ! s ! r ' 97v chosen to give maximal period. As we outline below, it is possible to vectorize the generation of a single stream of random numbers. In his recent papers =-=[22, 24] Marsaglia-=- recommends a new class of generators, termed "very long period" (VLP) generators. These are similar to generalized Fibonacci generators but can achieve periods close to 2 rw , whereas the g... |

4 | A parallel Monte Carlo transport algorithm using a pseudo-random tree to guarantee reproducibility", Parallel Computing 4 - Frederickson, Hiromoto, et al. - 1987 |

4 |
A current view of random number generators", Computer Science and Statistics: The Interface (edited by L
- Marsaglia
- 1985
(Show Context)
Citation Context ... sequence ([16], ex. 3.2.2.2). Attempts have been made to generalize the Fibonacci recurrence to obtain "generalized Fibonacci " or "lagged Fibonacci" random number generators [12,=-= 16, 30]. Marsaglia [20] considers-=- generators F (r; s; `) which satisfy Un = Un\Gammar `Un\Gammas for fixed "lags" r and s (r ? s ? 0) and nsr. Here ` is some binary operator, e.g. addition (mod m), subtraction (mod m), mult... |

4 |
Tausworthe, "Random numbers generated by linear recurrence modulo two
- C
- 1965
(Show Context)
Citation Context ...od m) or "exclusive or" (mod m = 2 w ). We abbreviate these operators by +; \Gamma;sand \Phi respectively. Generators using \Phi are also called "shift register" generators or &quo=-=t;Tausworthe" generators [11, 18, 31]-=-. If ` is + or \Gamma (mod m) then a theory of generalized Fibonacci generators can be based on the generating function G(x) = 1 X n=0 Unx n which is given by G(x) = P (x)=Q(x) mod m; where Q(x) = 1 \... |

3 |
An exhaustive analysis of multiplicative conguential random number generators with modulus 2 \Gamma 1
- Fishman, Moore
- 1986
(Show Context)
Citation Context ...s not occur in the sequence. There is much theory regarding the best choice of multiplier a for linear congruential generators [16, 27], and an exhaustive search has been performed for certain moduli =-=[8]-=-. Marsaglia [19] pointed out the fundamental weakness of the class of linear congruential generators. If d-tuples (u dn ; u dn+1 ; : : : ; u dn+d\Gamma1 ) of normalized numbers are considered as point... |

3 |
RAND/VP Users Guide
- Haan
- 1992
(Show Context)
Citation Context ...rator F (97; 33; \Gamma). Marsaglia [22] now prefers his VLP generators (described below) . The algorithm used in RAND/VP "has been modified to generate several streams of random numbers in paral=-=lel" [13]-=-. Presumably this means that the recurrence un = un\Gamma97 \Gamma un\Gamma33 mod 1 is applied to vectors of length v ? 1 rather than to single real numbers un . This is equivalent to using the genera... |

3 | Parallelization of the Ising model and its performance evaluation", Parallel Computing 13 - Heermann, Burkitt - 1990 |

3 | Tsay, "Matrices and the structure of random number sequences", Linear Algebra and Applications 67 - Marsaglia, H - 1985 |

2 | Enison, "Quasi-random number sequences from a long-period TLP generator with remarks on application to cryptography", Computing Surveys 11 - Bright, L - 1979 |

2 | Parallelization of the Ising model and its performance evaluation, Parallel Comput - Heermann, Burkitt - 1990 |