B. D. Ripley, Thoughts on pseudorandom number generators, Journal of Computational and Applied Mathematics, 31 (1990), pp. 153--163.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....every year in (supposedly serious) journal articles. One of my favorite exercises for students when I teach a simulation course is to have them test a bad generator recently proposed in a journal or available on a popular computer. For more on bad generators, see [41, 50, 79, 83] As Ripley [84] said: Random number generation seems to be one of the most misunderstood subjects in computer science . On the surface, it looks easy and attractive. This is probably why so many new generators are proposed by people from so many different fields (mathematics, computer science, physics, ....

....properties, but which are also slower. The question of empirical statistical testing is treated in x11. 2. What is a Good generator We summarize in this section the major requirements for a good random number generator, for general purpose simulation. These requirements are also discussed in [10, 41, 79, 84], and we do not always share the views of all these authors. 2.1. STATISTICAL UNIFORMITY AND UNPREDICTABILITY As we said, the sequence of observations from a generator should behave as if it was the realization of a sequence of independent random variables, uniformly distributed over the set U . ....

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B. D. Ripley, Thoughts on Pseudorandom Number Generators, J. of Computational and Applied Mathematics 31 (1990) 153--163.

....accuracy) with all standard compilers and on all reasonable computers. Being able to reproduce the same sequence of random numbers on a given computer or on different computers (repeata bility) is important for program verification and for variance reduction (Bratley, Fox, and Schrage 1987; Ripley 1990). Repeatability is a major advantage of pseudorandom sequences with respect to sequences generated by physical devices. Of course, for the latter, one could store an extremely long sequence on a disk and reuse it as needed. But this is not as convenient as a good pseudorandom number generator ....

Ripley, B. D. 1990. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31:153--163.

....package write the integers to the ASCII file in the order that they are generated, and that this order be preserved in the conversion to hexadecimal. All the packages reviewed herein have reproducible RNGs. Whether any of the packages discussed herein meets the remaining desiderata of an RNG (Ripley 1990) cannot be determined from the user manuals. SAS cites Fishman and Moore (1982) for its RNG, but o#ers no further details. SPSS and S Plus hardcopy manuals provide no details of their RNGs. None of the vendors provides the algorithm, its period, or the statistical tests it has passed. This is a ....

Ripley, B. D. (1990), "Thoughts on Pseudorandom Number Generators," Journal of Computational and Applied Mathematics, 31, 153--163.

....this TABLE 1.2. List of selected generators. G1. LCG with m = 2 31 Gamma 1 and a = 742938285 (see [FM86] G2. LCG with m = 2 31 Gamma 1 and a = 630360016 (see [LK91] G3. LCG with m = 2 31 Gamma 1 and a = 16807 (see [BFS87, LK91] G4. LCG with m = 2 32 , a = 69069, and c = 1 (see [LK91, Rip90]) G5. LCG with m = 2 31 and a = 65539 (RANDU, see [LK91] G6. LCG with m = 2 31 and a = 452807053 (see [DvdMST95] G7. LCG with m = 2 31 , a = 1103515245, c = 12345 (see [Pla92] G8. Implicit inv. with m = 2 31 Gamma 1, a 1 = a 2 = 1 (see [Eic92] G9. Explicit inv. with m = 2 31 ....

B. D. Ripley. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31:153--163, 1990.

.... a 3 = a 4 = 0, proposed by L Ecuyer, Blouin, and Couture (1993) G9 is an explicit inversive generator of the form x i = ai b) mod M , z i = x Gamma1 i mod M = x M Gamma2 i mod M , u i = z i =M (EichenauerHerrmann 1992, Hellekalek 1995) G10 is the GFSR generator given in the Appendix of Ripley (1990). G11, G12, and G13 are the combined LCG of L Ecuyer (1988) the combined MRG given in Figure 1 of L Ecuyer (1996a) and the combined Tausworthe generator in Figure 1 of L Ecuyer (1996b) respectively. G14 to G16 are double precision versions of G11 to G13. Each call to G14 makes two calls to G11, ....

....1. G5. LCG with M = 2 31 and a = 65539. G6. LCG with M = 2 31 and a = 452807053. G7. LCG with M = 2 31 , a = 1103515245, c = 12345. G8. MRG of order 5, from L Ecuyer, Blouin, and Couture (1993) G9. Explicit inversive generator with M = 2 31 and a = b = 1. G10. GFSR 521 in the Appendix of Ripley (1990). G11. Combined LCG in Fig. 3 of L Ecuyer (1988) G12. Combined MRG in Fig. 1 of L Ecuyer (1996a) G13. Combined Tausworthe generator in Fig. 1 of L Ecuyer (1996b) G14. A double precision version of G11. G15. A double precision version of G12. G16. A double precision version of G13. 4.2 Test ....

Ripley, B. D. 1990. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31, 153--163.

....been tested. In addition, we study shortcomings of linear congruential generators in parallel applications. A summary of generators used from the 1960s to 1980s is given in Park and Miller [91, Sect. 4] and Dudewicz and Ralley [29, Chapter. 1] Surveys on pseudorandom numbers are contained in [61, 88, 90, 67, 65, 86, 83, 96, 63, 49, 57, 42] ; surveys for parallel PRNGs are [6, 31] Linear congruential generators (LCGs) are the best analyzed and most widely used PRNGs. LCGs allow an easy (number ) theoretical analysis based on the lattice structure formed by s dimensional vectors x n = x n ; x n s Gamma1 ) n 0 of generated ....

....low bits of the numbers generated are not very random. Spectral test for dim. 2 s 8: 0.84 0.52 0.63 0.49 0.68 0.43 0.54 E: See [36, 76, 73, 110, 40] I : Source code in [93, p. 276 (not recommended ) L: Times for various generators in C (incl. ANSIC) are given in Ripley [96, p. 160] Quote[96]: The Unix generator rand has been replaced by drand48 (see 1.14) which is far too slow and by a feedback generator random of the type F (r; s; 1.2 LCG(2 31 Gamma1; a = 7 5 = 16807; 0; 1) MINSTD 0 0.0005 0 0.0005 Quote[93] First suggested by Lewis, Goodman, and Miller in 1969 [78] ....

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B.D. Ripley. Thoughts on pseudorandom number generators. J. Comput. Appl. Math., 31:153--163, 1990.

....Weyl with ff = p 2 (see Holian et al. 1994) WEY2. Shuffled nested Weyl with ff = p 2 (see Holian et al. 1994) CLCG4. Combined LCG of L Ecuyer and Andres (1997) CMRG96. Combined MRG in Fig. 1 of L Ecuyer (1996a) books and papers (e.g. Bratley, Fox, and Schrage 1987, Park and Miller 1988, Ripley 1990). LCG6 is used in the VAX VMS operating system and on Convex computers. LCG5 and LCG9 are the rand and rand48 functions in the standard libraries of the C programming language (Plauger 1992) LCG7 is taken from Fishman (1996) and LCG8 is used in the CRAY system library. LCG1 to LCG4 have period ....

Ripley, B. D. 1990. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31, 153--163.

....inversive with m = 2 31 Gamma 1 and a = b = 1. G10. Implicit inversive with m = 2 32 , a = b = 1, and z 0 = 5. G11. Explicit inversive of [12] with m = 2 32 , a = 6, and b = 1. G12. Modified explicit inversive of [10] with m = 2 32 , a = 6, and b = 1. G13. GFSR 521 in the Appendix of [36]. G14. GFSR proposed in [17] G15. Combined LCG in Fig. 3 of [20] G16. Combined MRG in Fig. 1 of [23] We must say that G1 to G12 are rather baby generators from our point of view: their period lengths are too short for current needs, so none of them can be recommended for general use. G13 to ....

....G1 to G7 are well known linear congruential generators (LCGs) based on a recurrence of the form x i = ax i Gamma1 c) mod m, with output u i = x i =m at step i. G1 and G2 are recommended by Fishman and Moore [13] and Law and Kelton [19] respectively. G3 and G4 are in several software packages [1, 36], G5 is the infamous RANDU, G6 corresponds to the URN12 generator of [7] and G7 is the LCG implemented in the rand function of the standard library of the C programming language [33] The next five generators are inversive generators modulo m. Their output at step i is always u i = z i =m. G8 ....

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B. D. Ripley. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31:153--163, 1990.

....Explicit inversive with M = 2 31 Gamma 1, a = 1, and b = 12345. G11. Implicit inversive with M = 2 32 , a = b = 1, and z 0 = 5. G12. Explicit inversive with M = 2 32 , a = 6, and b = 1. G13. Modified explicit inversive with M = 2 32 , a = 6, and b = 1. G14. GFSR 521 in the Appendix of Ripley (1990). G15. GFSR proposed in Kirkpatrick and Stoll (1981) G16. Combined LCG in Fig. 3 of L Ecuyer (1988) G17. Combined MRG in Fig. 1 of L Ecuyer (1996a) G18. Combined Tausworthe generator in Fig. 1 of L Ecuyer (1996b) G1 to G7 are linear congruential generators (LCGs) based on a recurrence of the ....

....are linear congruential generators (LCGs) based on a recurrence of the form x i = ax i Gamma1 c) mod M , with output u i = x i =M at step i. G1 is recommended in Fishman and Moore III (1986) G2 in Law and Kelton (1991) G3 and G4 are in many software packages (Bratley, Fox, and Schrage 1987, Ripley 1990), G5 is the infamous RANDU (Law and Kelton 1991) G6 is the URN12 generator of Dudewicz et al. 1995) and G7 is the rand function in the standard library of the C language (Plauger 1992) The next six are inversive generators modulo M , whose output at step i is u i = z i =M . G8 is an implicit ....

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Ripley, B. D. 1990. Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, 31, 153--163.

....cross correlations between the single streams, one might for example start with a single initial generator which is split into non overlapping substreams. In the case of linear congruential generators (LCGs) the substreams themselves can be realized as LCGs by a simple parameter substitution [11, 18]. In this paper we will consider the leap frog technique for splitting. Another method to obtain parallel streams of LCGs is the so called consecutive blocks splitting which is known to suffer from long range correlations within the original sequence that imply correlations between the parallel ....

B.D. Ripley. Thoughts on pseudorandom number generators. J. Comput. Appl.

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B. D. Ripley, Thoughts on pseudorandom number generators, Journal of Computational and Applied Mathematics, 31 (1990), pp. 153--163.

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Ripley, B. D. (1990). Thoughts on pseudorandom number generators. Journal of Computational and Applied Mathematics, Vol. 31, pp. 153--163.

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Ripley, B. D. 1990. Thoughts on Pseudorandom Number Generators. J. of Computational and Applied Mathematics, 31:153--163.

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