Extensions of Forsythe's method for random sampling from the normal distribution

Authors:
J. H. Ahrens and U. Dieter

Journal:
Math. Comp. **27** (1973), 927-937

MSC:
Primary 65C10

DOI:
https://doi.org/10.1090/S0025-5718-1973-0329190-8

MathSciNet review:
0329190

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Abstract | References | Similar Articles | Additional Information

Abstract: This article is an expansion of G. E. Forsythe's paper "Von Neumann's comparison method for random sampling from the normal and other distributions" [5]. It is shown that Forsythe's method for the normal distribution can be adjusted so that the average number of uniform deviates required drops to 2.53947 in spite of a shorter program. In a further series of algorithms, is reduced to values close to 1 at the expense of larger tables. Extensive computational experience is reported which indicates that the new methods compare extremely well with known sampling algorithms for the normal distribution.

**[1]**J. H. Ahrens, U. Dieter & A. Grube, "Pseudo-random numbers: A new proposal for the choice of multiplicators,"*Computing*(*Arch. Elektron. Rechnen*), v. 6, 1970, pp. 121-138. MR**43**#5679. MR**0279958 (43:5679)****[2]**J. H. Ahrens & U. Dieter, "Computer methods for sampling from the exponential and normal distributions,"*Comm. Assoc. Comput. Mach.*, v. 15, 1972, pp. 873-881. MR**0336955 (49:1728)****[3]**G. E. Box & M. E. Muller, "A note on the generation of random normal deviates,"*Ann. Math. Statist.*, v. 29, 1958, pp. 610-611.**[4]**U. Dieter & J. H. Ahrens, "A combinatorial method for the generation of normally distributed random numbers,"*Computing*, v. 10, 1973. MR**0388727 (52:9561)****[5]**G. E. Forsythe, "Von Neumann's comparison method for random sampling from the normal and other distributions,"*Math Comp.*, v. 26, 1972, pp. 817-826. MR**0315863 (47:4412)****[6]**D. E. Knuth,*The Art of Computer Programming*, Vol. 2:*Seminumerical Algorithms*, Addison-Wesley, Reading, Mass., 1969. MR**44**#3531. MR**0286318 (44:3531)****[7]**G. Marsaglia, "Generating a variable from the tail of the normal distribution,"*Technometrics*, v. 6, 1964, pp. 101-102.**[8]**M. D. MacLaren, G. Marsaglia & T. A. Bray, "A fast procedure for generating normal random variables,"*Comm. Assoc. Comput. Mach.*, v. 7, 1964, pp. 4-10.**[9]**J. von Neumann, "Various techniques used in connection with random digits,"*Collected Works*, Vol. 5, Pergamon Press, New York, 1963, pp. 768-770.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1973-0329190-8

Keywords:
Random number generation,
normal distribution

Article copyright:
© Copyright 1973
American Mathematical Society