11.1.1.3 Arrays with Pseudo-Random Numbers

LUX offers various functions that return pseudo-random numbers. These numbers are determined by a fixed deterministic algorithm, so the numbers are not random in the sense that nobody can know which number will be generated next. A main advantage of using a deterministic algorithm is that of reproducibility: If you restart the algorithm with the same starting conditions, then it produces the same sequence of numbers.

Of course, to be useful, the pseudo-random numbers generated by the algorithm must share a sufficient number of properties with truly random numbers. For instance, the number of pseudo-random numbers in any given interval of values must be statistically indistinguishable from those in a truly random sequence, and the correlation between any two pseudo-random numbers at a fixed distance of each other in the pseudo-random sequence should be statistically zero.

The method used by LUX to generate pseudo-random numbers (other than random bits) is the MT19937 “Mersenne Twister” generator of Makoto Matsumoto and Takuji Nishimura, which has a period of 2^19937 - 1 and passes most statistical tests. LUX generates random bits using a primitive polynomial modulo 2.

The pseudo-random number functions LUX offers are:

RANDOMB( [SEED = seed,] dimensions)
RANDOMD( [SEED = seed,] distribution, dimensions)
RANDOME( scale, dimensions)
RANDOMU( [SEED = seed, PERIOD = period,] dimensions)
RANDOMN( [SEED = seed,] dimensions)
RANDOM( [SEED = seed, PERIOD = period,] dimensions
        [, /UNIFORM, /NORMAL, /SAMPLE, /SHUFFLE, /BITS])

The randomb function returns random bits. The randomu function returns uniformly distributed integer or floating-point numbers. The randomn function returns numbers normally distributed with zero average and unit variance. The randomd function returns integer numbers distributed according to a user-specified distribution. The randome function returns numbers exponentially distributed across the whole range of representable numbers. The random function selects a distribution by keyword. randomu is an alias of RANDOM,/UNIFORM, and randomn is an alias of RANDOM,/NORMAL.

See also: random, randomb, randomd, randome, randomn, randomn

Bibliography:

• - D.E. Knuth: "Seminumerical Algorithms", 2nd Edition, Addison-Wesley, Reading, MA (1981).
• - Stephen K. Park and Keith W. Miller: "Random Number Generators: Good Ones Are Hard To Find", Communications of the ACM, volume 31, pp. 1192–1201 (1988).
• - W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery: "Numerical Recipes in C", 2nd edition, Cambridge University Press: chapter 7 (1992)
• - R.C. Tausworthe: "Random Numbers Generated by Linear Recurrence Modulo Two", Mathematics of Computation, volume 19, p. 201 (1965)
• - E.J. Watson: "Primitive Polynomials (Mod 2)", Mathematics of Computation, volume 16, p. 368 (1962)
• - Makoto Matsumoto and Takuji Nishimura: "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator", ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1 (Jan. 1998), Pages 3–30