RANDU

Three-dimensional plot of 100,000 values generated with RANDU. Each point represents 3 consecutive pseudorandom values. It is clearly seen that the points fall in 15 two-dimensional planes.

RANDU is a linear congruential pseudorandom number generator of the Park–Miller type, which has been used since the 1960s.^[1] It is defined by the recurrence:

$V_{j+1} = 65539\cdot V_j\, \bmod\, 2^{31}\,$

with the initial seed number, $\scriptstyle V_0$ as an odd number. It generates pseudorandom integers $\scriptstyle V_j$ which are uniformly distributed in the interval [1, 2³¹ − 1], but in practical applications are often mapped into pseudorandom rationals $\scriptstyle X_j$ in the interval (0, 1), by the formula:

$X_j = \frac{V_{j}}{2^{31}}$.

IBM's RANDU is widely considered to be one of the most ill-conceived random number generators ever designed.^[2] It fails the spectral test badly for dimensions greater than 2, and every integer result is odd. However, at least eight low-order bits are dropped when converted to single-precision (32 bit, 24 bit mantissa) floating-point.

The reason for choosing these particular values is that with a 32-bit-integer word size, the arithmetic of mod 2³¹ and $65539$ $(\text{i.e., }2^{16} + 3)$ calculations could be done quickly, using special features of some computer hardware.

Problems with multiplier and modulus

To show the problem with these values, of multiplier 65539 and modulus 2³¹, consider the following calculation where every term should be taken mod 2³¹. Start by writing the recursive relation as:

$x_{k+2}=(2^{16}+3) x_{k+1}=(2^{16}+3 )^2 x_{k}\,$

which becomes, after expanding the quadratic factor:

$x_{k+2}=(2^{32}+6 \cdot2^{16} +9 )x_{k}=[6 \cdot (2^{16}+3)-9]x_{k}\,$

because 2³² mod 2³¹ = 0

and allows us to show the correlation between three points as:

$x_{k+2}=6x_{k+1}-9x_{k}\,$

As a result of this correlation, the points in three-dimensional space (mod 2³¹) fall in 15 planes.^[3] As a result of the wide use of RANDU in the early 1970s, many results from that time are seen as suspicious.^[4]

This misbehavior was already detected in 1963^[5] on a 36-bit computer, and carefully reimplemented on the 32-bit IBM System/360.

Sample output

The start and end of the RANDU’s output period for the initial seed $\scriptstyle V_0 = 1$ is:

1, 65539, 393225, 1769499, 7077969, 26542323, … (sequence A096555 in the OEIS)

References

1. Entacher, Karl (June 2000). "A collection of classical pseudorandom number generators with linear structures - advanced version". Retrieved 8 August 2013.
2. Knuth D.E. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Second Edition. Addison-Wesley, 1981. ISBN 0-201-03822-6. Section 3.3.4, p. 104. [Extensive coverage of statistical tests for non-randomness.]
3. Marsaglia, George (1968). "Random Numbers Fall Mainly in the Planes". Proc. Natl. Acad. Sci. U.S.A. 61 (1): 25–28. doi:10.1073/pnas.61.1.25.
4. Press, William H.; et al. (1992). Numerical Recipes in Fortran 77: The Art of Scientific Computing (2nd ed.). ISBN 0-521-43064-X.
5. ref. 7 of http://portal.acm.org/citation.cfm?id=363827

External links