In mathematics, a Weyl sequence is a sequence from the equidistribution theorem proven by Hermann Weyl:
The sequence of all multiples of an irrational α,
0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.2In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval [0, 1).
In computing
In computing, an integer version of this sequence is often used to generate a discrete uniform distribution rather than a continuous one. Instead of using an irrational number, which cannot be calculated on a digital computer, the ratio of two integers is used in its place. An integer k is chosen, relatively prime to an integer modulus m. In the common case that m is a power of 2, this amounts to requiring that k is odd.
The sequence of all multiples of such an integer k,
0, k, 2k, 3k, 4k, … is equidistributed modulo m.That is, the sequence of the remainders of each term when divided by m will be uniformly distributed in the interval [0, m).
The term appears to originate with George Marsaglia’s paper "Xorshift RNGs".3 The following C code generates what Marsaglia calls a "Weyl sequence":
d += 362437;In this case, the odd integer is 362437, and the results are computed modulo m = 232 because d is a 32-bit quantity. The results are equidistributed modulo 232.
See also
References
Weyl, H. (September 1916). "Über die Gleichverteilung von Zahlen mod. Eins" [On the uniform distribution of numbers modulo one]. Mathematische Annalen (in German). 77 (3): 313–352. doi:10.1007/BF01475864. S2CID 123470919. /wiki/Hermann_Weyl ↩
Kuipers, L.; Niederreiter, H. (2006) [1974]. Uniform Distribution of Sequences. Dover Publications. ISBN 0-486-45019-8. 0-486-45019-8 ↩
Marsaglia, George (July 2003). "Xorshift RNGs". Journal of Statistical Software. 8 (14). doi:10.18637/jss.v008.i14. /wiki/George_Marsaglia ↩