In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Weyl sequence is a sequence from the
equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequence
:''a'', 2''a'', 3''a'', ... mod 1
is uniformly distributed on the circle \mathbb/\mathbb, when ''a'' is an irrational number. It is a special case of the ergodic ...
proven by
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
:
The sequence of all multiples of an irrational ''α'',
::0, ''α'', 2''α'', 3''α'', 4''α'', ...
:is
equidistributed modulo 1.
In other words, the sequence of the fractional parts of each term will be uniformly distributed in the interval
, 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'', 2''k'', 3''k'', 4''k'', …
: 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
, ''m'').
The term appears to originate with George Marsaglia’s pape
"Xorshift RNGs"
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 because d is a 32-bit quantity. The results are equidistributed modulo 2
32.
See also
* List of things named after Hermann Weyl
References
Mathematical series
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