Van der Corput sequence
Encyclopedia
A van der Corput sequence is a low-discrepancy sequence
over the unit interval
first published in 1935 by the Dutch
mathematician J. G. van der Corput
. It is constructed by reversing the base n representation of the sequence of natural number
s (1, 2, 3, …). For example, the decimal
van der Corput sequence begins:
whereas the binary
van der Corput sequence can be written as:
or, equivalently, as:
The elements of the van der Corput sequence (in any base) form a dense set
in the unit interval: for any real number in [0, 1] there exists a subsequence
of the van der Corput sequence that converges
towards that number. They are also equidistributed over the unit interval.
Low-discrepancy sequence
In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy....
over the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
first published in 1935 by the Dutch
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
mathematician J. G. van der Corput
Johannes van der Corput
Johannes Gualtherus van der Corput was a Dutch mathematician, working in the field of analytic number theory....
. It is constructed by reversing the base n representation of the sequence of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (1, 2, 3, …). For example, the decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
van der Corput sequence begins:
- 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.11, 0.21, 0.31, 0.41, 0.51, 0.61, 0.71, 0.81, 0.91, 0.02, 0.12, 0.22, 0.32, …
whereas the binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
van der Corput sequence can be written as:
- 0.12, 0.012, 0.112, 0.0012, 0.1012, 0.0112, 0.1112, 0.00012, 0.10012, 0.01012, 0.11012, 0.00112, 0.10112, 0.01112, 0.11112, …
or, equivalently, as:
The elements of the van der Corput sequence (in any base) form a dense set
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the unit interval: for any real number in [0, 1] there exists a subsequence
Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
of the van der Corput sequence that converges
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
towards that number. They are also equidistributed over the unit interval.
External links
- Van der Corput sequence at MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...