Constructions Of Low-discrepancy Sequences

THE VAN DER CORPUT SEQUENCE

''See main article Van Der Corput Sequence ''

Let

:

n=\sum_{k=0}^{L-1}d_k(n)b^k

be the ''b''-ary representation of the positive integer ''n'' ≥ 1, i.e. 0 ≤ ''d''_k(''n'') < ''b''. Set

:

g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.

Then there is a constant ''C'' depending only on ''b'' such that (''g''_''b''(''n''))_{''n'' ≥ 1} satisfies

:

_N(g_b(1),\dots,g_b(N))\leq Crac{\log N}{N}.

THE HALTON SEQUENCE

''See main article Halton Sequences ''

The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let ''s'' be an arbitrary dimension and ''b''₁, ..., ''b''_''s'' be arbitrary Coprime integers greater than 1. Define

:

x(n)=(g_{b_1}(n),\dots,g_{b_s}(n)).

Then there is a constant ''C'' depending only on ''b''₁, ..., ''b''_''s'', such that (''x''(''n''))_''n''≥1 is a ''s''-dimensional sequence with

:

_N(x(1),\dots,x(N))\leq C'rac{(\log N)^s}{N}.

THE HAMMERSLEY SET

Let ''b''₁,...,''b''_s-1 be Coprime positive integers greater that 1. For given ''s'' and ''N'', the ''s''-dimensional Hammersley set of size ''N'' is defined by

:

x(n)=(g_{b_1}(n),\dots,g_{b_{s-1}}(n),rac{n}{N})

for ''n'' = 1, ..., ''N''. Then

:

_N(x(1),\dots,x(N))\leq Crac{(\log N)^{s-1}}{N}

where ''C'' is a constant depending only on ''b''₁, ..., ''b''_''s''−1.

	CATEGORIES ABOUT CONSTRUCTIONS OF LOW-DISCREPANCY SEQUENCES

	numerical analysis

	quasirandomness

	diophantine approximation

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Information About

Constructions Of Low-discrepancy Sequences