Invariant subspace: Difference between revisions

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There are some standard '''constructions of [[low-discrepancy sequence]]s'''.


==The van der Corput sequence==
{{Main|van der Corput sequence}}


Let
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:<math>
n=\sum_{k=0}^{L-1}d_k(n)b^k
</math>
 
be the ''b''-ary representation of the positive integer ''n'' ≥ 1, i.e. 0 ≤ ''d''<sub>k</sub>(''n'') < ''b''. Set
 
:<math>
g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.
</math>
 
Then there is a constant ''C'' depending only on ''b'' such that (''g''<sub>''b''</sub>(''n''))<sub>''n'' ≥ 1</sub> satisfies
 
:<math>
D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N},
</math>
 
where ''D''<sup>*</sup><sub>''N''</sub> is the 
'''[[Low-discrepancy sequence#Definition of discrepancy|star discrepancy]]'''.
 
==The Halton sequence==
[[Image:Halton sequence 2D.svg|thumb|right|First 256 points of the (2,3) Halton sequence]]
{{Main|Halton sequence}}
 
The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let ''s'' be an arbitrary dimension and ''b''<sub>1</sub>, ..., ''b''<sub>''s''</sub> be arbitrary [[coprime]] integers greater than 1. Define
 
:<math>
x(n)=(g_{b_1}(n),\dots,g_{b_s}(n)).
</math>
 
Then there is a constant ''C'' depending only on ''b''<sub>1</sub>, ..., ''b''<sub>''s''</sub>, such that sequence {''x''(''n'')}<sub>''n''≥1</sub> is a ''s''-dimensional sequence with
 
:<math>
D^*_N(x(1),\dots,x(N))\leq C'\frac{(\log N)^s}{N}.
</math>
 
==The Hammersley set==
 
[[File:Hammersley set 2D.svg|thumb|right|2D Hammersley set of size 256]]
Let ''b''<sub>1</sub>,...,''b''<sub>s-1</sub> be [[coprime]] positive integers greater than 1. For given ''s'' and ''N'', the ''s''-dimensional [[John Hammersley|Hammersley]] set of size ''N'' is defined by
 
:<math>
x(n)=(g_{b_1}(n),\dots,g_{b_{s-1}}(n),\frac{n}{N})
</math>
 
for ''n'' = 1, ..., ''N''. Then
 
:<math>
D^*_N(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N}
</math>
 
where ''C'' is a constant depending only on ''b''<sub>1</sub>, ..., ''b''<sub>''s''&minus;1</sub>.
 
== Poisson disk sampling ==
 
Poisson disk sampling is popular in video games to rapidly placing objects in a way that appears random-looking
but guarantees that every two points are separated by at least the specified minimum distance.<ref>
Herman Tulleken.
[http://devmag.org.za/2009/05/03/poisson-disk-sampling/ "Poisson Disk Sampling"].
''Dev.Mag'' Issue 21, March 2008.
</ref>
 
==References==
{{reflist}}
* ''Quasi-Monte Carlo Simulations'', http://www.puc-rio.br/marco.ind/quasi_mc.html
 
[[Category:Numerical analysis]]
[[Category:Quasirandomness]]
[[Category:Diophantine approximation]]

Latest revision as of 04:04, 31 May 2014


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