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 | |||
:<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''−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]] |
Revision as of 18:52, 20 October 2013
There are some standard constructions of low-discrepancy sequences.
The van der Corput sequence
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Let
be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ dk(n) < b. Set
Then there is a constant C depending only on b such that (gb(n))n ≥ 1 satisfies
where D*N is the star discrepancy.
The Halton sequence
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The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let s be an arbitrary dimension and b1, ..., bs be arbitrary coprime integers greater than 1. Define
Then there is a constant C depending only on b1, ..., bs, such that sequence {x(n)}n≥1 is a s-dimensional sequence with
The Hammersley set
Let b1,...,bs-1 be coprime positive integers greater than 1. For given s and N, the s-dimensional Hammersley set of size N is defined by
for n = 1, ..., N. Then
where C is a constant depending only on b1, ..., bs−1.
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.[1]
References
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- Quasi-Monte Carlo Simulations, http://www.puc-rio.br/marco.ind/quasi_mc.html
- ↑ Herman Tulleken. "Poisson Disk Sampling". Dev.Mag Issue 21, March 2008.