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<html>
<head>
<title>
VAN_DER_CORPUT_DATASET - Generate van der Corput Datasets
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
VAN_DER_CORPUT_DATASET <br> Generate van der Corput Datasets
</h1>
<hr>
<p>
<b>VAN_DER_CORPUT_DATASET</b>
is a MATLAB program which
creates a van der Corput sequence dataset and writes it to a file.
</p>
<p>
The program is interactive, and allows the user to choose
the parameters that define the sequence.
</p>
<p>
The NDIM-dimensional Halton sequence is derived
from the 1-dimensional van der Corput sequence by using a set of
different (usually distinct prime) bases for each dimension,
and the Hammersley sequence
is derived in almost the same way.
</p>
<p>
The van der Corput sequence is often used to generate a "subrandom"
sequence of points which have a better covering property
than pseudorandom points.
</p>
<p>
The van der Corput sequence generates a sequence of points in [0,1]
which (theoretically) never repeats. Except for SEED = 0, the
elements of the van der Corput sequence are strictly between 0 and 1.
</p>
<p>
The van der Corput sequence writes an integer in a given base B,
and then its digits are "reflected" about the decimal point.
This maps the numbers from 1 to N into a set of numbers in [0,1],
which are especially nicely distributed if N is one less
than a power of the base.
</p>
<p>
Hammersley suggested generating a set of N nicely distributed
points in two dimensions by setting the first component of the
Ith point to I/N, and the second to the van der Corput
value of I in base 2.
</p>
<p>
Halton suggested that in many cases, you might not know the number
of points you were generating, so Hammersley's formulation was
not ideal. Instead, he suggested that to generate a nicely
distributed sequence of points in M dimensions, you simply
choose the first M primes, P(1:M), and then for the J-th component of
the I-th point in the sequence, you compute the van der Corput
value of I in base P(J).
</p>
<p>
Thus, to generate a Halton sequence in a 2 dimensional space,
it is typical practice to generate a pair of van der Corput sequences,
the first with prime base 2, the second with prime base 3.
Similarly, by using the first K primes, a suitable sequence
in K-dimensional space can be generated.
</p>
<p>
The generation is quite simple. Given an integer SEED, the expansion
of SEED in base BASE is generated. Then, essentially, the result R
is generated by writing a decimal point followed by the digits of
the expansion of SEED, in reverse order. This decimal value is actually
still in base BASE, so it must be properly interpreted to generate
a usable value.
</p>
<p>
Here is an example in base 2:
<table border="1">
<tr>
<th>SEED (decimal)</th>
<th>SEED (binary)</th>
<th>VDC (binary)</th>
<th>VDC (decimal)</th>
</tr>
<tr>
<td>0</td><td>0</td><td>.0</td><td>0.0</td>
</tr>
<tr>
<td>1</td><td>1</td><td>.1</td><td>0.5</td>
</tr>
<tr>
<td>2</td><td>10</td><td>.01</td><td>0.25</td>
</tr>
<tr>
<td>3</td><td>11</td><td>.11</td><td>0.75</td>
</tr>
<tr>
<td>4</td><td>100</td><td>.001</td><td>0.125</td>
</tr>
<tr>
<td>5</td><td>101</td><td>.101</td><td>0.625</td>
</tr>
<tr>
<td>6</td><td>110</td><td>.011</td><td>0.375</td>
</tr>
<tr>
<td>7</td><td>111</td><td>.111</td><td>0.875</td>
</tr>
<tr>
<td>8</td><td>1000</td><td>.0001</td><td>0.0625</td>
</tr>
</table>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<i>r</i> = <b>van_der_corput_dataset</b> ( <i>base</i>, <i>seed</i>, <i>n</i> )
</blockquote>
where
<ul>
<li>
<i>base</I> is the base of the sequence;
<li>
<li>
<i>seed</i> is the index of the first element to be computed;
</li>
<li>
<i>n</i> is the number of points to generate.
</li>
<li>
<i>r</i> is the computed sequence of <i>n</i> values.
</li>
</ul>
The program generates the data and writes it to the file
<b>van_der_corput_<i>base</i>_<i>seed</i>_<i>n</i>.txt</b>.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>VAN_DER_CORPUT_DATASET</b> is available in
<a href ="../../cpp_src/van_der_corput_dataset/van_der_corput_dataset.html">a C++ version</a> and
<a href ="../../f_src/van_der_corput_dataset/van_der_corput_dataset.html">a FORTRAN90 version</a> and
<a href ="../../m_src/van_der_corput_dataset/van_der_corput_dataset.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/faure_dataset/faure_dataset.html">
FAURE_DATASET</a>,
a MATLAB program which
creates a Faure quasirandom dataset;
</p>
<p>
<a href = "../../m_src/grid_dataset/grid_dataset.html">
GRID_DATASET</a>,
a MATLAB program which
creates a grid sequence and writes it to a file.
</p>
<p>
<a href = "../../m_src/latin_center_dataset/latin_center_dataset.html">
LATIN_CENTER_DATASET</a>,
a MATLAB program which
creates a Latin Center Hypercube dataset;
</p>
<p>
<a href = "../../m_src/latin_edge_dataset/latin_edge_dataset.html">
LATIN_EDGE_DATASET</a>,
a MATLAB program which
creates a Latin Edge Hypercube dataset;
</p>
<p>
<a href = "../../m_src/latin_random_dataset/latin_random_dataset.html">
LATIN_RANDOM_DATASET</a>,
a MATLAB program which
creates a Latin Random Hypercube dataset;
</p>
<p>
<a href = "../../m_src/niederreiter2_dataset/niederreiter2_dataset.html">
NIEDERREITER2_DATASET</a>,
a MATLAB program which
creates a Niederreiter quasirandom dataset with base 2;
</p>
<p>
<a href = "../../m_src/normal_dataset/normal_dataset.html">
NORMAL_DATASET</a>,
a MATLAB program which
generates a dataset of multivariate normal pseudorandom values and writes them to a file.
</p>
<p>
<a href = "../../m_src/sobol_dataset/sobol_dataset.html">
SOBOL_DATASET</a>,
a MATLAB program which
computes a Sobol quasirandom sequence and writes it to a file.
</p>
<p>
<a href = "../../m_src/uniform_dataset/uniform_dataset.html">
UNIFORM_DATASET</a>,
a MATLAB program which
generates a dataset of uniform pseudorandom values and writes them to a file.
</p>
<p>
<a href = "../../m_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a MATLAB library which
computes elements of a van der Corput sequence.
</p>
<p>
<a href = "../../datasets/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a dataset directory which
contains van der Corput sequences.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Johannes van der Corput,<br>
Verteilungsfunktionen I & II,<br>
Nederl. Akad. Wetensch. Proc.,<br>
Volume 38, 1935, pages 813-820, pages 1058-1066.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "van_der_corput_dataset.m">van_der_corput_dataset.m</a>,
the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "../../datasets/van_der_corput/van_der_corput_2_0_10.txt">
van_der_corput_2_0_10.txt</a>,
the results for <b>van_der_corput_dataset ( 2, 0, 10 )</b>.
</li>
<li>
<a href = "../../datasets/van_der_corput/van_der_corput_2_5_10.txt">
van_der_corput_2_5_10.txt</a>,
the results for <b>van_der_corput_dataset ( 2, 5, 10 )</b>.
</li>
<li>
<a href = "../../datasets/van_der_corput/van_der_corput_3_0_10.txt">
van_der_corput_3_0_10.txt</a>,
the results for <b>van_der_corput_dataset ( 3, 0, 10 )</b>.
</li>
<li>
<a href = "../../datasets/van_der_corput/van_der_corput_10_0_10.txt">
van_der_corput_10_0_10.txt</a>,
the results for <b>van_der_corput_dataset ( 10, 0, 10 )</b>.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 09 December 2009.
</i>
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