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<html>
<head>
<title>
VAN_DER_CORPUT - The van der Corput Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
VAN_DER_CORPUT <br> The van der Corput<br> Quasirandom Sequence
</h1>
<hr>
<p>
<b>VAN_DER_CORPUT</b>
is a C++ library which
computes the van der Corput quasirandom sequence.
</p>
<p>
<b>VAN_DER_CORPUT</b> includes several subroutines to make it easy
to manipulate this computation, to compute the next N entries, to
compute a particular entry, or to restart the sequence at a
particular point.
</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">
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</b> is available in
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">a C++ version</a> and
<a href = "../../f_src/van_der_corput/van_der_corput.html">a FORTRAN90 version</a> and
<a href = "../../m_src/van_der_corput/van_der_corput.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/box_behnken/box_behnken.html">
BOX_BEHNKEN</a>,
a C++ library which
computes a Box-Behnken design,
that is, a set of arguments to sample the behavior
of a function of multiple parameters;
</p>
<p>
<a href = "../../cpp_src/cvt/cvt.html">
CVT</a>,
a C++ library which
computes points in
a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes Faure
sequences.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes points on a grid.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes Halton sequences.
</p>
<p>
<a href = "../../cpp_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a C++ library which
computes Hammersley sequences.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ library which
computes sets of points in a 2D hexagonal grid.
</p>
<p>
<a href = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes improved Latin Hypercube datasets.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes Latin square data choosing the center value.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes Latin square data choosing the edge value.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes Latin square data choosing a random value in the square.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes Niederreiter sequences with base 2.
</p>
<p>
<a href = "../../cpp_src/normal/normal.html">
NORMAL</a>,
a C++ library which
computes a sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../m_src/sequence_streak_display/sequence_streak_display.html">
SEQUENCE_STREAK_DISPLAY</a>,
a MATLAB program which
makes a "streak file" of a van der Corput sequence.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes Sobol sequences.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes uniform random values.
</p>
<p>
<a href = "../../datasets/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a dataset directory which
contains datasets of van der Corput sequences.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
John Halton,<br>
On the efficiency of certain quasi-random sequences of points
in evaluating multi-dimensional integrals,<br>
Numerische Mathematik,<br>
Volume 2, pages 84-90, 1960.
</li>
<li>
John Hammersley,<br>
Monte Carlo methods for solving multivariable problems,<br>
Proceedings of the New York Academy of Science,<br>
Volume 86, pages 844-874, 1960.
</li>
<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.cpp">van_der_corput.cpp</a>, the source code.
</li>
<li>
<a href = "van_der_corput.hpp">van_der_corput.hpp</a>, the include file.
</li>
<li>
<a href = "van_der_corput.sh">van_der_corput.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "van_der_corput_prb.cpp">van_der_corput_prb.cpp</a>,
a sample problem.
</li>
<li>
<a href = "van_der_corput_prb.sh">van_der_corput_prb.sh</a>,
commands to compile, link and run the sample problem.
</li>
<li>
<a href = "van_der_corput_prb_output.txt">van_der_corput_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CIRCLE_UNIT_VAN_DER_CORPUT</b> picks a van der Corput point on the unit circle.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>I4_LOG_2</b> returns the integer part of the logarithm base 2 of an I4.
</li>
<li>
<b>I4_TO_VAN_DER_CORPUT</b> computes an element of a van der Corput sequence.
</li>
<li>
<b>I4_TO_VAN_DER_CORPUT_SEQUENCE:</b> next N elements of a van der Corput sequence.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 round off unit.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VAN_DER_CORPUT</b> computes the next element in the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_BASE_GET</b> gets the base for a van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_BASE_SET</b> sets the base for a van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEED_GET</b> gets the "seed" for the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEED_SET</b> sets the "seed" for the van der Corput sequence.
</li>
<li>
<b>VAN_DER_CORPUT_SEQUENCE:</b> next N elements in the van der Corput sequence.
</li>
<li>
<b>VDC_NUMERATOR_SEQUENCE:</b> van der Corput numerator sequence base 2.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 03 February 2011.
</i>
<!-- John Burkardt -->
</body>
</html>