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<html>
<head>
<title>
FAURE - The Faure Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FAURE <br> The Faure Quasirandom Sequence
</h1>
<hr>
<p>
<b>FAURE</b>
is a C++ library which
computes elements of the Faure quasirandom sequence.
</p>
<p>
A quasirandom or low discrepancy sequence, such as the Faure,
Halton, Hammersley, Niederreiter or Sobol sequences, is
"less random" than a pseudorandom number sequence, but
more useful for such tasks as approximation of integrals in
higher dimensions, and in global optimization.
This is because low discrepancy sequences tend to sample
space "more uniformly" than random numbers. Algorithms
that use such sequences may have superior convergence.
Faure sequences, in particular, seem to have become popular
in mathematical finance simulations.
</p>
<p>
<b>FAURE</b> is adapted from code in ACM TOMS Algorithm 647.
The original, true, correct version of ACM TOMS Algorithm 647
is available in the TOMS subdirectory of
<a href = "http://www.netlib.org/">the NETLIB web site</a>.
</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>FAURE</b> is available in
<a href = "../../cpp_src/faure/faure.html">a C++ version</a> and
<a href = "../../f_src/faure/faure.html">a FORTRAN90 version</a> and
<a href = "../../m_src/faure/faure.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 elements of a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../dataset/faure/faure.html">
FAURE</a>,
a dataset directory which
contains files of sample Faure datasets.
</p>
<p>
<a href = "../../cpp_src/faure_dataset/faure_dataset.html">
FAURE_DATASET</a>,
a C++ program which
can create a Faure quasirandom dataset.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes elements of a grid sequence.
</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 elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes elements of a Latin Hypercube dataset, choosing points at random.
</p>
<p>
<a href = "../../cpp_src/lcvt/lcvt.html">
LCVT</a>,
a C++ library which
computes a latinized Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes elements of a Niederreiter quasirandom sequence using base 2.
</p>
<p>
<a href = "../../cpp_src/normal/normal.html">
NORMAL</a>,
a C++ library which
computes elements of a sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes Sobol sequences.
</p>
<p>
<a href = "../../f77_src/toms647/toms647.html">
TOMS647</a>,
a FORTRAN77 library which
is a version of ACM TOMS algorithm 647,
for evaluating Faure, Halton and Sobol quasirandom sequences.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a C++ library which
computes elements of a 1D van der Corput sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Paul Bratley, Bennett Fox, Harald Niederreiter,<br>
Implementation and Tests of Low Discrepancy Sequences,<br>
ACM Transactions on Modeling and Computer Simulation,<br>
Volume 2, Number 3, July 1992, pages 195-213.
</li>
<li>
Henri Faure,<br>
Discrepance de suites associees a un systeme de numeration
(en dimension s),<br>
Acta Arithmetica,<br>
Volume 41, 1982, pages 337-351.
</li>
<li>
Henri Faure,<br>
Good permutations for extreme discrepancy,<br>
Journal of Number Theory,<br>
Volume 42, 1992, pages 47-56.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "faure.cpp">faure.cpp</a>, the source code;
</li>
<li>
<a href = "faure.hpp">faure.hpp</a>, the include file;
</li>
<li>
<a href = "faure.sh">faure.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "faure_prb.cpp">faure_prb.cpp</a>,
the sample test code;
</li>
<li>
<a href = "faure_prb.sh">faure_prb.sh</a>,
commands to compile the test code;
</li>
<li>
<a href = "faure_prb_output.txt">faure_prb_output.txt</a>,
output from a run of the test code;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BINOMIAL_TABLE</b> computes a table of bionomial coefficients MOD QS.
</li>
<li>
<b>FAURE</b> generates a new quasirandom Faure vector with each call.
</li>
<li>
<b>FAURE_GENERATE</b> generates a Faure dataset.
</li>
<li>
<b>I4_LOG_I4</b> returns the logarithm of an I4 to an I4 base.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>PRIME_GE</b> returns the smallest prime greater than or equal to N.
</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>
</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 11 December 2009.
</i>
<!-- John Burkardt -->
</body>
</html>