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<html>
<head>
<title>
SOBOL - The Sobol Quasirandom Sequence
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SOBOL <br> The Sobol Quasirandom Sequence
</h1>
<hr>
<p>
<b>SOBOL</b>
is a C++ library which
computes elements of the Sobol quasirandom sequence,
by Bennett Fox.
</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.
</p>
<p>
<b>SOBOL</b> is an adapation of the INSOBL and GOSOBL routines
in ACM TOMS Algorithm 647 and ACM TOMS Algorithm 659. The original
code can only compute the "next" element of the sequence. The
revised code allows the user to specify the index of any desired element.
</p>
<p>
A remark by Joe and Kuo shows how to extend the algorithm from
the original maximum spatial dimension of 40 up to a maximum
spatial dimension of 1111. The FORTRAN90 and C++ versions of this
program have been updated in this way. In particular, the extra data
in the C++ version of the program was kindly formatted and
supplied by Steffan Berridge.
</p>
<p>
The routines with a prefix of <b>I8_</b> use 64 bit integers,
and use the <i>long int</i> to get this. On some systems,
a <i>long int</i> is simply 32 bits. In that case, try
using the <i>long long int</i> datatype instead.
</p>
<p>
The original, true, correct versions of ACM TOMS Algorithm 647
and ACM TOMS Algorithm 659
are available in the TOMS subdirectory of
<a href = "http://www.netlib.org/">the NETLIB web site</a>.
The version displayed here has been converted to FORTRAN90,
and other internal changes have been made to suit me.
</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>SOBOL</b> is available in
<a href = "../../cpp_src/sobol/sobol.html">a C++ version</a> and
<a href = "../../f_src/sobol/sobol.html">a FORTRAN90 version</a> and
<a href = "../../m_src/sobol/sobol.html">a MATLAB version</a> and
<a href = "../../py_src/sobol/sobol.html">a Python 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/gsl/gsl.html">
GSL</a>,
a C++ library which
includes routines to compute elements of the Sobol 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 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 elements of a
sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../cpp_src/sobol_dataset/sobol_dataset.html">
SOBOL_DATASET</a>,
a C++ program which
computes a Sobol quasirandom sequence and writes it to a file.
</p>
<p>
<a href = "../../cpp_src/sobol_old/sobol_old.html">
SOBOL_OLD</a>,
a C++ library which
implements the ACM TOMS algorithm,
and is restricted to a maximal spatial dimension of 40.
</p>
<p>
<a href = "../../f_src/toms647/toms647.html">
TOMS647</a>
a FORTRAN90 library which
evaluates Faure, Halton and Sobol sequences.
</p>
<p>
<a href = "../../f77_src/toms659/toms659.html">
TOMS659</a>
a FORTRAN77 library which
evaluates 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 = "../../cpp_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a C++ library which
computes a van der Corput sequences.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Original FORTRAN77 version by Bennett Fox;
C++ version by John Burkardt.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
IA Antonov, VM Saleev,<br>
An Economic Method of Computing LP Tau-Sequences,<br>
USSR Computational Mathematics and Mathematical Physics,<br>
Volume 19, 1980, pages 252-256.
</li>
<li>
Paul Bratley, Bennett Fox,<br>
Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,<br>
ACM Transactions on Mathematical Software,<br>
Volume 14, Number 1, March 1988, pages 88-100.
</li>
<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>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</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>
<li>
Stephen Joe, Frances Kuo,<br>
Remark on Algorithm 659:
Implementing Sobol's Quasirandom Sequence Generator,<br>
ACM Transactions on Mathematical Software,<br>
Volume 29, Number 1, March 2003, pages 49-57.
</li>
<li>
Harald Niederreiter,<br>
Random Number Generation and quasi-Monte Carlo Methods,<br>
SIAM, 1992,<br>
ISBN13: 978-0-898712-95-7,<br>
LC: QA298.N54.
</li>
<li>
William Press, Brian Flannery, Saul Teukolsky, William Vetterling,<br>
Numerical Recipes in FORTRAN: The Art of Scientific Computing,<br>
Second Edition,<br>
Cambridge University Press, 1992,<br>
ISBN: 0-521-43064-X,<br>
LC: QA297.N866.
</li>
<li>
Ilya Sobol,<br>
Uniformly Distributed Sequences with an Additional Uniform
Property,<br>
USSR Computational Mathematics and Mathematical Physics,<br>
Volume 16, 1977, pages 236-242.
</li>
<li>
Ilya Sobol, YL Levitan,<br>
The Production of Points Uniformly Distributed in a Multidimensional
Cube (in Russian),<br>
Preprint IPM Akademii Nauk SSSR,<br>
Number 40, Moscow 1976.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "sobol.cpp">sobol.cpp</a>, the source code;
</li>
<li>
<a href = "sobol.hpp">sobol.hpp</a>, the include file;
</li>
<li>
<a href = "sobol.sh">sobol.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "sobol_prb.cpp">sobol_prb.cpp</a>,
the sample test code;
</li>
<li>
<a href = "sobol_prb.sh">sobol_prb.sh</a>,
commands to compile the test code;
</li>
<li>
<a href = "sobol_prb_output.txt">sobol_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_BIT_HI1</b> returns the position of the high 1 bit base 2 in an integer.
</li>
<li>
<b>I4_BIT_LO0</b> returns the position of the low 0 bit base 2 in an integer.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two integers.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two integers.
</li>
<li>
<b>I4_SOBOL</b> generates a new quasirandom Sobol vector with each call.
</li>
<li>
<b>I4_SOBOL_GENERATE</b> generates a Sobol dataset.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>I8_BIT_HI1</b> returns the position of the high 1 bit base 2 in an integer.
</li>
<li>
<b>I8_BIT_LO0</b> returns the position of the low 0 bit base 2 in an integer.
</li>
<li>
<b>I8_SOBOL</b> generates a new quasirandom Sobol vector with each call.
</li>
<li>
<b>I8_SOBOL_GENERATE</b> generates a Sobol dataset.
</li>
<li>
<b>I8_MAX</b> returns the maximum of two I8's.
</li>
<li>
<b>I8_MIN</b> returns the smaller of two I8's.
</li>
<li>
<b>I8_UNIFORM</b> returns a scaled pseudorandom I8.
</li>
<li>
<b>R4_ABS</b> returns the absolute value of an R4.
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</li>
<li>
<b>R4_UNIFORM_01</b> returns a unit pseudorandom R4.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>TAU_SOBOL</b> defines favorable starting seeds for Sobol sequences.
</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 12 December 2009.
</i>
<!-- John Burkardt -->
</body>
</html>