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<html>
<head>
<title>
NIEDERREITER - The Niederreiter Quasirandom Sequence [Arbitrary base]
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NIEDERREITER <br> The Niederreiter Quasirandom Sequence [Arbitrary base]
</h1>
<hr>
<p>
<b>NIEDERREITER</b>
is a C++ library which
implements the Niederreiter
quasirandom sequence, using an "arbitrary" base; more correctly,
the code is not restricted to using a base of 2, but can instead
use a base that is a prime or a power of a prime.
</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>NIEDERREITER</b> is an adaptation of the INLO and GOLO routines
in ACM TOMS Algorithm 738. The original code can only
compute the "next" element of the sequence. The revised code
allows the user to specify the index of the desired element.
</p>
<p>
The original, true, correct version of ACM TOMS Algorithm 738
is 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>NIEDERREITER</b> is available in
<a href = "../../cpp_src/niederreiter/niederreiter.html">a C++ version</a> and
<a href = "../../f_src/niederreiter/niederreiter.html">a FORTRAN90 version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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 = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes elements of a Halton quasirandom sequence.
</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 a Niederreiter sequence for a base of 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 elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../f_src/toms738/toms738.html">
TOMS738</a>,
a FORTRAN90 library which
is a version of ACM TOMS algorithm 738,
for evaluating Niederreiter 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 van der Corput pseudorandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<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, 1988, pages 88-100.
</li>
<li>
Paul Bratley, Bennett Fox, Harald Niederreiter,<br>
Algorithm 738:
Programs to Generate Niederreiter's Low-Discrepancy Sequences,<br>
ACM Transactions on Mathematical Software,<br>
Volume 20, Number 4, 1994, pages 494-495.
</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, 1992, pages 195-213.
</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, 1986, pages 362-376.
</li>
<li>
Rudolf Lidl, Harald Niederreiter,<br>
Finite Fields,<br>
Second Edition,<br>
Cambridge University Press, 1997,<br>
ISBN: 0521392314,<br>
LC: QA247.3.L53
</li>
<li>
Harald Niederreiter,<br>
Low-discrepancy and low-dispersion sequences,<br>
Journal of Number Theory,<br>
Volume 30, 1988, pages 51-70.
</li>
<li>
Harald Niederreiter,<br>
Random Number Generation and quasi-Monte Carlo Methods,<br>
SIAM, 1992,<br>
ISBN13: 978-0-898712-95-7.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<b>GFARIT</b> must be run first, to set up a tables
of addition and multiplication.
<ul>
<li>
<a href = "gfarit.cpp">gfarit.cpp</a>, the source code;
</li>
<li>
<a href = "gfarit.sh">gfarit.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "gfarit_output.txt">gfarit_output.txt</a>,
output from a run of GFARIT;
</li>
<li>
<a href = "gfarit.txt">gfarit.txt</a>,
the data file created by the run;
</li>
</ul>
</p>
<p>
<b>GFPLYS</b> must be run second, to set up a table
of irreducible polynomials.
<ul>
<li>
<a href = "gfplys.cpp">gfplys.cpp</a>, the source code;
</li>
<li>
<a href = "gfplys.sh">gfplys.sh</a>,
commands to compile the source code;
</li>
<li>
<a href = "gfplys_output.txt">gfplys_output.txt</a>,
output from a run of GFPLYS;
</li>
<li>
<a href = "gfplys.txt">gfplys.txt</a>,
the data file created by the run;
</li>
</ul>
</p>
<p>
Once <b>GFARIT</b> and <b>GFPLYS</b> have been run to set up the tables,
the <b>NIEDERREITER</b> routines can be used.
<ul>
<li>
<a href = "niederreiter.cpp">niederreiter.cpp</a>, the source code;
</li>
<li>
<a href = "niederreiter.hpp">niederreiter.hpp</a>, the include file;
</li>
<li>
<a href = "niederreiter.sh">niederreiter.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "niederreiter_prb.cpp">niederreiter_prb.cpp</a>,
the sample test code;
</li>
<li>
<a href = "niederreiter_prb.sh">niederreiter_prb.sh</a>,
commands to compile the test code;
</li>
<li>
<a href = "niederreiter_prb_output.txt">niederreiter_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>CALCC</b> calculates the value of the constants C(I,J,R).
</li>
<li>
<b>CALCV</b> calculates the constants V(J,R).
</li>
<li>
<b>GOLO</b> generates a new quasi-random vector on each call.
</li>
<li>
<b>I4_CHARACTERISTIC</b> gives the characteristic for an integer.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</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>INLO</b> calculates the values of C(I,J,R).
</li>
<li>
<b>NIEDERREITER</b> returns an element of a Niederreiter sequence for base BASE.
</li>
<li>
<b>NIEDERREITER_GENERATE</b> generates a set of Niederreiter values.
</li>
<li>
<b>NIEDERREITER_WRITE</b> writes a set of Niederreiter values to a file.
</li>
<li>
<b>PLYMUL</b> multiplies one polynomial by another.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>SETFLD</b> sets up the arithmetic tables for a finite field.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</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 24 September 2007.
</i>
<!-- John Burkardt -->
</body>
</html>