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<html>
<head>
<title>
NIEDERREITER2 - The Niederreiter Quasirandom Sequence [Base 2]
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NIEDERREITER2 <br> The Niederreiter Quasirandom Sequence [Base 2]
</h1>
<hr>
<p>
<b>NIEDERREITER2</b>
is a MATLAB library which
produces elements of the Niederreiter
quasirandom sequence, using a base of 2.
</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>NIEDERREITER2</b> is an adapation of the INLO2 and GOLO2 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 any 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>.
</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>NIEDERREITER2</b> is available in
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">a C++ version</a> and
<a href = "../../f_src/niederreiter2/niederreiter2.html">a FORTRAN90 version</a> and
<a href = "../../m_src/niederreiter2/niederreiter2.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cvt/cvt.html">
CVT</a>,
a MATLAB library which
computes elements of a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../m_src/faure/faure.html">
FAURE</a>,
a MATLAB library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../m_src/grid/grid.html">
GRID</a>,
a MATLAB library which
computes elements of a grid sequence.
</p>
<p>
<a href = "../../m_src/halton/halton.html">
HALTON</a>,
a MATLAB library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../m_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a MATLAB library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../m_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a MATLAB library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../m_src/hex_grid_angle/hex_grid_angle.html">
HEX_GRID_ANGLE</a>,
a MATLAB library which
computes elements of an angled hexagonal grid dataset.
</p>
<p>
<a href = "../../m_src/ihs/ihs.html">
IHS</a>,
a MATLAB library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../m_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../m_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../m_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a MATLAB library which
computes elements of a Latin Hypercube dataset, choosing points at random.
</p>
<p>
<a href = "../../m_src/lcvt/lcvt.html">
LCVT</a>,
a MATLAB library which
computes a latinized Centroidal Voronoi Tessellation.
</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/sobol/sobol.html">
SOBOL</a>,
a MATLAB library which
computes elements of a Sobol quasirandom sequence.
</p>
<p>
<a href = "../../f77_src/toms738/toms738.html">
TOMS738</a>,
a FORTRAN77 library which
evaluates Niederreiter's quasirandom sequence;<br>
this is ACM TOMS algorithm 738;
</p>
<p>
<a href = "../../m_src/uniform/uniform.html">
UNIFORM</a>,
a MATLAB library which
computes elements of a uniform pseudorandom sequence.
</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 1D van der Corput sequence.
</p>
<h3 align = "center">
Author:
</h3>
<p>
MATLAB version by John Burkardt;
performance enhancements by Jeremy Dewar, Tulane University.
</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, pages 88-100, 1988.
</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, pages 494-495, 1994.
</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, pages 195-213, 1992.
</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, pages 362-376, 1986.
</li>
<li>
R Lidl, Harald Niederreiter, <br>
Finite Fields,<br>
Cambridge University Press, 1984, page 553.
</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.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "calcc2.m">calcc2.m</a>,
computes the constants C(I,J,R).
</li>
<li>
<a href = "calcv2.m">calcv2.m</a>,
computes the constants V(J,R).
</li>
<li>
<a href = "exor.m">exor.m</a>,
computes the bitwise exclusive OR of two integers.
</li>
<li>
<a href = "get_seed.m">get_seed.m</a>,
returns a seed for the uniform random number generator.
</li>
<li>
<a href = "i4_uniform.m">i4_uniform.m</a>,
returns a random integer in a given range.
</li>
<li>
<a href = "niederreiter2.m">niederreiter2.m</a>,
returns an element of the Niederreiter sequence base 2.
</li>
<li>
<a href = "niederreiter2_generate.m">niederreiter2_generate.m</a>,
generates a set of Niederreiter values.
</li>
<li>
<a href = "plymul2.m">plymul2.m</a>,
multiplies two polynomials in the field of order 2.
</li>
<li>
<a href = "r8mat_write.m">r8mat_write.m</a>,
writes an R8MAT file.
</li>
<li>
<a href = "setfld2.m">setfld2.m</a>,
sets up arithmetic tables for the finite field of order 2.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "niederreiter2_test.m">niederreiter2_test.m</a>,
runs all the tests;
</li>
<li>
<a href = "niederreiter2_test01.m">niederreiter2_test01.m</a>,
tests <b>setfld2</b>;
</li>
<li>
<a href = "niederreiter2_test02.m">niederreiter2_test02.m</a>,
tests <b>plymul2</b>;
</li>
<li>
<a href = "niederreiter2_test03.m">niederreiter2_test03.m</a>,
tests <b>exor</b>;
</li>
<li>
<a href = "niederreiter2_test04.m">niederreiter2_test04.m</a>,
tests <b>niederreiter2</b>;
</li>
<li>
<a href = "niederreiter2_test05.m">niederreiter2_test05.m</a>,
shows how the seed may be used to manipulate the output of
<b>niederreiter2</b>;
</li>
<li>
<a href = "niederreiter2_test_output.txt">niederreiter2_test_output.txt</a>,
output from the tests;
</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 17 November 2008.
</i>
<!-- John Burkardt -->
</body>
</html>