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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 FORTRAN90 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 = "../../f_src/cvt/cvt.html">
CVT</a>,
a FORTRAN90 library which
computes elements of a Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../f_src/faure/faure.html">
FAURE</a>,
a FORTRAN90 library which
computes elements of a Faure quasirandom sequence.
</p>
<p>
<a href = "../../f_src/grid/grid.html">
GRID</a>,
a FORTRAN90 library which
computes elements of a grid dataset.
</p>
<p>
<a href = "../../f_src/halton/halton.html">
HALTON</a>,
a FORTRAN90 library which
computes elements of a Halton quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a FORTRAN90 library which
computes elements of a Hammersley quasirandom sequence.
</p>
<p>
<a href = "../../f_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a FORTRAN90 library which
computes elements of a hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/hex_grid_angle/hex_grid_angle.html">
HEX_GRID_ANGLE</a>,
a FORTRAN90 library which
computes elements of an angled hexagonal grid dataset.
</p>
<p>
<a href = "../../f_src/ihs/ihs.html">
IHS</a>,
a FORTRAN90 library which
computes elements of an improved distributed Latin hypercube dataset.
</p>
<p>
<a href = "../../f_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../f_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../f_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing
points at random.
</p>
<p>
<a href = "../../f_src/lcvt/lcvt.html">
LCVT</a>,
a FORTRAN90 library which
computes a latinized Centroidal Voronoi Tessellation.
</p>
<p>
<a href = "../../f_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a FORTRAN90 library which
computes a Niederreiter sequence for a base of 2.
</p>
<p>
<a href = "../../f_src/normal/normal.html">
NORMAL</a>,
a FORTRAN90 library which
computes elements of a
sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../f_src/sobol/sobol.html">
SOBOL</a>,
a FORTRAN90 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 = "../../f_src/uniform/uniform.html">
UNIFORM</a>,
a FORTRAN90 library which
computes elements of a uniform pseudorandom sequence.
</p>
<p>
<a href = "../../f_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a FORTRAN90 library which
computes elements of a van der Corput pseudorandom sequence.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Paul Bratley and 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>
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.f90">gfarit.f90</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.f90">gfplys.f90</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.f90">niederreiter.f90</a>, the source code;
</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.f90">niederreiter_prb.f90</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>,
the output file;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CALCC</b> computes the constants C(I,J,R).
</li>
<li>
<b>CALCV</b> calculates the constants V(J,R).
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>NIEDERREITER</b> returns an element of the Niederreiter sequence.
</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 two polynomials in the field.
</li>
<li>
<b>SETFLD</b> sets up arithmetic tables for the finite field.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 07 June 2010.
</i>
<!-- John Burkardt -->
</body>
</html>