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<html>
<head>
<title>
DIAPHONY - Compute the Diaphony of an M-dimensional Pointset
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
DIAPHONY <br> the "Diaphony" (dispersion) <br>
of an M-dimensional Pointset
</h1>
<hr>
<p>
<b>DIAPHONY</b>
is a FORTRAN90 program which
computes the "diaphony" of an M-dimensional pointset.
</p>
<p>
The "diaphony" of an M-dimensional pointset is a numeric measure
of the uniformity of the dispersion of the points throughout
the unit hypercube.
</p>
<p>
Regarded as a random variable itself, the diaphony of a set of N points
has an expected value of 1/sqrt(N).
</p>
<p>
For the Halton datasets in 2D, 7D and 16D, here is the table
of the number of points versus the diaphony:
<table "border=1">
<tr><th>Diaphony(M,N)</th><th>M=2D</th><th>M=7D</th><th>M=16D</th>
<th>1/Sqrt(N)</th></tr>
<tr><th> N=10</th><td>0.246</td><td>0.316</td>
<td>0.316</td><td>0.316</td></tr>
<tr><th> N=100</th><td>0.043</td><td>0.099</td>
<td>0.099</td><td>0.100</td></tr>
<tr><th> N=1000</th><td>0.006</td><td>0.031</td>
<td>0.031</td><td>0.031</td></tr>
<tr><th>N=10000</th><td>0.001</td><td>0.009</td>
<td>0.009</td><td>0.010</td></tr>
</table>
</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>DIAPHONY</b> is available in
<a href = "../../c_src/diaphony/diaphony.html">a C version</a> and
<a href = "../../cpp_src/diaphony/diaphony.html">a C++ version</a> and
<a href = "../../f77_src/diaphony/diaphony.html">a FORTRAN77 version</a> and
<a href = "../../f_src/diaphony/diaphony.html">a FORTRAN90 version</a> and
<a href = "../../m_src/diaphony/diaphony.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/star_discrepancy/star_discrepancy.html">
STAR_DISCREPANCY</a>,
a C++ program which
reads a <b>TABLE file</b> of points (presumed to lie in the
unit hypercube) and computes bounds on the star discrepancy,
a measure of dispersion, by Eric Thiemard.
</p>
<p>
<a href = "../../f_src/table_latinize/table_latinize.html">
TABLE_LATINIZE</a>,
a FORTRAN90 program which
reads a file of
points and creates a "latinized" version by adjusting the data.
</p>
<p>
<a href = "../../f_src/table_quality/table_quality.html">
TABLE_QUALITY</a>,
a FORTRAN90 program which
reads a file of
points and computes the quality of dispersion.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Peter Heelekalek,<br>
On Correlation Analysis of Pseudorandom Numbers,
in Monte Carlo and Quasi-Monte Carlo Methods 1996,<br>
edited by Harald Niederreiter, Peter Hellekalek,
Gerhard Larcher, and Peter Zinterhof,<br>
Volume 127 of Lecture Notes in Statistics,<br>
Springer Verlag, 1997, pages 251-265.
</li>
<li>
Peter Heelekalek, Harald Niederreiter,<br>
The Weighted Spectral Test: Diaphony,<br>
ACM Transactions on Modeling and Computer Simulation,<br>
Volume 8, Number 1, January 1998, pages 43-60.
</li>
<li>
Peter Heelekalek, Hannes Leeb,<br>
Dyadic Diaphony,<br>
Acta Arithmetica,<br>
Volume 80, Number 2, 1997, pages 187-196.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "diaphony.f90">diaphony.f90</a>, the source code.
</li>
<li>
<a href = "diaphony.sh">diaphony.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for DIAPHONY.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DIAPHONY</b> evaluates the diaphony of a N-dimensional point set.
</li>
<li>
<b>F</b> evaluates the function used to calculate the diaphony.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>R8_MODP</b> returns the nonnegative remainder of real division.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 25 January 2012.
</i>
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