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<html>
<head>
<title>
LINE_CVT_LLOYD - Lloyd's CVT Method for a Line Segment
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LINE_CVT_LLOYD <br> Lloyd's CVT Method for a Line Segment
</h1>
<hr>
<p>
<b>LINE_CVT_LLOYD</b>
is a C library which
carries out Lloyd's iteration for computing a
Centroidal Voronoi Tesselation (CVT) of points
over the interior of a line segment in 1D.
</p>
<p>
A constraint has been added to the computation, which forces the first
and last points to be fixed to the endpoints of the line segment.
This is not actually a requirement for Lloyd's method.
</p>
<p>
At least for the uniform density case, the exact solution of this problem
is known in advance, and is simply a set of equally spaced points.
For instance, for N = 4, the solution over the interval [0,1] would be
0, 1/3, 2/3, 1, if the endpoint constraint is imposed. If the endpoint
constraint is NOT imposed, then the exact solution is: 1/8, 3/8, 5/8, 7/8.
</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>LINE_CVT_LLOYD</b> is available in
<a href = "../../c_src/cvt_1d_lloyd/cvt_1d_lloyd.html">a C version</a> and
<a href = "../../cpp_src/cvt_1d_lloyd/cvt_1d_lloyd.html">a C++ version</a> and
<a href = "../../f77_src/cvt_1d_lloyd/cvt_1d_lloyd.html">a FORTRAN77 version</a> and
<a href = "../../f_src/cvt_1d_lloyd/cvt_1d_lloyd.html">a FORTRAN90 version</a> and
<a href = "../../m_src/cvt_1d_lloyd/cvt_1d_lloyd.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/line_grid/line_grid.html">
LINE_GRID</a>,
a C++ library which
computes a grid of points
over the interior of a line segment in 1D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
SIAM Review, Volume 41, 1999, pages 637-676.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "line_cvt_lloyd.c">line_cvt_lloyd.c</a>, the source code.
</li>
<li>
<a href = "line_cvt_lloyd.h">line_cvt_lloyd.h</a>, the include file.
</li>
<li>
<a href = "line_cvt_lloyd.sh">line_cvt_lloyd.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "line_cvt_lloyd_prb.c">line_cvt_lloyd_prb.c</a>,
a sample calling program.
</li>
<li>
<a href = "line_cvt_lloyd_prb.sh">line_cvt_lloyd_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "line_cvt_lloyd_prb_output.txt">line_cvt_lloyd_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>TEST01</b> applies the unconstrained iteration, with random unsorted
starting values.
<ul>
<li>
<a href = "test01_energy_commands.txt">test01_energy_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test01_energy_data.txt">test01_energy_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test01_energy.png">test01_energy.png</a>
a PNG image of the "energy" of the partition.
</li>
<li>
<a href = "test01_evolution_commands.txt">test01_evolution_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test01_evolution_data.txt">test01_evolution_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test01_evolution.png">test01_evolution.png</a>
a PNG image of the evolution of the generators.
</li>
<li>
<a href = "test01_motion_commands.txt">test01_motion_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test01_motion_data.txt">test01_motion_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test01_motion.png">test01_motion.png</a>
a PNG image of the "motion" of the generators.
</li>
</ul>
</p>
<p>
<b>TEST02</b> applies the constrained iteration, with random unsorted
starting values.
<ul>
<li>
<a href = "test02_energy_commands.txt">test02_energy_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test02_energy_data.txt">test02_energy_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test02_energy.png">test02_energy.png</a>
a PNG image of the "energy" of the partition.
</li>
<li>
<a href = "test02_evolution_commands.txt">test02_evolution_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test02_evolution_data.txt">test02_evolution_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test02_evolution.png">test02_evolution.png</a>
a PNG image of the evolution of the generators.
</li>
<li>
<a href = "test02_motion_commands.txt">test02_motion_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test02_motion_data.txt">test02_motion_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test02_motion.png">test02_motion.png</a>
a PNG image of the "motion" of the generators.
</li>
</ul>
</p>
<p>
<b>TEST03</b> applies the unconstrained iteration, with random sorted
starting values.
<ul>
<li>
<a href = "test03_energy_commands.txt">test03_energy_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test03_energy_data.txt">test03_energy_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test03_energy.png">test03_energy.png</a>
a PNG image of the "energy" of the partition.
</li>
<li>
<a href = "test03_evolution_commands.txt">test03_evolution_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test03_evolution_data.txt">test03_evolution_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test03_evolution.png">test03_evolution.png</a>
a PNG image of the evolution of the generators.
</li>
<li>
<a href = "test03_motion_commands.txt">test03_motion_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test03_motion_data.txt">test03_motion_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test03_motion.png">test03_motion.png</a>
a PNG image of the "motion" of the generators.
</li>
</ul>
</p>
<p>
<b>TEST04</b> applies the constrained iteration, with random sorted
starting values.
<ul>
<li>
<a href = "test04_energy_commands.txt">test04_energy_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test04_energy_data.txt">test04_energy_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test04_energy.png">test04_energy.png</a>
a PNG image of the "energy" of the partition.
</li>
<li>
<a href = "test04_evolution_commands.txt">test04_evolution_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test04_evolution_data.txt">test04_evolution_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test04_evolution.png">test04_evolution.png</a>
a PNG image of the evolution of the generators.
</li>
<li>
<a href = "test04_motion_commands.txt">test04_motion_commands.txt</a>
GNUPLOT commands to create a plot.
</li>
<li>
<a href = "test04_motion_data.txt">test04_motion_data.txt</a>
data needed to create a plot.
</li>
<li>
<a href = "test04_motion.png">test04_motion.png</a>
a PNG image of the "motion" of the generators.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ENERGY_PLOT</b> plots the energy as a function of the iterations.
</li>
<li>
<b>EVOLUTION_PLOT</b> plots all points as a function of the iterations.
</li>
<li>
<b>LINE_CCVT_LLOYD</b> carries out the constrained Lloyd algorithm.
</li>
<li>
<b>LINE_CCVT_LLOYD_STEP</b> takes one step of Lloyd's constrained CVT algorithm.
</li>
<li>
<b>LINE_CVT_ENERGY</b> computes the CVT energy for a given set of generators.
</li>
<li>
<b>LINE_CVT_LLOYD</b> carries out the Lloyd algorithm.
</li>
<li>
<b>LINE_CVT_LLOYD_STEP</b> takes one step of Lloyd's unconstrained CVT algorithm.
</li>
<li>
<b>MOTION_PLOT</b> plots the motion as a function of the iterations.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_SORT_INSERT_A</b> ascending sorts an R8VEC using an insertion sort.
</li>
<li>
<b>R8VEC_UNIFORM_AB</b> returns a scaled pseudorandom R8VEC.
</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 = "../c_src.html">
the C source codes</a>.
</p>
<hr>
<i>
Last revised on 30 July 2014
</i>
<!-- John Burkardt -->
</body>
</html>