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<html>
<head>
<title>
TRIANGULATION_QUAD - Integral Estimate Over a Triangulated Region
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGULATION_QUAD <br>
Integral Estimate Over a Triangulated Region
</h1>
<hr>
<p>
<b>TRIANGULATION_QUAD</b>
is a C++ program which
reads information defining a triangulation, and estimates the integral of
a function whose values are given at the nodes.
</p>
<p>
Note that this program only expects to recieve values of the function <b>f(x,y)</b>
at the nodes of the triangulation, that is, as a simple list of values. No formula
for <b>f</b> is expected or used. A much better estimate for the integral might
be possible if a formula for <b>f(x,y)</b> were available, in which case a higher
order quadrature scheme could be employed.
</p>
<p>
This program should be able to compute exactly the integrals of 1, x, y, and any
linear combination of these. It will only be able to approximate the integrals
of other functions, and the accuracy of the approximation will depend in part
on the size of the triangles in the triangulation.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>triangulation_quad</b> <i>prefix</i>
</blockquote>
where <i>'prefix'</i> is the common prefix for the node, element, and value files
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b>, the node coordinates;
</li>
<li>
<i>prefix</i><b>_elements.txt</b>, the elements of the triangulation.
</li>
<li>
<i>prefix</i><b>_values.txt</b>, the value (which may be a scalar or vector)
of some function at each node.
</li>
<li>
<i>quad</i> is the (scalar or vector) result of the integration estimate, which
is computed and printed out by the program.
</li>
</ul>
</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>TRIANGULATION_QUAD</b> is available in
<a href = "../../cpp_src/triangulation_quad/triangulation_quad.html">a C++ version</a> and
<a href = "../../f_src/triangulation_quad/triangulation_quad.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangulation_quad/triangulation_quad.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature rules for a variety of multidimensional reqions.
</p>
<p>
<a href = "../../cpp_src/tet_mesh_quad/tet_mesh_quad.html">
TET_MESH_QUAD</a>,
a C++ program which
estimates the integral of a function over a region defined by a tetrahedral mesh.
</p>
<p>
<a href = "../../f77_src/toms706/toms706.html">
TOMS706</a>,
a FORTRAN77 library which
estimates the integral of a function
over a triangulated region.
</p>
<p>
<a href = "../../cpp_src/triangulation/triangulation.html">
TRIANGULATION</a>,
a C++ library which
carries out various operations on order 3 ("linear") or order 6
("quadratic") triangulations.
</p>
<p>
<a href = "../../cpp_src/triangulation_node_to_element/triangulation_node_to_element.html">
TRIANGULATION_NODE_TO_ELEMENT</a>,
a C++ program which
reads files describing a set of nodes, their triangulation, and the
value of one or more quantities at each node, and outputs a file
that averages the quantities for each element. This operation
in effect creates an "order1" finite element model of the data.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangulation_quad.cpp">triangulation_quad.cpp</a>,
the program.
</li>
<li>
<a href = "triangulation_quad.sh">triangulation_quad.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>EXAMPLE</b> is a set of nodes in the unit square, which have
been arranged into an order 3 triangulation.
<ul>
<li>
<a href = "example_nodes.txt">example_nodes.txt</a>,
the node file.
</li>
<li>
<a href = "example_elements.txt">example_elements.txt</a>,
the element file.
</li>
<li>
<a href = "example_values.txt">example_values.txt</a>,
the values file, containing 4 values at each node.
(These values are actually the values of 1, x, y, and x*y,
which have exact integrals of 1, 1/2, 1/2, and 1/4.)
Only the first three integrals will be estimated exactly.
</li>
<li>
<a href = "example_output.txt">example_output.txt</a>,
the output file from the command<br>
triangulation_quad example
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TRIANGULATION_QUAD.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the 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>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>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT file.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT</b> prints an I4MAT, transposed.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>MESH_BASE_ZERO</b> ensures that the element definition is zero-based.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads the data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 15 October 2009.
</i>
<!-- John Burkardt -->
</body>
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</html>