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<html>
<head>
<title>
PIECEWISE_LINEAR_PRODUCT_INTEGRAL - Piecewise Linear Product Integral
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PIECEWISE_LINEAR_PRODUCT_INTEGRAL <br> Piecewise Linear Product Integral
</h1>
<hr>
<p>
<b>PIECEWISE_LINEAR_PRODUCT_INTEGRAL</b>
is a C++ library which
calculates the exact value of the integral of the product of two
piecewise linear functions F(X) and G(X).
</p>
<p>
The piecewise linear function F(X) is defined by
<ul>
<li>
<b>F_NUM</b>, the number of nodes;
</li>
<li>
<b>F_X</b>(1:F_NUM), the X coordinates of the nodes (in ascending order);
</li>
<li>
<b>F_V</b>(1:F_NUM), the value of F at each node;
</li>
</ul>
</p>
<p>
The piecewise linear function G(X) is defined similarly, and there is
no requirement that G(X) uses the same nodes as F(X), or the same number of nodes.
</p>
<p>
The task, then, is to determine the value of
<pre>
Integral ( A <= X <= B ) F(X) * G(X) dx
</pre>
This value can be determined exactly, by breaking the interval ot integration
into subintervals over which F(X) and G(X) are simple linear functions.
</p>
<p>
In finite element programs over 1-dimensional geometries, integrals like
this may occur when assembling the stiffness matrix, but these integrals
are generally treated using quadrature.
</p>
<p>
A more appropriate use for this function occurs when coarsening a finite
element solution, or constructing a piecewise linear least squares
finite element approximant to data that is regarded as a piecewise linear
function.
</p>
<p>
In case 1, we treat the original finite element solution as the function F(X),
and in case 2, we treat the data as a finite element function whose mesh
is implicit in the X coordinates of the given data.
</p>
<p>
Our G_X is, in case 1, the coarse mesh and in case 2 the given mesh.
To determine the finite element coefficient U(I) on this mesh, we
must integrate our data function F(X) against the I-th basis function,
which in this case is simply the I-th "hat function. So to use
our formula for G(X), we set G_V(1:G_NUM) to 0, except that G_V(I) = 1,
and compute
<pre>
U(I) = Integral ( G_X(I-1) <= X <= G_X(I+1) ) F(X) * G(X) dx
</pre>
Doing this for I from 1 to G_NUM allows us to compute the coefficients
of the coarsened solution (case 1) or the least squares approximant to
the piecewise linear function representing the data F.
</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>PIECEWISE_LINEAR_PRODUCT_INTEGRAL</b> is available in
<a href = "../../c_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a C version</a> and
<a href = "../../cpp_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a C++ version</a> and
<a href = "../../f77_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a FORTRAN77 version</a> and
<a href = "../../f_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a FORTRAN90 version</a> and
<a href = "../../m_src/piecewise_linear_product_integral/piecewise_linear_product_integral.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../data/fem/fem.html">
FEM</a>,
a data directory which
contains a description and examples of files that describe a finite element model.
</p>
<p>
<a href = "../../cpp_src/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a 1D linear two point boundary value problem.
</p>
<p>
<a href = "../../cpp_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a C++ program which
applies the finite element method to a 1D linear two point boundary value problem
using adaptive refinement to improve the solution.
</p>
<p>
<a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a 1D nonlinear two point boundary value problem.
</p>
<p>
<a href = "../../cpp_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a C++ program which
applies the p-method version of the finite element method to a linear
two point boundary value problem in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">
FEM1D_PROJECT</a>,
a C++ program which
projects data into a finite element space, including the least squares
approximation of data, or the projection of a finite element solution
from one mesh to another.
</p>
<p>
<a href = "../../cpp_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a C++ program which
samples a scalar or vector finite element function of one variable,
defined by FEM files,
returning interpolated values at the sample points.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
TA640.2.Z54.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "piecewise_linear_product_integral.cpp">
piecewise_linear_product_integral.cpp</a>, the source code.
</li>
<li>
<a href = "piecewise_linear_product_integral.hpp">
piecewise_linear_product_integral.hpp</a>, the include file.
</li>
<li>
<a href = "piecewise_linear_product_integral.sh">
piecewise_linear_product_integral.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "piecewise_linear_product_integral_prb.cpp">
piecewise_linear_product_integral_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "piecewise_linear_product_integral_prb.sh">
piecewise_linear_product_integral_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "piecewise_linear_product_integral_prb_output.txt">
piecewise_linear_product_integral_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_INTEGRAL:</b> piecewise linear product integral.
</li>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_QUAD:</b> estimate piecewise linear product integral.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</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>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</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 30 April 2009.
</i>
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