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<html>
<head>
<title>
CAUCHY_PRINCIPAL_VALUE - Estimate Singular Integrals
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
CAUCHY_PRINCIPAL_VALUE <br> Estimate Singular Integrals
</h1>
<hr>
<p>
<b>CAUCHY_PRINCIPAL_VALUE</b>
is a C++ library which
uses Gauss-Legendre quadrature to estimate the Cauchy Principal Value
of certain singular integrals.
</p>
<p>
The singular integrals to be considered will have the form:
<pre>
Integral ( a <= t <= b ) f(t) / ( t - x ) dt
</pre>
The Cauchy Principal Value is defined as
<pre>
CPV = limit ( s --> x ) Integral ( a <= t <= s ) f(t) / ( t - x ) dt
+ limit ( x <-- s ) Integral ( s <= t <= b ) f(t) / ( t - x ) dt
</pre>
</p>
<p>
We suppose that our singular integral is posed on an interval that is
symmetric with respect to the location of the singularity:
<pre>
Integral ( x-d <= t <= x+d ) f(t) / ( t - x ) dt
</pre>
and we propose to estimate the integral using a Gauss-Legendre rule of
even order N:
<pre>
CPV approx sum ( 1 <= i <= N ) w(i) * f(xi(i)*d+x) / xi(i)
</pre>
where xi(i) and w(i) are the points and weights, respectively, of
the Gauss-Legendre rule.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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>CAUCHY_PRINCIPAL_VALUE</b> is available in
<a href = "../../c_src/cauchy_principal_value/cauchy_principal_value.html">a C version</a> and
<a href = "../../cpp_src/cauchy_principal_value/cauchy_principal_value.html">a C++ version</a> and
<a href = "../../f77_src/cauchy_principal_value/cauchy_principal_value.html">a FORTRAN77 version</a> and
<a href = "../../f_src/cauchy_principal_value/cauchy_principal_value.html">a FORTRAN90 version</a> and
<a href = "../../m_src/cauchy_principal_value/cauchy_principal_value.html">a MATLAB version</a> and
<a href = "../../py_src/cauchy_principal_value/cauchy_principal_value.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for approximating an integral over a 1D domain.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Julian Noble,<br>
Gauss-Legendre Principal Value Integration,<br>
Computing in Science and Engineering,<br>
Volume 2, Number 1, January-February 2000, pages 92-95.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cpv.cpp">cpv.cpp</a>, the source code.
</li>
<li>
<a href = "cpv.hpp">cpv.hpp</a>, the include file.
</li>
<li>
<a href = "cpv.sh">cpv.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cpv_prb.cpp">cpv_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "cpv_prb.sh">cpv_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "cpv_prb_output.txt">cpv_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CPV</b> estimates the Cauchy Principal Value of an integral.
</li>
<li>
<b>LEGENDRE_SET</b> sets abscissas and weights for Gauss-Legendre quadrature.
</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 modified on 01 April 2015.
</i>
<!-- John Burkardt -->
</body>
</html>