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<html>
<head>
<title>
CIRCLE_RULE - Quadrature Rules for the Unit Circle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CIRCLE_RULE <br> Quadrature Rules for the Unit Circle
</h1>
<hr>
<p>
<b>CIRCLE_RULE</b>
is a C++ library which
computes quadrature rules
over the circumference of the unit circle in 2D.
</p>
<p>
The user specifies the value NT, the number of equally spaced angles.
The program returns vectors T(1:NT) and W(1:NT), which define the rule Q(f).
</p>
<p>
Given NT and the vectors T and W, the integral I(f) of
a function f(x,y) is estimated by Q(f) as follows:
<pre>
q = 0.0
for i = 1, nt
x = cos ( t(i) )
y = sin ( t(i) )
q = q + w(j) * f ( x, y )
end
</pre>
</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>CIRCLE_RULE</b> is available in
<a href = "../../c_src/circle_rule/circle_rule.html">a C version</a> and
<a href = "../../cpp_src/circle_rule/circle_rule.html">a C++ version</a> and
<a href = "../../f77_src/circle_rule/circle_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/circle_rule/circle_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/circle_rule/circle_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/circle_arc_grid/circle_arc_grid.html">
CIRCLE_ARC_GRID</a>,
a C++ program which
computes points equally spaced along a circular arc;
</p>
<p>
<a href = "../../cpp_src/circle_integrals/circle_integrals.html">
CIRCLE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the circumference of the unit circle in 2D.
</p>
<p>
<a href = "../../cpp_src/circle_monte_carlo/circle_monte_carlo.html">
CIRCLE_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
on the circumference of the unit circle in 2D;
</p>
<p>
<a href = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/disk_rule/disk_rule.html">
DISK_RULE</a>,
a C++ library which
computes quadrature rules
over the interior of the unit disk in 2D.
</p>
<p>
<a href = "../../cpp_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/pyramid_rule/pyramid_rule.html">
PYRAMID_RULE</a>,
a C++ program which
computes a quadrature rule
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a C++ library which
computes Lebedev quadrature rules
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature rules for a variety of M-dimensional regions,
including the interior of the square, cube and hypercube, the pyramid,
cone and ellipse, the hexagon, the M-dimensional octahedron,
the circle, sphere and hypersphere, the triangle, tetrahedron and simplex,
and the surface of the circle, sphere and hypersphere.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "circle_rule.cpp">circle_rule.cpp</a>, the source code.
</li>
<li>
<a href = "circle_rule.hpp">circle_rule.hpp</a>, the include file.
</li>
<li>
<a href = "circle_rule.sh">circle_rule.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "circle_rule_prb.cpp">circle_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "circle_rule_prb.sh">circle_rule_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "circle_rule_prb_output.txt">circle_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CIRCLE_RULE</b> computes a quadrature rule for the unit circle.
</li>
<li>
<b>CIRCLE01_MONOMIAL_INTEGRAL:</b> integral on circumference of unit circle in 2D.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</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 06 April 2014.
</i>
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