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<html>
<head>
<title>
PYRAMID_RULE - Generate a Quadrature Rule for a Pyramid
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PYRAMID_RULE <br> Generate a Quadrature Rule for a Pyramid
</h1>
<hr>
<p>
<b>PYRAMID_RULE</b>
is a C++ program which
generates a quadrature rule
over the interior of a pyramid in 3D.
</p>
<p>
The quadrature rules generated by <b>PYRAMID_RULE</b> are all
examples of <i>conical product rules</i>, and involve a kind of
direct product of the form:
<blockquote>
Legendre rule in X * Legendre rule in Y * Jacobi rule in Z
</blockquote>
where the Jacobi rule includes a factor of (1-Z)^2.
</p>
<p>
The integration region is:
<pre>
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
</pre>
When Z is zero, the integration region is a square lying in the (X,Y)
plane, centered at (0,0,0) with "radius" 1. As Z increases to 1, the
radius of the square diminishes, and when Z reaches 1, the square has
contracted to the single point (0,0,1).
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>pyramid_rule</b> <i>legendre_order</i> <i>jacobi_order</I> <i>filename</i>
</blockquote>
where
<ul>
<li>
<i>legendre_order</i> is the order of the 1D Legendre quadrature rule
to be used. The X and Y dimensions will use a product of this rule.
</li>
<li>
<i>jacobi_order</i> is the order of the 1D Jacobi quadrature rule to
be used. The Z dimension will use this rule which will include a
factor of (1-X)^2 which accounts for the narrowing of the pyramid.
</li>
<li>
<i>filename</i> is the common prefix for the files containing the region, weight
and abscissa information of the quadrature rule;
</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>PYRAMID_RULE</b> is available in
<a href = "../../c_src/pyramid_rule/pyramid_rule.html">a C version</a> and
<a href = "../../cpp_src/pyramid_rule/pyramid_rule.html">a C++ version</a> and
<a href = "../../f77_src/pyramid_rule/pyramid_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pyramid_rule/pyramid_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/pyramid_rule/pyramid_rule.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of the unit pyramid in 3D.
</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_grid/pyramid_grid.html">
PYRAMID_GRID</a>,
a C++ library which
computes a grid of points
over the interior of the unit pyramid in 3D;
</p>
<p>
<a href = "../../cpp_src/pyramid_integrals/pyramid_integrals.html">
PYRAMID_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/pyramid_monte_carlo/pyramid_monte_carlo.html">
PYRAMID_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate integrals of a function
over the interior of the unit pyramid in 3D;
</p>
<p>
<a href = "../../datasets/quadrature_rules_pyramid/quadrature_rules_pyramid.html">
QUADRATURE_RULES_PYRAMID</a>,
a dataset directory which
contains quadrature rules
over the interior of the unit pyramid 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 unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</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>
Carlos Felippa,<br>
A compendium of FEM integration formulas for symbolic work,<br>
Engineering Computation,<br>
Volume 21, Number 8, 2004, pages 867-890.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "pyramid_rule.cpp">pyramid_rule.cpp</a>, the source code.
</li>
<li>
<a href = "pyramid_rule.sh">pyramid_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for PYRAMID_RULE.
</li>
<li>
<b>DTABLE_WRITE0</b> writes a DTABLE file with no header.
</li>
<li>
<b>JACOBI_COMPUTE</b> computes a Jacobi quadrature rule.
</li>
<li>
<b>JACOBI_RECUR</b> evaluates a Jacobi polynomial.
</li>
<li>
<b>JACOBI_ROOT</b> improves an approximate root of a Jacobi polynomial.
</li>
<li>
<b>LEGENDRE_COMPUTE</b> computes a Legendre quadrature rule.
</li>
<li>
<b>PYRAMID_HANDLE</b> computes the requested pyramid rule and outputs it.
</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_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 23 July 2009.
</i>
<!-- John Burkardt -->
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