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<html>
<head>
<title>
PYRAMID_EXACTNESS - Precision Test for Pyramid Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PYRAMID_EXACTNESS <br> Precision Test for Pyramid Quadrature Rules
</h1>
<hr>
<p>
<b>PYRAMID_EXACTNESS</b>
is a C++ program which
measures the precision of a quadrature rule
over the interior of a pyramid in 3D.
</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_exactness</b> <i>filename</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>filename</i> is the common prefix of the filenames containing the
abscissas and the weights of the quadrature rule.
</li>
<li>
<i>degree_max</i> is the maximum degree of the monomials to be checked.
</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_EXACTNESS</b> is available in
<a href = "../../c_src/pyramid_exactness/pyramid_exactness.html">a C version</a> and
<a href = "../../cpp_src/pyramid_exactness/pyramid_exactness.html">a C++ version</a> and
<a href = "../../f77_src/pyramid_exactness/pyramid_exactness.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pyramid_exactness/pyramid_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/pyramid_exactness/pyramid_exactness.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cube_exactness/cube_exactness.html">
CUBE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/hypercube_exactness/hypercube_exactness.html">
HYPERCUBE_EXACTNESS</a>,
a C++ program which
measures the monomial exactness of an M-dimensional quadrature rule
over the interior of the unit hypercube in M dimensions.
</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 = "../../cpp_src/pyramid_rule/pyramid_rule.html">
PYRAMID_RULE</a>,
a C++ program which
can compute a quadrature rule
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/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of a quadrature rule
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../cpp_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a rectangle in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_exactness/wedge_exactness.html">
WEDGE_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
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_exactness.cpp">pyramid_exactness.cpp</a>, the source code.
</li>
<li>
<a href = "pyramid_exactness.sh">pyramid_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>PYRAMID_L3X3_J3</b> is a pyramid rule formed by a conical product of a
3x3 Legendre rule and an order 3 Jacobi rule.
<ul>
<li>
<a href = "pyramid_l3x3_j3_w.txt">pyramid_l3x3_j3_w.txt</a>,
the weight file.
</li>
<li>
<a href = "pyramid_l3x3_j3_x.txt">pyramid_l3x3_j3_x.txt</a>,
the abscissa file.
</li>
<li>
<a href = "pyramid_l3x3_j3_exactness.txt">pyramid_l3x3_j3_exactness.txt</a>,
the exactness file produced by the program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for PYRAMID_EXACTNESS.
</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>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads the data from a DTABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a DTABLE file.
</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>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>PYRA_UNIT_MONOMIAL:</b> monomial integral in a unit pyramid.
</li>
<li>
<b>PYRA_UNIT_VOLUME:</b> volume of a unit pyramid with square base.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8VEC_DOT_PRODUCT</b> computes the dot product of a pair of R8VEC's.
</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_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 28 July 2009.
</i>
<!-- John Burkardt -->
</body>
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</html>