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<html>
<head>
<title>
PYRAMID_INTEGRALS - Integrals Inside the Unit Pyramid in 3D
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PYRAMID_INTEGRALS <br> Integrals Inside the Unit Pyramid in 3D
</h1>
<hr>
<p>
<b>PYRAMID_INTEGRALS</b>
is a MATLAB library which
returns the exact value of the integral of any monomial x^i y^j z^k
over the interior of the unit pyramid in 3D.
</p>
<p>
The unit pyramid has a square base of area 4, and a height of 1.
Specifically, the integration region is:
<pre>
- ( 1 - Z ) <= X <= 1 - Z
- ( 1 - Z ) <= Y <= 1 - Z
0 <= Z <= 1.
</pre>
The volume of the unit pyramid is 4/3.
</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_INTEGRALS</b> is available in
<a href = "../../c_src/pyramid_integrals/pyramid_integrals.html">a C version</a> and
<a href = "../../cpp_src/pyramid_integrals/pyramid_integrals.html">a C++ version</a> and
<a href = "../../f77_src/pyramid_integrals/pyramid_integrals.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pyramid_integrals/pyramid_integrals.html">a FORTRAN90 version</a>.
<a href = "../../m_src/pyramid_integrals/pyramid_integrals.html">a MATLAB version</a> and
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/ball_integrals/ball_integrals.html">
BALL_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit ball in 3D.
</p>
<p>
<a href = "../../m_src/circle_integrals/circle_integrals.html">
CIRCLE_INTEGRALS</a>,
a MATLAB 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 = "../../m_src/cube_integrals/cube_integrals.html">
CUBE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit cube in 3D.
</p>
<p>
<a href = "../../m_src/disk_integrals/disk_integrals.html">
DISK_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit disk in 2D.
</p>
<p>
<a href = "../../m_src/hyperball_integrals/hyperball_integrals.html">
HYPERBALL_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit hyperball in M dimensions.
</p>
<p>
<a href = "../../m_src/hypercube_integrals/hypercube_integrals.html">
HYPERCUBE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../m_src/hypersphere_integrals/hypersphere_integrals.html">
HYPERSPHERE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the surface of the unit hypersphere in M dimensions.
</p>
<p>
<a href = "../../m_src/line_integrals/line_integrals.html">
LINE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the unit line segment in 1D.
</p>
<p>
<a href = "../../m_src/polygon_integrals/polygon_integrals.html">
POLYGON_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of a polygon in 2D.
</p>
<p>
<a href = "../../m_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../m_src/pyramid_grid/pyramid_grid.html">
PYRAMID_GRID</a>,
a MATLAB library which
computes a grid of points
over the interior of the unit pyramid in 3D;
</p>
<p>
<a href = "../../m_src/pyramid_monte_carlo/pyramid_monte_carlo.html">
PYRAMID_MONTE_CARLO</a>,
a MATLAB 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 = "../../m_src/simplex_integrals/simplex_integrals.html">
SIMPLEX_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/sphere_integrals/sphere_integrals.html">
SPHERE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/square_integrals/square_integrals.html">
SQUARE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit square in 2D.
</p>
<p>
<a href = "../../m_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_integrals/triangle_integrals.html">
TRIANGLE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_integrals/wedge_integrals.html">
WEDGE_INTEGRALS</a>,
a MATLAB library which
returns the exact value of the integral of any monomial
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "monomial_value.m">monomial_value.m</a>,
evaluates a monomial.
</li>
<li>
<a href = "pyramid01_integral.m">pyramid01_integral.m</a>,
monomial integral in a unit pyramid.
</li>
<li>
<a href = "pyramid01_sample.m">pyramid01_sample.m</a>,
sample the unit pyramid.
</li>
<li>
<a href = "pyramid01_volume.m">pyramid01_volume.m</a>,
volume of a unit pyramid with square base.
</li>
<li>
<a href = "r8_choose.m">r8_choose.m</a>,
computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<a href = "r8_mop.m">r8_mop.m</a>,
returns the I-th power of -1 as an R8.
</li>
<li>
<a href = "r8mat_transpose_print.m">r8mat_transpose_print.m</a>,
prints an R8MAT, transposed.
</li>
<li>
<a href = "r8mat_transpose_print_some.m">r8mat_transpose_print_some.m</a>,
prints some of an R8MAT, transposed.
</li>
<li>
<a href = "r8mat_uniform_01.m">r8mat_uniform_01.m</a>,
fills an R8MAT with unit pseudorandom numbers.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "pyramid_integrals_test.m">pyramid_integrals_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "pyramid_integrals_test_output.txt">pyramid_integrals_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "pyramid_integrals_test01.m">pyramid_integrals_test01.m</a>,
compares the exact integrals to Monte Carlo estimates.
</li>
<li>
<a href = "pyramid_integrals_test02.m">pyramid_integrals_test02.m</a>,
prints some sample points from the unit pyramid.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 13 April 2014.
</i>
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</html>