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<html>
<head>
<title>
TETRAHEDRON_MONTE_CARLO - Monte Carlo Integral Estimates over the Unit Tetrahedron in 3D
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_MONTE_CARLO <br> Monte Carlo Integral Estimates over the Unit Tetrahedron in 3D
</h1>
<hr>
<p>
<b>TETRAHEDRON_MONTE_CARLO</b>
is a MATLAB library which
uses the Monte Carlo method to estimate the integral of a function F(X,Y,Z)
over the interior of the unit tetrahedron in 3D.
</p>
<p>
The interior of the unit tetrahedron in 3D is defined by the constraints:
<pre>
0 <= X
0 <= Y
0 <= Z
X + Y + Z <= 1
</pre>
The functions F(X,Y,Z) are monomials, having the form
<pre>
F(X,Y,Z) = X^E(1) * Y^E(2) * Z^E(3)
</pre>
where the exponents are nonnegative integers.
</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>TETRAHEDRON_MONTE_CARLO</b> is available in
<a href = "../../c_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">a C version</a> and
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">a C++ version</a> and
<a href = "../../f77_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">a FORTRAN77 version</a> and
<a href = "../../f_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/ball_monte_carlo/ball_monte_carlo.html">
BALL_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate integrals of a function
over the interior of the unit ball in 3D;
</p>
<p>
<a href = "../../m_src/circle_monte_carlo/circle_monte_carlo.html">
CIRCLE_MONTE_CARLO</a>,
a MATLAB 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 = "../../m_src/cube_monte_carlo/cube_monte_carlo.html">
CUBE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit cube in 3D.
</p>
<p>
<a href = "../../m_src/disk_monte_carlo/disk_monte_carlo.html">
DISK_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit disk in 2D;
</p>
<p>
<a href = "../../m_src/ellipse_monte_carlo/ellipse_monte_carlo.html">
ELLIPSE_MONTE_CARLO</a>
a MATLAB library which
uses the Monte Carlo method to estimate the value of integrals
over the interior of an ellipse in 2D.
</p>
<p>
<a href = "../../m_src/ellipsoid_monte_carlo/ellipsoid_monte_carlo.html">
ELLIPSOID_MONTE_CARLO</a>
a MATLAB library which
uses the Monte Carlo method to estimate the value of integrals
over the interior of an ellipsoid in M dimensions.
</p>
<p>
<a href = "../../m_src/hyperball_monte_carlo/hyperball_monte_carlo.html">
HYPERBALL_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit hyperball in M dimensions;
</p>
<p>
<a href = "../../m_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">
HYPERBALL_VOLUME_MONTE_CARLO</a>,
a MATLAB program which
applies a Monte Carlo method to estimate the volume
of the unit hyperball in M dimensions;
</p>
<p>
<a href = "../../m_src/hypercube_monte_carlo/hypercube_monte_carlo.html">
HYPERCUBE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../m_src/hypersphere_monte_carlo/hypersphere_monte_carlo.html">
HYPERSPHERE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
on the surface of the unit sphere in M dimensions;
</p>
<p>
<a href = "../../m_src/line_monte_carlo/line_monte_carlo.html">
LINE_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the length of the unit line in 1D.
</p>
<p>
<a href = "../../m_src/polygon_monte_carlo/polygon_monte_carlo.html">
POLYGON_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of a polygon in 2D.
</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_monte_carlo/simplex_monte_carlo.html">
SIMPLEX_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/sphere_triangle_monte_carlo/sphere_triangle_monte_carlo.html">
SPHERE_TRIANGLE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over a spherical triangle on the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../m_src/square_monte_carlo/square_monte_carlo.html">
SQUARE_MONTE_CARLO</a>,
a MATLAB library which
applies a Monte Carlo method to estimate the integral of a function
over the interior of the unit square in 2D.
</p>
<p>
<a href = "../../m_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a MATLAB library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a MATLAB program which
investigates the monomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</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/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a MATLAB library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_monte_carlo/wedge_monte_carlo.html">
WEDGE_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Claudio Rocchini, Paolo Cignoni,<br>
Generating Random Points in a Tetrahedron,<br>
Journal of Graphics Tools,<br>
Volume 5, Number 4, 2000, pages 9-12.
</li>
<li>
Reuven Rubinstein,<br>
Monte Carlo Optimization, Simulation and Sensitivity of
Queueing Networks,<br>
Krieger, 1992,<br>
ISBN: 0894647644,<br>
LC: QA298.R79.
</li>
<li>
Greg Turk,<br>
Generating Random Points in a Triangle,<br>
in Graphics Gems I,<br>
edited by Andrew Glassner,<br>
AP Professional, 1990,<br>
ISBN: 0122861663,<br>
LC: T385.G697
</li>
</ol>
</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 = "r8vec_uniform_01.m">r8vec_uniform_01.m</a>
returns a unit pseudorandom R8VEC.
</li>
<li>
<a href = "tetrahedron01_monomial.m">tetrahedron01_monomial.m</a>,
monomial integrals in the interior of the unit tetrahedron in 3D.
</li>
<li>
<a href = "tetrahedron01_sample.m">tetrahedron01_sample.m</a>
samples points from the interior of the unit tetrahedron in 3D.
</li>
<li>
<a href = "tetrahedron01_volume.m">tetrahedron01_volume.m</a>
computes the volume of the unit tetrahedron in 3D.
</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 = "tetrahedron_monte_carlo_test.m">tetrahedron_monte_carlo_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "tetrahedron_monte_carlo_test01.m">tetrahedron_monte_carlo_test01.m</a>,
uses TETRAHEDRON01_SAMPLE to estimate integrals over the interior of the
unit tetrahedron in 3D.
</li>
<li>
<a href = "tetrahedron_monte_carlo_test_output.txt">tetrahedron_monte_carlo_test_output.txt</a>,
the output file.
</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 14 January 2014.
</i>
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