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<html>
<head>
<title>
HYPERBALL_VOLUME_MONTE_CARLO - M-dimensional Sphere Volume by Monte Carlo
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HYPERBALL_VOLUME_MONTE_CARLO <br> M-dimensional Sphere Volume by Monte Carlo
</h1>
<hr>
<p>
<b>HYPERBALL_VOLUME_MONTE_CARLO</b>
is a C++ program which
investigates the behavior of a Monte Carlo procedure when it is applied
to compute the integral of a discontinuous function. In particular,
our integration region is the M-dimensional unit hypercube and our function
f(x) is 1 if the point x is inside the unit hyperball of radius 1,
and 0 otherwise.
</p>
<p>
The program uses the Monte Carlo method to estimate the volume.
Estimates are made starting with 2^0 (=1) points and doubling
repeatedly up to 2^25 points.
</p>
<p>
Because the integrand is discontinuous, any quadrature rule based on
the idea of interpolation will probably be unable to do a good job.
A family of quadrature rules, which rely on increasing the order of
interpolation to improve accuracy, will probably get increasingly
bad answers.
</p>
<p>
By contrast, a basic Monte Carlo rule, which assumes nothing about
the function, integrates this function just as well as it integrates
most any other square-integrable function. (That's both the strength
and weakness of the blunt instrument we call Monte Carlo integration.)
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>hyperball_volume_monte_carlo</b> <i>dim_num</i> <i>seed</i>
</blockquote>
where
<ul>
<li>
<i>dim_num</i> is the spatial dimension.
</li>
<li>
<i>seed</i> is an optional seed for the random number generator.
If it is not specified on the command line, a default value is used.
</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>HYPERBALL_VOLUME_MONTE_CARLO</b> is available in
<a href = "../../c_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">a C version</a> and
<a href = "../../cpp_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">a C++ version</a> and
<a href = "../../f77_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hyperball_volume_monte_carlo/hyperball_volume_monte_carlo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/ball_monte_carlo/ball_monte_carlo.html">
BALL_MONTE_CARLO</a>,
a C++ 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 = "../../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_monte_carlo/cube_monte_carlo.html">
CUBE_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/disk_monte_carlo/disk_monte_carlo.html">
DISK_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/ellipse_monte_carlo/ellipse_monte_carlo.html">
ELLIPSE_MONTE_CARLO</a>
a C++ 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 = "../../cpp_src/ellipsoid_monte_carlo/ellipsoid_monte_carlo.html">
ELLIPSOID_MONTE_CARLO</a>
a C++ 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 = "../../cpp_src/hyperball_integrals/hyperball_integrals.html">
HYPERBALL_INTEGRALS</a>,
a C++ library which
defines test functions for integration
over the interior of the unit hyperball in M dimensions.
</p>
<p>
<a href = "../../cpp_src/hyperball_monte_carlo/hyperball_monte_carlo.html">
HYPERBALL_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/hypersphere_monte_carlo/hypersphere_monte_carlo.html">
HYPERSPHERE_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/line_monte_carlo/line_monte_carlo.html">
LINE_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the length of the unit line in 1D.
</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/simplex_monte_carlo/simplex_monte_carlo.html">
SIMPLEX_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a C++ library which
applies a Monte Carlo method to estimate the integral of a function
over the surface of the unit sphere in 3D;
</p>
<p>
<a href = "../../cpp_src/sphere_triangle_monte_carlo/sphere_triangle_monte_carlo.html">
SPHERE_TRIANGLE_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/square_monte_carlo/square_monte_carlo.html">
SQUARE_MONTE_CARLO</a>,
a C++ 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 = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals over the unit tetrahedron.
</p>
<p>
<a href = "../../cpp_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_monte_carlo/wedge_monte_carlo.html">
WEDGE_MONTE_CARLO</a>,
a C++ 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>
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 = "hyperball_volume_monte_carlo.cpp">hyperball_volume_monte_carlo.cpp</a>, the source code.
</li>
<li>
<a href = "hyperball_volume_monte_carlo.sh">hyperball_volume_monte_carlo.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hyperball_volume_monte_carlo_output.txt">hyperball_volume_monte_carlo_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for HYPERBALL_VOLUME_MONTE_CARLO.
</li>
<li>
<b>HYPERBALL01_INDICATOR</b> evaluates the unit hyperball indicator function.
</li>
<li>
<b>HYPERBALL01_VOLUME</b> computes the volume of a unit hyperball.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from 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 03 January 2014.
</i>
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