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<html>
<head>
<title>
BALL_VOLUME_MONTE_CARLO - M-dimensional Sphere Volume by Monte Carlo
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BALL_VOLUME_MONTE_CARLO <br> M-dimensional Sphere Volume by Monte Carlo
</h1>
<hr>
<p>
<b>BALL_VOLUME_MONTE_CARLO</b>
is a FORTRAN90 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 and our function
f(x) is 1 if the point x is inside the hypersphere 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>ball_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">
Example Output:
</h3>
<p>
Here are the results for dimension 4:
</p>
<pre>
Log(N) N Estimate Error
0 1 0.0000000 4.9
1 2 0.0000000 4.9
2 4 4.0000000 0.93
3 8 4.0000000 0.93
4 16 3.0000000 1.9
5 32 4.0000000 0.93
6 64 5.5000000 0.57
7 128 5.3750000 0.44
8 256 5.2500000 0.32
9 512 5.0625000 0.13
10 1024 5.0625000 0.13
11 2048 4.9140625 0.21E-01
12 4096 4.9648438 0.30E-01
13 8192 4.9394531 0.47E-02
14 16384 4.9990234 0.64E-01
15 32768 4.9746094 0.40E-01
16 65536 4.9738770 0.39E-01
17 131072 4.9431152 0.83E-02
18 262144 4.9287109 0.61E-02
19 524288 4.9505920 0.16E-01
20 1048576 4.9356537 0.85E-03
21 2097152 4.9341202 0.68E-03
22 4194304 4.9353104 0.51E-03
23 8388608 4.9374676 0.27E-02
24 16777216 4.9353895 0.59E-03
25 33554432 4.9356027 0.80E-03
oo oo 4.9348022 0.0
</pre>
<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>BALL_VOLUME_MONTE_CARLO</b> is available in
<a href = "../../c_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">a C version</a> and
<a href = "../../cpp_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">a C++ version</a> and
<a href = "../../f77_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">a FORTRAN77 version</a> and
<a href = "../../f_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">a FORTRAN90 version</a> and
<a href = "../../m_src/ball_volume_monte_carlo/ball_volume_monte_carlo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/ball_volume_quad/ball_volume_quad.html">
BALL_VOLUME_QUAD</a>,
a FORTRAN90 program which
applies a quadrature rule to estimate the volume of the unit 6D sphere;
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../f_src/sphere_monte_carlo/sphere_monte_carlo.html">
SPHERE_MONTE_CARLO</a>,
a FORTRAN90 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 = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for a variety of unusual areas, surfaces
and volumes in 2D, 3D and multiple dimensions.
</p>
<p>
<a href = "../../f_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a tetrahedron.
</p>
<p>
<a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a triangle.
</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 = "ball_volume_monte_carlo.f90">ball_volume_monte_carlo.f90</a>, the source code.
</li>
<li>
<a href = "ball_volume_monte_carlo.sh">ball_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 = "ball_volume_monte_carlo_output.txt">ball_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 BALL_VOLUME_MONTE_CARLO.
</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>SPHERE_INDICATOR</b> evaluates the unit sphere indicator function.
</li>
<li>
<b>SPHERE_UNIT_VOLUME_ND</b> computes the volume of a unit sphere in multi-dimensions.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 25 September 2010.
</i>
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