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<html>
<head>
<title>
HYPERSPHERE_INTEGRALS - Integrals on the Surface of the Unit Hypersphere in M Dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HYPERSPHERE_INTEGRALS <br> Integrals on the Surface of the Unit Hypersphere in M Dimensions
</h1>
<hr>
<p>
<b>HYPERSPHERE_INTEGRALS</b>
is a C++ library which
returns the exact value of the integral of any monomial
over the surface of the unit hypersphere in M dimensions.
</p>
<p>
The surface of the unit hypersphere in M dimensions is defined by
<pre>
sum ( 1 <= i <= m ) x(i)^2 = 1
</pre>
</p>
<p>
The integrands are all of the form
<pre>
f(x) = product ( 1 <= m <= m ) x(i) ^ e(i)
</pre>
where the exponents e are nonnegative integers.
If any exponent is an odd integer, the integral will be zero.
Thus, the "interesting" results occur when all exponents are even.
</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>HYPERSPHERE_INTEGRALS</b> is available in
<a href = "../../c_src/hypersphere_integrals/hypersphere_integrals.html">a C version</a> and
<a href = "../../cpp_src/hypersphere_integrals/hypersphere_integrals.html">a C++ version</a> and
<a href = "../../f77_src/hypersphere_integrals/hypersphere_integrals.html">a FORTRAN77 version</a> and
<a href = "../../f_src/hypersphere_integrals/hypersphere_integrals.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hypersphere_integrals/hypersphere_integrals.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/ball_integrals/ball_integrals.html">
BALL_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/circle_integrals/circle_integrals.html">
CIRCLE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the surface of the unit circle in 2D.
</p>
<p>
<a href = "../../cpp_src/cube_integrals/cube_integrals.html">
CUBE_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/disk_integrals/disk_integrals.html">
DISK_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/hyperball_integrals/hyperball_integrals.html">
HYPERBALL_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/hypercube_integrals/hypercube_integrals.html">
HYPERCUBE_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/hypersphere_monte_carlo/hypersphere_monte_carlo.html">
HYPERSPHERE_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate the integral of a function
over the surface of the unit hypersphere in M dimensions.
</p>
<p>
<a href = "../../cpp_src/hypersphere_properties/hypersphere_properties.html">
HYPERSPHERE_PROPERTIES</a>,
a C++ library which
carries out various operations for an M-dimensional hypersphere, including
converting between Cartesian and spherical coordinates,
stereographic projection, sampling the surface of the sphere, and
computing the surface area and volume.
</p>
<p>
<a href = "../../cpp_src/line_integrals/line_integrals.html">
LINE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the length of the unit line in 1D.
</p>
<p>
<a href = "../../cpp_src/polygon_integrals/polygon_integrals.html">
POLYGON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of a polygon in 2D.
</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/simplex_integrals/simplex_integrals.html">
SIMPLEX_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/sphere_integrals/sphere_integrals.html">
SPHERE_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/square_integrals/square_integrals.html">
SQUARE_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/triangle_integrals/triangle_integrals.html">
TRIANGLE_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/wedge_integrals/wedge_integrals.html">
WEDGE_INTEGRALS</a>,
a C++ 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">
Reference:
</h3>
<p>
<ol>
<li>
Gerald Folland,<br>
How to Integrate a Polynomial Over a Sphere,<br>
American Mathematical Monthly,<br>
Volume 108, Number 5, May 2001, pages 446-448.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "hypersphere_monte_carlo.cpp">hypersphere_monte_carlo.cpp</a>, the source code.
</li>
<li>
<a href = "hypersphere_monte_carlo.hpp">hypersphere_monte_carlo.hpp</a>, the include file.
</li>
<li>
<a href = "hypersphere_monte_carlo.sh">hypersphere_monte_carlo.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "hypersphere_monte_carlo_prb.cpp">hypersphere_monte_carlo_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "hypersphere_monte_carlo_prb.sh">hypersphere_monte_carlo_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hypersphere_monte_carlo_prb_output.txt">hypersphere_monte_carlo_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HYPERSPHERE01_AREA</b> returns the surface area of the unit hypersphere.
</li>
<li>
<b>HYPERSPHERE01_MONOMIAL_INTEGRAL:</b> monomial integrals on the unit hypersphere.
</li>
<li>
<b>HYPERSPHERE01_SAMPLE</b> uniformly samples the surface of the unit hypersphere.
</li>
<li>
<b>I4VEC_UNIFORM_AB</b> returns a scaled pseudorandom I4VEC.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>R8_GAMMA</b> evaluates Gamma(X) for a real argument.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</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 07 January 2014.
</i>
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