HYPERSPHERE_INTEGRALS
Integrals on the Surface of the Unit Hypersphere in M Dimensions

HYPERSPHERE_INTEGRALS 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.

The surface of the unit hypersphere in M dimensions is defined by

        sum ( 1 <= i <= m ) x(i)^2 = 1
      

The integrands are all of the form

        f(x) = product ( 1 <= m <= m ) x(i) ^ e(i)
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

HYPERSPHERE_INTEGRALS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BALL_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_MONTE_CARLO, 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.

HYPERSPHERE_PROPERTIES, 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.

LINE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a C++ library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Reference:

1. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.

Source Code:

• hypersphere_monte_carlo.cpp, the source code.
• hypersphere_monte_carlo.hpp, the include file.
• hypersphere_monte_carlo.sh, BASH commands to compile the source code.

Examples and Tests:

• hypersphere_monte_carlo_prb.cpp, a sample calling program.
• hypersphere_monte_carlo_prb.sh, BASH commands to compile and run the sample program.
• hypersphere_monte_carlo_prb_output.txt, the output file.

List of Routines:

• HYPERSPHERE01_AREA returns the surface area of the unit hypersphere.
• HYPERSPHERE01_MONOMIAL_INTEGRAL: monomial integrals on the unit hypersphere.
• HYPERSPHERE01_SAMPLE uniformly samples the surface of the unit hypersphere.
• I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
• MONOMIAL_VALUE evaluates a monomial.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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