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

HYPERSPHERE_INTEGRALS is a FORTRAN90 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)

where the exponents 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.

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 FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK01_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 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 FORTRAN90 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a FORTRAN90 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 FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Reference:

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.f90, the source code.

Examples and Tests:

hypersphere_monte_carlo_test.f90, a sample calling program.
hypersphere_monte_carlo_test.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.

You can go up one level to the FORTRAN90 source codes.

Last revised on 07 January 2014.

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