hyperball_integrals


hyperball_integrals. a FORTRAN77 code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

The interior of the unit hyperball 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 <= i <= 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:

hyperball_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

hyperball_integrals_test

ball_integrals, a FORTRAN77library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

circle_integrals, a FORTRAN77 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a FORTRAN77 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 FORTRAN77 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN77 library which returns the exact value of the integral of any monomial over the interior of a triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN77 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:


Last revised on 21 October 2023.