tetrahedron_integrals

tetrahedron_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

The interior of the unit tetrahedron in 3D is defined by

        0 <= x
        0 <= y
        0 <= z
             x + y + z <= 1
      

The integrands are all of the form

        f(x,y,z) = x^e1 * y^e2 * z^e3
      

Licensing:

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

Languages:

tetrahedron_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 Python code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

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

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

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

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

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

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

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

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

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

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

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

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

TETRAHEDRON_MONTE_CARLO, a Python code which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

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

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

Reference:

• Jean Lasserre, Konstantin Avrachenkov,
  The multidimensional version of the integral from A to B of X to the P,
  American Mathematics Monthly,
  Volume 108, Number 2, 2001, pages 151-154.

Source Code:

• tetrahedron_integrals.py, the source code.
• tetrahedron_integrals.sh, runs all the tests.
• tetrahedron_integrals.txt, the output file.