TETRAHEDRON_MONTE_CARLO
Monte Carlo Integral Estimates over the Unit Tetrahedron in 3D

TETRAHEDRON_MONTE_CARLO is a C++ library which uses the Monte Carlo method to estimate the integral of a function F(X,Y,Z) over the interior of the unit tetrahedron in 3D.

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

        0 <= X
        0 <= Y
        0 <= Z
             X + Y + Z <= 1
      

        F(X,Y,Z) = X^E(1) * Y^E(2) * Z^E(3)
      

Licensing:

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

Languages:

TETRAHEDRON_MONTE_CARLO 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_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit ball in 3D;

CIRCLE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function on the circumference of the unit circle in 2D;

CUBE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit cube in 3D;

DISK_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2D;

ELLIPSE_MONTE_CARLO a C++ library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipse in 2D.

ELLIPSOID_MONTE_CARLO a C++ library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipsoid in M dimensions.

HYPERBALL_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions;

HYPERBALL_VOLUME_MONTE_CARLO, a C++ program which applies a Monte Carlo method to estimate the volume of the unit hyperball in M dimensions;

HYPERCUBE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function on the surface of the unit sphere in M dimensions;

LINE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the length of the unit line in 1D.

POLYGON_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

PYRAMID_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;

SIMPLEX_GM_RULE, a C++ library which defines Grundmann-Moeller quadrature rules over the interior of a triangle in 2D, a tetrahedron in 3D, or over the interior of the simplex in M dimensions.

SIMPLEX_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the interior of the unit simplex in M dimensions.

SPHERE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the surface of the unit sphere in 3D.

SPHERE_TRIANGLE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over a spherical triangle on the surface of the unit sphere in 3D;

SQUARE_MONTE_CARLO, a C++ library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2D.

TETRAHEDRON_ARBQ_RULE, a C++ library which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_EXACTNESS, a C++ program which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TETRAHEDRON_FELIPPA_RULE, a C++ library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

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.

TETRAHEDRON_KEAST_RULE, a C++ library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCC_RULE, a C++ library which defines Newton-Cotes Closed (NCC) quadrature rules over the interior of a tetrahedron in 3D.

TETRAHEDRON_NCO_RULE, a C++ library which defines Newton-Cotes Open (NCO) quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

WEDGE_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the interior of the unit wedge in 3D.

Reference:

1. Claudio Rocchini, Paolo Cignoni,
   Generating Random Points in a Tetrahedron,
   Journal of Graphics Tools,
   Volume 5, Number 4, 2000, pages 9-12.
2. Reuven Rubinstein,
   Monte Carlo Optimization, Simulation and Sensitivity of Queueing Networks,
   Krieger, 1992,
   ISBN: 0894647644,
   LC: QA298.R79.
3. Greg Turk,
   Generating Random Points in a Triangle,
   in Graphics Gems I,
   edited by Andrew Glassner,
   AP Professional, 1990,
   ISBN: 0122861663,
   LC: T385.G697

Source Code:

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

Examples and Tests:

• tetrahedron_monte_carlo_prb.cpp, a sample calling program.
• tetrahedron_monte_carlo_prb.sh, commands to compile and run the sample program.
• tetrahedron_monte_carlo_prb_output.txt, the output file.

List of Routines:

• MONOMIAL_VALUE evaluates a monomial.
• R8VEC_SUM returns the sum of an R8VEC.
• R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
• TETRAHEDRON01_MONOMIAL_INTEGRAL: integrals in the unit tetrahedron in 3D.
• TETRAHEDRON01_SAMPLE samples the unit tetrahedron in 3D.
• TETRAHEDRON01_VOLUME returns the volume of the unit tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.