TETRAHEDRON_MONTE_CARLO
Monte Carlo Integral Estimates over a Tetrahedron

TETRAHEDRON_MONTE_CARLO is a FORTRAN77 library which estimates the integral of a function over a tetrahedron using the Monte Carlo method.

The library makes it relatively easy to compare different methods of producing sample points in the tetrahedron, and to vary the tetrahedron over which integration is carried out.

Licensing:

The computer code and data files 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 FORTRAN77 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 FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function on the circumference of the unit circle in 2D;

DISK_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2D;

HYPERBALL_MONTE_CARLO, a FORTRAN77 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 FORTRAN77 program which applies a Monte Carlo method to estimate the volume of the unit hyperball in M dimensions;

HYPERSPHERE_MONTE_CARLO, a FORTRAN77 library which applies a Monte Carlo method to estimate the integral of a function on the surface of the unit sphere in M dimensions;

SIMPLEX_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the interior of the unit simplex in M dimensions.

SPHERE_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the surface of the unit sphere in 3D.

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_MONTE_CARLO, a FORTRAN77 library which uses the Monte Carlo method to estimate integrals over the unit triangle.

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.f, the source code.
• tetrahedron_monte_carlo.sh, commands to compile the source code.

Examples and Tests:

• tetrahedron_monte_carlo_prb.f, 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:

• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• REFERENCE_TO_PHYSICAL_TET4 maps TET4 reference points to physical points.
• TETRAHEDRON_INTEGRAND_01 evaluates 1 integrand function.
• TETRAHEDRON_INTEGRAND_02 evaluates 3 integrand functions.
• TETRAHEDRON_INTEGRAND_03 evaluates 6 integrand functions.
• TETRAHEDRON_INTEGRAND_04 evaluates 10 integrand functions.
• TETRAHEDRON_INTEGRAND_05 evaluates 15 integrand functions.
• TETRAHEDRON_MONTE_CARLO applies the Monte Carlo rule to integrate a function.
• TETRAHEDRON_UNIT_SAMPLE_01 selects points from the unit tetrahedron.
• TETRAHEDRON_UNIT_SAMPLE_02 selects points from the unit tetrahedron.
• TETRAHEDRON_UNIT_SAMPLE_03 selects points from the unit tetrahedron.
• TETRAHEDRON_UNIT_SAMPLE_04 selects points from the unit tetrahedron.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron in 3D.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.