TETRAHEDRON_MONTE_CARLO
Monte Carlo Integral Estimates over a Tetrahedron

TETRAHEDRON_MONTE_CARLO, a MATLAB library which estimates the integral of a function over a general 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 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

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

SPHERE_MONTE_CARLO, a MATLAB library which applies a Monte Carlo method to estimate the integral of a function over the surface of the unit sphere in 3D;

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

tetrahedron_monte_carlo_test

TRIANGLE_MONTE_CARLO, a MATLAB 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:

• r8mat_det_4d.m computes the determinant of a 4x4 R8MAT.
• r8mat_transpose_print.m prints an R8MAT, transposed.
• r8mat_transpose_print_some.m prints some of an R8MAT, transposed.
• r8vec_uniform_01.m returns a unit pseudorandom R8VEC.
• reference_to_physical_t3.m maps reference tetrahedron points to physical points.
• s_len_trim.m returns the length of a string to the last nonblank.
• tetrahedron_integrand_01.m evaluates 1 integrand function.
• tetrahedron_integrand_02.m evaluates 3 integrand functions.
• tetrahedron_integrand_03.m evaluates 6 integrand functions.
• tetrahedron_integrand_04.m evaluates 10 integrand functions.
• tetrahedron_integrand_05.m evaluates 15 integrand functions.
• tetrahedron_monte_carlo.m applies the Monte Carlo rule to integrate a function.
• tetrahedron_unit_sample_01.m selects points from the unit tetrahedron.
• tetrahedron_unit_sample_02.m selects points from the unit tetrahedron.
• tetrahedron_unit_sample_03.m selects points from the unit tetrahedron.
• tetrahedron_unit_sample_04.m selects points from the unit tetrahedron.
• tetrahedron_volume.m computes the volume of a tetrahedron.
• timestamp.m prints the current YMDHMS date as a time stamp.