TETRAHEDRON_MONTE_CARLO
Monte Carlo Integral Estimates over a Tetrahedron

TETRAHEDRON_MONTE_CARLO is a MATLAB 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 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 FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULE, a MATLAB library which defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

KEAST, a MATLAB library which defines a number of quadrature rules for a tetrahedron.

NCC_TETRAHEDRON, a MATLAB library which defines Newton-Cotes Closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON, a MATLAB library which defines Newton-Cotes Open quadrature rules on a tetrahedron.

RANDOM_DATA, a MATLAB library which generates sample points for various probability distributions, spatial dimensions, and geometries;

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;

STROUD, a MATLAB library which defines quadrature rules for a variety of multidimensional reqions.

TETRAHEDRON_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a 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.

Examples and Tests:

• tetrahedron_monte_carlo_test.m, a sample calling program.
• tetrahedron_monte_carlo_test01.m, uses TETRAHEDRON_UNIT_SAMPLE_01 to integrate on the unit tetrahedron.
• tetrahedron_monte_carlo_test02.m, uses TETRAHEDRON_UNIT_SAMPLE_02 to integrate on the unit tetrahedron.
• tetrahedron_monte_carlo_test03.m, uses TETRAHEDRON_UNIT_SAMPLE_03 to integrate on the unit tetrahedron.
• tetrahedron_monte_carlo_test04.m, uses TETRAHEDRON_UNIT_SAMPLE_04 to integrate on the unit tetrahedron.
• tetrahedron_monte_carlo_test05.m, uses TETRAHEDRON_UNIT_SAMPLE_01 to integrate on a general tetrahedron.
• tetrahedron_monte_carlo_test06.m, uses TETRAHEDRON_UNIT_SAMPLE_02 to integrate on a general tetrahedron.
• tetrahedron_monte_carlo_test07.m, uses TETRAHEDRON_UNIT_SAMPLE_03 to integrate on a general tetrahedron.
• tetrahedron_monte_carlo_test08.m, uses TETRAHEDRON_UNIT_SAMPLE_04 to integrate on a general tetrahedron.
• tetrahedron_monte_carlo_test_output.txt, the output file.

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