TRIANGLE_MONTE_CARLO
Monte Carlo Integral Estimates over a Triangle

TRIANGLE_MONTE_CARLO is a MATLAB library which estimates the integral of a function over a triangle using the Monte Carlo method.

The library makes it relatively easy to compare different methods of producing sample points in the triangle, and to vary the triangle 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:

TRIANGLE_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;

DUNAVANT, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a MATLAB library which defines Fekete rules for interpolation or quadrature on a triangle.

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.

LYNESS_RULE, a MATLAB library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes Closed quadrature rules on a triangle.

NCO_TRIANGLE, a MATLAB library which defines Newton-Cotes Open quadrature rules on a triangle.

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.

TEST_TRI_INT, a MATLAB library which defines test functions for quadrature over the triangle.

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

WANDZURA, a MATLAB library which defines Wandzura rules for quadrature on 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_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 T3 reference points to physical points.
• s_len_trim.m returns the length of a string to the last nonblank.
• timestamp.m prints the current YMDHMS date as a time stamp.
• triangle_area.m computes the area of a triangle.
• triangle_integrand_01.m evaluates 1 integrand function.
• triangle_integrand_02.m evaluates 2 integrand functions.
• triangle_integrand_03.m evaluates 3 integrand functions.
• triangle_integrand_04.m evaluates 4 integrand functions.
• triangle_integrand_05.m evaluates 5 integrand functions.
• triangle_monte_carlo.m applies the Monte Carlo rule to integrate a function.
• triangle_unit_sample_01.m selects points from the unit triangle.
• triangle_unit_sample_02.m selects points from the unit triangle.
• triangle_unit_sample_03.m selects points from the unit triangle.
• triangle_unit_sample_04.m selects points from the unit triangle.

Examples and Tests:

• triangle_monte_carlo_test.m, a sample calling program.
• triangle_monte_carlo_test01.m, uses TRIANGLE_UNIT_SAMPLE_01 to integrate on the unit triangle.
• triangle_monte_carlo_test02.m, uses TRIANGLE_UNIT_SAMPLE_02 to integrate on the unit triangle.
• triangle_monte_carlo_test03.m, uses TRIANGLE_UNIT_SAMPLE_03 to integrate on the unit triangle.
• triangle_monte_carlo_test04.m, uses TRIANGLE_UNIT_SAMPLE_04 to integrate on the unit triangle.
• triangle_monte_carlo_test05.m, uses TRIANGLE_UNIT_SAMPLE_01 to integrate on a general triangle.
• triangle_monte_carlo_test06.m, uses TRIANGLE_UNIT_SAMPLE_02 to integrate on a general triangle.
• triangle_monte_carlo_test07.m, uses TRIANGLE_UNIT_SAMPLE_03 to integrate on a general triangle.
• triangle_monte_carlo_test08.m, uses TRIANGLE_UNIT_SAMPLE_04 to integrate on a general triangle.
• triangle_monte_carlo_test_output.txt, the output file.

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