tetrahedron_monte_carlo

tetrahedron_monte_carlo, an Octave code 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 MIT license

Languages:

tetrahedron_monte_carlo is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

tetrahedron_monte_carlo_test

annulus_monte_carlo an Octave code which uses the Monte Carlo method to estimate the integral of a function over the interior of a circular annulus in 2d.

ball_monte_carlo, an Octave code which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit ball in 3d;

circle_monte_carlo, an Octave code 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, an Octave code 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, an Octave code which uses the Monte Carlo method to estimate integrals over the interior of the general disk in 2D.

disk01_monte_carlo, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2d;

disk01_quarter_monte_carlo, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit quarter disk in 2D;

ellipse_monte_carlo an Octave code which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipse in 2D.

ellipsoid_monte_carlo an Octave code which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipsoid in M dimensions.

hyperball_monte_carlo, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in m dimensions;

hypercube_monte_carlo, an Octave code 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, an Octave code 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, an Octave code which uses the Monte Carlo method to estimate integrals over the length of the unit line in 1D.

polygon_monte_carlo, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

pyramid_monte_carlo, an Octave code which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;

simplex_monte_carlo, an Octave code which uses the Monte Carlo method to estimate integrals over the interior of the unit simplex in m dimensions.

sphere_monte_carlo, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the surface of the unit sphere in 3d;

sphere_triangle_monte_carlo, an Octave code 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, an Octave code which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2D.

tetrahedron_integrals, an Octave code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3d.

tetrahedron01_monte_carlo, an Octave code which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

triangle_monte_carlo, an Octave code which uses the Monte Carlo method to estimate integrals over the interior of a triangle.

triangle01_monte_carlo, an Octave code which uses the Monte Carlo method to estimate integrals over the interior of the unit triangle in 2D.

wedge_monte_carlo, an Octave code 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:

• 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.