TRIANGLE_MONTE_CARLO
Monte Carlo Integral Estimates over a Triangle

TRIANGLE_MONTE_CARLO is a C library which estimates the integral of a function over a general 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 C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

ANNULUS_MONTE_CARLO a C library 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, a C 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 C library 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, a C library 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, a C library which uses the Monte Carlo method to estimate integrals over the interior of the general disk in 2D.

DISK01_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate integrals over the interior of the unit disk in 2D.

DISK01_QUARTER_MONTE_CARLO, a C library 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 a C library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipse in 2D.

ELLIPSOID_MONTE_CARLO a C library which uses the Monte Carlo method to estimate the value of integrals over the interior of an ellipsoid in M dimensions.

HYPERBALL_MONTE_CARLO, a C library 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, a C library 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, a C library 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, a C library which uses the Monte Carlo method to estimate integrals over the length of the unit line in 1D.

POLYGON_MONTE_CARLO, a C library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

PYRAMID_MONTE_CARLO, a C library which applies a Monte Carlo method to estimate integrals of a function over the interior of the unit pyramid in 3D;

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

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

SPHERE_TRIANGLE_MONTE_CARLO, a C library 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, a C library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit square in 2D.

TETRAHEDRON_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate integrals over the interior of a general tetrahedron.

TETRAHEDRON01_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

TRIANGLE_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

TRIANGLE01_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate integrals over the interior of the unit triangle in 2D.

WEDGE_MONTE_CARLO, a C library 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:

• triangle_monte_carlo.c, the source code.
• triangle_monte_carlo.h, the include file.
• triangle_monte_carlo.sh, commands to compile the source code.

Examples and Tests:

• triangle_monte_carlo_test.c, a sample calling program.
• triangle_monte_carlo_test.sh, commands to compile and run the sample program.
• triangle_monte_carlo_test.txt, the output file.

List of Routines:

• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8VEC_SUM returns the sum of an R8VEC.
• R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA computes the area of a triangle in 2D.
• TRIANGLE_INTEGRAND_01 evaluates 1 integrand function.
• TRIANGLE_INTEGRAND_02 evaluates 2 integrand functions.
• TRIANGLE_INTEGRAND_03 evaluates 3 integrand functions.
• TRIANGLE_INTEGRAND_04 evaluates 4 integrand functions.
• TRIANGLE_INTEGRAND_05 evaluates 5 integrand functions.
• TRIANGLE_MONTE_CARLO applies the Monte Carlo rule to integrate a function.
• TRIANGLE_UNIT_SAMPLE_01 selects points from the unit triangle.
• TRIANGLE_UNIT_SAMPLE_02 selects points from the unit triangle.
• TRIANGLE_UNIT_SAMPLE_03 selects points from the unit triangle.
• TRIANGLE_UNIT_SAMPLE_04 selects points from the unit triangle.

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