TRIANGLE01_MONTE_CARLO
Monte Carlo Integral Estimates over the Unit Triangle in 2D

TRIANGLE01_MONTE_CARLO is a MATLAB library which uses the Monte Carlo method to estimate the integral of a function F(X,Y) over the interior of the unit triangle in 2D.

The interior of the unit triangle in 2D is defined by the constraints:

        0 <= X
        0 <= Y
             X + Y <= 1
      

        F(X,Y) = X^E(1) * Y^E(2)
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TRIANGLE01_MONTE_CARLO is available in a Python version.

Related Data and Programs:

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

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

DISK01_QUARTER_MONTE_CARLO, a Python 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 Python 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 Python 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 Python library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit ball in M dimensions;

HYPERCUBE_MONTE_CARLO, a Python 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 Python 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 Python library which applies a Monte Carlo method to estimate the integral of a function over the length of the unit line segment in 1D;

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

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

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

TRIANGLE_INTEGRALS, a Python library which returns the exact value of the integral of any polynomial over the interior of an arbitrary triangle in 2D.

TRIANGLE01_INTEGRALS, a Python library which returns the exact value of the integral of any polynomial over the interior of the unit triangle in 2D.

WEDGE_MONTE_CARLO, a Python 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:

• monomial_value.py, evaluates a monomial.
• r8mat_print.py prints an R8MAT.
• r8mat_print_some.py prints some of an R8MAT.
• r8mat_print.py prints an R8MAT, transposed.
• r8mat_print_some.py prints some of an R8MAT, transposed.
• r8vec_uniform_01.py returns a unit pseudorandom R8VEC.
• timestamp.py prints the current YMDHMS date as a time stamp.
• triangle01_area.py computes the area of the unit triangle in 2D.
• triangle01_monomial_integral.py, monomial integrals in the interior of the unit triangle in 2D.
• triangle01_monte_carlo.py, uses the Monte Carlo method to estimate the integral of a function over the unit triangle in 2D.
• triangle01_sample.py samples points from the interior of the unit triangle in 2D.

Examples and Tests:

• triangle01_monte_carlo_tests.py, a sample calling program.
• triangle01_monte_carlo_tests.txt, the output file.

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