Monte Carlo Integral Estimates over the Unit Simplex in M Dimensions

**SIMPLEX_MONTE_CARLO**
is a MATLAB library which
uses the Monte Carlo method to estimate the integral of a function F(X)
over the interior of the unit simplex in M dimensions.

The interior of the unit simplex in M dimensions is defined by the constraints:

0 <= X(1:M) sum ( 1 <= I <= M ) X(I) <= 1The functions F(X) are monomials, having the form

F(X) = product ( 1 <= I <= M ) X(I)^E(I)where the exponents are nonnegative integers.

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

**SIMPLEX_MONTE_CARLO** is available in
a C version and
a C++ version and
a FORTRAN77 version and
a FORTRAN90 version and
a MATLAB version and
a Python version.

BALL_MONTE_CARLO, a MATLAB 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 MATLAB 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 MATLAB 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 MATLAB library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit disk in 2D;

DISK_QUARTER_MONTE_CARLO, a MATLAB 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 MATLAB 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 MATLAB 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 MATLAB library which applies a Monte Carlo method to estimate the integral of a function over the interior of the unit hyperball in M dimensions;

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

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

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

SIMPLEX_GM_RULE, a MATLAB library which defines Grundmann-Moeller quadrature rules over the interior of a triangle in 2D, a tetrahedron in 3D, or over the interior of the simplex in M dimensions.

SIMPLEX_GRID, a MATLAB library which generates a regular grid of points over the interior of an arbitrary simplex in M dimensions.

SIMPLEX_INTEGRALS, a MATLAB library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_MONTE_CARLO, a MATLAB library which uses the Monte Carlo method to estimate integrals over the surface of the unit sphere in 3D.

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

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

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

WEDGE_MONTE_CARLO, a MATLAB library which uses the Monte Carlo method to estimate integrals over the interior of the unit wedge in 3D.

- monomial_value.m, evaluates a monomial.
- r8vec_uniform_01.m returns a unit pseudorandom R8VEC.
- simplexn01_monomial.m, monomial integrals in the interior of the unit simplex in M dimensions.
- simplex01_sample.m samples points from the interior of the unit simplex in M dimensions.
- simplex01_volume.m computes the volume of the unit simplex in M dimensions.
- timestamp.m prints the current YMDHMS date as a time stamp.

- simplex_monte_carlo_test.m, a sample calling program.
- simplex_monte_carlo_test01.m, uses SIMPLEX01_SAMPLE to estimate integrals over the interior of the unit simplex in 3D.
- simplex_monte_carlo_test02.m, uses SIMPLEX01_SAMPLE to estimate integrals over the interior of the unit simplex in 6D.
- simplex_monte_carlo_test_output.txt, the output file.

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