simplex_integrals

simplex_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

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

        0 <= X(1:M)
        sum ( 1 <= I <= M ) X(I) <= 1
      

The integrands are all of the form

        f(x) = product ( 1 <= I <= M) X(I)^E(I)
      

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

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

Related Data and Programs:

python_integrals, a Python code which returns the exact value of the integral of any monomial over the surface or interior of some geometric object, including a line, quadrilateral, box, circle, disk, sphere, ball and others.

simplex_grid, a Python code which generates a regular grid of points over the interior of an arbitrary simplex in m dimensions.

simplex_monte_carlo, a Python code which uses the monte carlo method to estimate integrals over the interior of the unit simplex in m dimensions.

Reference:

• Jean Lasserre, Konstantin Avrachenkov,
  The multidimensional version of the integral from A to B of X to the P,
  American Mathematics Monthly,
  Volume 108, Number 2, 2001, pages 151`-154.

Source Code:

• simplex_integrals.py, the source code.
• simplex_integrals.sh, runs all the tests.
• simplex_integrals.txt, the output file.