SIMPLEX_INTEGRALS
Integrals Inside the Unit Simplex in M Dimensions

SIMPLEX_INTEGRALS is a FORTRAN90 library 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

SIMPLEX_INTEGRALS 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:

BALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK01_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_GM_RULE, a FORTRAN90 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 FORTRAN90 library which generates a regular grid of points over the interior of an arbitrary simplex in M dimensions.

SIMPLEX_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

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.f90, the source code.

Examples and Tests:

• simplex_integrals_test.f90, a sample calling program.
• simplex_integrals_test.txt, the output file.

List of Routines:

• I4VEC_UNIFORM_AB returns a scaled pseudorandom I4VEC.
• MONOMIAL_VALUE evaluates a monomial.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• SIMPLEX01_MONOMIAL_INTEGRAL: integral in unit simplex in M dimensions.
• SIMPLEX01_SAMPLE samples the unit simplex in M dimensions.
• SIMPLEX01_VOLUME computes the volume of the unit simplex in M dimensions.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.