# simplex_integrals

simplex_integrals, a MATLAB 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)
```
where the exponents are nonnegative integers.

### 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 MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit ball in 3d.

circle_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit circle in 2d.

cube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit cube in 3d.

disk01_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit disk in 2d.

hexagon_integrals, a MATLAB code which returns the exact value of the integral of a monomial over the interior of a hexagon in 2d.

hyperball_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hyperball in m dimensions.

hypercube_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit hypercube in m dimensions.

hypersphere_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in m dimensions.

line_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the length of the unit line in 1d.

polygon_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of a polygon in 2d.

pyramid_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3d.

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

simplex_monte_carlo, a MATLAB code 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 MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3d.

square_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit square in 2d.

tetrahedron_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3d.

triangle_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2d.

wedge_integrals, a MATLAB code 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:

Last revised on 15 March 2019.