SimplicialCubature

This package is a port of the R package SimplicialCubature, written by John P. Nolan, and which contains R translations of some Matlab and Fortran code written by Alan Genz.

A simplex is a triangle in dimension 2, a tetrahedron in dimension 3. This package provides two main functions: integrateOnSimplex, to integrate an arbitrary function on a simplex, and integratePolynomialOnSimplex, to get the exact value of the integral of a multivariate polynomial on a simplex.

A n-dimensional simplex must be given by n+1 vectors of length n, which represent the simplex vertices. For example, the 3-dimensional unit simplex is encoded as follows:

S = [[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]]

Or you can get it by running CanonicalSimplex(3).

Suppose you want to integrate the function $$f(x, y ,z) = x + yz$$ on the unit simplex. To use integrateOnSimplex, you have to define $f$ as a function of a 3-dimensional vector:

function f(x)
  return x[1] + x[2]*x[3]
end

using SimplicialCubature
I = integrateOnSimplex(f, S)

Then the value of the integral is given in I.integral.

Since the function $f$ of this example is polynomial, you can use integratePolynomialOnSimplex:

using SimplicialCubature
using TypedPolynomials

@polyvar x y z
P = x + y*z
integratePolynomialOnSimplex(P, S)

Be careful if your polynomial does not involve one of the variables. For example if $P(x, y, z) = x + y$, you have to encode as a polynomial depending on z: type P = x + y + 0*z.

In addition, on this example where the vertex coordinates of S and the coefficients of P are integer numbers, there is a more clever way to proceed: while integratePolynomialOnSimplex implements an exact procedure, it is not free of (small) numerical errors, but the returned value in this situation will be really exact if you use a polynomial with rational coefficients:

@polyvar x y z
P = 1//1*x + y*z
integratePolynomialOnSimplex(P, S)

References

A. Genz and R. Cools. An adaptive numerical cubature algorithm for simplices. ACM Trans. Math. Software 29, 297-308 (2003).
Jean B. Lasserre. Simple formula for the integration of polynomials on a simplex. BIT Numerical Mathematics 61, 523-533 (2021).