Welcome to pysimplicialcubature’s documentation!

This package is a port of a part of the R package SimplicalCubature, written by John P. Nolan, and which contains R translations of some Matlab and Fortran code written by Alan Genz. In addition it provides a function for the exact computation of the integral of a polynomial over a simplex.

Indices and tables

Members functions

pysimplicialcubature.simplicialcubature.integrateOnSimplex(f, S, dim=1, maxEvals=10000, absError=0.0, tol=1e-05, rule=3, info=False)

Integration on a simplex.

Parameters:

  • f (function) – The function to be integrated.

  • S (array-like) – Simplex or simplices; a n-dimensional simplex is given as n+1 vectors of length n, the vertices.

  • dim (integer) – The dimension of the values of f.

  • maxEvals (integer) – Maximum number of calls to f.

  • absError (number) – Desired absolute error.

  • tol (number) – Desired relative error.

  • rule (integer) – Integer between 1 and 4 corresponding to the integration rule; a 2*rule+1 degree rule will be applied.

  • info (Boolean) – Whether to return more info.

Returns:

The value of the integral is in the field “integral” of the returned value.

Return type:

dictionary

Examples

>>> from pysimplicialcubature.simplicialcubature import integrateOnSimplex
>>> from math import exp
>>> # simplex vertices
>>> v1 = [0.0, 0.0, 0.0]
>>> v2 = [1.0, 1.0, 1.0]
>>> v3 = [0.0, 1.0, 1.0]
>>> v4 = [0.0, 0.0, 1.0]
>>> # simplex
>>> S = [v1, v2, v3, v4]
>>> # function to integrate
>>> f = lambda x : exp(x[0] + x[1] + x[2])
>>> # integral of f on S
>>> I_f = integrateOnSimplex(f, S)
>>> I_f["integral"]
pysimplicialcubature.simplicialcubature.integratePolynomialOnSimplex(P, S)

Integration of a polynomial on a simplex.

Parameters:

  • P (polynomial) – The polynomial to be integrated.

  • S (array-like) – Simplex; a n-dimensional simplex is given as n+1 vectors of length n, the vertices.

Returns:

The value of the integral of the polynomial over the simplex.

Return type:

number

Examples

>>> from pysimplicialcubature.simplicialcubature import integratePolynomialOnSimplex
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> # simplex vertices
>>> v1 = [0.0, 0.0, 0.0]
>>> v2 = [1.0, 1.0, 1.0]
>>> v3 = [0.0, 1.0, 1.0]
>>> v4 = [0.0, 0.0, 1.0]
>>> # simplex
>>> S = [v1, v2, v3, v4]
>>> # polynomial to integrate
>>> P = Poly(x**4 + y + 2*x*y**2 - 3*z, x, y, z, domain = "RR")
>>> # integral of P on S
>>> integratePolynomialOnSimplex(P, S)

References

    1. 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).