Welcome to PyPolyhedralCubature’s documentation!

This package allows to evaluate the integral of a multivariate function over a convex polytope, given by the linear combinations of the variables defining the bounds of the integrals.

See the README file for usage examples.

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Members functions

polyhedralcubature.getAb(inequalities, symbols)

Get the matrix-vector representation of a set of linear inequalities.

Parameters:

• inequalities (list) – list of symbolic inequalities

• symbols (list) – list of symbols

Returns:

• matrix – The matrix of the coefficients of the inequalities.

• vector – The vector made of the bounds of the inequalities.

Examples

>>> from pypolyhedralcubature.polyhedralcubature import getAb
>>> from sympy.abc import x, y, z
>>> # linear inequalities
>>> i1 = (x >= -5) & (x <= 4)
>>> i2 = (y >= -5) & (y <= 3 - x)
>>> i3 = (z >= -10) & (z <= 6 - x - y)
>>> # get matrix-vector representation of these inequalities
>>> A, b = getAb([i1, i2, i3], [x, y, z])

polyhedralcubature.integrateOnPolytope(f, A, b, dim=1, maxEvals=10000, absError=0.0, tol=1e-05, rule=3)

Integration a function over a convex polytope.

Parameters:

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

• A (array-like) – matrix of the coefficients of the linear inequalities (see README for an example)

• b (vector-like) – vector of the upper bounds of the linear inequalities

• 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.

Returns:

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

Return type:

dictionary

Examples

>>> from pypolyhedralcubature.polyhedralcubature import *
>>> from sympy.abc import x, y, z
>>> # integral bounds
>>> i1 = (x >= -5) & (x <= 4)
>>> i2 = (y >= -5) & (y <= 3 - x)
>>> i3 = (z >= -10) & (z <= 6 - x - y)
>>> # get matrix-vector representation of these inequalities
>>> A, b = getAb([i1, i2, i3], [x, y, z])
>>> # function to integrate
>>> f = lambda x, y, z : x*(x+1) - y*z**2
>>> # integral of f on the polytope defined by the bounds
>>> g = lambda v : f(v[0], v[1], v[2])
>>> I_f = integrateOnPolytope(g, A, b)
>>> I_f["integral"]

polyhedralcubature.integratePolynomialOnPolytope(P, A, b)

Integration a polynomial over a convex polytope.

Parameters:

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

• A (array-like) – matrix of the coefficients of the linear inequalities (see README for an example)

• b (vector-like) – vector of the upper bounds of the linear inequalities

Returns:

The exact value of the integral of P over the polytope defined by A and b.

Return type:

number

Examples

>>> from pypolyhedralcubature.polyhedralcubature import *
>>> from sympy import Poly
>>> from sympy.abc import x, y, z
>>> # integral bounds
>>> i1 = (x >= -5) & (x <= 4)
>>> i2 = (y >= -5) & (y <= 3 - x)
>>> i3 = (z >= -10) & (z <= 6 - x - y)
>>> # get matrix-vector representation of these inequalities
>>> A, b = getAb([i1, i2, i3], [x, y, z])
>>> # polynomial to integrate
>>> P = Poly(x*(x+1) - y*z**2, domain = "RR")
>>> # integral of P over the polytope defined by the bounds
>>> integratePolynomialOnPolytope(P, A, b)