polynomials_on_simplices.polynomial.polynomials_simplex_base module

Abstract base class and basic functionality for polynomials defined on a simplex.

class PolynomialSimplexBase(coeff, vertices, r=None)[source]

Bases: polynomials_on_simplices.polynomial.polynomials_base.PolynomialBase, abc.ABC

Abstract base class for a polynomial defined on an m-dimensional simplex T.

Parameters:
  • coeff – Coefficients for the polynomial in the chosen basis for \(\mathcal{P}_r (T)\) (see basis()). If p is expressed in the chosen basis \(\{ b_{\nu, r} \}\) as \(p(x) = \sum_{\nu} a_{\nu} b_{\nu, r}(x)\) then \(\text{coeff}[i] = a_{\nu(i)}\), where \(\nu(i)\) is the i:th multi-index in the sequence of all multi-indices of dimension m with norm \(\leq r\) (see polynomials_on_simplices.algebra.multiindex.generate() function). Array of scalars for a scalar valued polynomial (n = 1) and array of n-dimensional vectors for a vector valued polynomial (\(n \geq 2\)).
  • vertices – Vertices of the simplex T ((m + 1) x m matrix where row i contains the i:th vertex of the simplex).
  • r (int) – Degree of the polynomial space. Optional, will be inferred from the number of polynomial coefficients if not specified.
multiply_with_constant(c)[source]

Multiplication of this polynomial with a constant scalar or a vector (only for a scalar valued polynomial), self * c.

Parameters:c (Union[float, Numpy array]) – Scalar or vector we should multiply this polynomial with.
Returns:Product of this polynomial with the constant.
Return type:Instance of self.__class__
simplex_vertices()[source]

Get the vertices of the simplex T on which this polynomial is defined.

Returns:Vertices of the simplex T ((m + 1) x m matrix where row i contains the i:th vertex of the simplex).
polynomials_equal_on_simplex(p1, p2, r, vertices, rel_tol=1e-09, abs_tol=1e-07)[source]

Check if two polynomials p1 and p2 are approximately equal by comparing their values on a given n-dimensional simplex T.

For scalar valued polynomials, the two polynomials are considered equal if

math.isclose(p1(xi), p2(xi), rel_tol=rel_tol, abs_tol=abs_tol)

is true for a set of random points \(\{ x_i \}_{i = 0}^{d - 1}\) in the given n-dimensional simplex T, where \(d\) is the dimension of the polynomial space p1 and p2 belongs to (as given by the get_dimension() function).

For vector valued polynomials the same check is done component wise.

Parameters:
  • p1 (Callable p1(x)) – First polynomial.
  • p2 (Callable p2(x)) – Second polynomial.
  • r (int) – Degree of the polynomials.
  • vertices – Vertices of the simplex T ((n + 1) x n matrix where row i contains the i:th vertex of the simplex).
  • rel_tol (float) – Tolerance for the relative error. See math.isclose for details.
  • abs_tol (float) – Tolerance for the absolute error. See math.isclose for details.
Returns:

Whether or not the two polynomials are approximately equal.
Return type:

bool