polynomials_on_simplices.polynomial.polynomials_base module

Abstract base class and basic functionality for polynomials.

class PolynomialBase(coeff, r=None, m=1)

Bases: abc.ABC

Abstract base class for a polynomial. The space of polynomials \(\mathcal{P} = \mathcal{P}(\mathbb{R}^m, \mathbb{R}^n)\) is defined as

\[\mathcal{P} = \{ p : \mathbb{R}^m \to \mathbb{R}^n | p(x) = \sum_{\nu} a_{\nu} x^{\nu}, \nu \in \mathbb{N}_0^m, a_{\nu} \in \mathbb{R}^n \}.\]

The domain dimension m and the target dimension n of the polynomial is given by the domain_dimension() and target_dimension() functions respectively.

For a computable polynomial we must have \(a_{\nu} = 0\) for all but finitely many \(\nu\), and then the degree of \(p\) is defined as \(r = \deg{p} = \max_{\nu : a_{\nu} \neq 0} |\nu|\). The degree is given by the degree() method.

There are many common bases for the space of polynomials. The basis used for a specific polynomial is given by the basis() method.

This class also defines the basic algebraic and differentiable structures of the space of polynomials.

Ring structure:

Addition: \(+ : \mathcal{P} \times \mathcal{P} \to \mathcal{P}, (p_1 + p_2)(x) = p_1(x) + p_2(x)\).

Multiplication: \(\cdot : \mathcal{P} \times \mathcal{P} \to \mathcal{P}, (p_1 \cdot p_2)(x) = p_1(x) \cdot p_2(x)\).

Vector space structure:

Scalar multiplication: \(\cdot : \mathbb{R} \times \mathcal{P} \to \mathcal{P}, (s \cdot p)(x) = s \cdot p(x)\).

Differentiable structure:

i:th partial derivative: \(\partial_i : \mathcal{P} \to \mathcal{P}, (\partial_i p)(x) = \frac{\partial p(x)}{\partial x^i}\).

Parameters:

• coeff – Coefficients for the polynomial in the chosen basis for \(\mathcal{P}_r (\mathbb{R}^m)\) (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 (\(n \geq 2\)).
• r (int) – Degree of the polynomial space. Optional, will be inferred from the number of polynomial coefficients if not specified.
• m (int) – Dimension of the domain of the polynomial.

basis()

Get basis for the space \(\mathcal{P}_r (\mathbb{R}^m)\) used to express this polynomial.

Returns:Unique identifier for the basis used. Return type:str

code_str(fn_name)

Generate a function code string for evaluating this polynomial.

Parameters:fn_name (str) – Name for the function in the generated code. Returns:Code string for evaluating this polynomial. Return type:str

degree()

Get degree of the polynomial.

Returns:Polynomial degree. Return type:int

degree_elevate(s)

Express the polynomial using a higher degree basis.

Let \(p(x) = \sum_{\substack{\nu \in \mathbb{N}_0^m \\ |\nu| \leq r}} a_{\nu} b_{\nu, r}(x)\) be this polynomial, where \(\{ b_{\nu, r} \}_{\substack{\nu \in \mathbb{N}_0^m \\ |\nu| \leq r}}\) is the chosen basis for \(\mathcal{P}_r (\mathbb{R}^m)\). Let \(\{ b_{\nu, s} \}_{\substack{\nu \in \mathbb{N}_0^m \\ |\nu| \leq s}}, s \geq r\) be the corresponding basis for \(\mathcal{P}_s (\mathbb{R}^m)\). Then this function returns a polynomial \(q(x)\)

\[\begin{split}q(x) = \sum_{\substack{\nu \in \mathbb{N}_0^m \\ |\nu| \leq s}} \tilde{a}_{\nu} b_{\nu, s}(x),\end{split}\]

such that \(p(x) = q(x) \, \forall x \in \mathbb{R}^m\).

Parameters:s (int) – New degree for the polynomial basis the polynomial should be expressed in. Returns:Elevation of this polynomial to the higher degree basis.

domain_dimension()

Get dimension of the polynomial domain.

Returns:Dimension of the domain of the polynomial. Return type:int

latex_str()

Generate a Latex string for this polynomial.

Returns:Latex string for this polynomial. Return type:str

multiply_with_constant(c)

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__

partial_derivative(i=0)

Compute the i:th partial derivative of the polynomial.

Parameters:i (int) – Index of partial derivative. Returns:i:th partial derivative of this polynomial.

target_dimension()

Get dimension of the polynomial target.

Returns:Dimension of the target of the polynomial. Return type:int

to_monomial_basis()

Compute the monomial representation of this polynomial.

Returns:This polynomial expressed in the monomial basis.

class PolynomialComponentsIterator(p)

Bases: object

Iterator for iterating over the components of a vector valued polynomial.

Parameters:p (Instance of PolynomialBase) – Vector valued polynomial.

get_degree_from_dimension(dim, n)

Get the maximum polynomial degree r for a polynomial in the space of all polynomials on an n-dimensional domain with given dimension. In a sense this is the inverse of the get_dimension() function.

Parameters:

• dim (int) – Dimension of the polynomial space.
• n (int) – Dimension of the domain.

Returns:

Maximum polynomial degree.

Return type:

int

get_dimension(r, n)

Get the dimension of the space of polynomials of degree <= r on an n-dimensional domain.

Parameters:

• r (int) – Maximum polynomial degree.
• n (int) – Dimension of the domain.

Returns:

Dimension of the space of polynomials (number of basis functions needed to span the space).

Return type:

int

polynomials_equal(p1, p2, r, m, rel_tol=1e-09, abs_tol=1e-07)

Check if two polynomials p1 and p2 are approximately equal.

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 \(\{ x_i \}_{i = 0}^{d - 1}\) from the m-dimensional unit cube, 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.
• m (int) – Dimension of the domain of the polynomials.
• 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