polynomials_on_simplices.polynomial.code_generation.generate_monomial_polynomial_functions module

Functionality for generating Python code used to evaluate monomial polynomials.

generate_function_eval_specific_scalar_valued(m, r, a, prettify_coefficients=False)

Generate code for evaluating a specific scalar valued degree r polynomial on an m-dimensional domain, expressed in the monomial basis.

\[ \begin{align}\begin{aligned}p : \mathbb{R}^m \to \mathbb{R},\\p_{\nu, r}(x)=\sum_{i = 0}^{\dim(\mathcal{P}_r(\mathbb{R}^m)) - 1} a_{\nu_i} x^{\nu_i},\end{aligned}\end{align} \]

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).

Parameters:

• m (int) – Dimension of the domain.
• r (int) – Degree of the polynomial space.
• a (Iterable[float]) – Coefficients for the polynomial in the monomial basis for \(\mathcal{P}_r (\mathbb{R}^m)\). \(\text{a}[i] = a_{\nu_i}\), 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).
• prettify_coefficients (bool) – Whether or not coefficients in the a array should be prettified in the generated code (e.g. converting 0.25 -> 1 / 4).

Returns:

Python code for evaluating the monomial base polynomial as specified by m, r and a.

Return type:

str

generate_function_eval_specific_vector_valued(m, r, a)

Generate code for evaluating a specific vector valued degree r polynomial on an m-dimensional domain, expressed in the monomial basis.

\[ \begin{align}\begin{aligned}p : \mathbb{R}^m \to \mathbb{R}^n, n > 1,\\p_{\nu, r}(x)=\sum_{i = 0}^{\dim(\mathcal{P}_r(\mathbb{R}^m)) - 1} a_{\nu_i} x^{\nu_i},\end{aligned}\end{align} \]

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).

Parameters:

• m (int) – Dimension of the domain.
• r (int) – Degree of the polynomial space.
• a (Iterable[n-dimensional vector]) – Coefficients for the polynomial in the monomial basis for \(\mathcal{P}_r (\mathbb{R}^m)\). \(\text{a}[i] = a_{\nu_i}\), 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).

Returns:

Python code for evaluating the monomial base polynomial as specified by m, r and a.

Return type:

str

generate_function_general(m, r)

Generate code for evaluating a general degree r monomial polynomial on an m-dimensional domain.

\[p(x) = \sum_{i = 0}^{\dim(\mathcal{P}_r(\mathbb{R}^m)) - 1} a_{\nu_i} x^{\nu},\]

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).

Parameters:

• m (int) – Dimension of the domain of the polynomial.
• r (int) – Degree of the polynomial space.

Returns:

Python code for evaluating the polynomial

Return type:

str

generate_function_specific(m, r, a)

Generate code for evaluating the degree r monomial polynomial on an m-dimensional domain with given basis coefficients a.

\[p(x) = \sum_{i = 0}^{\dim(\mathcal{P}_r(\mathbb{R}^m)) - 1} a_{\nu_i} x^{\nu},\]

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).

Parameters:

• m (int) – Dimension of the domain of the polynomial.
• r (int) – Degree of the polynomial space.
• a (Union[Iterable[float], Iterable[n-dimensional vector]]) – Coefficients for the polynomial in the monomial basis for \(\mathcal{P}_r (\mathbb{R}^m)\). \(\text{a}[i] = a_{\nu_i}\), 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).

Returns:

Python code for evaluating the polynomial

Return type:

str

generate_monomial_basis(m, r)

Generate code for evaluating all monomial base polynomials for the space \(\mathcal{P}_r(\mathbb{R}^m)\).

Parameters:

• m (int) – Dimension of the domain.
• r (int) – Degree of the polynomial space.

Returns:

List of codes for evaluating each of the base polynomials.

Return type:

List[str]

generate_monomial_basis_fn(nu)

Generate code for evaluating a monomial basis polynomial on an m-dimensional domain, \(p_{\nu} (x) = x^{\nu}\), where m is equal to the length of nu.

Parameters:nu (int or MultiIndex or Tuple[int, …]) – Multi-index indicating which monomial basis polynomial code should be generated for. Returns:Python code for evaluating the monomial base polynomial as specified by nu. Return type:str

Examples

>>> generate_monomial_basis_fn(0)
'1'
>>> generate_monomial_basis_fn(1)
'x'
>>> generate_monomial_basis_fn((1, 1))
'x[0] * x[1]'