polynomials_on_simplices.polynomial.code_generation.generate_lagrange_polynomial_functions_simplex module

Functionality for generating Python code used to evaluate Lagrange polynomials.

generate_function_eval_general_scalar_valued(m, r)[source]

Generate code for evaluating a general scalar valued degree r polynomial on the m-dimensional unit simplex (\(\Delta_c^m\)), expressed in the Lagrange basis.

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

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).

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

Python code for evaluating a general degree r Lagrange polynomial on an m-dimensional domain.
Return type:

str
generate_function_eval_specific_scalar_valued(m, r, a, prettify_coefficients=False)[source]

Generate code for evaluating a specific scalar valued degree r polynomial on the m-dimensional unit simplex (\(\Delta_c^m\)), expressed in the Lagrange basis.

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

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 Lagrange 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 Lagrange polynomial as specified by m, r and a.
Return type:

str
generate_function_eval_specific_vector_valued(m, r, a)[source]

Generate code for evaluating a specific vector valued degree r polynomial on the m-dimensional unit simplex (\(\Delta_c^m\)), expressed in the Lagrange basis.

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

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 Lagrange 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 Lagrange base polynomial as specified by m, r and a.
Return type:

str
generate_function_general(m, r)[source]

Generate code for evaluating a general degree r Lagrange polynomial on the m-dimensional unit simplex.

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

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)[source]

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

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

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 Lagrange basis for \(\mathcal{P}_r (\Delta_c^m)\). \(\text{a}[i] = a_{\nu_i}\), 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_lagrange_basis(m, r)[source]

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

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_lagrange_basis_fn(nu, r)[source]

Generate code for evaluating a Lagrange basis polynomial on the m-dimensional unit simplex, where m is equal to the length of nu.

Parameters:
  • nu (int or MultiIndex or Tuple[int, …]) – Multi-index indicating which Lagrange basis polynomial code should be generated for.
  • r (int) – Degree of polynomial.
Returns:

Python code for evaluating the Lagrange base polynomial as specified by nu.
Return type:

str

Examples

>>> generate_lagrange_basis_fn(1, 1)
'x'
>>> generate_lagrange_basis_fn((1, 1), 2)
'4 * x[0] * x[1]'