:py:mod:`symfem.elements.bernstein`
===================================

.. py:module:: symfem.elements.bernstein

.. autoapi-nested-parse::

   Bernstein elements on simplices.

   This element's definition appears in
   https://doi.org/10.1007/s00211-010-0327-2 (Kirby, 2011) and
   https://doi.org/10.1137/11082539X (Ainsworth, Andriamaro, Davydov, 2011)



Module Contents
---------------

Classes
~~~~~~~

.. autoapisummary::

   symfem.elements.bernstein.BernsteinFunctional
   symfem.elements.bernstein.Bernstein



Functions
~~~~~~~~~

.. autoapisummary::

   symfem.elements.bernstein.single_choose
   symfem.elements.bernstein.choose
   symfem.elements.bernstein.bernstein_polynomials



.. py:function:: single_choose(n: int, k: int) -> sympy.core.expr.Expr

   Calculate choose function of a set of powers.

   :param n: Number of items
   :param k: Number to select

   :returns: Number of ways to pick k items from n items (ie n choose k)


.. py:function:: choose(n: int, powers: List[int]) -> sympy.core.expr.Expr

   Calculate choose function of a set of powers.

   :param n: Number of items
   :param k: Numbers to select

   :returns: A multichoose function


.. py:function:: bernstein_polynomials(n: int, d: int, vars: symfem.symbols.AxisVariablesNotSingle = x) -> List[sympy.core.expr.Expr]

   Return a list of Bernstein polynomials.

   :param n: The polynomial order
   :param d: The topological dimension
   :param vars: The variables to use

   :returns: Bernstein polynomials


.. py:class:: BernsteinFunctional(reference: symfem.references.Reference, integral_domain: symfem.references.Reference, index: int, degree: int, entity: Tuple[int, int])


   Bases: :py:obj:`symfem.functionals.BaseFunctional`

   Functional for a Bernstein element.

   .. py:method:: dof_point() -> symfem.geometry.PointType

      Get the location of the DOF in the cell.

      :returns: Location of the DOF


   .. py:method:: _eval_symbolic(function: symfem.functions.AnyFunction) -> symfem.functions.AnyFunction

      Apply the functional to a function.

      :param function: The function

      :returns: Evaluation of the functional


   .. py:method:: get_tex() -> Tuple[str, List[str]]

      Get a representation of the functional as TeX, and list of terms involved.

      :returns: TeX representation



.. py:class:: Bernstein(reference: symfem.references.Reference, order: int)


   Bases: :py:obj:`symfem.finite_element.CiarletElement`

   Bernstein finite element.

   .. py:attribute:: names
      :value: ['Bernstein', 'Bernstein-Bezier']

      

   .. py:attribute:: references
      :value: ['interval', 'triangle', 'tetrahedron']

      

   .. py:attribute:: min_order
      :value: 0

      

   .. py:attribute:: continuity
      :value: 'C0'

      

   .. py:attribute:: last_updated
      :value: '2023.05'