:py:mod:`symfem.polynomials.legendre`
=====================================

.. py:module:: symfem.polynomials.legendre

.. autoapi-nested-parse::

   Orthogonal (Legendre) polynomials.



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


Functions
~~~~~~~~~

.. autoapisummary::

   symfem.polynomials.legendre._jrc
   symfem.polynomials.legendre.orthogonal_basis_interval
   symfem.polynomials.legendre.orthogonal_basis_triangle
   symfem.polynomials.legendre.orthogonal_basis_quadrilateral
   symfem.polynomials.legendre.orthogonal_basis_tetrahedron
   symfem.polynomials.legendre.orthogonal_basis_hexahedron
   symfem.polynomials.legendre.orthogonal_basis_prism
   symfem.polynomials.legendre.orthogonal_basis_pyramid
   symfem.polynomials.legendre.orthogonal_basis
   symfem.polynomials.legendre.orthonormal_basis



.. py:function:: _jrc(a, n) -> Tuple[sympy.core.expr.Expr, sympy.core.expr.Expr, sympy.core.expr.Expr]

   Get the Jacobi recurrence relation coefficients.

   :param a: The parameter a
   :param n: The parameter n

   :returns: The Jacobi coefficients


.. py:function:: orthogonal_basis_interval(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = [x[0]]) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_triangle(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = [x[0], x[1]]) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_quadrilateral(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = [x[0], x[1]]) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_tetrahedron(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_hexahedron(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_prism(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis_pyramid(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthogonal_basis(cell: str, order: int, derivs: int, variables: Optional[symfem.symbols.AxisVariablesNotSingle] = None) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthogonal polynomials.

   :param cell: The cell type
   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthogonal polynomials


.. py:function:: orthonormal_basis(cell: str, order: int, derivs: int, variables: Optional[symfem.symbols.AxisVariablesNotSingle] = None) -> List[List[symfem.functions.ScalarFunction]]

   Create a basis of orthonormal polynomials.

   :param cell: The cell type
   :param order: The maximum polynomial degree
   :param derivs: The number of derivatives to include
   :param variables: The variables to use

   :returns: A set of orthonormal polynomials