symfem.polynomials.legendre

Orthogonal (Legendre) polynomials.

Module Contents

Functions

_jrc(→ Tuple[sympy.core.expr.Expr, ...)	Get the Jacobi recurrence relation coefficients.
orthogonal_basis_interval(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_triangle(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_quadrilateral(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_tetrahedron(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_hexahedron(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_prism(...)	Create a basis of orthogonal polynomials.
orthogonal_basis_pyramid(...)	Create a basis of orthogonal polynomials.
orthogonal_basis(...)	Create a basis of orthogonal polynomials.
orthonormal_basis(...)	Create a basis of orthonormal polynomials.

symfem.polynomials.legendre._jrc(a, n) → Tuple[sympy.core.expr.Expr, sympy.core.expr.Expr, sympy.core.expr.Expr]

Get the Jacobi recurrence relation coefficients.

Parameters:

• a – The parameter a

• n – The parameter n

Returns:

The Jacobi coefficients

symfem.polynomials.legendre.orthogonal_basis_interval(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = [x[0]]) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.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.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.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.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthogonal_basis_tetrahedron(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthogonal_basis_hexahedron(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthogonal_basis_prism(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthogonal_basis_pyramid(order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle = x) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthogonal_basis(cell: str, order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle | None = None) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthogonal polynomials.

Parameters:

• cell – The cell type

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthogonal polynomials

symfem.polynomials.legendre.orthonormal_basis(cell: str, order: int, derivs: int, variables: symfem.symbols.AxisVariablesNotSingle | None = None) → List[List[symfem.functions.ScalarFunction]]

Create a basis of orthonormal polynomials.

Parameters:

• cell – The cell type

• order – The maximum polynomial degree

• derivs – The number of derivatives to include

• variables – The variables to use

Returns:

A set of orthonormal polynomials