puncturedfem.Polynomial

Polynomial in two variables, represented as a list of Monomials.

Attributes:

Name	Type	Description
monos	list[Monomial]	List of Monomials in the Polynomial.

__add__(other)

Define the addition operation self + other.

Parameters:

Name	Type	Description	Default
other	object	Object to add to self. Must be either a Polynomial or a scalar.	required

__call__(x, y)

Evaluate the Polynomial at the point (x, y).

Parameters:

Name	Type	Description	Default
x	FloatLike	x-coordinate of the point.	required
y	FloatLike	y-coordinate of the point.	required

Returns:

Type	Description
FloatLike	Value(s) of the Polynomial at the point(s) (x, y). If x and y are arrays, returns an array of values of the same shape.

__eq__(other)

Test equality between self and other.

Parameters:

Name	Type	Description	Default
other	object	Object to compare with self. Must be a Polynomial.	required

__iadd__(other)

Define the increment operation self += other.

Parameters:

Name	Type	Description	Default
other	object	Object to add to self. Must be either a Polynomial or a scalar.	required

__init__(coef_multidx_pairs=None)

Polynomial in two variables, represented as a list of Monomials.

Parameters:

Name	Type	Description	Default
coef_multidx_pairs	list[tuple[float, int, int]], optional	List of coefficient / multi-index pairs, by default None	None

__mul__(other)

Define the multiplication operator self * other.

Parameters:

Name	Type	Description	Default
other	object	Object to multiply with self. Must be either a Polynomial or a scalar.	required

__neg__()

Negate the Polynomial.

__pow__(exponent)

Define the exponentiation operator self ** exponent.

Parameters:

Name	Type	Description	Default
exponent	int	Nonnegative integer exponent.	required

__radd__(other)

Define the addition operator other + self.

Parameters:

Name	Type	Description	Default
other	object	Object to add to self. Must be either a Polynomial or a scalar.	required

__repr__()

Get a string representation of the Polynomial.

Returns:

Type	Description
str	String representation of the Polynomial.

__rmul__(other)

Define the multiplication operator other * self.

Parameters:

Name	Type	Description	Default
other	object	Object to multiply with self. Must be either a Polynomial or a scalar.	required

__rsub__(other)

Define the subtraction operator other - self.

Parameters:

Name	Type	Description	Default
other	object	Object to subtract self from. Must be either a Polynomial or a scalar.	required

__str__()

Get a string representation of the Polynomial.

Returns:

Type	Description
str	String representation of the Polynomial.

__sub__(other)

Define the subtraction operator self - other.

Parameters:

Name	Type	Description	Default
other	object	Object to subtract from self. Must be either a Polynomial or a scalar.	required

__truediv__(other)

Divide the Polynomial by a scalar.

Parameters:

Name	Type	Description	Default
other	object	Scalar to divide the Polynomial by.	required

add_monomial(m)

Add a Monomial to the Polynomial.

Parameters:

Name	Type	Description	Default
m	Monomial	Monomial to add to the Polynomial.	required

add_monomial_with_idx(coef, idx)

Add a Monomial with a given lexicographical index idx.

Parameters:

Name	Type	Description	Default
coef	float	Coefficient of the Monomial.	required
idx	int	Lexicographical index of the Monomial.	required

See Also

MultiIndex

add_monomials(monos=None)

Add a list of Monomials to the Polynomial.

Parameters:

Name	Type	Description	Default
monos	list[Monomial], optional	List of Monomials to add to the Polynomial, by default None	None

add_monomials_with_idxs(coef_list, idx_list)

Add list of Monomials with given lexicographical indices.

Parameters:

Name	Type	Description	Default
coef_list	list[float]	List of coefficients of the Monomials.	required
idx_list	list[int]	List of lexicographical indices of the Monomials.	required

Raises:

Type	Description
PolynomialError	If the number of coefficients and multi-indices are not equal.

See Also

MultiIndex

anti_laplacian()

Polynomial anti-Laplacian of the Polynomial.

Returns:

Type	Description
Polynomial	Polynomial P such that Delta P = self.

compose(q1, q2)

Compose the Polynomial with two other Polynomials.

Parameters:

Name	Type	Description	Default
q1	Polynomial	Polynomial in x and y.	required
q2	Polynomial	Polynomial in x and y.	required

Returns:

Type	Description
Polynomial	Composed Polynomial new(x,y) = self(q1(x,y), q2(x,y)).

consolidate()

Consolidate the coefficients of repeated indices.

copy()

Get a copy of the Polynomial.

Returns:

Type	Description
Polynomial	A copy of the Polynomial.

get_weighted_normal_derivative(K)

Compute the weighted normal derivative of the Polynomial.

Parameters:

Name	Type	Description	Default
K	MeshCell	MeshCell on whose boundary to compute the normal derivative.	required

Returns:

Type	Description
np.ndarray	Values of the weighted normal derivative of the Polynomial.

get_weighted_tangential_derivative(K)

Compute the weighted normal derivative of the Polynomial.

Parameters:

Name	Type	Description	Default
K	MeshCell	MeshCell on whose boundary to compute the tangential derivative.	required

Returns:

Type	Description
np.ndarray	Values of the weighted tangential derivative of the Polynomial.

grad()

Gradient of the Polynomial.

Returns:

Type	Description
tuple[Polynomial, Polynomial]	Pair of the partial derivatives of the Polynomial with respect to x and y.

is_zero()

Is True if the Polynomial is zero.

Returns:

Type	Description
bool	True if the Polynomial is zero.

laplacian()

Laplacian of the Polynomial.

Returns:

Type	Description
Polynomial	Laplacian of the Polynomial: Delta f = f_{xx} + f_{yy}.

partial_deriv(var)

Partial derivative of the Polynomial with respect to the variable var.

Parameters:

Name	Type	Description	Default
var	str	Variable with respect to which the derivative is taken. Must be either 'x' or 'y'.	required

Returns:

Type	Description
Polynomial	Partial derivative of the Polynomial with respect to the variable var.

pow(exponent)

Raise the Polynomial to a nonnegative integer power.

Parameters:

Name	Type	Description	Default
exponent	int	Nonnegative integer exponent.	required

Returns:

Type	Description
Polynomial	Polynomial raised to the exponent.

remove_zeros()

Remove terms with zero coefficients.

set(coef_multidx_pairs=None)

Set the Polynomial to the list of coefficient / multi-index pairs.

Parameters:

Name	Type	Description	Default
coef_multidx_pairs	list[tuple[float, int, int]], optional	List of coefficient / multi-index pairs, by default None	None

set_to_zero()

Set the Polynomial to zero.

Notes

This is done by removing all Monomials from the Polynomial.

sort()

Sort the Monomials according to multi-index id.

Notes

Using Insertion Sort algorithm since Monomial list is assumed be be short.