Hermite Functions

evaluate_hermite_expansion(x, coeff=array([0.]))

Evaluate at x a function determined by its coefficients in the Hermite expansion.

Parameters:

Name	Type	Description	Default
x	np.ndarray	numpy array containing the points where the expansion is evaluated.	required
coeff	np.ndarray, optionnal	numpy array containing the coefficients in the Hermite expansion (the n-th element is the coefficient associated to the n-th Hermite polynomial).	array([0.])

Returns:

Type	Description
np.ndarray	numpy array containing the approximation of the function in a truncated Hermite expansion at x (the function is vectorized).

hermite_coefficients_piecewise_linear(n, x, y, extension=<ExtrapolationMethod.CONSTANT: 'CONSTANT'>, sort=True, error=False)

Computes the Hermite coefficients of a piece-wise linear function continuous in an interval, with a desired extension outside the interval.

Parameters:

Name	Type	Description	Default
n	int	non-negative integer indicating the order up to which the Hermite coefficients are obtained.	required
x	np.ndarray	numpy array of points in the real line indicating the breakpoints for the pice-wise linear function. Hence the function is continuous within the interval [x[0] , x[x.size-1]]. The elements of x must all be different with respect to each other.	required
y	np.ndarray	numpy array indicating the images of the breakpoints x of the piece-wise linear function. It must have the same number of elements as x.	required
extension	ExtrapolationMethod, optionnal	string indicating the definition of the function outside the pice-wise linear intervale. Two options available: "constant" and "null". "constant": (default option) the function is continuously extended having constant values over the two sides of the interval. "null": the function is null outside the interval. Thus may imply that the resulting function is not continuous over the whole real line.	<ExtrapolationMethod.CONSTANT: 'CONSTANT'>
sort	bool, optionnal	boolean (by default True). If True, it is supposed that the values in x are sorted. If false, the arrays x and y are sorted according to x's sorting order.	True
error	bool, optionnal	boolean (by default False). If True, the approximation error (L^2 norm with Gaussian weight) will be retourned as a second element of the output list.	False

Returns:

Type	Description
np.ndarray	numpy array containing the first coefficients in the Hermite expansion of the function. If Error=True, a (n,2) array, the first element containing the coefficients (previous output) and the second containing the error of the Hermite approximation.

Exceptions:

Type	Description
ValueError	Number of given polynomials must be positive.
ValueError	Real and Gaussian datasets must have the same size.
ValueError	Real values must be unique.

hermite_polynomials(x, n)

Function to compute the Hermite polynomials up to order n.

Parameters:

Name	Type	Description	Default
x	np.ndarray	numpy array containing the points where the polynomials are evaluated (this function is vectorized).	required
n	int	non-negative integer indicating the maximal order of the desired polynomials.	required

Returns:

Type	Description
np.ndarray	Numpy matrix of shape (n+1 , x.size) whose k-th row contains the Hermite polynomial of order k evaluated at x.