Hermite Functions🍋
evaluate_hermite_expansion(x, coeff=np.zeros(1))
🍋
Evaluate at x a function determined by its coefficients in the Hermite expansion.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
x |
ndarray
|
numpy array containing the points where the expansion is evaluated. |
required |
coeff |
ndarray
|
numpy array containing the coefficients in the Hermite expansion (the n-th element is the coefficient associated to the n-th Hermite polynomial). |
zeros(1)
|
Returns:
| Type | Description |
|---|---|
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, 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 |
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 |
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
|
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. |
CONSTANT
|
sort |
bool
|
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
|
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 |
|---|---|
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. |
Raises:
| Type | Description |
|---|---|
ValueError
|
|
hermite_polynomials(x, n)
🍋
Function to compute the Hermite polynomials up to order n.
Parameters:
| Name | Type | Description | Default |
|---|---|---|---|
x |
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 |
|---|---|
ndarray
|
Numpy matrix of shape (n+1 , x.size) whose k-th row contains the Hermite polynomial of order k evaluated at x. |