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Module « scipy.special »

Fonction roots_hermitenorm - module scipy.special

Signature de la fonction roots_hermitenorm

def roots_hermitenorm(n, mu=False) 

Description

help(scipy.special.roots_hermitenorm)

Gauss-Hermite (statistician's) quadrature.

Compute the sample points and weights for Gauss-Hermite
quadrature. The sample points are the roots of the nth degree
Hermite polynomial, :math:`He_n(x)`. These sample points and
weights correctly integrate polynomials of degree :math:`2n - 1`
or less over the interval :math:`[-\infty, \infty]` with weight
function :math:`w(x) = e^{-x^2/2}`. See 22.2.15 in [AS]_ for more
details.

Parameters
----------
n : int
    quadrature order
mu : bool, optional
    If True, return the sum of the weights, optional.

Returns
-------
x : ndarray
    Sample points
w : ndarray
    Weights
mu : float
    Sum of the weights

See Also
--------
scipy.integrate.fixed_quad
numpy.polynomial.hermite_e.hermegauss

Notes
-----
For small n up to 150 a modified version of the Golub-Welsch
algorithm is used. Nodes are computed from the eigenvalue
problem and improved by one step of a Newton iteration.
The weights are computed from the well-known analytical formula.

For n larger than 150 an optimal asymptotic algorithm is used
which computes nodes and weights in a numerical stable manner.
The algorithm has linear runtime making computation for very
large n (several thousand or more) feasible.

References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

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