Module « scipy.special »

Fonction roots_hermite - module scipy.special

Signature de la fonction roots_hermite

def roots_hermite(n, mu=False) 

Description

roots_hermite.__doc__

Gauss-Hermite (physicist'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:`H_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}`. See 22.2.14 in [AS]_ for
    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

    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 applied
    which computes nodes and weights in a numerically stable manner.
    The algorithm has linear runtime making computation for very
    large n (several thousand or more) feasible.

    See Also
    --------
    scipy.integrate.quadrature
    scipy.integrate.fixed_quad
    numpy.polynomial.hermite.hermgauss
    roots_hermitenorm

    References
    ----------
    .. [townsend.trogdon.olver-2014]
        Townsend, A. and Trogdon, T. and Olver, S. (2014)
        *Fast computation of Gauss quadrature nodes and
        weights on the whole real line*. :arXiv:`1410.5286`.
    .. [townsend.trogdon.olver-2015]
        Townsend, A. and Trogdon, T. and Olver, S. (2015)
        *Fast computation of Gauss quadrature nodes and
        weights on the whole real line*.
        IMA Journal of Numerical Analysis
        :doi:`10.1093/imanum/drv002`.
    .. [AS] Milton Abramowitz and Irene A. Stegun, eds.
        Handbook of Mathematical Functions with Formulas,
        Graphs, and Mathematical Tables. New York: Dover, 1972.