hermite.py 文件源码

def hermroots(c):
    """
    Compute the roots of a Hermite series.

    Return the roots (a.k.a. "zeros") of the polynomial

    .. math:: p(x) = \\sum_i c[i] * H_i(x).

    Parameters
    ----------
    c : 1-D array_like
        1-D array of coefficients.

    Returns
    -------
    out : ndarray
        Array of the roots of the series. If all the roots are real,
        then `out` is also real, otherwise it is complex.

    See Also
    --------
    polyroots, legroots, lagroots, chebroots, hermeroots

    Notes
    -----
    The root estimates are obtained as the eigenvalues of the companion
    matrix, Roots far from the origin of the complex plane may have large
    errors due to the numerical instability of the series for such
    values. Roots with multiplicity greater than 1 will also show larger
    errors as the value of the series near such points is relatively
    insensitive to errors in the roots. Isolated roots near the origin can
    be improved by a few iterations of Newton's method.

    The Hermite series basis polynomials aren't powers of `x` so the
    results of this function may seem unintuitive.

    Examples
    --------
    >>> from numpy.polynomial.hermite import hermroots, hermfromroots
    >>> coef = hermfromroots([-1, 0, 1])
    >>> coef
    array([ 0.   ,  0.25 ,  0.   ,  0.125])
    >>> hermroots(coef)
    array([ -1.00000000e+00,  -1.38777878e-17,   1.00000000e+00])

    """
    # c is a trimmed copy
    [c] = pu.as_series([c])
    if len(c) <= 1:
        return np.array([], dtype=c.dtype)
    if len(c) == 2:
        return np.array([-.5*c[0]/c[1]])

    m = hermcompanion(c)
    r = la.eigvals(m)
    r.sort()
    return r