Function hermcompanion

numpy/polynomial/hermite.py:1490–1534

Return the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when `c` is a Hermite basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be

(c)

def hermcompanion(c):
    """Return the scaled companion matrix of c.

    The basis polynomials are scaled so that the companion matrix is
    symmetric when `c` is a Hermite basis polynomial. This provides
    better eigenvalue estimates than the unscaled case and for basis
    polynomials the eigenvalues are guaranteed to be real if
    `numpy.linalg.eigvalsh` is used to obtain them.

    Parameters
    ----------
    c : array_like
        1-D array of Hermite series coefficients ordered from low to high
        degree.

    Returns
    -------
    mat : ndarray
        Scaled companion matrix of dimensions (deg, deg).

    Examples
    --------
    >>> from numpy.polynomial.hermite import hermcompanion
    >>> hermcompanion([1, 0, 1])
    array([[0.        , 0.35355339],
           [0.70710678, 0.        ]])

    """
    # c is a trimmed copy
    [c] = pu.as_series([c])
    if len(c) < 2:
        raise ValueError('Series must have maximum degree of at least 1.')
    if len(c) == 2:
        return np.array([[-.5 * c[0] / c[1]]])

    n = len(c) - 1
    mat = np.zeros((n, n), dtype=c.dtype)
    scl = np.hstack((1., 1. / np.sqrt(2. * np.arange(n - 1, 0, -1))))
    scl = np.multiply.accumulate(scl)[::-1]
    top = mat.reshape(-1)[1::n + 1]
    bot = mat.reshape(-1)[n::n + 1]
    top[...] = np.sqrt(.5 * np.arange(1, n))
    bot[...] = top
    mat[:, -1] -= scl * c[:-1] / (2.0 * c[-1])
    return mat


def hermroots(c):