Function polyfromroots

numpy/polynomial/polynomial.py:152–213

Generate a monic polynomial with given roots. Return the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), where the :math:`r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. F

(roots)

def polyfromroots(roots):
    """
    Generate a monic polynomial with given roots.

    Return the coefficients of the polynomial

    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

    where the :math:`r_n` are the roots specified in `roots`.  If a zero has
    multiplicity n, then it must appear in `roots` n times. For instance,
    if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
    then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear
    in any order.

    If the returned coefficients are `c`, then

    .. math:: p(x) = c_0 + c_1 * x + ... +  x^n

    The coefficient of the last term is 1 for monic polynomials in this
    form.

    Parameters
    ----------
    roots : array_like
        Sequence containing the roots.

    Returns
    -------
    out : ndarray
        1-D array of the polynomial's coefficients If all the roots are
        real, then `out` is also real, otherwise it is complex.  (see
        Examples below).

    See Also
    --------
    numpy.polynomial.chebyshev.chebfromroots
    numpy.polynomial.legendre.legfromroots
    numpy.polynomial.laguerre.lagfromroots
    numpy.polynomial.hermite.hermfromroots
    numpy.polynomial.hermite_e.hermefromroots

    Notes
    -----
    The coefficients are determined by multiplying together linear factors
    of the form ``(x - r_i)``, i.e.

    .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)

    where ``n == len(roots) - 1``; note that this implies that ``1`` is always
    returned for :math:`a_n`.

    Examples
    --------
    >>> from numpy.polynomial import polynomial as P
    >>> P.polyfromroots((-1,0,1))  # x(x - 1)(x + 1) = x^3 - x
    array([ 0., -1.,  0.,  1.])
    >>> j = complex(0,1)
    >>> P.polyfromroots((-j,j))  # complex returned, though values are real