Function lagfromroots

numpy/polynomial/laguerre.py:251–304

Generate a Laguerre series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Laguerre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appe

(roots)

def lagfromroots(roots):
    """
    Generate a Laguerre series with given roots.

    The function returns the coefficients of the polynomial

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

    in Laguerre form, where the `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 * L_1(x) + ... +  c_n * L_n(x)

    The coefficient of the last term is not generally 1 for monic
    polynomials in Laguerre form.

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

    Returns
    -------
    out : ndarray
        1-D array of coefficients.  If all roots are real then `out` is a
        real array, if some of the roots are complex, then `out` is complex
        even if all the coefficients in the result are real (see Examples
        below).

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

    Examples
    --------
    >>> from numpy.polynomial.laguerre import lagfromroots, lagval
    >>> coef = lagfromroots((-1, 0, 1))
    >>> lagval((-1, 0, 1), coef)
    array([0.,  0.,  0.])
    >>> coef = lagfromroots((-1j, 1j))
    >>> lagval((-1j, 1j), coef)
    array([0.+0.j, 0.+0.j])

    """
    return pu._fromroots(lagline, lagmul, roots)


def lagadd(c1, c2):