Function chebfromroots

numpy/polynomial/chebyshev.py:514–566

Generate a Chebyshev 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 Chebyshev form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must ap

(roots)

def chebfromroots(roots):
    """
    Generate a Chebyshev 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 Chebyshev 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 * T_1(x) + ... +  c_n * T_n(x)

    The coefficient of the last term is not generally 1 for monic
    polynomials in Chebyshev 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.laguerre.lagfromroots
    numpy.polynomial.hermite.hermfromroots
    numpy.polynomial.hermite_e.hermefromroots

    Examples
    --------
    >>> import numpy.polynomial.chebyshev as C
    >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
    array([ 0.  , -0.25,  0.  ,  0.25])
    >>> j = complex(0,1)
    >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
    array([1.5+0.j, 0. +0.j, 0.5+0.j])

    """
    return pu._fromroots(chebline, chebmul, roots)


def chebadd(c1, c2):