# numpy.polynomial.chebyshev.chebfromroots

polynomial.chebyshev.chebfromroots(roots)

  Generate a Chebyshev series with given roots.

  The function returns the coefficients of the polynomial

  \[p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),\]

  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

  \[p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)\]

  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])