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.


Sequence containing the roots.


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).


>>> 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])