Generate a Chebyshev series with the given roots.
Return the array of coefficients for the C-series whose roots (a.k.a. “zeros”) are given by roots. The returned array of coefficients is ordered from lowest order “term” to highest, and zeros of multiplicity greater than one must be included in roots a number of times equal to their multiplicity (e.g., if 2 is a root of multiplicity three, then [2,2,2] must be in roots).
Parameters : | roots : array_like
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Returns : | out : ndarray
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See also
polyfromroots
Notes
What is returned are the c_i such that:
\sum_{i=0}^{n} c_i*T_i(x) = \prod_{i=0}^{n} (x - roots[i])
where n == len(roots) and T_i(x) is the i-th Chebyshev (basis) polynomial over the domain [-1,1]. Note that, unlike polyfromroots, due to the nature of the C-series basis set, the above identity does not imply c_n = 1 identically (see Examples).
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+0.j, 0.5+0.j])