where \(c_0^{\ast}=\ c_{0\ }-\ln R\ \) and \(c_i^{\ast}=c_i\) for \(i\ \ge1\). The roots \(x^{\ast}\) are the values for which Eq. \ref{eq:ChebyshevRoot} is satisfied.
The key steps in the Numpy Chebyshev_fit root finding routine for the evaluation of \(T\left(R\right)\) are these:
- make a copy of the Chebyshev results so as not to modify the original
- replace \(c_0\) with \(c_0^{\ast}\) so that 0 is returned when series is evaluated for correct value of T
- find all roots, then isolate the (ideally) one real root within the temperature domain of the fit
The key parts of the code corresponding to each of these step are listed here: