Fast methods for computing the values of the derivatives of a rational function (Q1196841)

A new fast method is given for the computation of the first \(\ell\) derivatives of a rational function at a given point. The problem is reduced to the solution of a lower triangular \((\ell+1)\times(\ell+1)\) Toeplitz system which may be solved by a fast method whose order of complexity is \(O((\ell+1)\log_ 2(\ell+1))\). For large \(\ell\) the method is better than the one given by \textit{J. Rokne} [J. Comput. Appl. Math. 14, 447-449 (1986; Zbl 0588.65019)] and for small \(\ell\) it is equally good.