Stable evaluation of derivatives for barycentric and continued fraction representations of rational functions

Fast algorithms for approximation by rational functions exist for both barycentric and Thiele continued fraction (TCF) representations. We present the first numerically stable methods for derivative evaluation in the barycentric representation, including an $O(n)$ algorithm for all derivatives. We also extend an earlier $O(n)$ algorithm for evaluation of the TCF first derivative to higher orders. Numerical experiments confirm the robustness and efficiency of the proposed methods.