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

TL;DR

This paper introduces the first numerically stable algorithms for derivative evaluation in barycentric and Thiele continued fraction representations of rational functions, enabling efficient computation of derivatives.

Contribution

It develops stable, efficient algorithms for derivative evaluation in barycentric and TCF representations, including all derivatives and higher orders, with proven robustness.

Findings

Algorithms are numerically stable and efficient.

Methods outperform previous unstable approaches.

Numerical experiments confirm robustness and speed.

Abstract

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.