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

TL;DR

The paper addresses numerically stable derivative evaluation for rational approximants in both the barycentric representation and the Thiele continued fraction (TCF) form. For barycentric interpolation it proposes a stable $O(n)$ algorithm for all derivatives by replacing the unstable standard formula with a modified formulation that avoids dominant cancellation; it also presents an $O(n^2)$ double-sum alternative as a baseline. For TCF, the authors show that the existing first-derivative $O(n)$ iteration can be extended to higher orders via recurrences for $p_k^{(m)}$ and $q_k^{(m)}$, preserving floating-point stability. Numerical experiments in Julia demonstrate robustness across intervals and the unit circle for a variety of test functions, with the proposed methods achieving stable derivative evaluation and practical performance. Software implementing the methods is available, enabling practitioners to compute derivatives of rational approximants with controlled numerical stability.

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.

Figures (1)

• Figure 1: Absolute error near a node using \ref{['eq:sw1']} to evaluate the first derivative of an AAA rational approximation $r(x)$ to $f(x) = e^x$ on $[-1,1]$. The approximation to $f$ is of type $(5,5)$ and accurate to within $5\times 10^{-13}$, but due to subtractive cancellation, the evaluation of $r'(x)$ grows when approaching the node $0.75$.

Theorems & Definitions (2)

• Theorem 1
• proof