Greedy Thiele continued-fraction approximation on continuum domains in the complex plane

TL;DR

The paper tackles adaptive rational approximation on continuum domains in the complex plane by employing a greedy Thiele continued-fraction (TCF) representation $r_n(z)$ with node selection analogous to AAA. It develops a continuum adaptive refinement strategy along the domain boundary and introduces new algorithms for evaluating $r_n$, along with derivatives and residues, using a single division per step. Empirical results on functions on $[-1,1]$ and the unit circle show that greedy TCF offers substantial speedups over AAA (approximately $2.5$–$12\times$) while maintaining comparable accuracy. The work also discusses stability considerations and practical aspects, including underflow, and argues for the continued development of TCF as a simple, fast alternative or complement to barycentric rational approximants in complex-domain settings.

Abstract

We describe an adaptive greedy algorithm for Thiele continued-fraction approximation of a function defined on a continuum domain in the complex plane. The algorithm iteratively selects interpolation nodes from an adaptively refined set of sample points on the domain boundary. We also present new algorithms for evaluating Thiele continued fractions and their accessory weights using only a single floating-point division. Numerical experiments comparing the greedy TCF method with the AAA algorithm on several challenging functions defined on the interval $[-1,1]$ and on the unit circle show that continuum TCF is consistently 2.5 to 8 times faster than AAA.

Figures (3)

• Figure 1: Error in the interpolation of $f(x)=\arctan(500x)$ using 1001 sample points in the interval $[-1, 1]$. The discretization is too coarse near the origin, where the error is of size $2\times 10^{-4}$.
• Figure 1: Comparison of TCF and AAA approximation on the interval $[-1,1]$ for six challenging functions. Each plot shows the max-norm error on a validation set described in the text as a function of the denominator degree. The black rings indicate the best approximation found for each run. Any approximation that has a genuine pole within the interval is marked with a red x. The dashed lines indicate the best approximation found by discrete variants of the algorithms.
• Figure 2: Comparison of greedy TCF and AAA approximation on the unit circle for six challenging functions. Each plot shows the max-norm error on a validation set described in the text as a function of the denominator degree. The black rings indicate the best approximation found for each run. Any approximation that has a genuine pole on the circle is marked with a red x. The dashed lines indicate the best approximation found by discrete variants of the algorithms.