# Rational minimax approximation via adaptive barycentric representations

## TL;DR

This paper introduces adaptive barycentric representations for rational minimax approximation, enabling robust algorithms that outperform traditional methods, especially near singularities, demonstrated by high-degree approximations in standard floating point.

## Contribution

It develops a new adaptive barycentric approach for rational minimax approximation, combining classical and iterative methods for improved robustness and efficiency.

## Key findings

- Able to compute high-degree rational approximations in standard precision
- Outperforms previous methods requiring extended precision
- Achieves quadratic convergence with combined algorithms

## Abstract

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.