Applications of AAA rational approximation

TL;DR

The paper surveys the AAA adaptive barycentric rational approximation framework and demonstrates its applicability across a wide range of numerical analysis problems, from on-the-fly function evaluation to analytic continuation, pole/zero localization, and advanced PDE/ODE solutions. By representing rational functions in a stable barycentric form and employing greedy selection of sampling points, AAA achieves near-best accuracy efficiently and integrates naturally with derivatives, integrals, and inverse problems. It also connects to potential theory, quadrature, and conformal mapping, while offering variants (AAALS, Lawson, continuum, periodic, type-(m,n), vector-valued) to address specific challenges like best approximation, singularities, and multivariate extensions. The practical impact spans complex resonances, model order reduction, Laplace and Helmholtz problems, and geometric conformal tasks, highlighting AAA as a versatile tool for modern computational mathematics with substantial room for theoretical development. Overall, AAA shifts the paradigm toward fast, data-driven rational approximation as a unifying method across disparate numerical disciplines, enabling new insights and efficient computations where traditional polynomial techniques falter.

Abstract

The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis.

Figures (64)

• Figure 3.1: Fifty scattered points in the complex plane at which a meromorphic function $f(z)$ is sampled. The aim is to estimate the value at $z=2$ (the red circle). For $f(z) = \tan(z)/\tan(2)$, AAA approximation gets a result accurate to 11 digits in a few milliseconds.
• Figure 3.2: AAA approximation of $\Gamma(z)$ based on samples at $50$ Chebyshev points in $[-1.5,1.5]$ (left) and $50$ equispaced points on $|z|=1.5$ (right). From inside out, the contours represent relative errors $|r(z)-\Gamma(z)|/|\Gamma(z)| = 10^{-13}, 10^{-12}, \dots,10^{-1}$.
• Figure 3.3: Error curves for AAA approximation of $\Gamma(z)$ on the circle $|z|=1.5$. On the left, straight AAA approximation. On the right, AAA-Lawson approximation with 20 Lawson steps. The error curve has winding number $21$ but is ragged because the errors are close to the level of machine precision (compare Figure \ref{['gamma2fig']}). In this paper we use magenta to distinguish error curves from AAA-Lawson best approximation.
• Figure 4.1: Repetition of the experiment of Figure \ref{['fig3.1']}, now with the poles and zeros of $r(z)$ shown as red and blue circles, respectively. The inner poles and zeros closely match those of the target function $f(z) = \tan(z)/\tan(2)$ (faint red and blue dots).
• Figure 4.2: On the left, poles (red circles) and zeros (blue circles) of the degree $29$ AAA approximation of the Riemann zeta function $\zeta(z)$ based on sample values at 100 points with $\hbox{\rm Re} z = 4$ (black dots). The pole and the zeros of $\zeta(z)$ are marked by red and blue dots, respectively. The pole of $\zeta(z)$ is captured to 11 digits of accuracy, whereas the other poles of $r(z)$ are artifacts of the approximation (red circles). The first few zeros of $\zeta(z)$ on the critical line $\hbox{\rm Re} z = 1/2$ are also highly accurate. This figure shows the upper half of a symmetric configuration in the complex plane. On the right, the same except with $50$ sample points in the right half-plane and also their reflections by (\ref{['poles-reflec']}) in the left half-plane. The rational approximation is now of degree $32$; the accuracy is similar.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 10.1
• Theorem 10.2
• Theorem 11.1
• Theorem 11.2
• Theorem 12.1
• Theorem 21.1
• Theorem 24.1