Applications of AAA rational approximation (2510.16237v1)

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

• The paper introduces the AAA algorithm that robustly delivers high-accuracy rational approximations through a barycentric representation and a greedy descent strategy.
• It demonstrates efficient methods for analytic continuation, pole-zero localization, and reliable derivative and integral computations, achieving up to 14 digits of accuracy.
• The work extends AAA to diverse areas including PDEs, model order reduction, and nonlinear eigenproblems, while outlining 20 open problems for future research.

Applications of AAA Rational Approximation: Theory, Algorithms, and Practice

Introduction and Algorithmic Foundations

The AAA (Adaptive Antoulas-Anderson) algorithm provides a robust, flexible, and highly accurate approach to rational approximation of functions on arbitrary real or complex domains. The central innovation of AAA is its use of a barycentric representation for rational functions, which circumvents the numerical instability inherent in the classical quotient form $p(z)/q(z)$, especially when poles and zeros cluster near singularities. The algorithm proceeds via a greedy descent strategy, adaptively selecting support points to minimize the maximum error over a discrete sample set, and solving a sequence of linearized least-squares problems using the Loewner matrix and SVD.

The barycentric form for a rational function of degree $n$ is

$$r(z) = \frac{\sum_{k=0}^n \frac{f_k \beta_k}{z-t_k}}{\sum_{k=0}^n \frac{\beta_k}{z-t_k}},$$

where $\{t_k\}$ are support points (nodes), $\{f_k\}$ are function values at those nodes, and $\{\beta_k\}$ are barycentric weights. The AAA algorithm adaptively selects the $t_k$ by maximizing the current error at each step, and computes the weights via the right singular vector of the Loewner matrix corresponding to the smallest singular value.

The computational complexity is $O(m n^3)$, where $m$ is the number of sample points and $n$ the degree, with $m$ typically scaling with $n$ for adequate resolution. For $n < 150$ and $m < 3000$, computation is typically sub-second on commodity hardware.

Function Approximation and Analytic Continuation

AAA excels in both offline (library construction) and online (on-the-fly) function approximation. For meromorphic functions sampled at scattered points, AAA can recover values at arbitrary locations with high accuracy, often exceeding 10 digits with moderate sample sizes. For example, approximating $f(z) = \tan(z)/\tan(2)$ at 50 random points yields 11–14 digits of accuracy at $z=2$.

Figure 1: AAA recovers the value of a meromorphic function at $z=2$ from 50 scattered samples in the complex plane, achieving 11 digits of accuracy.

For analytic continuation, AAA rational approximants can extend function values far beyond the original data domain, outperforming polynomial approximants, which are limited by the nearest singularity (as dictated by Bernstein ellipses and potential theory). The "one-wavelength principle" is observed: the number of wavelengths over which analytic continuation is successful scales linearly with the number of digits of accuracy in the rational approximation.

Figure 2: AAA rational approximation of $\tanh(z)$ on $[-1,1]$ enables analytic continuation well beyond the interval, unlike polynomial minimax fitting.

Pole and Zero Localization

AAA provides a practical and accurate method for locating poles and zeros of analytic and meromorphic functions from sampled data, generalizing the classical Padé approach to arbitrary sample sets. Poles and zeros are computed a posteriori via a generalized eigenvalue problem derived from the barycentric representation. In practice, poles and zeros within the data cloud are typically accurate to the approximation tolerance (e.g., 13 digits), and those further away degrade gracefully.

Figure 3: Poles (red) and zeros (blue) of the AAA approximant to $f(z) = \tan(z)/\tan(2)$ closely match the true singularities, even outside the sample region.

This capability extends to rootfinding for analytic functions, where AAA-based "proxy rootfinding" can outperform bespoke polynomial-based methods, and to the identification of branch points via rational approximation of the logarithmic derivative.

Derivatives, Integrals, and Quadrature

AAA rational approximants can be differentiated and integrated efficiently. The derivative in barycentric form is computed via divided differences, and for higher derivatives, recursive formulas are available. For integration, conversion to partial fractions enables termwise integration, and for certain classes of integrals (e.g., over the real line), residue calculus applied to the rational approximant yields highly accurate results.

Figure 4: Comparison of polynomial and rational approximations for differentiating $f(x) = \tanh(8x)$; the rational approach achieves similar accuracy with an order-of-magnitude lower degree.

AAA-based quadrature formulas are constructed by approximating the Cauchy transform of the weight function, with the quadrature nodes corresponding to the poles of the rational approximant. This approach generalizes classical Gaussian quadrature and enables the construction of optimal quadrature rules for nonstandard domains and weights.

Equispaced Interpolation and Data Imputation

AAA provides a robust solution to the classical Runge and Gibbs phenomena in equispaced interpolation. Unlike polynomials and trigonometric polynomials, which suffer from divergence or endpoint artifacts, AAA rational approximants achieve steady convergence and high accuracy for analytic and smooth functions, even in the presence of missing data (imputation).

Figure 5: AAA rational approximation outperforms polynomial and trigonometric interpolation for equispaced data, avoiding the Runge and Gibbs phenomena.

Figure 6: AAA fills in large gaps in analytic data with high accuracy, even when 27.5% of samples are missing.

Inverse Functions and Multivariate Extension

AAA can be used to compute numerical inverses of functions, including those with singularities, by simply swapping the roles of input and output in the sample set. This is particularly effective for functions with ill-conditioned inverses, such as $\sin(x)$ or solutions to ODEs.

Figure 7: AAA computes the inverse of a numerically solved ODE initial value problem, enabling rapid evaluation of the inverse function.

For multivariate problems, while no direct analog of AAA exists due to the lack of a multivariate barycentric formula, a practical workaround is to apply AAA along one-dimensional slices (e.g., lines or circles), enabling high-accuracy extension and imputation in higher dimensions.

Figure 8: AAA-based imputation along horizontal slices reconstructs missing data in a bivariate analytic function.

PDEs, Laplace, and Helmholtz Problems

The AAALS (AAA-least squares) method, introduced by Costa, extends AAA to the solution of Laplace and related PDEs by fitting boundary data with the real part of a rational function whose poles are adaptively chosen outside the domain. This approach achieves high accuracy, even in domains with corners or nonconvexity, where polynomial methods fail due to slow convergence.

Figure 9: Poles of the AAA approximant to the Schwarz function of a closed curve cluster near regions of high curvature, revealing the analytic structure of the domain.

For the Helmholtz equation and scattering problems, AAA is used to place singularities (e.g., Hankel functions) optimally, enabling efficient and accurate solution of high-frequency wave problems.

Model Order Reduction, Eigenvalue Problems, and Nonlinear Eigenproblems

In model order reduction (MOR) and reduced order modeling (ROM), AAA is used to approximate high-degree transfer functions by lower-degree rational functions, often achieving 10+ digits of accuracy with a dramatic reduction in degree. The method is competitive with, and often superior to, classical approaches such as vector fitting and the Loewner framework, especially in the SISO case.

For large-scale eigenvalue problems, AAA approximates the scalarized resolvent $(zI-A)^{-1}$, with the poles of the rational approximant corresponding to eigenvalues of $A$ within the region of interest. This approach is scalable, parallelizable, and effective for both linear and nonlinear eigenvalue problems, including those arising in photonics and resonance computations.

Figure 10: AAA accurately locates poles (resonances) of the Riemann zeta function from sampled data on a line in the complex plane.

Zolotarev Problems, Best Approximation, and Algorithmic Extensions

AAA, together with the AAA-Lawson extension, enables the computation of best (minimax) rational approximations in the $\infty$-norm, both for real and complex data. This is critical for applications such as digital filter design, spectral slicing, and divide-and-conquer eigenvalue algorithms, where Zolotarev sign and ratio problems arise. The paper introduces modifications to AAA (e.g., the 'sign' and 'damping' options) to address convergence issues in these challenging settings.

Figure 11: Error curves for AAA and AAA-Lawson approximations to $\Gamma(z)$ on $|z|=1.5$; AAA-Lawson achieves a near-circular error curve, indicating near-optimality.

Theoretical Insights and Open Problems

The paper provides a comprehensive potential-theoretic framework for understanding the convergence of rational approximations, including the role of Hermite integrals, equilibrium measures, and the classification of poles (approximation, pole, branch cut, and spurious). It highlights the exponential and root-exponential convergence rates achievable by rational functions, in contrast to the algebraic rates of polynomials in the presence of singularities.

Figure 12: The "one-wavelength principle" in analytic continuation: AAA achieves approximately one wavelength of accurate extension per 13 digits of accuracy.

A set of 20 open problems is articulated, spanning algorithmic, theoretical, and multivariate challenges, including the development of a multivariate AAA, robust best-approximation algorithms, and the extension of AAA to other function classes (e.g., Gaussians, RBFs).

Conclusion

The AAA algorithm and its extensions have transformed the landscape of rational approximation, providing a practical, reliable, and theoretically grounded tool for a wide range of applications in numerical analysis, scientific computing, and engineering. The method's adaptability, accuracy, and ability to handle singularities and complex geometries position it as a central component in modern computational mathematics. The paper's comprehensive treatment, extensive numerical evidence, and identification of open problems set the stage for further advances in rational approximation theory, algorithms, and applications.