Applications of AAA rational approximation

Lloyd N. Trefethen; Yuji Nakatsukasa

REVIEW 3 major objections 4 minor 1 cited by

Applications of AAA rational approximation

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The AAA algorithm makes rational approximation a routine, near-best tool across numerical analysis.

desk verdict A sweeping, honest review of AAA rational approximation by the people who built it: genuinely useful as a reference, but treat its pole-inference promises as conditional — Section 14 shows the poles can be wrong while the fit is perfect. read the letter →

arxiv 2510.16237 v1 pith:OH44KBYV submitted 2025-10-17 math.NA cs.NA

Yuji Nakatsukasa , Lloyd N. Trefethen This is my paper · ORCID

classification math.NAcs.NA MSC 41A2065D1565E05

keywords AAAalgorithmrationalapproximationbarycentricrepresentationpolesandzerosanalyticcontinuationquadraturemodelorderreductionsingularitylocation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that the AAA algorithm—a greedy, SVD-based method that represents rational functions in barycentric form—has become a routine, near-best tool that can be applied in 26 different areas of numerical analysis. The central claim is that rational functions, whose poles are free to migrate to singularities, remove the main weakness of polynomial approximation whenever the target function has poles, branch points, or other nearby singularities, and that AAA makes working with them as easy as working with polynomials. If the paper is right, many numerical tasks that once required high-degree polynomials, bespoke algorithms, or expensive optimisation can be done in milliseconds by fitting a rational function to sample values, with machine-precision accuracy on the data set and surprisingly accurate estimates of singularities beyond it.

What carries the argument

The central object is the barycentric rational approximant r(z) = [Σ f_k β_k/(z−t_k)] / [Σ β_k/(z−t_k)] computed by the AAA algorithm: at each greedy step the sample point with largest error becomes a new support point, and the weights β_k are chosen by minimising a linearised least-squares residual—the smallest singular vector of a divided-difference matrix. This representation is stable where a numerator/denominator quotient would be unstable, and it makes poles and zeros of r directly computable as eigenvalues of a small generalised eigenproblem. The same machinery, with small modifications, underlies extensions to best approximation, continuum domains, periodic functions, and vector-valu

What would settle it

Take a family of functions with exactly known poles and branch points, sample each on a fixed grid that does not concentrate points near the singularities, and compare AAA-extracted singularities with the exact ones as the tolerance is tightened from 10^-6 to 10^-13. If, as in the Burgers example the paper reproduces, the pole sets remain accurate only for the singularities closest to the data and diverge completely for all others at every tolerance, then the claim that AAA poles can be used for reliable singularity inference would be refuted.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that one algorithm—the AAA greedy iteration with barycentric representation and a tall-skinny SVD at each step—can compute rational approximations that are close to machine precision on almost any discrete sample set in the complex plane, and that these approximations behave like near-best rational approximants. Because the barycentric form avoids the instability of representing the quotient of two polynomials, poles and zeros can be extracted as eigenvalues of a small generalised eigenproblem, turning rational approximation into a black-box tool. The paper demonstrates the reach of this tool across function approximation, pole and zero location, de

Load-bearing premise

The load-bearing premise is that the greedy SVD-based AAA iteration, although lacking a theorem guaranteeing near-best convergence on a continuum, nevertheless delivers trustworthy approximations—and trustworthy poles, zeros, and extrapolated values—across the wide range of applications considered.

Editorial extensions

If this is right

For functions with branch points on the approximation domain, rational approximations converge root-exponentially, while polynomials converge only algebraically; the paper reports cases where degrees in the hundreds replace degrees in the millions.
Poles and zeros of a meromorphic function can be recovered from samples away from the singularities, often with accuracy that extends well beyond the data set, enabling fast singularity location in ODE, PDE, and scattering problems.
Numerical analytic continuation by rational approximation can extend a function beyond its data domain by roughly one wavelength of its oscillatory behaviour, with the number of wavelengths growing with the working precision.
Quadrature rules can be viewed as rational approximations of a Cauchy transform, so AAA can generate quadrature formulas adapted to arbitrary contours and weight functions; the paper shows classical Gauss-type rules reproduced this way.
For model order reduction, a transfer function of degree 1000 can be reduced to degree about 30 by fitting samples on the imaginary axis, with ten-digit accuracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If AAA's near-best behaviour is as general as the examples suggest, numerical libraries that currently rely on polynomial spectral methods could switch to AAA-based rational methods for stiff or singular problems, with order-of-magnitude reductions in degree and no loss of accuracy.
The roughly linear relation between working precision and the number of wavelengths of analytic continuation suggests a testable law: doubling the digits should roughly double the achievable extension distance; a systematic experiment across functions with known singularities could turn the one-wavelength principle into a theorem or refute it.
The equivalence between quadrature nodes and poles of rational approximations of a Cauchy transform suggests a design principle: for any new kernel or contour, one can derive a quadrature rule by solving a rational approximation problem, which may be more direct than constructing conformal maps.
The paper's own Burgers example—where AAA poles agree with exact poles only near the real axis—implies that pole-based singularity inference has a finite horizon; a careful study of how this horizon depends on tolerance, sample clustering, and singularity strength would delineate where AAA can be trusted.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, this is the AAA paper: Nakatsukasa and Trefethen's 26-application tour of rational approximation, with MATLAB snippets for everything. It will be the standard citation for "rational functions can do that too" for years. Second, it is more honest than most reviews of this type. It repeatedly says where AAA has no theorem, where it fails (real intervals, sign functions, noise), and where spurious poles appear. The section that matters most for the broad thesis is Section 14: a reproduction from VandenHeuvel et al. shows AAA pole sets for a Burgers solution that are completely different from the true poles even though the approximant matches the solution to 1e-13 on the real axis. The paper warns the reader, but it does not quantify how often this happens or give a diagnostic. Since pole/residue extraction is a selling point in Sections 4, 16–18 and 22, this is the soft underbelly.

What is actually new? Not much by way of theorems. The log-derivative branch point procedure in Section 6 appears to be new and works to 10 digits. Section 22's quadrature-rational connection (Horning-Trefethen) is striking: Gauss nodes as poles of a rational approximation to a Cauchy transform. Section 21 gives a reproducible Zolotarev ratio computation checked to 0.1% against the analytic answer. The 20 open problems in the Discussion are useful.

The self-citation is heavy, but it is also fair: these people invented AAA, and several of the supporting papers are their own. It only matters that some flagships — Bruno-Santana-Trefethen, Horning-Trefethen — are still submitted, so a referee cannot check everything. That is a legitimate but minor worry.

The bigger issue is the gap between "we have a fast algorithm that often works" and the opening promise of a routine near-best tool. The paper is appropriately hedged; it says the theoretical challenge is open and that AAA is least reliable on real intervals. The examples are selected, and there is tolerance tuning per experiment. But as a review, that is acceptable. It consolidates a fragmented literature and does not oversell.

Verdict: send it to referees. It deserves the round. Cite it if you work anywhere near rational approximation or complex singularity analysis. The pole-inference caveat belongs in the referee report; it should not sink the paper.

Referee Report

3 major / 4 minor

Summary. This invited review demonstrates the AAA algorithm for rational approximation across 26 areas of numerical analysis, including function approximation, pole/zero and branch-point location, differentiation and integration, analytic continuation, ODE/PDE extension into the complex plane, eigenvalues and resonances, model order reduction, Zolotarev sign and ratio problems, quadrature, Cauchy/Wiener-Hopf/Riemann–Hilbert problems, and conformal mapping. The paper is experiment-driven: each section contains MATLAB snippets, figures, and a candid discussion of what is known, conjectured, or open. The central claim, stated in Section 1, is that AAA-type rational approximation is a fast, near-best, general-purpose tool that outperforms polynomial methods for functions with singularities and for extrapolation. The authors are explicit that parts of the theory are missing, and they list twenty open problems.

Significance. If the demonstrations are correct, this is an important and timely review that makes a strong case for rational approximation as a routine numerical tool. The paper's strengths include reproducible code snippets, external analytic checks (the two-disk Zolotarev ratio is within 0.1% of the known value in Section 21; resonances in the closed disk match Bessel roots in Section 16), and unusually candid disclosure of limitations and failure modes, such as the Section 14 pole discrepancy and the Section 20 sign-problem difficulties. The review will be a valuable resource for practitioners and researchers. However, the broad-utility thesis is conditional on the empirical reliability of AAA's poles and extrapolated values, and two of the paper's own examples show that this reliability has limits. Those limits are acknowledged but not quantified, and they affect several of the showcased applications, so the manuscript would benefit from a dedicated discussion of reliability and diagnostics.

major comments (3)

[Section 14; also Sections 4, 16–18, 22] The manuscript reproduces a VandenHeuvel et al. experiment in which AAA poles for a Burgers solution differ completely from the exact poles beyond the pair closest to the real axis, despite a 1e-13 fit on the real axis. This is a documented counterexample to the inference that a good data fit implies correct singularity structure—an inference used in pole/zero finding (Section 4), resonance and eigenvalue computation via poles of scalarized resolvents (Sections 16–18), and quadrature-node interpretation (Section 22). The paper warns about this in Section 14 but does not provide a diagnostic, a frequency estimate, or a stability check. Since the broad-utility claim is supported by selected examples, this gap is load-bearing. I ask for a short 'reliability of pole extraction' discussion: for instance, sensitivity of computed poles to sample perturbation, or an a posteriori consistency test
[Section 20, Eqs. (20.2)–(20.3); Section 21] The paper states that standard AAA 'almost invariably runs into trouble' on Zolotarev sign problems, and the 'sign' and 'damping' fixes are explicitly 'not bulletproof' and have no theoretical justification. The Zolotarev ratio problem of Section 21 is solved through these fixes, and the Batakci-spiral example is admitted to be 'only approximate' and not fully converged. Thus the paper's own headline examples in Sections 20–21, and their downstream applications (quadrature, MOR), are outside the regime of routine AAA reliability. Please add a quantitative statement of when the sign modification succeeds or fails (for example, over a small corpus of E/F geometries) and clarify which of the 26 applications inherit this caveat.
[Section 10; Section 2] The paper's extrapolation claims are central, especially for analytic continuation and extension of ODE/PDE solutions into C. Yet Section 10's one-wavelength principle is explicitly a rule of thumb, not a theorem, and Section 2 states that no theorem ensures AAA is near-best in the relevant sense. The paper gives no general guidance beyond 'about one wavelength' for when extrapolated values can be trusted. Since Sections 13–15 and 24–28 build on continuation of computed solutions, this is a load-bearing heuristic. Please either provide a more precise accuracy-versus-distance tradeoff or mark the affected applications as heuristic in the relevant sections.

minor comments (4)

[Section 21, code after Eq. (21.7)] The code defines rstar without the factor sqrt(sigma) that appears in Eq. (21.4). This is harmless for pole/zero plots but should be noted if the normalized value is needed.
[Section 16, table] The entry 'same' for mode 3 in the closed-circle column would be clearer if the numerical value 3.8317060 were repeated.
[Section 12, Theorem 12.1] The theorem is attributed to Stahl (1989, Lemma 2) but no proof or precise statement of the specialization is given. Adding a sentence on how the stated version follows would help readers.
[Section 4, Eq. (4.1)] The display of the generalized eigenvalue problem is visually hard to parse in the arXiv version; a clearer matrix layout or a reference to Corless (2004) with the exact eigenvalue problem would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the paper's demonstrations are largely independent numerical experiments verified against exact or external data; self-citations are references to documented algorithms and prior work, not load-bearing reductions.

full rationale

The central claim is an empirical advocacy thesis: AAA rational approximation is fast, flexible, and near-best across many applications. The supporting evidence is not a derivation that assumes its conclusion. For example, Section 3 evaluates a AAA fit at z=2 and checks against the known value of tan(2)/tan(2); Section 4 compares computed poles and zeros of tan and zeta against the exact values; Section 17 checks eigenvalue poles against exactly known diagonal entries; Section 21 checks the Zolotarev ratio against the analytic value 2^-60 within 0.1%. Section 2 explicitly disclaims any theorem that AAA is near-best: "The true theoretical challenge of AAA approximation is to establish a theorem to ensure that, under appropriate assumptions, AAA finds not just accurate but near-best approximations." The paper also documents failures, notably Section 14's reproduction of VandenHeuvel et al. showing AAA pole sets for a Burgers solution differing from exact poles while the functions agree to 10^-13 on the real axis. This is a reliability limitation, not a circular step. The many self-citations (Nakatsukasa-Sète-Trefethen 2018; Trefethen-Wilber 2025; Bruno-Santana-Trefethen submitted; Horning-Trefethen submitted; Driscoll-Nakatsukasa-Trefethen 2024) point to documented algorithms and prior work whose contents are independently reproducible (MATLAB/Chebfun code, external verification, and mathematical derivations reproduced in the text, e.g. Section 22 equations (22.4)-(22.9)). None of these citations is used in place of a missing derivation, and the central experimental results are checked against external benchmarks rather than against the paper's own fit. No claim is a fitted input renamed as a prediction; no ansatz is smuggled in via citation; no uniqueness theorem from the authors is invoked to forbid alternatives, and the paper explicitly acknowledges other methods such as Thiele continued fractions. The limitation flagged in Section 14 weakens the pole-inference applications but does not make the paper's reasoning circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper is a review: the only 'free parameters' are user-set AAA controls (tolerances, degrees, Lawson steps, damping, sample sets) chosen per experiment; none is fitted to data to manufacture agreement with a target result, but the reported digit counts depend on them. The axioms are standard complex analysis, imports from the cited literature (Walsh, Stahl, Istace-Thiran), and the unproven empirical premise that AAA's greedy fit is near-best in practice — a premise the paper itself flags. No new physical or mathematical entities are postulated; the 'sign'/'damping' modifications and the branch-point log-derivative method are algorithmic ingredients with no independent falsifiable handle beyond the experiments shown.

free parameters (4)

AAA stopping tolerance = 10^-13 default; 10^-7 to 10^-10 in several experiments
The convergence tolerance is loosened per experiment to make examples succeed (sin^-1 at 1e-7 in Section 7; missing data at 1e-10 in Section 9; BVPs at 1e-6 in Section 13; zeta/potential theory at 1e-10 in Section 12). The reported digit counts depend on these choices.
Rational degree n or degree limits = varies (e.g., 30, 35, 80, 150)
Degrees are fixed by hand in several demonstrations (Zolotarev problems at degree 30/35/80 in Sections 20-21; aaax deg=150 in Section 13; mmax=200 in Section 7). The paper's efficiency comparisons (e.g., Table 12.1) depend on achieving these degrees.
Lawson iteration count and damping factor delta = 20-600 steps; delta = 0.95, 0.75
AAA-Lawson and damping parameters (Sections 3, 20-21) are user-chosen; Section 21 admits convergence is 'only approximate' for the Batakci spiral despite 600 iterations.
Sample set Z (geometry and size) = chosen per experiment (50-3000 points)
Sample locations (random, equispaced, Chebyshev, exponentially clustered, log-spaced) materially affect measured accuracy; e.g., Section 3's digit counts (10-14 digits) are for specific random seeds.

assumptions (6)

standard math Standard complex analysis: Cauchy integral formula, residue theorem, Sokhotski-Plemelj formulas, Schwarz reflection principle (Theorems 11.1-11.2)
Invoked throughout Sections 5, 11, 22-23; the paper states or cites the relevant theorems.
domain assumption Walsh Hermite integral formula (12.1)-(12.2) and the Gonchar-Stahl rho^2 2n+1 interpolation-point version (Theorem 12.1)
Imported from Walsh (1969) and Stahl (1989, Lemma 2); the paper uses Theorem 12.1 to quantitatively explain AAA accuracy in Section 12 but does not prove it.
ad hoc to paper AAA (as implemented in Chebfun aaa.m) genuinely achieves the reported accuracy, and sample-set convergence implies the claimed continuum accuracy
The paper acknowledges no proof exists: Section 2 calls Theorem 2.1 a 'Yogiism' and says 'There is no guarantee that this will be achieved in the presence of rounding errors'. All accuracy claims in the review rest on this empirical premise.
domain assumption Istace-Thiran equivalence between Zolotarev sign and ratio problems (Theorem 21.1)
Quoted from Istace and Thiran (1995); used in Section 21 to convert sign solutions into ratio solutions.
domain assumption Known analytic benchmarks: two-disk Zolotarev solution (21.7)-(21.8) from Starke (1992); Bessel-root eigenvalues for the closed circular cavity (Section 16); Chapman's asymptotic formulas (16.3)-(16.5)
Used to validate AAA results; assumed correct as external benchmarks.
domain assumption Numerical environment: IEEE double precision and correct behavior of MATLAB/Chebfun functions (aaa.m, minimax, chebfun)
All experiments run in MATLAB with Chebfun; reproducibility depends on these tools behaving as stated.

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Cite this review

Pith. "Pith review of Applications of AAA rational approximation." pith.science (2026). https://pith.science/paper/OH44KBYV

@misc{pith2026251016237,
  author       = {Pith},
  title        = {Pith review of: Applications of AAA rational approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OH44KBYV}},
  note         = {Machine review of arXiv:2510.16237}
}

read the original abstract

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

Figures

Figures reproduced from arXiv: 2510.16237 by the authors.

**Figure 3.1.** Fifty scattered points in the complex plane at which a mero [PITH_FULL_IMAGE:figures/full_fig_p011_3_1.png] view at source ↗

**Figure 3.2.** AAA approximation of Γ(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)−Γ(z)|/|Γ(z)| = 10−13 , 10−12 , . . . , 10−1 . Rational approximation is not the only way in which one might approximate f, but its simplicity and accuracy are remarkable. The reader might find it interesting to consider t… view at source ↗

**Figure 3.3.** Error curves for AAA approximation of Γ( [PITH_FULL_IMAGE:figures/full_fig_p013_3_3.png] view at source ↗

Figures from the paper (61 more)

**Figure 4.1.** Figure 4.1: Repetition of the experiment of Figure [PITH_FULL_IMAGE:figures/full_fig_p018_4_1.png]

**Figure 4.2.** Figure 4.2: On the left, poles (red circles) and zeros (blue circles) of the [PITH_FULL_IMAGE:figures/full_fig_p020_4_2.png]

**Figure 5.1.** Figure 5.1: Numerical computation of the derivative of [PITH_FULL_IMAGE:figures/full_fig_p023_5_1.png]

**Figure 6.1.** Figure 6.1: Zeros (blue) and poles (red) of a AAA approximation to the [PITH_FULL_IMAGE:figures/full_fig_p028_6_1.png]

**Figure 7.1.** Figure 7.1: Solution of the ODE (7.1) and its inverse function computed with AAA. for comparison, suppose we want the inverse of sin(x) for x ∈ [−π/2, π/2]. Since f(x) has derivative zero at x = ±π/2, f −1 (t) = sin−1 (t) has square root singularities at t = ±1. Because of ill-c…

**Figure 7.2.** Figure 7.2: 64 equispaced points on a curve determined by a numerical [PITH_FULL_IMAGE:figures/full_fig_p032_7_2.png]

**Figure 7.3.** Figure 7.3: Sixteen points on the boundary of the field of values of a 5 [PITH_FULL_IMAGE:figures/full_fig_p033_7_3.png]

**Figure 8.1.** Figure 8.1: For interpolation of nonperiodic equispaced data, polynomials [PITH_FULL_IMAGE:figures/full_fig_p035_8_1.png]

**Figure 8.2.** Figure 8.2: Illustration from (Huybrechs and Trefethen (2023)) of six different methods of equispaced interpolation applied to the function f(x) = exp(−1/x2 ) for x ∈ [−1, 1]. AAA approximation shows steady convergence down to accuracy 10−13 as the number of sample points n inc…

**Figure 9.1.** Figure 9.1: Use of AAA approximation to fill in gaps in an analytic function. [PITH_FULL_IMAGE:figures/full_fig_p037_9_1.png]

**Figure 9.2.** Figure 9.2: A similar experiment to that of Figure [PITH_FULL_IMAGE:figures/full_fig_p038_9_2.png]

**Figure 9.3.** Figure 9.3: Filling in missing data in the bivariate MATLAB “peaks” func [PITH_FULL_IMAGE:figures/full_fig_p039_9_3.png]

**Figure 10.1.** Figure 10.1: Numerical analytic continuation of tanh( [PITH_FULL_IMAGE:figures/full_fig_p041_10_1.png]

**Figure 10.2.** Figure 10.2: Illustration of the “one-wavelength principle” by six examples [PITH_FULL_IMAGE:figures/full_fig_p042_10_2.png]

**Figure 11.1.** Figure 11.1: On the left, the Schwarz reflection principle (Theorem [PITH_FULL_IMAGE:figures/full_fig_p044_11_1.png]

**Figure 11.2.** Figure 11.2: Numerically computed Schwarz functions for closed curves Γ [PITH_FULL_IMAGE:figures/full_fig_p046_11_2.png]

**Figure 11.3.** Figure 11.3: Similar plots for a curve Γ consisting of two pieces joined at [PITH_FULL_IMAGE:figures/full_fig_p047_11_3.png]

**Figure 12.1.** Figure 12.1: Approximation of f(z) = e 4z on the unit circle. The plot on the left shows superexponential convergence of AAA approximations as a function of degree n down to tolerance 10−10 with n = 9, with slower superexponential convergence for polynomials. On the right, level…

**Figure 12.2.** Figure 12.2: Approximation of f(z) = e 4z on the unit circle again. Now n more interpolation points have been added to the image, shown as green dots, and the potential function has been approximately squared from (12.2) to (12.3). This roughly doubles the span of amplitudes sho…

**Figure 12.3.** Figure 12.3: Approximation of f(z) = tan(z 2 ) on [−1, 1]2 . For a meromorphic function like this, polynomial approximations converge exponentially whereas rational approximations converge superexponentially [PITH_FULL_IMAGE:figures/full_fig_p053_12_3.png]

**Figure 12.4.** Figure 12.4: Approximation of f(z) = p 1 + 25(z − 0.25)2 on [−1, 1]. For any function like this with branch points, both polynomial and rational approximations converge exponentially [PITH_FULL_IMAGE:figures/full_fig_p053_12_4.png]

**Figure 12.5.** Figure 12.5: Approximation of f(z) = log(−z) on a nonconvex region. Both polynomial and rational functions converge exponentially, but at vastly different rates (Trefethen (2024)) [PITH_FULL_IMAGE:figures/full_fig_p053_12_5.png]

**Figure 13.1.** Figure 13.1: Solutions to the six singularly perturbed ODE boundary-value [PITH_FULL_IMAGE:figures/full_fig_p055_13_1.png]

**Figure 14.1.** Figure 14.1: Solutions to the periodic Burgers equation ( [PITH_FULL_IMAGE:figures/full_fig_p059_14_1.png]

**Figure 14.2.** Figure 14.2: A trajectory u(t) of the Lorenz equations together with AAA poles in the complex t-plane. The poles line up along approximate branch cuts. poles at every time step in a numerical simulation. The same would be true for many other problems, indicating the exceptional …

**Figure 14.3.** Figure 14.3: A figure from (VandenHeuvel et al. (2023)) comparing the exact pole structure of a Burgers solution with its AAA approximation from data on the real axis. The poles closest to the axis are accurately captured, but not the poles further out. (Note that the real and i…

**Figure 15.1.** Figure 15.1: At the left, contours |f(x, y)| = 0, 0.2, 0.4, . . . , 1.4 of the bivariate function (15.1). The other two images show the same contours for the degree 2 and 4 rational approximations (15.2) and (15.3). No numerical approximation algorithm is known for finding nea…

**Figure 15.2.** Figure 15.2: Bivariate approximation of (15.1) by AAA approximation along vertical lines. On the left, approximation along vertical lines with data samples just above the singularity, marked by black dots. On the right, approximation along vertical lines with samples on both sid…

**Figure 15.3.** Figure 15.3: Reproduction of a figure from (Trefethen (2023)) showing continuation of a smooth function across a boundary by univariate AAA approximation on 60 radial lines. As in section 10, about one wavelength of accurate extension is achieved. particular value x = x0 , the f…

**Figure 16.1.** Figure 16.1: Images from (Bruno et al. (submitted)) of the first six complex resonances in the unit disk with a gap in the boundary of angle π/8. Images 2–3 and 4–5 are near-degenerate pairs that become exactly degenerate in the limit of zero gap size. wave at frequency κ scatte…

**Figure 17.1.** Figure 17.1: AAA calculation of the eigenvalues inside the circle [PITH_FULL_IMAGE:figures/full_fig_p071_17_1.png]

**Figure 18.1.** Figure 18.1: Calculation of 7 nonlinear eigenvalues of the “butterfly” prob [PITH_FULL_IMAGE:figures/full_fig_p074_18_1.png]

**Figure 18.2.** Figure 18.2: Image reproduced from (Betz et al. (2024)) suggesting the use of AAA approximation to calculate nonlinear eigenvalues in photonics applications. e = polyeig(c{:}); plot(e,'.k'), axis equal, hold on Z = 1+1i + 0.3*exp(2i*pi*(1:300)/300); plot(Z,'.b') u = randn(64,1);…

**Figure 19.1.** Figure 19.1: On the left, absolute values of the degree 1006 transfer func [PITH_FULL_IMAGE:figures/full_fig_p078_19_1.png]

**Figure 20.1.** Figure 20.1: Approximation of signE/F (z) for a pair of disks discretised by 80 equispaced points on each boundary. Standard AAA does poorly on such problems (left), whereas AAA with the 'sign' modification introduced in 2024 and described in (Trefethen and Wilber (2025)) does m…

**Figure 20.2.** Figure 20.2: Repetition of Figure [PITH_FULL_IMAGE:figures/full_fig_p082_20_2.png]

**Figure 20.3.** Figure 20.3: Four more examples of AAA-Lawson minimax solutions of the [PITH_FULL_IMAGE:figures/full_fig_p084_20_3.png]

**Figure 21.1.** Figure 21.1: Approximation to the degree n = 30 solution r ∗ n of the Zolotarev ratio problem for the two-disks configuration of Figures 20.1 and 20.2. This is a transformation of the function ˆrn of the right image of Figure 20.2 by the formula (21.4). Blue and red dots mark …

**Figure 21.2.** Figure 21.2: Degree 44 solution of the Zolotarev ratio problem for a pair [PITH_FULL_IMAGE:figures/full_fig_p087_21_2.png]

**Figure 21.3.** Figure 21.3: The Batakci lollipop. The image shows a degree 80 Zolotarev [PITH_FULL_IMAGE:figures/full_fig_p089_21_3.png]

**Figure 22.1.** Figure 22.1: On the left, nodes for the 20-point Gauss quadrature formula [PITH_FULL_IMAGE:figures/full_fig_p091_22_1.png]

**Figure 22.2.** Figure 22.2: Like Figure [PITH_FULL_IMAGE:figures/full_fig_p092_22_2.png]

**Figure 22.3.** Figure 22.3: On the left, a figure from (Hale et al. (2008)) showing quadrature nodes derived by conformal mapping for a problem of computing functions of matrices with eigenvalues in [m, M]. On the right, a figure adapted from (Horning and Trefethen (submitted)) showing nodes…

**Figure 23.1.** Figure 23.1: The Cauchy transform of h(t) = e 3t on Γ = [−1, 1] is computed by evaluating the integral (23.1) at 100 points on an ellipse by adaptive quadrature and then approximating the result by a AAA rational function. The approximation has degree 22, with little asymmetry i…

**Figure 23.2.** Figure 23.2: Poles of a AAA rational approximation to [PITH_FULL_IMAGE:figures/full_fig_p097_23_2.png]

**Figure 23.3.** Figure 23.3: Poles for a rational approximation to solve a Riemann-Hilbert [PITH_FULL_IMAGE:figures/full_fig_p098_23_3.png]

**Figure 24.1.** Figure 24.1: Illustration of AAA-least squares (AAALS) for a Laplace Di [PITH_FULL_IMAGE:figures/full_fig_p103_24_1.png]

**Figure 24.2.** Figure 24.2: Repetition of Figure [PITH_FULL_IMAGE:figures/full_fig_p106_24_2.png]

**Figure 25.1.** Figure 25.1: First row: conformal map computed by Chebfun [PITH_FULL_IMAGE:figures/full_fig_p109_25_1.png]

**Figure 25.2.** Figure 25.2: Conformal map onto a half-plane computed by AAALS. [PITH_FULL_IMAGE:figures/full_fig_p111_25_2.png]

**Figure 25.3.** Figure 25.3: Doubly-connected conformal map computed by AAALS. [PITH_FULL_IMAGE:figures/full_fig_p111_25_3.png]

**Figure 26.1.** Figure 26.1: Upper row: a AAA-Helmholtz scattering computation with [PITH_FULL_IMAGE:figures/full_fig_p114_26_1.png]

**Figure 26.2.** Figure 26.2: Another AAA-Helmholtz scattering computation involving a [PITH_FULL_IMAGE:figures/full_fig_p115_26_2.png]

**Figure 27.1.** Figure 27.1: Figure from (Brubeck and Trefethen (2022)) showing Stokes flow in a triangular driven cavity computed by the lightning Stokes method. No-slip boundary conditions are applied everywhere, with the top boundary moving at constant speed and the remainder of the boundary…

**Figure 27.2.** Figure 27.2: Stokes flow in another triangular driven cavity with a curved [PITH_FULL_IMAGE:figures/full_fig_p117_27_2.png]

**Figure 28.1.** Figure 28.1: Numerical computation of the Hilbert transform ( [PITH_FULL_IMAGE:figures/full_fig_p121_28_1.png]

**Figure 29.1.** Figure 29.1: Repetition of Figure [PITH_FULL_IMAGE:figures/full_fig_p123_29_1.png]

**Figure 29.2.** Figure 29.2: Error curves for two more degree 11 AAA-Lawson approxim [PITH_FULL_IMAGE:figures/full_fig_p124_29_2.png]

**Figure 29.3.** Figure 29.3: The error “curve” (discrete dots) for six trials of degree 10 [PITH_FULL_IMAGE:figures/full_fig_p125_29_3.png]

**Figure 30.1.** Figure 30.1: Continuum AAA near-best rational approximation of [PITH_FULL_IMAGE:figures/full_fig_p127_30_1.png]

**Figure 31.1.** Figure 31.1: Periodic AAA approximation of (31.1) computed by aaatrig. Three periods are shown. by transplanting every problem to Laurent series on the unit circle. If we ask ourselves why not, perhaps the first answer is that such a transplantation converts real problems to co…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Rational Minimax Approximations for Matrix-Valued Functions: Existence, Optimality and Algorithms
math.OC 2026-06 conditional novelty 6.0 of 10

Matrix-valued rational minimax approximants with a common denominator are proved to exist on dense point sets, with Kolmogorov/Ruttan optimality certificates and an equivalence to the m-d-Lawson dual conditions.

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Works this paper leans on

16 extracted references · 3 linked inside Pith · cited by 1 Pith paper

[1]

Adcock and D

B. Adcock and D. Huybrechs (2019), Frames and numerical approximation,SIAM Rev.61, 443–473. B. Alpert, L. Greengard and T. Hagstrom (2000), Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation,SIAM J. Numer. Anal.37, 1138–1164. A. C. Antoulas and B. D. Q. Anderson (1986), On the scalar rational interpolation problem,IMA J. M...

2019
[3]

Bai and J

Z. Bai and J. Demmel (1998), Using the matrix sign function to compute invariant subspaces,SIAM J. Matrix Anal. Applics.19(1), 205–225. Z. Bai, J. Demmel and M. Gu (1997), An inverse free parallel spectral divide and conquer algorithm for nonhermitian eigenproblems,Numer. Math.76, 279–308. G. A. Baker, Jr. (1961), Application of the Pad´ e approximant met...

1998
[7]

S. J. Harris and N. R. McDonald (2025a), Modelling wildfire spread and spot- fire merger using conformal mapping and AAA-least squares methods,Environ. Model. Softw.185, 106303. S. J. Harris and N. R. McDonald (2025b), Vortex equilibria using least-squares methods,Theor. Comput. Fluid Dyn.39, 1–17. S. J. Harris, R. A. Palmer and N. R. McDonald (2025), Mod...

2025
[14]

M. C. Brennan, M. Embree and S. Gugercin (2023), Contour integral methods for nonlinear eigenvalue problems: A systems theoretic approach,SIAM Rev.65, 439–470. P. D. Brubeck and L. N. Trefethen (2022), Lightning Stokes solver,SIAM J. Sci. Comput.44, A1205–A1226. P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen (2021), Vandermonde with Arnoldi,SIAM Rev.63...

arXiv 2023
[16]

I. V. Gosea and S. G¨ uttel (2021), Algorithms for the rational approximation of matrix-valued functions,SIAM J. Sci. Comput.43, A3033–A3054. A. Greenbaum and L. N. Trefethen (1994), GMRES/CR and Arnoldi/Lanczos as matrix approximation problems,SIAM J. Sci. Comp.15, 359–368. S. Gugercin, A. C. Antoulas and C. Beattie (2008),H 2 model reduction for large- ...

2021
[21]

Applications of AAA rational approximation145 A. C. Ionita (2013), Lagrange rational interpolation and its applications to approx- imation of large-scale dynamical systems, PhD thesis, Rice University. M.-P. Istace and J.-P. Thiran (1993), On computing best complex rational approx- imants,Numerical Algorithms5, 299–308. M.-P. Istace and J.-P. Thiran (1995...

2013
[25]

L. N. Trefethen (to appear), Numerical conformal mapping,Notices of the AMS. L. N. Trefethen and J. A. C. Weideman (2014), The exponentially convergent trapezoidal rule,SIAM Rev.56, 385–458. L. N. Trefethen and H. D. Wilber (2025), Computation of Zolotarev rational func- tions,SIAM J. Sci. Comput.47, A2205–A2220. L. N. Trefethen, A. Birkisson and T. A. Dr...

2014
[36]

Platte, L

R. Platte, L. N. Trefethen and A. B. J. Kuijlaars (2011), Impossibility of fast stable approximation of analytic functions from equispaced samples,SIAM Rev. 53, 308–318. E. Polizzi (2009), Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B79, 115112–1–115112–6. C. Pozrikidis (1992),Boundary Integral and Singularity Methods for Li...

2011

Show all 16 references

[150]

L. N. Trefethen (2023), Numerical analytic continuation,Jpn. J. Ind. Appl. Math. 40, 1587–1636. L. N. Trefethen (2024), Polynomial and rational convergence rates for Laplace problems on planar domains,Proc. Roy. Soc. A480, 20240178. Applications of AAA rational approximation14...

2023
[283]

J. H. Curtiss (1962), Interpolation by harmonic polynomials,J. SIAM10, 709–736. P. J. Davis (1974),The Schwarz Function and its Applications, Math. Assoc. Amer. P. J. Davis and H. Pollak (1958), On the analytic continuation of mapping func- tions,Trans. AMS87, 198–225. E. Deck...

1962 arXiv
[1004]

MacMillen (2024), BaryRational.jl,github.com/macd/BaryRational.jl

D. MacMillen (2024), BaryRational.jl,github.com/macd/BaryRational.jl. MathWorks Inc. (2020), ‘RF Toolbox’,https://www.mathworks.com. T. Matsuda and Y. Nakatsukasa (2025), Polynomial approximation of noisy func- tions,Numer. Math.157, 1285–1311. S. W. McCue, C. J. Lustri, D. J....

2024 arXiv
[1236]

A. C. Rodriguez, L. Balicki and S. Gugercin (2023), The p-AAA algorithm for data-driven modeling of parametric dynamical systems,SIAM J. Sci. Comput. 45, A1332–A1358. V. Rokhlin (1990), Rapid solution of integral equations of scattering theory in two dimensions,J. Comput. Phys...

2023
[1671]

Boull´ e, A

N. Boull´ e, A. Herremans and D. Huybrechs (2024), Multivariate rational approx- imation of functions with curves of singularities,SIAM J. Sci. Comput.46, A3401–A3426. N. Boull´ e, Y. Nakatsukasa and A. Townsend (2020), Rational neural networks, Adv. Neural Info. Proc. Syst.33...

2024
[1821]

A. P. Austin, M. Krishnamoorthy, S. Leyffer, S. Mrenna, J. M¨ uller and H. Schulz (2021), Practical algorithms for multivariate rational approximation,Computer Phys. Commun.261, 107663. S. Axler (1986), Harmonic functions from a complex analysis viewpoint,Amer. Math. Monthly93...

2021
[2025]

T. A. Driscoll and Y. Zhou (2025), Greedy Thiele continued-fraction approximation on continuum domains in the complex plane,arXiv preprint arXiv:2510.07295. T. A. Driscoll, F. Bornemann and L. N. Trefethen (2008), The chebop system for automatic solution of differential equati...

2025
[2523]

S. J. Harris and N. R. McDonald (2023), Penguin huddling: a continuum model, Acta Appl. Math.185,

2023

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Adcock and D

B. Adcock and D. Huybrechs (2019), Frames and numerical approximation,SIAM Rev.61, 443–473. B. Alpert, L. Greengard and T. Hagstrom (2000), Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation,SIAM J. Numer. Anal.37, 1138–1164. A. C. Antoulas and B. D. Q. Anderson (1986), On the scalar rational interpolation problem,IMA J. M...

2019

[2] [3]

Bai and J

Z. Bai and J. Demmel (1998), Using the matrix sign function to compute invariant subspaces,SIAM J. Matrix Anal. Applics.19(1), 205–225. Z. Bai, J. Demmel and M. Gu (1997), An inverse free parallel spectral divide and conquer algorithm for nonhermitian eigenproblems,Numer. Math.76, 279–308. G. A. Baker, Jr. (1961), Application of the Pad´ e approximant met...

1998

[3] [7]

S. J. Harris and N. R. McDonald (2025a), Modelling wildfire spread and spot- fire merger using conformal mapping and AAA-least squares methods,Environ. Model. Softw.185, 106303. S. J. Harris and N. R. McDonald (2025b), Vortex equilibria using least-squares methods,Theor. Comput. Fluid Dyn.39, 1–17. S. J. Harris, R. A. Palmer and N. R. McDonald (2025), Mod...

2025

[4] [14]

M. C. Brennan, M. Embree and S. Gugercin (2023), Contour integral methods for nonlinear eigenvalue problems: A systems theoretic approach,SIAM Rev.65, 439–470. P. D. Brubeck and L. N. Trefethen (2022), Lightning Stokes solver,SIAM J. Sci. Comput.44, A1205–A1226. P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen (2021), Vandermonde with Arnoldi,SIAM Rev.63...

arXiv 2023

[5] [16]

I. V. Gosea and S. G¨ uttel (2021), Algorithms for the rational approximation of matrix-valued functions,SIAM J. Sci. Comput.43, A3033–A3054. A. Greenbaum and L. N. Trefethen (1994), GMRES/CR and Arnoldi/Lanczos as matrix approximation problems,SIAM J. Sci. Comp.15, 359–368. S. Gugercin, A. C. Antoulas and C. Beattie (2008),H 2 model reduction for large- ...

2021

[6] [21]

Applications of AAA rational approximation145 A. C. Ionita (2013), Lagrange rational interpolation and its applications to approx- imation of large-scale dynamical systems, PhD thesis, Rice University. M.-P. Istace and J.-P. Thiran (1993), On computing best complex rational approx- imants,Numerical Algorithms5, 299–308. M.-P. Istace and J.-P. Thiran (1995...

2013

[7] [25]

L. N. Trefethen (to appear), Numerical conformal mapping,Notices of the AMS. L. N. Trefethen and J. A. C. Weideman (2014), The exponentially convergent trapezoidal rule,SIAM Rev.56, 385–458. L. N. Trefethen and H. D. Wilber (2025), Computation of Zolotarev rational func- tions,SIAM J. Sci. Comput.47, A2205–A2220. L. N. Trefethen, A. Birkisson and T. A. Dr...

2014

[8] [36]

Platte, L

R. Platte, L. N. Trefethen and A. B. J. Kuijlaars (2011), Impossibility of fast stable approximation of analytic functions from equispaced samples,SIAM Rev. 53, 308–318. E. Polizzi (2009), Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B79, 115112–1–115112–6. C. Pozrikidis (1992),Boundary Integral and Singularity Methods for Li...

2011