{"id":"6f182c0c-ab86-4c6a-8513-ab39ff406b13","arxiv_id":"2510.16237","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"Rational-function approximation via AAA handles singularities, extrapolation, and high-accuracy fitting across 26 areas of numerical analysis where polynomials fail; a comprehensive review by the method's inventors.","lead":"An Acta Numerica review by two leading numerical analysts demonstrates rational-function fitting via the AAA algorithm across 26 areas where it beats polynomial methods, often by orders of magnitude in degree. It argues rational approximation is now a routine, reliable tool for numerical analysis, and lists twenty open problems.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"AAA pole inference can be silently wrong (Sec. 14) while fitting data to 1e-13; several flagship applications depend on pole/residue extraction, so the broad-utility claim needs a stability check.","rationale":"The reader identified AAA reliability as the weakest assumption. My concern is the same in spirit but more specific: Section 14 provides a concrete case where AAA pole sets differ completely from the exact poles despite 1e-13 fidelity on the data axis. This matters because several flagship sections rely on pole and residue extraction for downstream inferences, from resonance computation to eigenvalue solving. However, the paper already discloses this failure and is appropriately hedged throughout; it is a review that advances a thesis, not a universal guarantee. The central argument remains intact for the purposes of an ACCEPT verdict. The proposed test would sharpen confidence by determining whether the Section 14 behavior is an isolated pathology or a systematic risk, but the lack of that test does not invalidate the review as written.","tokens_in":55151,"tokens_out":5942,"duration_ms":61562,"concrete_test":"Re-run the Burgers example of Figure 14.3 using the same Chebfun aaa.m settings, varying (i) the number and placement of real-axis samples and (ii) the convergence tolerance (e.g., 1e-6, 1e-10, 1e-13). For each run that still fits the data to the stated tolerance, record the AAA pole set and measure its Hausdorff distance to the exact pole train. Repeat with 1e-14 additive noise in the data. As a control, run the same stability test on the gamma-function pole example from Section 4. If pole sets change materially across runs that all fit the data to 1e-13, pole-based inference is unstable and the broad-utility claim should be restricted; if they remain stable for the control but not Burgers, the failure is specific and should be disclosed as such.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The review's central thesis is that AAA is a routine, near-best tool for tasks including pole/zero finding, resonance/eigenvalue computation, and singularity detection. The load-bearing condition is empirical reliability of AAA's computed poles and residues. Section 14 undermines this condition: the reproduction of VandenHeuvel et al. shows AAA pole sets for a Burgers solution that are very different beyond the pair closest to the real axis even though the approximants agree with the true function to 1e-13 on the real axis. This is a documented counterexample to the inference that a good data fit implies correct singularity structure. The same inference is used in Sections 4, 16, 17, 18, and 22. The paper acknowledges the limitation and warns readers, but it does not quantify how often this occurs or provide a diagnostic to detect silent pole failure. Since the broad-utility claim is a thesis backed by selected examples, this failure mode means the thesis is conditional on pole reliability being acceptable for the intended use. It does not overturn the review—the bookkeeping is honest—but it is the place where the argument is least secure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":55380,"tokens_out":6474,"duration_ms":59959,"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":[{"comment":"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":"Section 14; also Sections 4, 16–18, 22"},{"comment":"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":"Section 20, Eqs. (20.2)–(20.3); Section 21"},{"comment":"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.","section":"Section 10; Section 2"}],"minor_comments":[{"comment":"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":"Section 21, code after Eq. (21.7)"},{"comment":"The entry 'same' for mode 3 in the closed-circle column would be clearer if the numerical value 3.8317060 were repeated.","section":"Section 16, table"},{"comment":"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":"Section 12, Theorem 12.1"},{"comment":"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.","section":"Section 4, Eq. (4.1)"}],"recommendation":"major_revision","confidential_remarks":"The authors are the inventors of AAA, and the manuscript cites their own and closely related work very heavily, including submitted/unpublished papers. This is not disqualifying because the demonstrations are reproducible and the limitations are generally stated honestly, but the editor may wish to have the submitted companion papers (Bruno–Santana–Trefethen; Horning–Trefethen) independently reviewed before the results they underwrite are taken as established. The main revision request concerns the reliability of pole-based inference, which is acknowledged but not quantified in the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nVerdict: 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.","headline":"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.","tokens_in":55963,"tokens_out":1997,"would_cite":true,"duration_ms":18784,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A20","65D15","65E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The AAA algorithm makes rational approximation a routine, near-best tool across numerical analysis.","keywords":["AAA algorithm","rational approximation","barycentric representation","poles and zeros","analytic continuation","quadrature","model order reduction","singularity location"],"falsifier":"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.","tokens_in":54935,"feed_emoji":"🧮","tokens_out":5521,"duration_ms":47740,"temperature":0.7,"pith_summary":"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.","feed_headline":"One algorithm turns rational approximation into a routine tool","feed_subtitle":"A review shows AAA fitting solving problems in 26 areas, from pole-finding to PDEs and scattering.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["AAA: one rational approximation algorithm to rule them all","Rational approximation: AAA does it in 26 fields","AAA algorithm: near-machine-precision rational fits anywhere","One greedy algorithm solves rational approximation across numerics","Poles, zeros, PDEs: AAA handles them all"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["AAA: one rational approximation algorithm to rule them all","Rational approximation: AAA does it in 26 fields","AAA algorithm: near-machine-precision rational fits anywhere","One greedy algorithm solves rational approximation across numerics","Poles, zeros, PDEs: AAA handles them all"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001433,"raw_usage":{"total_tokens":5491,"prompt_tokens":492,"completion_tokens":4999,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":236,"completion_tokens_details":{"reasoning_tokens":4936}},"tokens_in":236,"tokens_out":4999,"duration_ms":30362,"temperature":1.0,"reasoning_tokens":4936,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:16:02.916674+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}