{"id":"babaa67b-282e-4e4f-92ad-41014945ea0c","arxiv_id":"1908.06001","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"AAA-Lawson combines the AAA algorithm with a Lawson-style iteratively reweighted least-squares phase to compute near-minimax rational approximations on arbitrary complex domains, with working code in Chebfun.","lead":"This paper introduces AAA-Lawson, a fast algorithm for computing near-optimal rational approximations of complex functions on arbitrary domains. It ships working code in Chebfun, making complex rational minimax approximation practical for the first time.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed 'within a few percent of minimax' is not established: the paper's own |x| example reports error 1.23e-4 vs 1.07e-4 minimax, a 15% gap, and no systematic optimality audit is provided.","rationale":"The paper introduces a genuinely useful algorithm with extensive experiments, honest documentation of failure modes, and a clear presentation of the barycentric IRLS construction. The reader's CONDITIONAL verdict is appropriate. The reader identified the discrete-grid proxy as the weakest assumption; that is real, but I see an even more immediately load-bearing issue: the central empirical claim of near-optimality is not quantified against independent minimax values, and one of the paper's own reported examples (|x|, 15% above the Chebfun minimax result) contradicts the 'within a few percent' wording. This is not a charge of dishonesty; it is a scope-of-claim problem. The default 20-step Lawson rule and the linear convergence acknowledged in Section 6 make the gap plausible. A systematic ratio audit would settle whether the claim is true for a broad class and should be reported. My concern does not change the reader's verdict, which already calls for qualification and additional support; it sharpens the reason.","tokens_in":17986,"tokens_out":15765,"duration_ms":155209,"concrete_test":"Reproduce all examples of Sections 4 and 5 with Chebfun's default aaa settings and compute a reference minimax error E* for each by an independent method (Chebfun minimax or cf for real intervals; high-resolution Remez, Ellacott-Williams, or Caratheodory-Fejer for complex problems where available). Tabulate the ratio ||f - r_AAA-Lawson||_inf / E*. In particular, rerun the |x| example with n=10 using both default settings and 1000 Lawson steps, and compare with Chebfun minimax. If the ratio for |x| remains above 1.05, or if several other examples exceed 1.05, the abstract's 'within a few percent' claim should be revised to a qualified statement with explicit exceptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in Section 1 is that AAA-Lawson converges 'to an approximation with an error within a few percent of the minimax value' across a wide range of problems. This is not supported by a systematic quantitative audit, and the paper's own results contain a concrete tension. In Section 5, Fig. 5.2, the approximation of |x| on [-1,1] with n=10 gives an AAA-Lawson error of 1.23e-4, while Chebfun minimax gives 1.07e-4; the ratio is 1.15, i.e. 15% above the minimax value, not 'a few percent'. For most complex examples in Section 4, no independent minimax error is reported; near-optimality is inferred visually from error curves and from near-circularity arguments that the paper itself describes as unproved. The default stopping rule of 20 Lawson steps (Section 3) is also admitted in Section 6 to be insufficient in at least one case (Fig. 4.4), where 'a few hundred' steps are needed to reach the converged optimum. Thus the headline claim that default-mode computation returns near-minimax errors across a wide range is not established as stated, even before the additional continuum-versus-discrete grid issue identified by the reader.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the AAA-Lawson algorithm for rational minimax approximation in the complex plane and on real intervals. The method first runs the AAA algorithm to select support points and obtain an initial barycentric rational approximant, then applies a Lawson/IRLS iteration to adjust the barycentric coefficients by solving a sequence of weighted linear least-squares problems (Eqs. (3.4)–(3.7)). The authors claim that, for a wide range of problems, the method converges in a fraction of a second on a laptop to an approximation whose error is within a few percent of the minimax value. The paper presents fourteen complex and eight real numerical examples, with comparisons to the Ellacott–Williams method, Chebfun's minimax command, and Carathéodory–Fejér lower bounds. Section 6 discusses six failure modes and explicitly states that no comprehensive convergence proof is available.","tokens_in":18288,"tokens_out":3910,"duration_ms":37625,"significance":"If the near-optimality claim is substantiated, the paper fills a genuine gap: there are few practical algorithms for complex rational minimax approximation, and the AAA-Lawson algorithm is simple, uses a numerically stable barycentric representation, and is available in Chebfun. The paper's strengths include the breadth of examples, honest discussion of failure modes, and external validation against independent benchmarks in several cases. The main weakness is that the headline quantitative claim is not systematically demonstrated, and at least one reported example contradicts the 'within a few percent' phrasing. This is a potentially important practical contribution, but the empirical evidence needs to be tightened before the central claim can be accepted as stated.","major_comments":[{"comment":"The central claim that AAA-Lawson converges 'to an approximation with an error within a few percent of the minimax value' is not supported as stated. In the |x| example of Fig. 5.2, the reported AAA-Lawson error is 1.23e-4 versus 1.07e-4 from Chebfun minimax, a 15% excess. For most complex examples in Section 4, no independent minimax error is reported; near-optimality is inferred visually from near-circular error curves, an argument that the paper itself describes as unproved in the final paragraph of Section 4. I request a systematic audit: a table listing, for every example, the computed error and the best available independent upper or lower bound (CF, EW, Chebfun minimax, or a refined-grid re-run), together with a revised wording of the claim calibrated to the observed ratios.","section":"Section 1 and Fig. 5.2"},{"comment":"The algorithm minimizes over a discrete sample set Z (Eq. (3.4)), while the paper claims to approximate the continuum minimax problem. Section 6 acknowledges 'Discretization too coarse' as a failure mode but provides neither an a priori condition nor an a posteriori check relating the grid to the gap between discrete and continuous minimax errors. In particular, the tanh-clustered grids used near singularities are chosen heuristically. Please add a grid-refinement check or an estimate of the discretization error, and state how the reported errors are verified with respect to the continuum.","section":"Section 3 and Section 6, failure mode 1"},{"comment":"The default termination criterion of 20 Lawson steps is admitted to be insufficient in at least one case: the second row of Fig. 4.4 states that the maximum error is attained at 20 points only if 'a few hundred' Lawson steps are taken, not with the default 20 steps. Since the headline claim concerns default-mode computation in a fraction of a second, the paper should either provide a principled stopping rule or explicitly document, for each example, the number of Lawson steps used and whether the default mode produced the reported error. This is necessary to support the claim that the method is near-optimal in default operation.","section":"Section 3 and Fig. 4.4"}],"minor_comments":[{"comment":"The text says that the |x| approximation shows '12 poles lining up along a branch cut,' but the degree n used for this example is not stated in the text; the number of poles should be consistent with the stated degree.","section":"Section 5, Fig. 5.2"},{"comment":"The phrase '10,000 times larger' in the discussion of the second row is used without an explicit reference to the first row's error; please make the comparison explicit.","section":"Section 4, Fig. 4.4"},{"comment":"There is a typographical spacing error in 'the re appear' in the abstract.","section":"Abstract"},{"comment":"The sentence 'The codes of this section and the next are available in the supplementary materials' should give a specific version or release date of the Chebfun implementation so that the numerical results are reproducible.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The paper is unusually honest about its limitations, which is to its credit. However, the discrepancy between the abstract's 'within a few percent' and the |x| example (15% excess) is a concrete issue that the authors should address with a quantitative audit. If the audit confirms near-optimality for most examples and the claim is revised accordingly, the paper could be accepted as an important algorithmic contribution despite the absence of a convergence proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is the paper that makes complex rational minimax approximation practical. The combination of AAA's greedy support-point selection with a Lawson/IRLS phase in a noninterpolatory barycentric form is genuinely new, and the implementation in Chebfun makes it easy to use. The evidence is strong: fourteen complex and eight real examples, including hard cases with singularities, conformal maps, model order reduction, and comparisons to the Ellacott-Williams method, CF lower bounds, and Chebfun minimax. The section on failure modes is unusually honest. I believe the core contribution: a fast, stable, general-purpose heuristic for problems that previously had no practical method.\n\nThe soft spots are real but not fatal. The headline quantitative claim \"within a few percent of the minimax value\" is not actually established by the paper. The |x| example in Figure 5.2 gives AAA-Lawson error 1.23e-4 vs. Chebfun minimax 1.07e-4, which is 15% above minimax. More generally, for most complex examples no independent minimax value is computed; near-optimality is inferred from near-circular error curves, and the paper admits the circularity-based bound is not proved. The default of 20 Lawson steps is also sometimes insufficient, as in Figure 4.4 where a few hundred are needed. So the abstract's \"within a few percent\" is too strong as a general statement. A systematic audit reporting ratios to known lower bounds would fix this. The discrete-grid adequacy issue is acknowledged but not analyzed; there is no a priori condition relating mesh to the gap between discrete and continuous minimax errors. That is a mild concern for a numerical paper, but it should be stated more carefully. And \"arbitrary domains\" oversells the method, since Section 6 lists six failure modes.\n\nNone of this undermines the utility. The algorithm works on a wide range of problems in fractions of a second, and the examples are reproducible via the Chebfun code. The lack of convergence theory is disclosed, not hidden. For anyone working in rational approximation, this is worth reading and citing.\n\nI would send it to a serious referee. The paper deserves referee time, with the expectation that the authors soften \"a few percent\" and ideally add a systematic comparison table or lower-bound audit. I would not desk-reject it; the practical contribution is real.","headline":"A practical algorithm for complex rational minimax approximation that deserves serious peer review, but the 'few percent of minimax' claim needs a systematic audit.","tokens_in":18798,"tokens_out":1878,"would_cite":true,"duration_ms":17793,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A20","65D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"AAA-Lawson, a two-step algorithm built from greedy support-point selection and iteratively reweighted least squares, computes near-minimax rational approximations for complex functions and arbitrary domains in fractions of a second.","keywords":["rational approximation","minimax approximation","barycentric formula","AAA algorithm","AAA-Lawson algorithm","iteratively reweighted least squares","complex approximation"],"falsifier":"Run AAA-Lawson on $e^z$ on the unit circle with degree 5 and 500 equispaced points, and compare the returned error with the Hankel singular-value lower bound $\\sigma_6 = 9.944144081\\times 10^{-11}$. The paper reports an error of $9.944364\\times 10^{-11}$; if a re-run yields an error more than a few percent above $\\sigma_6$, the central near-optimality claim fails for this representative case, and if it falls below $\\sigma_6$, the lower bound itself is contradicted.","tokens_in":17767,"feed_emoji":"🧮","tokens_out":11376,"duration_ms":102037,"temperature":0.7,"pith_summary":"Rational minimax approximation—the best rational function of a given degree in the supremum norm—has long been an established topic for real functions on real intervals, but almost no working algorithms existed for complex functions or complex domains. This paper introduces the AAA-Lawson algorithm to fill that gap. The method first runs the AAA algorithm to select support points and a near-best barycentric rational approximant, then applies a Lawson iteratively reweighted least-squares phase to adjust the barycentric coefficients toward the minimax solution. Experiments across disks, ellipses, polygons, arcs, annuli, unions of domains, and random point sets report convergence in a fraction of a second on a laptop, with errors within a few percent of the true minimax value.","feed_headline":"One algorithm computes complex rational minimax fits in under a second","feed_subtitle":"A two-phase method turns minimax rational approximation into routine laptop computation on arbitrary domains.","key_machinery":"The central object is the barycentric rational representation $r(z)=\\sum_{k=0}^n \\alpha_k/(z-t_k)\\big/\\sum_{k=0}^n \\beta_k/(z-t_k)$ in noninterpolatory mode, where the $t_k$ are support points chosen greedily by AAA and the coefficients $\\alpha_k,\\beta_k$ are free parameters. Every rational function of degree $n$ has such a representation, and it is numerically stable even when poles cluster exponentially near singularities. The optimization machinery is Lawson-type iteratively reweighted least squares: at each step the current residual $e_j$ at each sample point becomes a weight $w_j\\leftarrow w_j|e_j|$, and the next coefficients solve the minimal-singular-value problem for the weighted Cauchy matrix in (3.6). At the $n+1$ sample points that coincide with support points, the infinite terms are replaced by the L'Hôpital-style values $(f_j\\beta_{k_j}-\\alpha_{k_j})^2$. This combination turns the nonlinear minimax problem into a short sequence of linear algebra problems.","core_discovery":"The paper's central claim is that one algorithm, AAA-Lawson, can solve rational minimax approximation problems on essentially arbitrary discrete domains—complex or real, connected or disconnected—at laptop speed and near-minimax accuracy. The key step is a switch from the interpolatory barycentric representation used by AAA to a noninterpolatory alpha-beta mode in which numerator and denominator coefficients are freed from the support-point values and optimized directly. The optimization is then carried out by Lawson iteration: a sequence of weighted least-squares problems, each solved as a minimal singular value problem for a Cauchy matrix, with weights updated by the current pointwise error. The paper argues that this combination inherits the numerical stability of AAA while improving typical errors by factors of roughly 2 to 20, and it documents the claim on 22 worked examples spanning analytic, meromorphic, singular, and nonanalytic functions.","pith_inferences":["Editorial inference: the same barycentric-plus-IRLS pattern should transfer to weighted minimax problems and to type $(m,n)$ with $m\\neq n$, extensions the paper notes but does not implement.","Editorial inference: the near-circular error curves observed on analytic complex examples make winding number a cheap, automatic certificate of near-optimality, even where uniqueness theory is unavailable.","Editorial inference: period-2 oscillations in the weight update, one of the paper's named failure modes, might be suppressed by underrelaxation or momentum; this is a testable modification the paper mentions only in passing.","Editorial inference: for real-interval problems where established minimax methods work, AAA-Lawson is unlikely to replace them, but it may serve as a reliable fallback and as an initializer for harder cases such as endpoint singularities and unbounded intervals."],"forward_implications":["Complex rational minimax approximation becomes a routine computation: problems on disks, ellipses, squares, arcs, annuli, unions of domains, and random point clouds can be solved in fractions of a second.","Approximations with exponentially clustered poles near boundary singularities become computable, supporting applications in conformal mapping, model order reduction, and Laplace and Helmholtz solving.","For smooth real-interval problems, the established real-interval minimax solvers remain preferable, but AAA-Lawson extends minimax computation to unbounded intervals, disjoint intervals, and endpoint singularities where those methods struggle.","The algorithm is available as a one-line command with a degree parameter, so non-specialists can obtain near-minimax rational fits without implementing optimality conditions."],"supporting_citations":[{"why":"Supplies the AAA algorithm and the barycentric support-point selection used in the first phase.","marker":"[51]"},{"why":"Introduces the IRLS iteration whose weight update the Lawson phase adapts to the nonlinear barycentric setting.","marker":"[42]"},{"why":"Provides the earlier EW method and published complex rational minimax test values that AAA-Lawson reproduces and improves upon.","marker":"[20]"},{"why":"Establishes the stable adaptive barycentric representation and rational minimax context that motivates the algorithm.","marker":"[21]"},{"why":"Demonstrates how clustered rational poles accelerate Laplace solving, an application class AAA-Lawson targets.","marker":"[29]"},{"why":"Supplies the alternative Istace-Thiran algorithm, whose different approach is compared but not used.","marker":"[38]"},{"why":"Provides the clamped beam model-reduction benchmark with matrix resolvent data that AAA-Lawson approximates.","marker":"[11]"},{"why":"Gives the conformal mapping application where AAA-Lawson improves rational representation accuracy.","marker":"[27]"}],"fun_headline_variants":["Complex rational minimax fits now routine on a laptop","AAA-Lawson solves complex rational minimax in under a second","New algorithm computes complex minimax fits at laptop speed","Rational minimax on complex domains: one algorithm, one second","Fast complex rational minimax via AAA and Lawson iteration"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire method rests on the finite sample set standing in faithfully for the continuous domain; if the sample points are too sparse near a singularity, the computed minimax answer can miss the true one.","fun_headline_variants_meta":{"raw":{"variants":["Complex rational minimax fits now routine on a laptop","AAA-Lawson solves complex rational minimax in under a second","New algorithm computes complex minimax fits at laptop speed","Rational minimax on complex domains: one algorithm, one second","Fast complex rational minimax via AAA and Lawson iteration"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000499,"raw_usage":{"total_tokens":2387,"prompt_tokens":834,"completion_tokens":1553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":1482}},"tokens_in":450,"tokens_out":1553,"duration_ms":9070,"temperature":1.0,"reasoning_tokens":1482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:57:19.195560+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run AAA-Lawson on $e^z$ on the unit circle with degree 5 and 500 equispaced points, and compare the returned error with the Hankel singular-value lower bound $\\sigma_6 = 9.944144081\\times 10^{-11}$. The paper reports an error of $9.944364\\times 10^{-11}$; if a re-run yields an error more than a few percent above $\\sigma_6$, the central near-optimality claim fails for this representative case, and if it falls below $\\sigma_6$, the lower bound itself is contradicted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the IRLS iteration whose weight update the Lawson phase adapts to the nonlinear barycentric setting."},{"cited_title":"Ellacott and J","cited_arxiv_id":null,"evidence_quote":"Provides the earlier EW method and published complex rational minimax test values that AAA-Lawson reproduces and improves upon."},{"cited_title":"Filip, Y","cited_arxiv_id":null,"evidence_quote":"Establishes the stable adaptive barycentric representation and rational minimax context that motivates the algorithm."},{"cited_title":"Gopal and L","cited_arxiv_id":null,"evidence_quote":"Demonstrates how clustered rational poles accelerate Laplace solving, an application class AAA-Lawson targets."},{"cited_title":"Istace and J.-P","cited_arxiv_id":null,"evidence_quote":"Supplies the alternative Istace-Thiran algorithm, whose different approach is compared but not used."},{"cited_title":"Chahlaoui and P","cited_arxiv_id":null,"evidence_quote":"Provides the clamped beam model-reduction benchmark with matrix resolvent data that AAA-Lawson approximates."},{"cited_title":"Gopal and L","cited_arxiv_id":null,"evidence_quote":"Gives the conformal mapping application where AAA-Lawson improves rational representation accuracy."}],"review_version":1}