An algorithm for real and complex rational minimax approximation

Lloyd N. Trefethen; Yuji Nakatsukasa

REVIEW 3 major objections 4 minor 86 references

An algorithm for real and complex rational minimax approximation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A practical algorithm for complex rational minimax approximation that deserves serious peer review, but the 'few percent of minimax' claim needs a systematic audit. read the letter →

arxiv 1908.06001 v1 pith:AA5EL5HD submitted 2019-08-16 math.NA cs.NA

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

classification math.NAcs.NA MSC 41A2065D15

keywords rationalapproximationminimaxbarycentricformulaAAAalgorithmAAA-Lawsoniterativelyreweightedleastsquarescomplex

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Dear colleague,

This 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.

The 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.

None 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.

I 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Section 1 and Fig. 5.2] 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 3 and Section 6, failure mode 1] 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 3 and Fig. 4.4] 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.

minor comments (4)

[Section 5, Fig. 5.2] 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 4, Fig. 4.4] 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.
[Abstract] There is a typographical spacing error in 'the re appear' in the abstract.
[Section 4] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found: the AAA-Lawson claims are benchmarked against external CF lower bounds and Chebfun minimax, and the self-citations provide initialization rather than a forced conclusion.

full rationale

The derivation chain is self-contained in the sense required by the circularity analysis. Step (I) of the algorithm uses the AAA algorithm [51] only to obtain an initial rational approximant and support points; the near-optimality claim is not obtained from AAA alone. Step (II) solves the linearized IRLS problem (3.4)-(3.7) by a minimal singular value computation, and the weights are updated by the standard Lawson rule (3.7). No parameter is fitted to a target and then renamed as a prediction. The paper's key quantitative evidence is external: in Fig. 4.1 the AAA-Lawson error 9.944364e-11 is compared with the Caratheodory-Fejer lower bound sigma_6 = 9.944144081e-11, an independent benchmark computed from Taylor coefficients; in Fig. 5.2 the |x| result 1.23e-4 is honestly compared with Chebfun minimax's 1.07e-4, showing a gap rather than forcing agreement. The self-citations to [51] and [21] are references to prior algorithmic building blocks, not to a uniqueness theorem or an unstated ansatz that makes the conclusion true by definition. The paper explicitly disclaims a convergence theorem ('we do not have a result comprehensive enough to report here'), so no theorem is being imported from the authors' prior work. Concerns that the 'within a few percent' headline is not systematically established, or that the discrete-continuum gap is not quantified, are correctness and evidentiary risks, not circularity. Score 1 reflects the presence of minor self-referential context without any load-bearing circular step.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The algorithm introduces no new mathematical entities; it combines known barycentric representations, the AAA greedy selection, and Lawson's IRLS iteration. The ledger captures the hand-chosen termination and grid parameters, plus the two unproved assumptions: discrete grid fidelity and IRLS convergence for the nonlinear problem.

free parameters (3)

Number of Lawson steps = 20 default, user-specified via 'lawson' option
Termination condition for the Lawson phase. The paper states (Section 3) that Chebfun takes 20 steps by default and that more steps are used when more digits are needed (Section 4, Fig. 4.4 discussion). Results depend on this choice.
Grid clustering parameters = tanh(linspace(-12,12,n)) with n varying per example
The sample point distribution near singularities, corners, and endpoints is chosen per example (Sections 4 and 5). The paper emphasizes that such clustering is crucial for success with clustered poles, making this a hand-chosen input.
Underrelaxation parameter = not specified
Section 6, failure mode 6, mentions that convergence can be recovered by underrelaxation in the update (3.7), but no formula or value is given. If used, this would be a free parameter governing the iteration.

assumptions (3)

standard math Every degree n rational function can be represented in the barycentric form (3.2) with arbitrary distinct support points (Theorem 3.1).
This justifies fixing the support points chosen by AAA and only adjusting coefficients. Proved in Section 3 via partial fractions or linear independence of monic polynomials.
domain assumption The discrete sample set Z represents the continuum in the supremum norm accurately enough that the discrete minimax solution is close to the continuous one.
The algorithm optimizes over a finite set Z. Section 6, failure mode 1, states that too coarse a discretization near singularities causes failure, so this assumption is load-bearing and not guaranteed.
ad hoc to paper The IRLS/Lawson iteration for the linearized barycentric problem converges to a local minimum of the nonlinear minimax problem.
This is the core heuristic. Section 6 explicitly states 'we do not have a result comprehensive enough to report here' and that AAA-Lawson 'has little theoretical foundation at present'. All numerical success relies on this unproved convergence.

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Cite this review

Pith. "Pith review of An algorithm for real and complex rational minimax approximation." pith.science (2026). https://pith.science/paper/AA5EL5HD

@misc{pith2026190806001,
  author       = {Pith},
  title        = {Pith review of: An algorithm for real and complex rational minimax approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AA5EL5HD}},
  note         = {Machine review of arXiv:1908.06001}
}

read the original abstract

Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Such a method is introduced here, the {\em AAA-Lawson algorithm,} available in Chebfun. The new algorithm solves a wide range of problems on arbitrary domains in a fraction of a second of laptop time by a procedure consisting of two steps. First, the standard AAA algorithm is run to obtain a near-best approximation and a set of support points for a barycentric representation of the rational approximant. Then a "Lawson phase" of iteratively reweighted least-squares adjustment of the barycentric coefficients is carried out to improve the approximation to minimax.

Figures

Figures reproduced from arXiv: 1908.06001 by the authors.

**Figure 4.** begins with the basic example of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 4.** shows approximations on two noncircular domains. In the fi [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: turns to problems with singularities on the boundary, wher [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 4.** Figure 4: shows two approximations on domains that are just arcs, [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 5.** Figure 5: turns to functions with a singularity or near-singularity in t [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 5.** Figure 5: shows approximation of exp( [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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