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REVIEW 3 major objections 5 minor 62 references

Quadrature formulas from rational approximations

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For an integral with a weight function, a rational approximation to the Cauchy transform yields a quadrature rule whose nodes are the poles and whose weights are the residues, with error controlled by the approximation error on an…

desk verdict A clear, honest paper that turns a classical residue identity into a practical recipe; the main gap is the unproven near-optimality of AAA-generated rules, which the authors themselves flag. read the letter →

arxiv 2507.14971 v1 pith:PZLOXMT3 submitted 2025-07-20 math.NA cs.NA

Andrew Horning , Lloyd N. Trefethen This is my paper · ORCID
classification math.NAcs.NA MSC 41A2065D32
keywords rationalapproximationquadratureformulasCauchytransformAAAalgorithmpolesandresiduesbranchcutscontourintegralsmatrixfunctions
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

Solving an integral numerically usually means finding nodes and weights by some bespoke analysis. This paper argues that many such rules are the same object: a rational approximation to the Cauchy transform of the weight function on the integration arc. Once the transform is approximated, the poles of the rational function are the nodes and the residues are the weights, with an error bound proportional to the approximation error on an enclosing contour. The paper demonstrates the recipe on seven families of integrals, from classical polynomial-exact rules to singular weights, oscillatory integrands, inverse Laplace contours, and matrix-function Cauchy integrals. If the claim is right, specialized quadrature becomes a routine numerical operation rather than a case-by-case research project.

What carries the argument

The central object is the Cauchy transform $C(s)$ of the weight function, together with a rational approximant $r_n(s)$ whose poles and residues become the quadrature nodes and weights. The argument is carried by the residue theorem: approximating $C$ on an enclosing contour $\Gamma$ converts the integral into a sum of residues, so approximation accuracy in the supremum norm on $\Gamma$ directly controls quadrature error. The computational engine is the AAA algorithm, an adaptive rational approximation routine; the authors also use its sign and damping variants for two-branch target functions, which lets the poles trace approximate branch cuts separating regions where the Cauchy transform takes different values.

What would settle it

One concrete test: for a weight with a known orthogonal-polynomial rule, such as $w(z)=(1-z^2)^{-1/2}$ on $[-1,1]$, generate poles and residues by rational approximation on an enclosing ellipse for increasing degree $n$ and compare the resulting sums $\sum_k c_k f(z_k)$ with the known rule on integrands whose singularities approach the interval; if the error stops tracking the approximation bound or the nodes become ill-conditioned as the contour approaches the singularities, the claim that the method automatically yields near-optimal rules would be falsified.

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Extended reading notes

Core claim

For an integral $I=\int_\gamma f(z)w(z)\,dz$ over an arc $\gamma$, the paper writes $I$ by Cauchy's formula as a contour integral of $f(s)$ times the Cauchy transform $C(s)=\int_\gamma w(z)/(s-z)\,dz$ over an enclosing curve $\Gamma$. If a rational function $r_n$ approximates $C$ on $\Gamma$ to accuracy $\varepsilon$, then $I$ is approximated by $\sum_{k=1}^n c_k f(z_k)$, where $z_k$ are the poles and $c_k$ the residues of $r_n$, and the error is at most $(\varepsilon/2\pi)|\Gamma|\,\|f\|_\Gamma$. The paper's claim is that this identity is the hidden source behind a broad family of quadrature formulas: classical polynomial-exact nodes are poles arranged along an approximate branch cut of $C$, and the optimal placement of that cut is what rational approximation discovers automatically. The authors support the claim by recovering known rules, including the trapezoidal rule on a circle and contour-based inverse Laplace rules, and by generating new rules for weights with algebraic singularities, support on disjoint intervals, oscillatory phases, and matrix spectra, all in a fraction of a second of computation.

Load-bearing premise

The practical generality of the method rests on the assumption that the AAA-type rational approximation algorithm, with the sign and damping modifications, reliably finds a near-best rational approximant on whatever contour the user supplies, and that the resulting poles and residues are well-conditioned quadrature nodes; no theorem guarantees this, and the paper notes that hand-tuning is sometimes needed.

Editorial extensions

If this is right

  • The classical $n$-point rules that integrate polynomials of degree up to $2n-1$ exactly are reproduced, with poles and residues matching the standard nodes and weights, so the rational-approximation view supplies a derivation of them independent of orthogonal polynomials.
  • New rules for non-classical weights---algebraically singular, supported on disjoint intervals, oscillatory, or defined only through samples---can be generated automatically from the weight function alone.
  • For inverse Laplace transforms the method reaches the optimal convergence rate $(9.28903\ldots)^{-n}$, roughly double the rate of earlier parabolic, hyperbolic, and cotangent contour rules.
  • For functions of matrices defined by Cauchy integrals, rational approximation reproduces the fast contour-integral constructions without explicit conformal maps, and it extends naturally to non-normal spectra in rectangles.

Reading between the lines

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

  • One consequence the authors leave implicit is that any quadrature rule can be diagnosed by forming the rational function whose poles are its nodes and comparing the induced branch cut with the true Cauchy transform's cut; systematic disagreement would locate suboptimality.
  • A testable extension is to feed numerically sampled Cauchy transforms, not formulas, into the same recipe; weights defined only by data should still yield near-optimal performance without knowing their singularities in advance.
  • Read through potential theory, the optimal node strings should converge to minimal-energy charge distributions, so equilibrium measures of the domain would predict asymptotic node density and give a benchmark for the algorithm's output.
  • For oscillatory integrals, the angle at which pole strings leave endpoints or stationary points should match steepest-descent directions of the phase; checking this for a family of phases would test whether the algorithm rediscovers the contours asymptotic analysis prescribes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper develops a unified framework for deriving quadrature rules from rational approximation of the Cauchy transform of a weight function. For an integral I = ∫_γ f w dz over an arc γ with weight w, equation (2.5) rewrites I as (1/2πi)∫_Γ f C ds, where C is the Cauchy transform of w and Γ is a contour enclosing γ; if a rational function r_n of degree n approximates C on Γ with poles z_k and residues c_k, residue calculus yields the quadrature rule I_n = Σ c_k f(z_k), with error bounded by (ε/2π)|Γ|‖f‖_Γ for ε = ‖C−r_n‖_Γ. The paper applies this template in seven settings: Gauss–Legendre rules and contour-adaptive variants for nearby singularities (§§3–4), singular and nonclassical weights (§5), oscillatory integrals (§6), Hankel contours for inverse Laplace transforms (§7), trapezoidal rules on circles (§9), and contour integrals for matrix functions (§10), with §8 covering the Zolotarev-type sign-function approximations needed for two-contour cases. In each application the AAA algorithm supplies nodes (poles) and weights (residues), and the experiments reproduce established optimal rates or produce new rules for the variant problems. The conceptual message, stated in §11, is that strings of quadrature nodes correspond to approximate branch cuts of the Cauchy transform.

Significance. If the paper's conclusions hold, the Section 2 framework is a genuinely unifying principle: the identity (2.5) and error bound (2.10) convert a rational approximation problem on a user-chosen contour into a quadrature rule with a transparent error estimate, subsuming Gauss, Gauss–Jacobi, trapezoidal, Talbot/Hankel, and conformally transplanted rules as instances of one construction. The seven applications are convincing, reproducible demonstrations — code for every example is printed in the text, and the reported accuracies (10^{−10}–10^{−13}) are checked against known optimal rates such as the 9.28903^{−n} Talbot rate of Section 7 — a strength that any reader can verify directly. The paper is also commendably explicit about scope: Section 10 concedes that hand-tuning of AAA parameters is often needed, and Section 11 lists near-optimality theorems, weight positivity, and potential-theory links as open problems. The value is as an insight-and-tools contribution resting on the empirical reliability of AAA; the words 'routine' and 'near-optimal' in the Abstract and Section 11 outrun what is established, but the construction and the demonstrations are sound.

major comments (3)
  1. [Section 7] The paragraph after Eq. (7.1) asserts: 'Since e^z is analytic, it follows from (2.4) that the Cauchy transform C(s) is e^s itself.' This is not what (2.4) gives. With C(s) = ∫_γ w(z)/(s−z) dz as defined in (2.4), a deformation of the Hankel contour produces C(s) = ±2πi e^s, the sign depending on the contour orientation, and only under the alternative 1/(2πi) normalization — which Section 2 explicitly notes it is not using — would one obtain ±e^s. The factor cancels in the final quadrature sum only because of the 1/(2πi) in (2.5), but as written the identity is inconsistent with the paper's own convention. In addition, the test integrand f(z) = −e/(1+z) has a pole at z = −1 lying on the cut (−∞,0] that Γ is taken to be, so the reference value 'integral 1' requires a convention about indenting the contour or taking a principal value. Please supply the derivation with explicit orientation and normalization, and state how this pole on Γ is treated.
  2. [Abstract; Sections 10–11] The Abstract's claim that rational approximation is 'now a routine technology' providing 'an easy new method' is stronger than the body establishes. Section 10 states that 'hand-tuning is often needed to get the best results,' and the examples of §§3–10 use not the default AAA but the recent 'sign', 'lawson', and 'damping' options from the companion paper [59]; Section 11, in turn, lists as open whether best rational approximations match Gauss convergence rates. The demonstrations are compelling for the configurations shown, but the advertised ease and generality are exactly the points for which the evidence is heuristic. I recommend rewording the Abstract and Section 11 (e.g., 'for many standard configurations' and 'empirically near-optimal') so that the paper's central claim is consistent with its own stated limitations; this is a wording fix rather than a demand for a new theorem, but it concerns the paper's main message.
  3. [Section 2, around (2.8)] Equation (2.8) and the surrounding text assume that all n poles of r_n lie inside Γ, but the practical recipe never states what to do when this fails. AAA on a user-specified contour routinely produces some poles outside Γ or spurious near-canceling pole-zero pairs; such poles simply do not contribute to the residue sum, so the error bound (2.10) remains valid with I_n restricted to the interior poles, while a pole lying close to Γ inflates ε = ‖C−r_n‖_Γ and degrades the rule. Since the advertised use case is generating rules on arbitrary contours, a short paragraph in Section 2 stating this, together with how the examples verify the condition, would materially improve the practical exposition.
minor comments (5)
  1. [Figure 2 caption] The caption of Figure 2 contains an unmatched parenthesis in 'log((z+1)/(z−1)))'; the intended expression is C(s) = log((s+1)/(s−1)).
  2. [Section 9, code block] In the code for the circle example, the target values F = [zeros(size(S)); −ones(size(S))] differ in sign from the stated Cauchy values in (9.3), where C = 1 inside and C = 0 outside; a sentence explaining that the inner component carries the value −C because of the negative orientation of Γ in (9.4) would let the reader map the code to the equations.
  3. [Section 3] The text states that the AAA weights 'are also close to Gauss weights,' but no quantitative comparison is shown; a short table or the norm of the weight difference for a few degrees would let the reader verify this claim directly, since the displayed convergence plots only test the combined node-weight rule.
  4. [Sections 5–6] The Cauchy transform samples in Sections 5–6 are computed with MATLAB's quadgk, with tolerances stated only in Section 5; please state the tolerances used in Section 6 as well, and add a remark on the accuracy to which C must be sampled so that it does not dominate the claimed quadrature errors, which reach 10^{−10}–10^{−13}.
  5. [References] References [6] and [45] do not appear to be cited in the text; please check the reference list for uncited entries.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: quadrature rules are generated from rational approximation of the Cauchy transform and checked against independent benchmarks; the only mild concern is reliance on the authors' own AAA and sign/damping tooling, which is a robustness caveat, not a circular step.

full rationale

The core chain in Section 2 is non-circular. Equations (2.5)-(2.10) are exact identities: for any rational r_n approximating the Cauchy transform C(s) on Gamma, (2.8) is the residue expansion of the contour integral, and (2.10) bounds the quadrature error by ||C - r_n||_Gamma and ||f||_Gamma. The nodes z_k and weights c_k are poles and residues of this rational approximant, not parameters fitted to the integral values I; the test integrands in Sections 3-10 are independent of the construction. Recovering Gauss-Legendre, the periodic trapezoidal rule, and Talbot contours is a cross-check against established external results, not a definition of those rules. The paper does cite the authors' own AAA/AAA-Lawson algorithms ([42], [43]) and the 2024 sign/damping modifications ([59]) as the practical engine, and Section 10 concedes that hand-tuning is often needed and Section 11 lists as open the theorem that best rational approximants match Gauss rates; these are robustness caveats, not circularity. Some contours (e.g., the Bernstein ellipse through +/- i/sqrt(20) in Section 3) are chosen from knowledge of the test integrand's singularities, which is favorable selection but again not fitting of the rule to I. No equation or claim in the paper reduces to its own input, and no load-bearing uniqueness theorem is imported from self-citations.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The mathematical theorem in Section 2 is self-contained and uses only standard complex analysis. The practical method depends on user-chosen contours, degrees, and AAA tuning, plus an unproven near-optimality heuristic. No new physical or mathematical entities are introduced.

free parameters (3)
  • Rational approximation degree n = 14, 20, 31, 32, 40 (varies by example)
    User-specified or tolerance-driven; determines the number of quadrature nodes and the accuracy of the rule. The examples choose n to balance accuracy and conditioning (Sections 3, 7, 9, 10).
  • Contour parameters for Γ = rho = 1/sqrt(20)+sqrt(21/20), slits at ±0.1i, stadium radius 1/sqrt(20), sector angle pi/3
    The error bound (2.10) depends on Γ, and the examples set Γ from the known singularities of the test integrands. This is a user-supplied modeling choice rather than a constant fitted to the final result.
  • AAA tuning parameters (lawson iterations, damping) = lawson=20 or 50; damping=0.5
    Section 10 says 'for clean results... we modified the call to aaa.m'; the paper admits hand-tuning is often needed. These values affect the computed nodes and weights.
assumptions (4)
  • standard math Cauchy integral formula and residue theorem apply to the chosen contours
    Invoked in Section 2, equations (2.2) and (2.8), to convert the arc integral into a contour integral and then into a sum of residues.
  • domain assumption The integrand f is analytic in the region enclosed by Γ, and Γ avoids the singularities of C
    The error bound (2.10) assumes this; each application must arrange a suitable Γ. The paper does not fully analyze cases where Γ passes near singularities, though it uses them numerically.
  • domain assumption The AAA algorithm with sign and damping modifications provides near-best rational approximants
    The practical method depends on the reliability of AAA-Lawson and its sign/damping variants from the companion paper [59]; details are deferred to that paper in Section 8.
  • ad hoc to paper Near-optimal rational approximants produce near-optimal quadrature rules
    Asserted in Section 4 ('near-optimal rational approximations are always choosing a near-optimal contour') and implied throughout; no proof is supplied, and Section 11 lists it as an open question.

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Pith. "Pith review of Quadrature formulas from rational approximations." pith.science (2026). https://pith.science/paper/PZLOXMT3

@misc{pith2026250714971,
  author       = {Pith},
  title        = {Pith review of: Quadrature formulas from rational approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PZLOXMT3}},
  note         = {Machine review of arXiv:2507.14971}
}
read the original abstract

It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new method to derive all kinds of quadrature formulas as well as fundamental insight into the mathematics of quadrature. Intervals or curves of quadrature nodes correspond to near-optimal branch cuts of the Cauchy transform.

Figures

Figures reproduced from arXiv: 2507.14971 by the authors.

Figure 3
Figure 3. see Fig. 4.1 of [28], [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

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