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arxiv: 2209.02302 · v3 · pith:QY7X3O5Tnew · submitted 2022-09-06 · 🧮 math.NA · cs.NA· hep-lat· physics.comp-ph

Towards non-linear quadrature formulae

Georg M. von Hippel This is my paper

Pith reviewed 2026-05-24 10:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NAhep-latphysics.comp-ph
keywords quadrature formulaenumerical integrationNewton-Cotesnonlinear methodserror boundsaffine transformationsscaling transformationsexponential functions
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The pith

Nonlinear quadrature formulae can be explicitly constructed to exactly integrate families of functions related by scaling or affine transformations, matching Newton-Cotes accuracy on the same nodes.

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

The paper starts from the observation that integrals of functions like exponentials f(x) = λ e^{αx} suggest a need for more flexible integration rules. It shows that nonlinear generalizations of quadrature formulae can be built explicitly for any function family obtained from a base function by scaling or affine argument changes. These rules deliver the same accuracy as Newton-Cotes formulae using identical nodes, and the familiar linear Newton-Cotes rules appear as the special case inside the new formalism. Explicit error bounds are derived that reduce to the classical Newton-Cotes bounds when the nonlinear rule is linearized.

Core claim

The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as Newton-Cotes formulae based on the same nodes, with the latter emerging as the linear case of our general formalism. We also derive explicit bounds on the error of the nonlinear quadrature formulae, which in the linear case devolve into the well-known bounds for Newton-Cotes formulae.

What carries the argument

The nonlinear generalization of quadrature formulae constructed to integrate exactly any function family generated from a base function by scaling or affine transformations of the argument.

If this is right

  • The nonlinear formulae achieve identical accuracy to Newton-Cotes formulae on the same nodes.
  • Newton-Cotes formulae arise directly as the linear special case inside the general construction.
  • Explicit error bounds hold for the nonlinear formulae and reduce to the classical Newton-Cotes bounds in the linear limit.
  • The construction applies to families such as scaled exponentials f(x) = λ e^{αx} and similar transformed functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction might be tested on function families closed under other transformations such as inversion or exponentiation to see whether exact integration remains possible.
  • Numerical implementations could compare the new nonlinear rules against standard methods on exponential integrands arising in differential equations.
  • If the error bounds prove sharp, they could guide node placement for nonlinear rules in the same way classical bounds guide Newton-Cotes node choice.

Load-bearing premise

Nonlinear generalizations of quadrature formulae exist and can be explicitly constructed for the wide class of functions generated from a given function by scaling or affine transformations of the argument.

What would settle it

Finding even one function family generated by scaling or affine transformation for which no nonlinear quadrature formula exists that matches the accuracy of Newton-Cotes on the same nodes would falsify the central claim.

Figures

Figures reproduced from arXiv: 2209.02302 by Georg M. von Hippel.

Figure 1
Figure 1. Figure 1: Comparison between the non-linear exponential rule (blue) and the trapezoidal rule (red) of the relative error | ˆI[f] − I[f]|/|I[f]| (left) and the ratio of the errors between the two rules (right) on a range of integrands (top to bottom: e−x + 1 2 e −2x , [ex − 1]−1 , cosh x, sin x). See the text for details [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between the higher-order non-linear ex￾ponential rule (blue) and Simpson’s rule (red) of the relative error | ˆI[f]−I[f]|/|I[f]| (left) and the ratio of the errors between the two rules (right) on a range of integrands (top to bottom: e−x + 1 2 e −2x , [ex − 1]−1 , cosh x, sin x). See the text for details [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the results from multistep eval￾uation of integrals using the non-linear exponential rule (blue) and the trapezoidal rule (red) on the left, and the higher-order non￾linear exponential rule (blue) and Simpson’s rule (red) on the right. The integrands are (top to bottom) e−x+ 1 2 e −2x , [ex−1]−1 , cosh x, sin x. Note the differences in scale. See the text for details [PITH_FULL_IMAGE:fi… view at source ↗
read the original abstract

Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by affine transformations of the argument using nonlinear generalizations of quadrature formulae. The main result of this paper is that such formulae can be explicitly constructed for a wide class of functions, and have the same accuracy as Newton-Cotes formulae based on the same nodes, with the latter emerging as the linear case of our general formalism. We also derive explicit bounds on the error of the nonlinear quadrature formulae, which in the linear case devolve into the well-known bounds for Newton-Cotes formulae.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates nonlinear generalizations of quadrature rules that exactly integrate families of functions generated from a base function via scaling or affine argument transformations. Motivated by the exponential family f(x) = λ e^{αx}, the central claim is that such nonlinear formulae can be explicitly constructed for a wide class of functions, achieve the same accuracy (degree of exactness) as Newton-Cotes rules on identical nodes, reduce to the linear Newton-Cotes case, and admit explicit error bounds that recover the classical Newton-Cotes bounds.

Significance. If the explicit-construction claim holds uniformly for a genuinely broad class without function-specific nonlinear solves, the work would provide a systematic extension of quadrature theory beyond polynomials, with the linear reduction serving as a useful consistency check. The error-bound derivation is a positive feature when it correctly specializes.

major comments (2)
  1. [§3] §3 (construction of the nonlinear rule): the assertion that explicit constructions exist for a 'wide class' of base functions is load-bearing for the central claim, yet the procedure reduces to recovering the family parameters (e.g., α, λ) from nodal values. For the exponential example this is closed-form via logarithms of ratios, but the manuscript does not exhibit a uniform closed-form procedure that remains explicit for qualitatively different base functions whose parameter recovery requires solving a nonlinear system without elementary solutions. This gap directly affects whether the 'explicit' qualifier holds beyond the motivating family.
  2. [§4] §4 (accuracy statement): the claim that the nonlinear rule has 'the same accuracy as Newton-Cotes formulae based on the same nodes' is stated without a precise definition of accuracy for the nonlinear case (e.g., whether it means exactness on the full nonlinear family or only on the linear span). The reduction argument to the linear case is given, but the general accuracy statement needs an explicit theorem relating the two.
minor comments (2)
  1. Notation for the nonlinear weights and nodes is introduced without a consolidated table comparing them to the classical Newton-Cotes coefficients; adding such a comparison would improve readability.
  2. The error-bound derivation in the final section would benefit from an explicit statement of the remainder term before specializing to the linear case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the construction procedure and the accuracy claim. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the nonlinear rule): the assertion that explicit constructions exist for a 'wide class' of base functions is load-bearing for the central claim, yet the procedure reduces to recovering the family parameters (e.g., α, λ) from nodal values. For the exponential example this is closed-form via logarithms of ratios, but the manuscript does not exhibit a uniform closed-form procedure that remains explicit for qualitatively different base functions whose parameter recovery requires solving a nonlinear system without elementary solutions. This gap directly affects whether the 'explicit' qualifier holds beyond the motivating family.

    Authors: The manuscript demonstrates the explicit construction explicitly for the exponential family, where the parameters are recovered in closed form from nodal values. The phrase 'wide class' is intended to cover families closed under scaling and affine transformations for which the parameter recovery step admits an explicit algebraic solution (as illustrated by the motivating example). We acknowledge that the text does not provide a uniform closed-form procedure applicable to arbitrary base functions whose recovery would require solving general nonlinear systems. We will revise §3 to delineate more precisely the subclass of base functions for which the full construction remains explicit, thereby clarifying the scope of the central claim. revision: yes

  2. Referee: [§4] §4 (accuracy statement): the claim that the nonlinear rule has 'the same accuracy as Newton-Cotes formulae based on the same nodes' is stated without a precise definition of accuracy for the nonlinear case (e.g., whether it means exactness on the full nonlinear family or only on the linear span). The reduction argument to the linear case is given, but the general accuracy statement needs an explicit theorem relating the two.

    Authors: Accuracy for the nonlinear quadrature is understood as exactness on the full nonlinear family generated by the base function under the allowed transformations. The reduction to the linear Newton-Cotes case is shown by specialization, but we agree that an explicit theorem relating the two notions of accuracy is desirable. We will add a dedicated theorem in §4 that (i) defines the degree of exactness for the nonlinear rule and (ii) proves that this degree coincides with the classical Newton-Cotes degree when the nonlinear family is restricted to its linear span. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation anchored externally to Newton-Cotes

full rationale

The paper's central claim constructs nonlinear quadrature rules for function families closed under scaling/affine transforms, then shows that the linear case recovers Newton-Cotes formulae of matching accuracy. This reduction supplies an independent external benchmark rather than a self-referential loop. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the provided text. The explicit-construction claim for the wide class is presented as a result, not presupposed by definition. The error bounds likewise reduce to the known Newton-Cotes case without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that such nonlinear rules exist and match the accuracy of linear ones for the specified function class; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Nonlinear quadrature formulae can be explicitly constructed for a wide class of functions generated by scaling or affine transformations of a base function.
    This is the main result stated in the abstract.

pith-pipeline@v0.9.0 · 5636 in / 1251 out tokens · 30142 ms · 2026-05-24T10:45:17.941114+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Yeramian and P

    E. Yeramian and P. Claverie, Analysis of multiexponential functions without a hypothesis as to the number of components, Nature 326 (1987) 169–174

    work page 1987

  2. [2]

    Vector Correlators in Lattice QCD: methods and applications

    D. Bernecker and H. B. Meyer, Vector Correlators in Lattice QCD: Methods and applications, Eur. Phys. J. A 47 (2011) 148, doi:10.1140/epja/i2011-11148-6 [arXiv:1107.4388]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epja/i2011-11148-6 2011

  3. [3]

    Werner and L

    H. Werner and L. Wuytack, Nonlinear Quadrature Rules in the Presence of a Singularity, Comp. & Maths. with Appls. 4 (1978) 237–245

    work page 1978

  4. [4]

    Wuytack, Numerical integraton by using nonlinear techniques, J

    L. Wuytack, Numerical integraton by using nonlinear techniques, J. Comp. Appl. Math. 1 (1975) 267–272

    work page 1975

  5. [5]

    L. N. Trefethen, Exactness of Quadrature Formulas, SIAM Review 64 (2022) 132–150, doi:10.1137/20M1389522 [arXiv:2101.09501]. Institut f ¨ur Kernphysik & PRISMA + Cluster of Excellence, University of Mainz, 55099 Mainz, Germany 12 G. M. VON HIPPEL ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ 10-5 10-4 0.001 0.010 0.100 1 10-11...

    work page doi:10.1137/20m1389522 2022