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The order of convergence of an optimal quadrature formula with derivative in the space $W_2^{(2,1)}$

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Pith's one-line read The paper claims an optimal endpoint-corrected trapezoidal rule in the Sobolev space W_2^{(2,1)} with an O(h^4) error, but the proof of the error formula is algebraically inconsistent.

arxiv 1908.00450 v1 pith:5OHJ2H7E submitted 2019-08-01 math.NA cs.NA

Abdullo R. Hayotov , Rashidjon G. Rasulov This is my paper · ORCID
classification math.NAcs.NA
keywords formulaquadratureerrorobtainedoptimalspacecoefficientsderivative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

When approximating an integral, the trapezoidal rule uses function values at evenly spaced points. This paper adds corrections using the derivative of the function at the two endpoints. The authors work in a space of functions whose second derivative plus first derivative is square-integrable, called W_2^{(2,1)}. They use a technique from Sergei Sobolev to find the best possible endpoint correction coefficients. Their formula looks like the trapezoidal rule plus a small term proportional to the derivative difference at the endpoints, with the proportionality constant depending on the step size h.

The authors claim the error of this rule decreases like h^4, and they provide an explicit asymptotic expansion. They also say the error is smaller than the Euler-Maclaurin rule on a different function space.

The problems are in the proof. The paper prints an expression for the square of the error that includes terms like 1/(2h) and 1/(12h^2) that blow up as h shrinks, which cannot match the claimed h^4 expansion. The intermediate formulas for A1, A2, A3, and A4 do not add up to the claimed final value. For example, with h=1, the printed A values give a large negative number, while the claimed norm is small and positive. There is also a sign inconsistency between the error functional and the condition for exactness for e^{-x}. Finally, comparing the error in W_2^{(2,1)} with the Euler-Maclaurin error in L_2^{(2)} compares norms in different spaces, so the comparison does not mean what the abstract says.

Because of these issues, the central error-order claim is not supported by the derivation as written.

Extended reading notes

Core claim

The central claim is Theorem 4: the square of the norm of the error functional for the optimal quadrature formula with coefficients (2) and (33) is ||l||^2 = 1/720 h^4 - 1/30240 h^6 + O(h^8), giving an O(h^4) order of convergence and, per Remark 2, an error smaller than the Euler-Maclaurin formula on L_2^{(2)}.

Load-bearing premise

The load-bearing premise is that the algebra in the minimization and norm computation is correct, specifically that the Lagrange system (20)-(21) has the stated unique solution and that the expressions A1-A4 in Theorem 4 evaluate to the printed values. As printed, equation (7) has the wrong sign for the C1 terms relative to error functional (5), and the A1-A4 values for h=1 do not combine to the claimed norm. If these are not mere typos, the central error-order claim does not follow.

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

No new physical entities or fitted constants are introduced. The central claim rests on prior results about extremal functions and discrete analogs, plus an unproved uniqueness assertion for the Lagrange system.

assumptions (4)
  • domain assumption W_2^{(2,1)}(0,1) with pseudo-inner product <phi,psi>=∫(phi''+phi')(psi''+psi')dx is a Hilbert space after identifying functions differing by span{1,e^{-x}}.
    Stated at the start of Section 1 with citation to [1]; all norm computations depend on this choice.
  • domain assumption Theorem 1 of [16] gives the extremal function form and norm identity for the error functional in W_2^{(m,m-1)}.
    Quoted in Section 2 and used to obtain (15) and (17); the present paper does not prove it.
  • domain assumption Theorem 2 of [14] provides the discrete analog D1 of d^2/dx^2-1 with properties (24)-(25).
    Used in Section 3.1 to solve the coefficient system; no proof is repeated.
  • domain assumption The system (20)-(21) has a unique solution for every N that gives the minimum of (17).
    Explicitly stated as omitted in Section 3: 'Here we omit the proof of the existence and uniqueness.'

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Pith. "Pith review of The order of convergence of an optimal quadrature formula with derivative in the space $W_2^{(2,1)}$." pith.science (2026). https://pith.science/paper/5OHJ2H7E

@misc{pith2026190800450,
  author       = {Pith},
  title        = {Pith review of: The order of convergence of an optimal quadrature formula with derivative in the space $W_2^(2,1)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OHJ2H7E}},
  note         = {Machine review of arXiv:1908.00450}
}
abstract

The present work is devoted to extension of the trapezoidal rule in the space $W_2^{(2,1)}$. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of a integrand. Using the discrete analog of the operator $\frac{d^2}{dx^{2}}-1$ the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set $\mathbf{F}=\mathrm{span}\{1,x,e^{x},e^{-x}\}$. Finally, in the space $W_2^{(2,1)}$ the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space $L_2^{(2)}$.

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Works this paper leans on

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