Determining the Rolle function in Hermite interpolatory approximation by solving an appropriate differential equation

J. S. C. Prentice

arxiv: 2303.04882 · v1 · pith:XA4YF7BEnew · submitted 2023-03-08 · 🧮 math.NA · cs.NA

Determining the Rolle function in Hermite interpolatory approximation by solving an appropriate differential equation

J. S. C. Prentice This is my paper

Pith reviewed 2026-05-24 09:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Hermite interpolationRolle functionpointwise errordifferential equationnumerical solutionpolynomial approximationerror correctioninterpolatory approximation

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The pith

The pointwise error in Hermite interpolation is determined by numerically solving a differential equation derived from the error term itself.

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

The paper establishes that the Rolle function, which encodes the pointwise error of a Hermite interpolant, satisfies a differential equation obtained directly from the standard interpolation error formula. Solving this equation numerically supplies the error values at chosen points inside the interval. These values are then used to construct a polynomial that approximates the Rolle function. Adding the polynomial to the original Hermite polynomial produces a corrected approximation whose accuracy is demonstrably higher on the test example.

Core claim

By deriving a differential equation from the Hermite interpolation error term and solving it numerically, the pointwise values of the Rolle function are obtained. These values are then fitted by a polynomial which serves as an approximation to the error function. Adding this polynomial approximation to the original Hermite polynomial produces a more accurate interpolatory approximation, with the improvement shown to be significant on an example.

What carries the argument

The differential equation derived from the Hermite error term, whose numerical solution supplies the Rolle function.

If this is right

The Rolle function can be recovered pointwise by numerical solution of the derived differential equation.
A polynomial approximation to the error can be built from the solved values without further derivative information.
The corrected approximation formed by adding the polynomial to the Hermite interpolant has smaller pointwise error.
The procedure applies to Hermite interpolation of any fixed order on a given interval.

Where Pith is reading between the lines

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

The same DE-based recovery could be tried on other polynomial or piecewise-polynomial interpolants whose error terms involve an unknown function satisfying a similar relation.
Choice of solver tolerance and sampling points for the DE solution would need to be tuned to keep the fitted polynomial from introducing new oscillations.
If the underlying function is analytic, the degree of the correcting polynomial might be selected automatically from the rate at which the solved error values decay.

Load-bearing premise

The differential equation derived from the error term can be solved numerically with sufficient accuracy to capture the true pointwise error, and a low-degree polynomial fit to those values will produce a meaningful accuracy gain when added to the Hermite polynomial.

What would settle it

For a known smooth test function and fixed nodes, compute the exact pointwise interpolation error, solve the derived DE numerically, fit the polynomial, form the corrected approximation, and verify whether its maximum error on the interval is smaller than that of the uncorrected Hermite polynomial.

read the original abstract

We determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which is then added to the original Hermite polynomial to form a more accurate approximation. An example demonstrates that improvements in accuracy are significant.

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.

Desk Editor's Note private letter to a colleague

The paper tries to correct Hermite interpolation error by deriving and numerically solving a DE for the Rolle function ξ(x), then fitting a polynomial to the result, but the approach depends on differentiability and extra smoothness that standard theory does not guarantee.

read the letter

The new element here is the specific step of turning the error formula e(x) = f^{(n+1)}(ξ(x)) ω(x)/(n+1)! into a differential equation for ξ(x) itself and solving it numerically to build a polynomial correction that gets added to the Hermite interpolant. That route is not the usual way people estimate or reduce interpolation error. The example is said to show clear accuracy gains, which at least indicates the authors have checked the procedure on some test case. That is the part that earns credit: they are working directly from the error expression rather than generic bounds or asymptotic analysis. The main weakness is exactly the one flagged in the stress-test note. Forming the DE requires differentiating with respect to x, which assumes a differentiable selection of ξ(x) exists and that f has a continuous (n+2)th derivative. Standard Hermite theory only gives existence of some ξ(x) in the interval for C^{n+1} functions; it does not supply either of those extra properties. If the example uses a C^∞ function, the reported improvement does not test whether the DE is even well-posed for the functions the method is supposed to apply to. The abstract also gives no error tables, no comparison against other correction techniques, and no check that the numerically solved ξ matches the true error. Those omissions make the central claim hard to evaluate. This is narrow numerical-analysis work aimed at people who already care about high-order polynomial approximation and error control. A reader in that subfield might want to see the full derivation and the example details, but the regularity gap is load-bearing and the evidence presented is thin. I would not send it to referees in its current form; the authors would need to address the differentiability issue and supply quantitative verification before it is worth referee time.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes determining the pointwise error in Hermite interpolation by numerically solving a differential equation derived from the error formula involving the Rolle function ξ(x), then approximating that error by a low-degree polynomial which is added to the original Hermite interpolant to improve accuracy; an example is said to demonstrate significant improvement.

Significance. If the central construction were rigorously justified, the approach would supply a novel numerical correction technique for Hermite interpolants by recovering information about the unknown ξ(x). The significance is reduced by the absence of any quantitative error tables, convergence rates, or baseline comparisons, and by the fact that the method rests on differentiability assumptions not guaranteed by the standard C^{n+1} setting of Hermite interpolation theory.

major comments (2)

[Derivation of the differential equation (abstract and §2)] The derivation of the differential equation for ξ(x) begins from the standard error formula e(x) = f^{(n+1)}(ξ(x)) ω(x)/(n+1)! and differentiates both sides. This step presupposes that a differentiable selection ξ(x) exists and that f is at least C^{n+2} so that f^{(n+2)} appears. Standard interpolation theory only guarantees existence of some (not necessarily differentiable) ξ(x) ∈ (a,b) when f ∈ C^{n+1}; the paper supplies no additional hypotheses or existence proof that would make the DE well-posed under the stated regularity. This assumption is load-bearing because the subsequent steps (numerical solution of the DE, polynomial fit, and correction) cannot be performed if the DE cannot be formed.
[Numerical example (abstract and §4)] The abstract asserts that “an example demonstrates that improvements in accuracy are significant,” yet the description contains no error norms, comparison with the plain Hermite polynomial, verification that the numerically obtained ξ(x) reproduces the true pointwise error, or statement of the smoothness class of the test function. Without these data the central claim of practical improvement cannot be assessed.

minor comments (2)

Notation for the nodal polynomial ω(x) and the precise degree of the Hermite interpolant should be stated explicitly at the first appearance.
The manuscript should clarify whether the method is intended only for C^∞ data or whether it claims to work for the minimal C^{n+1} class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond point-by-point to the major comments below and outline the revisions we intend to make.

read point-by-point responses

Referee: [Derivation of the differential equation (abstract and §2)] The derivation of the differential equation for ξ(x) begins from the standard error formula e(x) = f^{(n+1)}(ξ(x)) ω(x)/(n+1)! and differentiates both sides. This step presupposes that a differentiable selection ξ(x) exists and that f is at least C^{n+2} so that f^{(n+2)} appears. Standard interpolation theory only guarantees existence of some (not necessarily differentiable) ξ(x) ∈ (a,b) when f ∈ C^{n+1}; the paper supplies no additional hypotheses or existence proof that would make the DE well-posed under the stated regularity. This assumption is load-bearing because the subsequent steps (numerical solution of the DE, polynomial fit, and correction) cannot be performed if the DE cannot be formed.

Authors: We agree that the standard Hermite error formula only guarantees a (possibly non-differentiable) ξ(x) under the C^{n+1} hypothesis. To make the differential equation well-posed, the revised manuscript will explicitly add the standing assumption that f belongs to C^{n+2} and that a differentiable selection ξ(x) exists on the interval. A short paragraph will be inserted in §2 discussing sufficient conditions for the existence of such a differentiable selection (for instance, when f is analytic or when the nodes are fixed and f^{(n+2)} is continuous). The abstract and the derivation will be updated to state these hypotheses clearly. revision: yes
Referee: [Numerical example (abstract and §4)] The abstract asserts that “an example demonstrates that improvements in accuracy are significant,” yet the description contains no error norms, comparison with the plain Hermite polynomial, verification that the numerically obtained ξ(x) reproduces the true pointwise error, or statement of the smoothness class of the test function. Without these data the central claim of practical improvement cannot be assessed.

Authors: We accept that the current description of the numerical example lacks the quantitative information needed for assessment. In the revised version we will augment §4 with tables reporting L^∞ and L^2 error norms for both the original Hermite interpolant and the corrected approximation, direct comparisons against the plain Hermite polynomial, a verification that the numerically recovered ξ(x) reproduces the observed pointwise error to within a prescribed tolerance, and an explicit statement of the C^∞ smoothness class of the test function. These additions will allow the reader to evaluate the claimed improvement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper starts from the standard Hermite error formula involving the Rolle function ξ(x) and derives a differential equation for ξ by differentiation of that formula. This is a direct mathematical step from the given error expression rather than a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. No steps reduce by construction to the paper's own inputs; the numerical solution and subsequent polynomial correction are applications of the derived DE, not circular. The derivation chain is self-contained against the external benchmark of classical interpolation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the method implicitly assumes the error term admits a usable differential equation and that numerical solution plus polynomial fitting is stable.

pith-pipeline@v0.9.0 · 5571 in / 1040 out tokens · 25018 ms · 2026-05-24T09:34:38.562632+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Prentice, Determining the Rolle function in Lagr ange interpo- latory approximation, arXiv.org, Cornell University Libr ary (2018)

J.S.C. Prentice, Determining the Rolle function in Lagr ange interpo- latory approximation, arXiv.org, Cornell University Libr ary (2018). [arXiv: 1810.09601]

work page arXiv 2018
[2]

Isaacson and H.B

E. Isaacson and H.B. Keller, Analysis of Numerical Methods , Dover, New York, 1994

work page 1994
[3]

Kincaid and W

D. Kincaid and W. Cheney, N umerical Analysis: Mathematics of Scien- tiﬁc Computing, 3rd ed. , Brooks/Cole, Paciﬁc Grove, 2002

work page 2002
[4]

Butcher, Numerical Methods for Ordinary Diﬀerential Equations , Wiley, Chichester, 2003

J.C. Butcher, Numerical Methods for Ordinary Diﬀerential Equations , Wiley, Chichester, 2003

work page 2003
[5]

Burden & J.D

R.L. Burden & J.D. Faires, Numerical Analysis 9th ed. , Brooks/Cole, Boston, 2011

work page 2011
[6]

Schultz, Spline Analysis, Prentice-Hall, Englewood Cliﬀs, NJ, 1973

M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliﬀs, NJ, 1973. 7

work page 1973

[1] [1]

Prentice, Determining the Rolle function in Lagr ange interpo- latory approximation, arXiv.org, Cornell University Libr ary (2018)

J.S.C. Prentice, Determining the Rolle function in Lagr ange interpo- latory approximation, arXiv.org, Cornell University Libr ary (2018). [arXiv: 1810.09601]

work page arXiv 2018

[2] [2]

Isaacson and H.B

E. Isaacson and H.B. Keller, Analysis of Numerical Methods , Dover, New York, 1994

work page 1994

[3] [3]

Kincaid and W

D. Kincaid and W. Cheney, N umerical Analysis: Mathematics of Scien- tiﬁc Computing, 3rd ed. , Brooks/Cole, Paciﬁc Grove, 2002

work page 2002

[4] [4]

Butcher, Numerical Methods for Ordinary Diﬀerential Equations , Wiley, Chichester, 2003

J.C. Butcher, Numerical Methods for Ordinary Diﬀerential Equations , Wiley, Chichester, 2003

work page 2003

[5] [5]

Burden & J.D

R.L. Burden & J.D. Faires, Numerical Analysis 9th ed. , Brooks/Cole, Boston, 2011

work page 2011

[6] [6]

Schultz, Spline Analysis, Prentice-Hall, Englewood Cliﬀs, NJ, 1973

M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliﬀs, NJ, 1973. 7

work page 1973