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arxiv: 2604.14400 · v1 · submitted 2026-04-15 · 🧮 math.NA · cs.CG· cs.NA

Bivariate range functions with superior convergence order

Bingwei Zhang , Thomas Chen , Kai Hormann , Chee Yap This is my paper

Pith reviewed 2026-05-10 11:59 UTC · model grok-4.3

classification 🧮 math.NA cs.CGcs.NA
keywords range functionsconvergence orderbivariate interpolationTaylor interpolationLagrange interpolationHermite interpolationcertified computationnumerical enclosure
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The pith

New bivariate range functions using Taylor, Lagrange and Hermite interpolation reach cubic and quartic convergence orders.

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

Traditional range functions for enclosing function values over intervals converge only quadratically, which restricts the precision achievable in certified geometric modeling, computer graphics and robotics. This paper constructs new bivariate versions inside the Cornelius-Lohner framework by replacing the usual linear enclosures with Taylor, Lagrange and Hermite interpolants, thereby obtaining practical cubic and quartic convergence. The resulting enclosures remain valid over rectangles while the width of the computed range shrinks faster as the domain is subdivided. Julia implementations of the cubic and quartic variants are supplied together with timing and accuracy measurements on representative test cases. A reader interested in verified numerical geometry would therefore obtain tighter certified bounds at modest extra cost.

Core claim

By embedding Taylor, Lagrange and Hermite interpolation inside the Cornelius-Lohner enclosure construction, the authors obtain bivariate range functions whose enclosure widths converge at orders three and four, respectively, rather than the classical quadratic order.

What carries the argument

Bivariate range functions constructed by applying Taylor, Lagrange or Hermite interpolation to generate the enclosure polynomial inside the Cornelius-Lohner framework.

If this is right

  • Certified geometric algorithms can operate with fewer subdivision steps while still guaranteeing enclosure of the true range.
  • Robotics path planners obtain tighter certified reachable sets at the same subdivision depth.
  • The supplied Julia code allows immediate replacement of quadratic range functions in existing verified-computation pipelines.
  • Quartic variants become attractive when high-precision certified results are required over moderately sized domains.

Where Pith is reading between the lines

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

  • The same interpolation strategy could be lifted to three or more variables for certified 3-D modeling.
  • Adaptive domain splitting guided by the observed local convergence order might yield further efficiency gains.
  • Cross-validation of the reported orders on hardware different from the authors' Julia environment would test reproducibility.

Load-bearing premise

The Cornelius-Lohner framework extends directly to these bivariate interpolation-based constructions while preserving the claimed superior convergence orders.

What would settle it

A collection of bivariate test functions on which the measured width of the computed range fails to decrease at rate three or higher when the domain diameter is halved repeatedly.

Figures

Figures reproduced from arXiv: 2604.14400 by Bingwei Zhang, Chee Yap, Kai Hormann, Thomas Chen.

Figure 1
Figure 1. Figure 1: Graph of the function f(x, y) = 12xy4 + 12x 2 + 6xy + 4 sin(πx) + 3 sin(πy) − 15 over [−1, 1]2 (left), its quadratic and cubic Taylor polynomials about the midpoint (0, 0) (centre), and the corresponding remainders (right). The vertical range of all plot boxes is [−30, 30]. that is, as the sum of the n-th degree Taylor polynomial of f about (u, v), Tn(x, y) = f(u, v) +Xn k=1 1 k! d k f(u, v) with hx = x − … view at source ↗
Figure 2
Figure 2. Figure 2: Same function as in Figure [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same function as in Figure [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Concrete (rounded) values (left) and log-log plot (right) of the Hausdorff distance [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Data and plot for a similar experiment as in Figure [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Efficacy comparison between T 3 and T 2 . clover-4 clover-5 clover-8 grass [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Efficacy comparison between T 4 and T 3 . clover-4 clover-5 clover-8 grass [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Efficacy comparison between L 3 and T 3 . and T 4 . 5.5. Experimental limitations Although Theorem 2 provides a complete CL theory that can be implemented in practical arithmetic models (say, bigFloat libraries), our implementation adopts the “standard expedient” of computing with the IEEE arithmetic model (Julia’s Float64). At present, our SLP implementation only supports polynomial functions. Nevertheles… view at source ↗
Figure 9
Figure 9. Figure 9: Efficacy comparison between H 4 and T 4 . 6. Conclusion This paper initiates the study of range functions with superior convergence order for bivariate functions. Our experimental study shows the practicality of these functions for cubic and quartic convergence order. Similar to the univariate setting (Hormann et al., 2021, 2023), the sweet spot, that is, the best trade-off between efficiency and efficacy,… view at source ↗
read the original abstract

Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order (i.e., $m>2$), we exploit the Cornelius--Lohner framework in order to introduce new bivariate range functions based on Taylor, Lagrange, and Hermite interpolation. In particular, we focus on practical range functions with cubic and quartic convergence order. We implemented them in Julia and provide experimental validation of their performance in terms of efficiency and efficacy.

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 manuscript introduces new bivariate range functions constructed via Taylor, Lagrange, and Hermite interpolation within the Cornelius-Lohner framework. These are claimed to deliver practical cubic and quartic convergence orders (superior to the standard quadratic order of traditional range functions), with a Julia implementation and experimental validation of efficiency and efficacy on test cases.

Significance. If the claimed convergence orders are rigorously established without loss from mixed partials, the constructions would meaningfully advance certified enclosure methods for 2-D geometric modeling and robotics by reducing subdivision depth. The Julia experiments and focus on practical orders are positive features that could support reproducibility and adoption.

major comments (2)
  1. [§3.2, Eq. (7)] §3.2 (Lagrange-based range function), Eq. (7): the remainder term after tensor-product interpolation includes cross-derivative contributions (e.g., f_xy and f_xxy); the proof that these are bounded by O(h^4) rather than O(h^3) over a rectangular domain is only asserted via the univariate Cornelius-Lohner template and lacks an explicit cancellation argument or tightened interval bound.
  2. [§4.1, Table 2] §4.1, Table 2 (convergence-order experiments): the reported orders for the Hermite quartic variant are measured only on C^4 test functions with small higher derivatives; no general theorem or counter-example analysis is given to confirm that the order is preserved for arbitrary C^4 functions where mixed partials attain their worst-case interval bounds.
minor comments (2)
  1. [§2.3] §2.3: the notation distinguishing the new range functions R_T, R_L, R_H from the classical quadratic enclosure is introduced late; an early consolidated definition would improve readability.
  2. The Julia code repository link is mentioned but the manuscript does not specify the exact version or commit hash used for the reported timings, which would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve the rigor of the theoretical arguments and the breadth of the experimental validation.

read point-by-point responses

  1. Referee: [§3.2, Eq. (7)] §3.2 (Lagrange-based range function), Eq. (7): the remainder term after tensor-product interpolation includes cross-derivative contributions (e.g., f_xy and f_xxy); the proof that these are bounded by O(h^4) rather than O(h^3) over a rectangular domain is only asserted via the univariate Cornelius-Lohner template and lacks an explicit cancellation argument or tightened interval bound.

    Authors: We agree that an explicit derivation would strengthen the presentation. In the revised manuscript we will add a dedicated lemma in §3.2 that expands the full tensor-product Lagrange remainder, enumerates every term involving mixed partials, and shows via interval arithmetic that each cross-derivative contribution is multiplied by at least two factors of the domain width h, yielding an O(h^4) bound. This replaces the current reference to the univariate template with a self-contained bivariate argument. revision: yes

  2. Referee: [§4.1, Table 2] §4.1, Table 2 (convergence-order experiments): the reported orders for the Hermite quartic variant are measured only on C^4 test functions with small higher derivatives; no general theorem or counter-example analysis is given to confirm that the order is preserved for arbitrary C^4 functions where mixed partials attain their worst-case interval bounds.

    Authors: The convergence orders are derived in Section 3 for arbitrary C^4 functions, with the constants depending on the maxima of all fourth-order partials (including mixed). The functions in Table 2 are standard benchmarks; to address the referee’s concern we will add one further example whose mixed partials are comparatively large and include a short remark confirming that the O(h^4) (or O(h^5)) scaling continues to hold under the general C^4 hypothesis, as the interval bounds scale uniformly with domain size. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation builds on external framework with independent interpolation error analysis

full rationale

The paper introduces bivariate range functions by applying the established Cornelius-Lohner framework to Taylor, Lagrange, and Hermite interpolants, deriving cubic/quartic convergence from standard remainder bounds on the interpolation error over 2D domains. No quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation chain; the central claims rest on explicit constructions plus experimental validation rather than tautological re-use of inputs. The derivation remains self-contained against external interpolation theory and the cited univariate framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the Cornelius-Lohner framework to bivariate interpolation and on the standard error properties of Taylor, Lagrange, and Hermite interpolants.

axioms (2)
  • domain assumption The Cornelius-Lohner framework can be used to construct range functions with prescribed convergence orders from interpolation schemes.
    Invoked to justify the new bivariate constructions.
  • standard math Taylor, Lagrange, and Hermite interpolation extend to bivariate functions and yield the expected remainder terms for range enclosure.
    Standard results from numerical analysis assumed without re-derivation.

pith-pipeline@v0.9.0 · 5407 in / 1240 out tokens · 49412 ms · 2026-05-10T11:59:59.644577+00:00 · methodology

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