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Quadrature rules for $C^0$ and $C^1$ splines, a recipe

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Pith's one-line read This paper claims a complete explicit recipe—closed formulas plus a recursion—for the nodes and weights of optimal quadrature rules on C^0 and C^1 spline spaces with arbitrary non-uniform, asymmetric partitions.

desk verdict A genuinely new explicit recipe for C0/C1 spline quadrature rules on non-uniform asymmetric partitions, but the paper's central validity claim—roots stay in [-1,1], weights stay positive—is asserted rather than proved, so treat it as a strong conditional result. read the letter →

arxiv 1908.06199 v1 pith:F4RWXCCD submitted 2019-08-16 math.NA cs.NA

Helmut Ruhland This is my paper · ORCID
classification math.NAcs.NA MSC 65D3265D0742C05
keywords optimalquadraturesplinespacesC0andC1splinessemi-classicalorthogonalpolynomialsJacobi-typeGauss-Legendrenon-uniformpartitionsisogeometricanalysis
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

This paper claims a complete, explicit recipe for the nodes and weights of optimal quadrature rules on spline spaces of continuity class 0 and 1: given any compact interval split into non-uniform, asymmetric subintervals and any number of nodes per subinterval, the formulas produce a rule with the minimum number of nodes that integrates the spline space exactly. The construction covers the even-degree $C^{0}$ classes $S_{2N,0}$ and odd-degree $C^{1}$ classes $S_{2N+1,1}$, plus the two previously missing “1/2-rule” classes $S_{2N-1,0}$ and $S_{2N,1}$. If the recipe is correct, it unifies scattered prior cases and replaces a nonlinear search for optimal rules with an explicit loop over subintervals. The paper is deliberately a recipe: formulas and algorithms are stated, proofs are omitted, and the author reports high-precision numerical checks.

What carries the argument

The load-bearing objects are the polynomial families $Q_n(\ldots,x)$ and $M_n(\ldots,x)$ together with the recursion maps that update their parameters. $Q_n$ is the boundary polynomial: its roots are the nodes of a non-middle subinterval, and its parameters $\alpha$ (for $c=0$) or $\alpha,\beta$ (for $c=1$) encode how the smoothness conditions at the neighbouring knots have shifted the orthogonality measure. $M_n$ is the middle polynomial, with parameters from both sides, and its $N+1$ (or $N$) roots give the central nodes. The recursion maps are fractional-linear for $c=0$ and rational for $c=1$, transforming the parameters from one subinterval to the next; a scaling factor $\lambda = L_i/L_{i-1}$ is inserted when subinterval lengths change. Iterating these maps from the boundary to the middle is what makes the rules explicit rather than the solution of a nonlinear system. The paper connects these polynomials to orthogonal polynomials associated with a Jacobi weight modified by endpoint Dirac masses and, in the $c=1$ case, endpoint derivative masses.

What would settle it

Run the Section 4 or Section 8 algorithm on an admissible partition chosen to stress the recursion—for example, subinterval lengths alternating by factors of 10 or 100 in an asymmetric pattern—and inspect whether any node falls outside $[-1,1]$ before scaling or whether any weight is negative; a single such case would show the claimed generality of the recipe fails.

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Extended reading notes

Core claim

The central claim is that for the four spline families $S_{2N,0}$, $S_{2N+1,1}$, $S_{2N-1,0}$, and $S_{2N,1}$ on any compact interval, all optimal (or Gaussian, where applicable) one-parameter quadrature rules have an explicit description. On each boundary subinterval the nodes are the roots of an explicitly written polynomial $Q_n(\ldots,x)$, built from Jacobi polynomials and their derivatives; on the middle subinterval the nodes are roots of an explicitly written polynomial $M_n(\ldots,x)$ built from Legendre polynomials, with weights given by Gauss–Legendre-style formulas involving derivatives of these polynomials. The parameters $\alpha$ (and $\beta$ for $C^1$), initially zero at the boundary, are propagated subinterval by subinterval through recursion maps, with a stretching-factor rescaling when subinterval lengths differ, until the middle parameters are reached. The paper identifies $Q_n$ and $M_n$ as semi-classical Jacobi-type orthogonal polynomials with endpoint mass terms, and notes that in the limit of a single subinterval the formulas reduce to classical Gauss–Legendre quadrature.

Load-bearing premise

The recipe assumes that the parameter recursion always keeps the roots of $Q_n$ and $M_n$ real and inside $[-1,1]$, with positive weights, for every non-uniform, asymmetric partition; the paper states the roots “have to lie” there but gives only numerical verification, not a proof.

Editorial extensions

If this is right

  • For any covered spline class and any non-uniform, asymmetric partition, the optimal quadrature rule can be written down by evaluating formulas and iterating the recursion, with no nonlinear system to solve.
  • The classical Gauss–Legendre rule reappears as the special case of one subinterval, so the construction is a genuine extension of the standard rule.
  • Previously known closed rules—such as the cubic and quintic $C^1$ rules and the $C^0$ quartic rule—are special cases of the same machinery, which covers arbitrary even and odd degrees rather than isolated low degrees.
  • The 1/2-rule formulas fill the previously missing $C^0$-odd and $C^1$-even classes, including cases where zeros from different subintervals are algebraically mixed through a larger Galois group.
  • In the infinite-interval limit, fixed points of the recursion produce the real-line optimal rules in explicit Gegenbauer-polynomial form, with the $C^1$ case exhibiting a second fixed point that yields an additional rule.

Reading between the lines

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

  • If the recipes are correct, the one-parameter freedom in each rule is a tunable handle: one could try to impose additional design constraints, such as forcing a node onto an interior knot, and the paper's own example shows at least one such constraint is feasible.
  • Viewing the recursion as a discrete dynamical system suggests a testable extension: one could study basins of attraction and convergence rates of the parameter map to classify which non-uniform partitions lead to well-behaved rules and which approach the boundary of admissibility.
  • The paper hints that $C^2$ splines with symmetric non-uniform subintervals may be attackable; a natural next test is to attempt analogues of $Q_n$ and $M_n$ with additional endpoint-mass parameters and see whether a finite recursion map still exists.
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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

4 major / 5 minor

Summary. The paper presents a set of recursive formulas and algorithms intended to generate Gaussian or optimal quadrature rules for spline spaces of continuity class 0 and 1 on a compact interval with arbitrary non-uniform, asymmetric subintervals. For the classes S_{2N,0} and S_{2N+1,1} it gives one-parameter families of rules, and for the two '1/2-rule' classes S_{2N-1,0} and S_{2N,1} it gives additional formulas. The construction is based on polynomials Q_n and M_n that generalize Legendre and Jacobi polynomials; the nodes are roots of these polynomials and the weights are computed by derivative formulas. The paper includes worked examples, comparisons with published rules and MATLAB code, high-precision numerical checks, and Maple code, but states explicitly in the introduction that no proofs are included.

Significance. If the claimed existence and positivity properties hold, the paper provides a genuinely useful unified recipe for families of spline quadrature rules that previously were known only in special cases. The reductions to Gauss-Legendre rules for S=1, the matches with the published rules by Bartoň et al., Nikolov, Hiemstra et al., and Johannessen's code, and the high-precision numerical verifications are concrete strengths. The Maple code in Appendix B and the Gegenbauer expansions in Appendix C make the formulas reproducible. However, the central assertion that the recursively generated polynomials always have real roots in [-1,1] with positive weights is not proved, and that is exactly the property that makes the output a quadrature rule.

major comments (4)
  1. [§4, §8, §11] The central claim that the algorithms produce quadrature rules for every non-uniform asymmetric partition is not established. Step 5 of Algorithm 1 and steps 6 and 10 of Algorithm 2 comment only that the roots 'have to lie' in [-1,1], and Section 11 states 'I did not add any proofs.' The finite numerical checks with S,N up to 20 and 'several non-uniform cases' do not cover the universal claim in the abstract. The paper should either supply a proof that the recursion keeps all roots in [-1,1] and all weights positive, or explicitly re-state the result as a conjecture restricted to verified cases.
  2. [§11, Eq. (10.3); §3, Eq. (3)] The identification of Q_n with the Arvesú–Marcellán–Álvarez-Nodarse polynomials P_n(2,0,0,8α+96β,0,-96β) is not reconciled with positivity. Starting from (α,β)=(0,0), one application of recursion (3) in Section 3 gives β_new = 1/[3(n+1)^2(n+2)^2] > 0, so the last coefficient -96β_new is negative. No argument is given that the standard zero-location and positivity theorems from [1] and [2] apply in this parameter regime. If the associated quadratic form loses positivity, the roots of Q_n may leave [-1,1] or some weights may become non-positive, which would invalidate the constructed rule.
  3. [§7.1, §7.3] The 1/2-rule construction depends on square roots whose realness is asserted without justification. In Section 7.1, ω is computed from ω = (-B ± sqrt(B^2 - 4AC))/(2A), and in Section 7.3 the middle-subinterval parameters are computed from SQ = B3^2 - 4 A3 C3. For a genuine quadrature rule on [a,b], these discriminants must be non-negative for all admissible inputs, including the strongly stretched and asymmetric cases; the paper gives no proof and no numerical exploration of discriminant failure. This is the same load-bearing existence gap as the root-location issue, specific to the 1/2-rule algorithms.
  4. [Abstract and §10] The abstract claims 'closed formulae for all Gaussian or optimal ... quadrature rules,' but Section 10 states 'It is not claimed that these are all node distributions that occur for C0/C1 optimal or suboptimal quadrature rules.' These statements need to be reconciled. Either the sense of 'all' must be defined precisely and the completeness of the listed node-distribution types proved, or the abstract must be weakened to describe the families actually constructed.
minor comments (5)
  1. [Abstract] The phrase 'parsing from the first/last subinterval to the middle' should be 'proceeding' rather than 'parsing'; elsewhere 'derivations' should be 'derivatives' (e.g., Section 1).
  2. [§2, §3] The equation numbering jumps from (3) to (5) in Sections 2 and 3, and several formulas such as the (5.1*) labels in Sections 6 and 7 are introduced with a star notation that is not summarized in one place. A consistent numbering and a table of the starred formulas would improve usability.
  3. [§5.2, §5.3] The real-line examples are presented as 'proof left for the reader' exercises. Since these examples are part of the claimed validation, either a concise derivation or an explicit pointer to the calculation in [14] should be included.
  4. [§4, §8] The algorithms contain minor inconsistencies: step 15 in Algorithm 1 appears to be missing, and in Algorithm 2 the comments refer to 'subinterval [SM]' while the text elsewhere says one or two middle subintervals; these should be aligned for reproducible implementation.
  5. [§11] The numerical verification paragraph reports tests for S=1,...,20 and N=1,...,20, but no details are given about which non-uniform/asymmetric length sequences were tested or how the 200-digit checks were distributed. Adding a precise test list would make the evidence easier to assess.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the quadrature construction is an explicit recursion checked against independent published codes, and the only self-reference is to the author's earlier real-line examples, which is not load-bearing.

full rationale

The paper's derivation chain is constructive rather than circular. The polynomials Q_n and M_n are defined directly from Jacobi or Legendre polynomials and the current parameters (section 2, eq. (1); section 3, eq. (1); eq. (5)), the recursion maps (3) and (3.1) deterministically update alpha and beta, and the weights are then computed by the Christoffel-type formulas (2) and (6). No parameter is fitted to the target nodes or weights, and no target quantity is reinserted as an input: for S = 1 the initial value alpha = 0 makes Q_n = P_n, which is the classical Gauss-Legendre reduction, not a definitional equivalence with the claimed output. The examples are checked against independent published codes ([9], [10], [12], [13]) and against the explicit rules of Nikolov [3] and Barton et al. [4], [5]. The sole self-citation, [14], is used in Sections 5.2, 5.3 and 9.2 only to identify limiting real-line analogues; it is not used to justify the compact-interval algorithms. The root-location and positivity assertions in Section 4 step 5 and Section 8 steps 6 and 10 are unproved, and Section 11 explicitly states "I did not add any proofs", but an unsupported assertion is a correctness risk, not a circular reduction. Accordingly, no load-bearing step reduces by construction or by self-citation to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No numerical constants are fitted to data. The only free quantities are the user-chosen parameters of the optimal families, omega and alpha_{R,M}. The central construction rests on unproved assumptions about root location, positivity, and the validity of the recursion, plus the cited semi-classical orthogonal polynomial theory.

free parameters (2)
  • omega (free parameter of the 1-parameter optimal families)
    Appears in M_{omega,n} in Section 2 and in Q_{omega,n} in Sections 6 and 7; it selects a member of the optimal family rather than being fitted to data.
  • alpha_{R,M} (free parameter of the C0 1/2-rule family)
    In Section 9.1 the middle-subinterval parameter is free and is chosen, for example, to place a node exactly at a subinterval boundary.
assumptions (4)
  • domain assumption The minimal node count for an optimal rule on S_{D,c} is N nodes per subinterval with one extra node in the middle subinterval.
    The degree-of-freedom counts in the algorithms in Sections 4 and 8 assume this minimality, but it is not proved in the paper.
  • domain assumption The semi-classical Jacobi-type orthogonal polynomials with endpoint masses (Koornwinder, Krall, Arvesu et al.) characterize the spline quadrature problem.
    Section 11 connects Q_n and M_n to these polynomials via references [1], [2], and [7], but the connection is asserted rather than derived.
  • ad hoc to paper For every admissible sequence of subinterval lengths, the recursion maps (3) and (3.2*) produce parameters for which the Q_n and M_n roots lie in [-1,1] and weights are positive.
    The algorithm comments in Sections 4 and 8 require roots to lie in [-1,1], but no proof is given; only numerical verification is reported.
  • standard math The Gegenbauer polynomial expansions in Appendix A are exact identities.
    These expansions are presented without derivation; they are consistent with standard Gegenbauer identities but are not formally verified in the paper.

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Pith. "Pith review of Quadrature rules for $C^0$ and $C^1$ splines, a recipe." pith.science (2026). https://pith.science/paper/F4RWXCCD

@misc{pith2026190806199,
  author       = {Pith},
  title        = {Pith review of: Quadrature rules for $C^0$ and $C^1$ splines, a recipe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4RWXCCD}},
  note         = {Machine review of arXiv:1908.06199}
}
abstract

Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N, 0}$ and $S_{2N+1, 1}$, i.e. even and odd degree are presented. Also rules for the 2 missing spline classes $S_{2N-1, 0}$ and $S_{2N, 1}$ (the so called 1/2-rules), i.e. odd and even degree are presented. These quadrature rules are explicit in the sense, that they compute the nodes and their weights in the first/last boundary subinterval and, via a recursion the other nodes/weights, parsing from the first/last subinterval to the middle of the interval. These closed formulae are based on the semi-classical Jacobi type orthogonal polynomials.

Figures

Figures reproduced from arXiv: 1908.06199 by the authors.

Figure 10.1
Figure 10.1. D1 (l, r) with S = l + r + 1 subintervals, n or n + 1 nodes per subinterval a building bloc (in red) consists of 1 subinterval with n nodes [PITH_FULL_IMAGE:figures/full_fig_p018_10_1.png] view at source ↗
Figure 10.2
Figure 10.2. D2 (l, r) with S = 2 (l + r) + 1 subintervals, n or n - 1 nodes per subinterval D3 (l, r) with n + 1 nodes in the subinterval SM a building bloc (in red) consists of 2 subintervals with n – 1 and n nodes [PITH_FULL_IMAGE:figures/full_fig_p018_10_2.png] view at source ↗
Figure 10.3
Figure 10.3. D4 (l, r) with S = 2 (l + r) + 2 subintervals, n or n - 1 nodes per subinterval a building bloc (in red) consists of 2 subintervals with n – 1 and n nodes [PITH_FULL_IMAGE:figures/full_fig_p018_10_3.png] view at source ↗
Figures from the paper (1 more)
Figure 10.4
Figure 10.4. Figure 10.4: D5 (l, m, r) with S = 2 (l + m + r) + 3 subintervals, n or n - 1 nodes per subinterval a building bloc (in red) consists of 2 subintervals with n – 1 and n nodes The green subintervals X mark positions for a modified “1/2-rules” algorithm (see 10.2), right of the le…

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

  1. [1]

    Koornwinder

    T.H. Koornwinder. Orthogonal polynomials with w eight function (1 — x) α (1 + x) ß + M δ(x + 1) + N δ(x “ 1), Canad. Math. Bull. 27 (1984), 205“214

    work page 1984

  2. [2]

    Arvesú, F

    J. Arvesú, F. Marcellán, R. Álvarez“Nodarse. On a Modification of the Jacobi Linear Functional: Asymptotic Properties an d Zeros of the Corresponding Orthogonal Polynomials Acta Applicandae Mathematicae 71: 127“158, 2002 21

    work page 2002

  3. [3]

    On certain definite quadrature fo rmulae Journal of Computational and Applied Mathematics 7 5 (1996) 329“343

    Geno Nikolov. On certain definite quadrature fo rmulae Journal of Computational and Applied Mathematics 7 5 (1996) 329“343

    work page 1996

  4. [4]

    Explicit Gaussian quadrature rules for cubic splines with non-uniform knot sequences

    Rajid Ait“Haddou, Michael Bartoň, Victor Manuel Calo. Explicit Gaussian quadrature rules for cubic splines with non“unifor m knot sequences. Preprint, arXiv:1410.7196 v1 [math.NA] 27 Oct 2014

    work page Pith review arXiv 2014

  5. [5]

    Gaussian quadrature rules for $C^1$ quintic splines

    Michael Bartoň, Rajid Ait“Haddou, Victor Manuel Calo. Gaussian quadrature rules for C 1 quintic splines. Preprint, arXiv:1503.00907 v1 [math.NA] 3 Mar 2015

    work page Pith review arXiv 2015

  6. [6]

    Optimal quadrature rules for isogeometric analysis

    Michael Bartoň, Victor Manuel Calo. Optimal rul es for isogeometric analysis Preprint, arXiv:1511.03882 v1 [math.NA] 12 Nov 2015

    work page Pith review arXiv 2015

  7. [7]

    H.L. Krall. On orthogonal polynomials satisfyin g a certain fourth order differential equation, The Pennsylvania State Coll ege Studies, No. 6, 1940, 1“24

    work page 1940

  8. [8]

    T.S. Chihara. Orthogonal polynomials and measur es with end point masses, Rocky Mountain Journal of Mathematics, Volume 15, Number 3, Summer 1985, 705“719

    work page 1985

Show all 14 references
  1. [9]

    Johannessen

    K.A. Johannessen. Optimal quadrature for univar iate and tensor product splines, Comput. Methods Appl. Mech. Engrg., Volume 316, (2 016), 84“99 http://dx.doi.org/10.1016/j.cma.2016.04.030

    2016 doi

  2. [10]

    Johannessen

    K.A. Johannessen. IGA“Quadrature, Open Source, MATLAB sources https://github.com/VikingScientist/IGA“quadrature

  3. [11]

    T. J. R. Hughes, A. Reali, and G. Sangalli. Effcient quadrature for NURBS“based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 199 (58): 301“313, 2010

    2010

  4. [12]

    Bartoň and V.M

    M. Bartoň and V.M. Calo. Optimal quadrature rules for odd“degree spline spaces and their application to tensor“product“based isogeome tric analysis. Computer Methods in Applied Mechanics and Engineering, 305 (2016), 217“240

    2016

  5. [13]

    R., Calabro, F., Schillinger, D., & Hughes, T

    Hiemstra, R. R., Calabro, F., Schillinger, D., & Hughes, T. J. (2017). Optimal and reduced quadrature rules for tensor product and hi erarchically refined splines in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 316, 966“1004

    2017

  6. [14]

    H. Ruhland. Quadrature rules for C 0, C 1 splines, the real line and the five (5) families, Preprint, arXiv:1801.03388 v4 [math.GM] 10 May 2018 22 Appendices Appendix A: The Q n (…, x) and M n (…, x) expressed by Gegenbauer polynomials A.1 c = 0, Q n (α, x) as sum of 2 Gegenba...

    2018 arXiv

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