Symmetric Triangle Quadrature Rules for Arbitrary Functions

Brian A. Freno; Brian F. Zinser; Salvatore Campione; William A. Johnson

REVIEW 3 major objections 4 minor 14 references

Symmetric Triangle Quadrature Rules for Arbitrary Functions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Symmetric triangle quadrature rules can be built to exactly integrate logarithmic singular functions, and on a representative electromagnetic integral they reduce relative error by two orders of magnitude compared with polynomial rules.

desk verdict Solid methods paper on symmetric triangle rules for singular integrands; the 'arbitrary functions' title oversells slightly, but the construction is sound and the gains on the EM test problem are real. read the letter →

arxiv 1909.01480 v3 pith:UF4HV3NF submitted 2019-09-03 math.NA cs.NA

Brian A. Freno , William A. Johnson , Brian F. Zinser , Salvatore Campione This is my paper · ORCID

classification math.NAcs.NA MSC 65D3265D30

keywords symmetricquadraturerulestrianglesingularintegrandsgeneralizedGaussianlogarithmicsingularitieselectromagneticintegrals

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 develops two ways to compute symmetric quadrature rules on triangles that exactly integrate user-selected sequences of functions, not just polynomials. The motivation is that polynomial rules converge slowly and non-monotonically for integrands with boundary singularities, such as the logarithmic singularities that arise in electromagnetic reaction integrals. Approach 1 solves a scaled nonlinear least-squares problem for the orbit points and weights, starting from known polynomial rules and replacing the highest-degree polynomial groups with singular functions; it suits moderate point counts. Approach 2 tensor-products one-dimensional generalized Gaussian rules and maps the unit square to three quadrilaterals inside the triangle, suiting larger point counts and giving monotonic error decrease. On the singular test integral $I_c$, both approaches generally beat polynomial rules, with Approach 1 often giving relative errors about two orders of magnitude smaller at the same number of points.

What carries the argument

The machinery is the orbit decomposition of symmetric triangle rules combined with function groups. A rule's points live in three orbit types: one centroid point (type 0), three points on medians (type 1), and six points off the medians (type 2); points in an orbit share one weight, so the unknowns are per orbit. The paper groups the target sequence into function groups $f_k$, each either a singularity or the monomials of a fixed degree, and requires the rule to integrate all preceding groups. Approach 1 solves for the orbit unknowns with a nonlinear least-squares objective in which each function's quadrature sum is compared with its exact integral; Approach 2 bypasses the multivariate optimization by building a one-dimensional generalized Gaussian rule for the singularity sequence and tensor-producting it, then mapping it to the triangle via a bilinear transformation of the three quadrilateral subdomains.

What would settle it

Take the same construction and apply it to an integrable boundary singularity outside the logarithmic family, such as $r^{-1/2}$ where $r$ is distance to a vertex, and check whether the relative error still stays two orders of magnitude below polynomial rules; if it does not, the method's demonstrated value is specific to the logarithmic singularity sequence rather than to arbitrary functions.

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Extended reading notes

Core claim

The central claim is that a symmetric triangle rule can be designed to integrate exactly a prescribed function sequence that mixes monomials with singular terms, e.g. $x^{2j-1}\ln x$ for $j\in\mathbb{N}^+$, while retaining the standard orbit structure of symmetric triangle rules (centroid, three median points, six off-median points). Approach 1 obtains such a rule by minimizing a scaled least-squares objective over orbit coordinates and weights, seeded with polynomial-rule points and weights, and accepting only interior points; the cost is a reduction in the maximum polynomial degree integrated exactly. Approach 2 instead forms the product of one-dimensional rules that integrate the same one-dimensional singularity sequence and maps the product rule to the triangle through a bilinear map of three quadrilateral subdomains, producing $n=3n'^2$ points. For the singular integral $I_c$, the resulting rules yield relative errors that are often two orders of magnitude below those of polynomial quadrature rules at comparable point counts, while for the smooth integral $I_s$ the polynomial rules remain best and Approach 1 remains comparable.

Load-bearing premise

The accuracy gains hinge on the assumption that the chosen logarithmic sequence $x^{2j-1}\ln x$ (or its two-dimensional counterpart) completely captures the singular behavior of the integrand on the triangle edges and vertices, since the paper offers no completeness proof for this representation.

Editorial extensions

If this is right

For the singular integral $I_c$, the new rules reach relative errors roughly two orders of magnitude smaller than polynomial rules at several point counts (e.g., $n=27$ for Approach 1).
Approach 2's relative error decreases monotonically with the number of points, guaranteeing that adding points improves accuracy for singular integrands.
Because both approaches preserve the symmetric orbit structure, the computed rules map to arbitrary triangles by an affine transformation with weights scaled by twice the area.
Because the function sequence is user-selected, a user can trade polynomial degree for the ability to integrate a chosen set of singular functions; the maximum polynomial degree drops as singular functions are added.
For a smooth test integrand ($I_s$), polynomial rules remain the most accurate, while Approach 1 retains comparable efficiency.

Reading between the lines

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

If the $x^{2j-1}\ln x$ sequence is assumed complete, the same construction should transfer to other integrable boundary singularities (e.g., $x^{-1/2}$ or $x^p\ln x$) by swapping the singularity sequence, but the paper does not test this.
Because Approach 2 builds triangle rules from one-dimensional rules and a bilinear quadrilateral map, its accuracy is largely set by the quality of the one-dimensional generalized Gaussian rule; tuning that rule for a different singularity family may be cheaper than rerunning Approach 1's optimization.
The reported gains for $I_c$ suggest that replacing polynomial triangle rules in boundary-element or method-of-moments codes with these rules could reduce the number of integration points needed for comparable accuracy, but that impact is an extrapolation beyond the paper's test integrals.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuinely useful methods paper. The two approaches are clearly described and the numerical results show real gains on a practical EM integral. The title and abstract promise "arbitrary functions," but the evidence only covers specific singular sequences, and there is no error analysis or completeness argument. Still, I think it deserves peer review.

What's new: Approach 1 generalizes the standard symmetric polynomial-rule optimization to arbitrary function sequences by grouping functions and replacing high-degree polynomial groups with singular functions. Approach 2 takes one-dimensional generalized Gaussian rules, takes an outer product, and maps to three quadrilaterals around the triangle; that gives symmetric rules that are monotone in n. Both are natural extensions of Ma/Rokhlin/Wandzura and Vipiana et al., but the symmetric triangle construction with function groups is new as far as I know.

What's done well: the paper is honest about the difficulties. It openly says the solver depends on initial guesses, uses random perturbations with 40 Latin hypercube samples, and that rules may not be unique. It explains the cross-term limitation of one-dimensional characterizations and offers three remedies. It computes to 64-digit precision with a residual tolerance of 1e-150, which is strong evidence the listed rules actually integrate the stated sequences. The test problem is relevant and the two-order-of-magnitude improvements in Figures 7a/8a are plausible.

Soft spots: no point/weight tables or code shipped, so the rules are not directly usable without re-running the optimization, which is a nontrivial step. The "arbitrary functions" claim is broader than demonstrated; the examples are limited to logarithmic singularities from EM, and there's no argument that the chosen sequences capture the full singularity structure. Section 4 explicitly acknowledges cross terms are omitted; the stress-test's concern about unsampled cross terms is legitimate, though it's a generalization gap, not an internal inconsistency. The reference solutions come from [10], which shares authors, so the benchmark isn't fully independent. No error analysis.

For whom: computational scientists working on singular integrals in EM or potential theory. I'd cite it if I were in that area. It deserves a serious referee; the referee should ask for reproducibility (tables or code) and a more modest claim about scope.

Referee Report

3 major / 4 minor

Summary. The paper proposes two numerical approaches for constructing symmetric quadrature rules on triangles that exactly integrate prescribed sets of functions, including functions with boundary singularities. Approach 1 solves a least-squares problem for orbit coordinates and weights, using polynomial rules as initial guesses and replacing some polynomial function groups with singular functions. Approach 2 builds tensor-product rules from one-dimensional generalized Gaussian rules and maps them to the triangle via a subdivision into three quadrilaterals. The methods are demonstrated on integrals with logarithmic singularities arising in electromagnetic simulations, where the new rules are reported to achieve relative errors up to two orders of magnitude smaller than standard polynomial rules for the singular test integral. The paper also discusses computational techniques, initial-guess strategies, and a uniqueness heuristic for Approach 1.

Significance. If the results are accepted, the paper provides a practical extension of symmetric triangle quadrature beyond polynomial integrands, addressing a real need in computational electromagnetics. Strengths include the clear description of both algorithms, the use of machine-precision residual checks for the constructed rules, and the direct comparison against established polynomial rules on a nontrivial singular test problem. The claim of 'arbitrary functions' is, however, broader than what is actually demonstrated: the construction depends on the user-supplied function sequence, and the numerical gains are shown only for a particular singular integrand with sequences borrowed from a companion reference. The paper would be more compelling with a reproducibility package containing the actual rules and with a more cautious statement of the scope.

major comments (3)

[Section 6 and Appendix A] The actual quadrature rules are not reported anywhere in the manuscript: Tables 4 and 5 give only orbit counts and maximum function-group indices, while Figures 4-6 show point distributions but no coordinates or weights. Since the paper's contribution is a set of symmetric triangle rules, this omission prevents readers from using or independently verifying the rules. Provide the computed points and weights as supplementary material or in an appendix.
[Abstract and Section 1] The phrase 'arbitrary functions' overstates the demonstrated scope. Approach 1 requires the user to prescribe a finite sequence of functions, and the reported accuracy for I_c relies on singular sequences taken from Reference [10] without a completeness or convergence analysis. Section 4 explicitly concedes that one-dimensional characterization omits cross terms, and Section 6.1 gives no argument that the chosen sequences capture all relevant singular behavior. The authors should either soften the title/abstract claims to 'prescribed function sequences' or provide a theoretical or systematic numerical justification for the generalization.
[Section 5 (Approach 2)] Approach 2 inherits the same completeness limitation in a more direct form: the tensor product of one-dimensional rules does not integrate cross terms such as x^p y^q ln x that can appear in edge and vertex expansions of the test integrand. The paper notes this issue for Approach 1 in Section 4 but does not discuss its effect on the monotonic convergence claimed for Approach 2 in Section 6.3. A comparison of Approach 2's error behavior with and without such cross terms, or at least a discussion of why they are negligible for the test cases, would strengthen the empirical claims.

minor comments (4)

[Equation (3)] The displayed objective function appears to have a typesetting error: the exponent on the relative error is missing or misplaced, and there is an unbalanced parenthesis. Please correct the formula.
[Table 1] For n = 1, the table lists ng = d = 1; while the centroid rule indeed integrates linear functions on the reference triangle, the entry would benefit from a short explanation because standard polynomial-degree conventions sometimes count only constants for a one-point rule.
[Figures 7 and 8] The legends and curves for 'Approach 1, 1D Singularities' and 'Approach 1, 2D Singularities' may be difficult to distinguish in grayscale; consider using distinct markers or line styles.
[Section 3.1] The sentence 'the ability to integrate f(α,β) indicates the ability to integrate f(β,α)' is correct for symmetric rules but would be clearer if the symmetry group of the equilateral triangle is explicitly invoked, since the reduction in the number of functions relies on it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quadrature rules are solved from the chosen function sequences by Equations (2)-(3), and the self-cited singular sequences and reference solutions are example inputs, not equations that define the rule.

full rationale

The derivation chain is self-contained. Section 2 defines an n-point rule as one satisfying Equation (1) for a sequence f, and Section 4 formulates the computation as the least-squares problem of Equations (2)-(3); the unknown weights and coordinates are solved so that the rule exactly integrates the chosen finite sequence. This is the defining equation of a quadrature rule, not a circular reduction: no target integral value I_c or I_s is used to determine the unknowns. The singular sequences in Section 6.1, such as s_j(x)=x^{2j-1} ln x, and the two-dimensional characterization are taken from the authors' previous work [10], but they are input assumptions for the numerical example, not outputs of the rule construction, and no parameter is fitted to the test integrals. The reference solutions for the error plots are also computed as described in [10]; although that reference shares authors, the comparison against polynomial quadrature rules is an external benchmark that does not feed back into the rule equations. Section 4 explicitly acknowledges that the one-dimensional singular characterization omits cross terms, and Section 7 notes limitations on optimality and uniqueness; these are scope caveats about the 'arbitrary functions' claim, not circularities. A possible generalization gap, namely the unproven completeness of the finite singular sequences for arbitrary integrands, is a correctness risk rather than a circularity by construction. Therefore no circular step meets the quoted-equation standard.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; function groups are an accounting device, not a postulated object. The main free choices are the degree-versus-singularity trade-off per point count and the unpublished solved node/weight values. All other inputs are standard numerical analysis assumptions or explicit external results.

free parameters (2)

Polynomial degree d and singular-function count n_s per point count in Approach 1 = Not single values; e.g., n=16, 1D characterization: d=5, n_s=3; details in Tables 4-5
The paper chooses, per n, how many function groups are reserved for singularities versus monomials. This trade-off is heuristic and directly controls which functions a rule integrates.
Orbit coordinates and weights for each Approach 1 rule = Not tabulated in the paper
These are the unknowns solved by Equation (2); they determine the rules but are not published, so any use requires recomputation.

assumptions (5)

domain assumption The chosen one- or two-dimensional singular function sequences in Section 6.1 faithfully represent the singular behavior of I_c over triangle edges and vertices.
The method guarantees exactness only for the listed functions; accuracy on I_c requires that the sequence captures the actual logarithmic edge and vertex singularities.
domain assumption A zero-residual solution to the nonlinear least squares problem in Equation (2) exists for the selected orbit counts and is found by Levenberg-Marquardt from perturbations of polynomial initial guesses.
No existence or global convergence proof is given; some point counts fail and are eliminated in Tables 4 and 5, so solver robustness is load-bearing.
standard math The one-dimensional generalized Gaussian rules from Reference [7] exactly integrate the one-dimensional sequences used in Approach 2.
Approach 2 inherits exactness on tensor products from the published continuation method, and the paper cites that method without re-deriving it.
standard math Symmetry enables reducing the monomial basis to floor((d+2)^2/4) functions for symmetric triangle rules.
This is a standard invariant-polynomial reduction used in the cited polynomial-rule literature and invoked in Section 2.3.
domain assumption The orbit counts from Table 1, taken from polynomial rules, are suitable starting structures for arbitrary function sequences.
The paper reuses the most efficient polynomial orbit counts rather than proving they are optimal or even workable for singular function sequences.

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Cite this review

Pith. "Pith review of Symmetric Triangle Quadrature Rules for Arbitrary Functions." pith.science (2026). https://pith.science/paper/UF4HV3NF

@misc{pith2026190901480,
  author       = {Pith},
  title        = {Pith review of: Symmetric Triangle Quadrature Rules for Arbitrary Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UF4HV3NF}},
  note         = {Machine review of arXiv:1909.01480}
}

read the original abstract

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.

Figures

Figures reproduced from arXiv: 1909.01480 by the authors.

**Figure 2.** Pictorial representation of Approach 1: The eight unknowns computed in Equation (2) for a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Pictorial representation of Approach 2: A triangle is split into three quadrilateral subdomains. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Approach 1: Points for n between 3 and 42 using one-dimensional characterization of the singularities. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Approach 1: Points for n between 3 and 52 using two-dimensional characterization of the singularities. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Approach 2: Points n between 3 and 108. 6.3. Results To assess the performance of the rules arising from these approaches, we compute reference solutions for Ic and Is, as described in Reference [10], for A being defined by the vertices (0, 0), (1/20, 1/20), and (−1/20…

**Figure 7.** Figure 7: Domain 1: Relative errors for Ic and Is when A0 = A. 0 20 40 60 80 100 120 n −7 −6 −5 −4 −3 −2 −1 log10 ε Polynomial Rules Approach 1, 1D Singularities Approach 1, 2D Singularities Approach 2 A A0 (a) Ic 0 20 40 60 80 100 120 n −16 −14 −12 −10 −8 −6 −4 −2 log10 ε Polyn…

**Figure 8.** Figure 8: Domain 2: Relative errors for Ic and Is when A0 6= A. and 8a for Ic, both approaches generally outperform the polynomial quadrature rules, and Approach 1 often outperforms the polynomial quadrature rules by orders of magnitude. In Figure 7a, for example, Approach 1 out…

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Reference graph

Works this paper leans on

14 extracted references · 7 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

B. A. Freno, W. A. Johnson, B. F. Zinser, D. F. Wilton, F. Vipiana, S. Campione, Symmetric numerical integration techniques for singular integrals in the method-of-moments implementation of the electric- ﬁeld integral equation, arXiv preprint arXiv:1911.02107 (2019)

work page Pith review arXiv 2019

[2] [1]

J. N. Lyness, D. Jespersen, Moderate degree symmetric quadrature rules for the triangle, IMA Journal of Applied Mathematics 15 (1) (1975) 19–32.doi:10.1093/imamat/15.1.19

work page doi:10.1093/imamat/15.1.19 1975

[3] [3]

Wandzura, H

S. Wandzura, H. Xiao, Symmetric quadrature rules on a triangle, Computers & Mathematics with Applications 45 (12) (2003) 1829–1840.doi:10.1016/S0898-1221(03)90004-6

work page doi:10.1016/s0898-1221(03)90004-6 2003

[4] [4]

S.-A. Papanicolopulos, Computation of moderate-degree fully-symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving, Computers & Mathematics with Applications 69 (7) (2015) 650–666.doi:10.1016/j.camwa.2015.02.014

doi:10.1016/j.camwa.2015.02.014 2015

[5] [5]

Analytical computation of moderate-degree fully-symmetric cubature rules on the triangle

S.-A. Papanicolopulos, List of zero-dimensional fully symmetric cubature rules on the triangle (Nov. 2011). URL https://arxiv.org/src/1111.3827v1/anc/allrules.pdf

work page Pith review arXiv 2011

[6] [6]

S. E. Mousavi, H. Xiao, N. Sukumar, Generalized Gaussian quadrature rules on arbitrary polygons, International Journal for Numerical Methods in Engineering 82 (1) (2010) 99–113.doi:10.1002/nme. 2759

doi:10.1002/nme 2010

[7] [7]

J. Ma, V. Rokhlin, S. Wandzura, Generalized Gaussian quadrature rules for systems of arbitrary func- tions, SIAM Journal on Numerical Analysis 33 (3) (1996) 971–996.doi:10.1137/0733048

doi:10.1137/0733048 1996

[8] [8]

Vipiana, D

F. Vipiana, D. R. Wilton, W. A. Johnson, Advanced numerical schemes for the accurate evaluation of 4-D reaction integrals in the method of moments, IEEE Transactions on Antennas and Propagation 61 (11) (2013) 5559–5566.doi:10.1109/TAP.2013.2277864

work page arXiv 2013