arXiv 22 Sep 2026 · Econometrics
arXiv:2609.26840 · PDF · Extracted main text
This paper contains new 80-digit positive-weight Gauss-Hermite and positive-weight interior-node Gauss-Legendre quadrature rules for up to five dimensions and varying polynomial degree accuracy (depending on quadrature type and dimension). Some of these rules improve on the best available rules in the literature and some offer rules where none (other than the tensor product) existed. The results were produced by combining methodology developed by previous researchers with new approaches. The full precision rules themselves are at https://doi.org/10.5281/zenodo.22881864. Software in Julia, Python, and R providing the rules is available via a package registry and/or GitHub: Quadriceps.jl (both 64-bit and 128-bit; https://github.com/NittanyLion/Quadriceps.jl), quadriceps-py (64-bit only; https://github.com/NittanyLion/quadriceps-py), and quadriceps-r (64-bit only; https://github.com/NittanyLion/quadriceps-r). The software used to create these rules is available via PositiveWeightQuadratureSolvers.jl (https://github.com/NittanyLion/PositiveWeightQuadratureSolvers.jl). The replication package is at QuadricepsReplicationPackage.jl (https://github.com/NittanyLion/QuadricepsReplicationPackage.jl). A snapshot of all five packages is archived at https://doi.org/10.5281/zenodo.22883240.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Moustapha Diallo and Zelalem Arega Worku (2026) High-order symmetric positive interior quadrature rules on two and three dimensional domains, 2026 | 0.693 | 12 | 2 | 50% |
| 2 | Ronald Cools and Ann Haegemans (1988) Another step forward in searching for cubature formulae with a minimal number of knots for the square | 0.644 | 3 | 2 | 67% |
| 3 | A. H. Stroud (1971) Approximate Calculation of Multiple Integrals | 0.575 | 7 | 2 | 29% |
| 4 | Mattia Festa and Alvise Sommariva (2012) Computing almost minimal formulas on the square | 0.511 | 4 | 2 | 25% |
| 5 | Ann Haegemans and Robert Piessens (1976) Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials | 0.511 | 4 | 2 | 25% |
| 6 | Hong Xiao and Zydrunas Gimbutas (2009) A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions | 0.511 | 3 | 2 | 33% |
| 7 | Vahid Keshavarzzadeh, Robert M. Kirby, and Akil Narayan (2018) Numerical integration in multiple dimensions with designed quadrature | 0.511 | 2 | 2 | 50% |
| 8 | S. I. Konyaev (1977) Kvadraturnye formuly 9-go poryadka, invariantnye otnositel'no gruppy ikosaèdra [Ninth-order quadrature formulas invariant with r… | 0.511 | 2 | 2 | 50% |
| 9 | H. M. Möller (1976) Kubaturformeln mit minimaler Knotenzahl | 0.511 | 2 | 2 | 50% |
| 10 | H. M. Möller (1979) Lower bounds for the number of nodes in cubature formulae | 0.511 | 2 | 2 | 50% |
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