GitHub

Credits

116 of the 142 rules were computed from scratch by the author. A further 11 (Legendre) rules were obtained by node elimination started from Diallo and Worku's published rules. Finally, 15 are rules from the literature (copied in, or found again by the author's search and recognized).

The origin of each rule (Stored rules, Quadriceps.ruleinfo, data/index.tsv, and the source_id in data/rules.bin) is one of:

  • own — computed by the author, with no published rule as its starting point;
  • transcribed: … — the published rule itself, copied in;
  • same-rule: … — the author's search converged to a rule identical to a published one (matched node for node; for GH up to a rotation, which the Gaussian weight permits). The rule belongs to the cited source;
  • derived: … — the author's node elimination started from a published rule for the same $(d, p)$ and went below its node count. The count is new; the starting point is theirs.

If you use a rule whose origin is not own, cite the source named there. The file NOTICE.md at the top of the repository carries the license notice that applies to the derived files.

Sources of the stored rules

  • R. Cools and A. Haegemans, Another step forward in searching for cubature formulae with a minimal number of knots for the square, Computing 40 (1988) 139–146; and Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller's lower bound, in Numerical Integration III, ISNM, Birkhäuser (1988) 25–36. — GH $d = 2$, $p = 13$.
  • M. Diallo and Z. A. Worku, High-order symmetric positive interior quadrature rules on two and three dimensional domains, arXiv:2601.14488 (2026); data at github.com/mdiallo-fula/SymmetricPositiveInteriorCubatures.jl (MIT license). — starting points of Le $d = 2$, $p = 77$ and $d = 3$, $p = 27, 29, …, 45$.
  • M. Festa and A. Sommariva, Computing almost minimal formulas on the square, J. Comput. Appl. Math. 236 (2012) 4296–4302. — Le $d = 2$, $p = 9, 13, 15, 25$.
  • A. Haegemans and R. Piessens, Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials, Numer. Math. 25 (1976) 139–148. — GH $d = 2$, $p = 11$.
  • A. Haegemans and R. Piessens, Construction of cubature formulas of degree seven and nine for symmetric planar regions, using orthogonal polynomials, SIAM J. Numer. Anal. 14 (1977) 492–508. — GH $d = 2$, $p = 9$.
  • S. I. Konyaev, Ninth-order quadrature formulas invariant with respect to the icosahedral group (in Russian), Dokl. Akad. Nauk SSSR 233 (1977) 784–787. — GH $d = 3$, $p = 9$.
  • J. Pinkse, Positive weight Hermite and Legendre quadrature rules (2026), arXiv:2609.26840, Zenodo, DOI 10.5281/zenodo.22904159; the rules to 80 digits are deposited at Zenodo, DOI 10.5281/zenodo.22881864. — Every rule whose origin is own (116 of the 142: all cells not listed under another source), and the eleven Le rules whose origin is derived — $d = 2$, $p = 77$ and $d = 3$, $p = 27, 29, …, 45$ — where the author's node elimination started from Diallo and Worku's published rule for the same cell and went below its node count: the count is the author's, the starting point is theirs, and both should be cited (see the Diallo and Worku entry and NOTICE.md).
  • A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall (1971). — GH $d = 2$, $p = 5$; $d = 3$, $p = 5, 7$; $d = 4$, $p = 7$; $d = 5$, $p = 3, 7$.
  • A. H. Stroud and D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (1963) 105–135. — GH $d = 5$, $p = 5$.

Lower bounds

  • H. M. Möller, Kubaturformeln mit minimaler Knotenzahl, Numer. Math. 25 (1976) 185–200.
  • H. M. Möller, Lower bounds for the number of nodes in cubature formulae, in Numerische Integration, ISNM 45, Birkhäuser (1979) 221–230.

One-dimensional rules

The one-dimensional Gauss–Hermite and Gauss–Legendre rules, used for $d = 1$ and as factors of tensor products, are computed by FastGaussQuadrature.jl.