Positive weight Hermite and Legendre quadrature rules - rules

Positive-weight cubature rules for two weights in dimensions d = 2, 3, 4, 5, at every odd degree of exactness p up to a ceiling that depends on the dimension: 57 rules for the Gaussian weight N(0, I_d) (Gauss-Hermite; p ≤ 33, 33, 23, 21 for d = 2, 3, 4, 5) and 85 rules for the uniform weight on the cube [0,1]^d (Gauss-Legendre; p ≤ 77, 45, 23, 21). A rule with N nodes integrates every polynomial of total degree at most p exactly. All weights are positive, and all nodes of the uniform-weight rules lie strictly inside the cube. The rules use far fewer nodes than the tensor product of one-dimensional Gauss rules; many improve on the smallest published node count for their cell, and many are for cells in which no rule other than the tensor product had been published.

Every rule is given twice: in double precision (rules/, 17 significant digits) and to 80 significant digits (rules_extended/, files ending in .mp.csv). The largest relative moment error of the 80-digit files, over all monomials of degree at most p, is 9.3e-69 for the Gaussian weight and 7.2e-70 for the uniform weight; each double-precision file is its 80-digit file rounded.

gh.zip / gh.tar.xz hold the Gaussian deposit and le.zip / le.tar.xz the uniform one; the two formats contain the same files. Each deposit has a README (format, normalization, how to verify a rule), a description of methods and credit, a cell-by-cell comparison with the published literature, a precision report, a one-row-per-rule summary.csv, and SHA-256 checksums of every file.

Not every rule is the author's. Eleven Gaussian cells and four uniform cells hold rules published by others (Stroud, Stroud and Secrest, Haegemans and Piessens, Cools and Haegemans, Konyaev; Festa and Sommariva), included so that the tables are complete, and eleven uniform cells were obtained by node elimination started from rules published by Diallo and Worku. summary.csv names the source of every such cell, and le/NOTICE_third_party.md carries the license notices that go with them.

The rules are described in J. Pinkse, "Positive weight Hermite and Legendre quadrature rules" (2026). The software that produced, serves and re-verifies them is archived in a separate record.