Positive weight Hermite and Legendre quadrature rules

Joris Pinkse

Papers from arXiv.org

Abstract: 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.

Date: 2026-09
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