Positive weight Hermite and Legendre quadrature rules
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.
Related
- Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature
- Symmetric Hermite quadrature-based balanced truncation for learning linear dynamical systems from derivative data
- Numerical evaluation of one-loop integrals using quadrature rules
- Conjectures of Bernstein and Erdős for weighted Lagrange interpolation with Hermite weight
- Fusion rules and rigidity for weight modules over the simple admissible affine sl(2) and N=2 superconformal vertex operator superalgebras
BibTeX
@misc{pinkse2026positive,
title = {{Positive weight Hermite and Legendre quadrature rules}},
author = {Joris Pinkse},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26840},
eprint = {2609.26840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26840},
}
RIS
TY - UNPB
TI - Positive weight Hermite and Legendre quadrature rules
AU - Joris Pinkse
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26840
UR - https://arxiv.org/abs/2609.26840
AB - 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.
ER -