# \[ANN\] Quadriceps.jl: Gauss-Hermite and Gauss-Legendre accuracy in several dimensions, at a fraction of the nodes, with positive weights

**URL:** https://discourse.julialang.org/t/ann-quadriceps-jl-gauss-hermite-and-gauss-legendre-accuracy-in-several-dimensions-at-a-fraction-of-the-nodes-with-positive-weights/139660
**Category:** Package Announcements
**Tags:** quadrature
**Created:** [September 25, 2026, 2:51am UTC](https://discourse.julialang.org/t/ann-quadriceps-jl-gauss-hermite-and-gauss-legendre-accuracy-in-several-dimensions-at-a-fraction-of-the-nodes-with-positive-weights/139660 "2026-09-25T02:51:29Z")
**Posts on this page:** 1
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### Author: ![Joris\_Pinkse](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/joris_pinkse/32/216398_2.png) [@Joris\_Pinkse](https://discourse.julialang.org/u/Joris_Pinkse)
#### Post date: [September 25, 2026, 2:51am UTC](https://discourse.julialang.org/t/ann-quadriceps-jl-gauss-hermite-and-gauss-legendre-accuracy-in-several-dimensions-at-a-fraction-of-the-nodes-with-positive-weights/139660/1 "2026-09-25T02:51:29Z")

</div>

If you integrate against a multivariate normal or uniform in two to five dimensions, you are probably using one of three things, and each has a problem.

- **A product of one-dimensional Gauss rules.** Exact, positive weights, but `qᵈ` nodes: a degree-9 rule in five dimensions costs 3125 evaluations, and that sits inside every likelihood evaluation, every optimization step, every bootstrap draw.
- **A sparse grid.** Far fewer nodes, but some weights are negative. A rule with negative weights is not a probability distribution: a positive integrand can integrate to a negative number, a log-likelihood can become the log of something negative, and rounding error is amplified instead of averaged away.
- **Monte Carlo or quasi-Monte Carlo.** Positive weights, any dimension, but the error falls slowly, and the integrand is evaluated at hundreds or thousands of points to get a few digits.

Quadriceps.jl offers a fourth: rules that integrate every polynomial up to a given degree exactly, like the product rule, whose weights are all positive, like the product rule, and which use a fraction of the nodes. The degree-9 rule in five dimensions has 244 nodes instead of 3125. For the cube at degree 21 in five dimensions, 10984 instead of 161051. They are the smallest positive-weight rules known to me; 116 of the 142 are new.

Because the weights are positive and sum to one, a rule is a discrete distribution, and anything you would do with a Gauss-Hermite grid you can do with it unchanged: mixed logit, random coefficients, Bayesian posteriors, expected utility, moments of a function of a normal vector.

The two functions follow `gausshermite(q)` and `gausslegendre(q)` from FastGaussQuadrature.jl, with the dimension in front. `q` is the size of the one-dimensional rule being replaced, so the degree is `2q − 1`.

```julia
using Quadriceps

X, w = ghpos(3, 4) # standard normal on ℝ³, degree 7: 27 nodes instead of 64
f(x) = x[1]^2 * x[2]^4
sum(w[i] * f(X[i, :]) for i in eachindex(w)) # E[Z₁² Z₂⁴] = 3.0

X, w = lepos(2, 5) # uniform on [0,1]², degree 9: 17 nodes instead of 25
sum(w .* X[:, 1] .^ 3 .* X[:, 2] .^ 2) # 1/4 · 1/3

ghpos(3; p = 7) == ghpos(3, 4) # true: ask for a degree instead of a q

```

`X` is `n × d`, one node per row; `w` has `n` positive weights. The number type is an optional first argument: `ghpos(Float128, 3, 4)` gives the rule in quadruple precision and `ghpos(BigFloat, 3, 4)` the underlying 80-digit rule, with no extra dependency. Rules are stored up to degree 33 for the normal and 77 for the cube in two dimensions, with lower ceilings in higher dimensions; `pragmatic = true` serves anything beyond that as the cheapest tensor product of stored rules.

Install with `Pkg.add("Quadriceps")`; Julia 1.10 or later, and the only dependency is FastGaussQuadrature.jl.

The rules and how they were found are in Joris Pinkse, _Positive weight Hermite and Legendre quadrature rules_ (2026), [arXiv:2609.26840](https://arxiv.org/abs/2609.26840). The rules that descend from published ones are marked as such by `Quadriceps.ruleinfo`, so you can cite the source.

- Repository: [GitHub - NittanyLion/Quadriceps.jl: Positive-weight cubature rules for the Gaussian weight and the cube (ghpos, lepos) · GitHub](https://github.com/NittanyLion/Quadriceps.jl)
- Documentation, with every stored rule and its provenance: [https://NittanyLion.github.io/Quadriceps.jl/dev/](https://nittanylion.github.io/Quadriceps.jl/dev/)
- The same rules for Python ([quadriceps-py](https://github.com/NittanyLion/quadriceps-py)), R ([quadriceps-r](https://github.com/NittanyLion/quadriceps-r)) and Stata ([quadriceps-stata](https://github.com/NittanyLion/quadriceps-stata)).

Requests for rules I have not stored are welcome.