Quadriceps.jl

Positive-weight cubature rules in several dimensions, for the Gaussian weight and for the uniform weight on the cube.

A cubature rule of degree $p$ for a weight $ω$ on $\mathbb{R}^d$ is a set of nodes $x_1, …, x_n$ and weights $w_1, …, w_n$ with

\[\sum_{i=1}^n w_i f(x_i) = \int f(x)\, ω(x)\, dx \qquad \text{for every polynomial } f \text{ of total degree} ≤ p .\]

Positive weights matter in practice: the rule is then a discrete probability distribution (up to scale), sums do not cancel, and an integrand that is nonnegative has a nonnegative approximate integral.

The product of one-dimensional Gauss rules is such a rule with $q^d$ nodes, where $q = (p+1)/2$. The rules in this package need far fewer:

The package exports two functions, ghpos and lepos, modeled on gausshermite(q) and gausslegendre(q) from FastGaussQuadrature.jl: q is the number of nodes of the one-dimensional Gauss rule, and the rule returned has its degree, $p = 2q - 1$.

using Quadriceps

X, w = ghpos(3, 4)          # 27 nodes for N(0, I₃), as exact as the 4×4×4 Gauss–Hermite grid (degree 7)
X, w = lepos(2, 5)          # 17 nodes for the uniform density on [0,1]², degree 9
X, w = ghpos(3; p = 7)      # the first rule again, requested by degree

• Guide: conventions, the two keyword arguments, accuracy.
• Stored rules: every rule, with its node count, Möller's bound, error and origin.
• Data format: the single binary file that holds the rules.
• Reference: docstrings.
• Credits: whose rules these are, and what to cite.