An order 10 Xiao–Gimbutas rule on a triangle.

An order 5 Gauss–Legendre rule on an interval.

An order 1 Sauter–Schwab rule on an edge-adjacent triangle and quadrilateral.

An order 4 Hammer–Marlowe–Stroud rule on a triangle.

Welcome to the online encyclopedia of quadrature rules, a reference website that lists a number of quadrature rules. Each quadrature rule is indexed using Q-index, for example Q000001.

You can:

• view the full list of quadrature rules by Q-index
• view the full alphabetical list of quadrature rules
• view the quadrature rules by domain
• view the quadrature rules by integral
• view the list of all pages in the encyclopedia

What is a quadrature rule?

Quadrature rules are sets of points and weights that are used to approximate integrals. If \(\{\mathbf{p}_0,\dots,\mathbf{p}_{n-1}\}\subset\mathbb{R}^d\) and \(\{w_0,\dots,w_{n-1}\}\subset\mathbb{R}\) are the points and weights (repectively) of the quadrature rule for a single integral, then:

$$\int f(x)\,\mathrm{d}x \approx \sum_{i=0}^{n-1}f(\mathbf{p}_i)w_i$$

The points that make up the quadrature rules in this encyclopedia are represented using barycentric coordinates.

Libraries

All of the quadrature rules included in the online encylopedia of quadrature rules are included in the quadraturerules library, which is available in the following languages:

• C++
• Python
• Rust