# Higher-order numerical integration on a triangle/tetrahedron/simplex

Let $T$ be a triangle and let $f$ be a smooth function on $T$.

We can use mid-point quadrature $\int f dx \approx |T|\cdot f(x_M)$, where $x_M$ is the middle-point of $T$.

Can you provide me with (a reference for) higher-order formulas on a simplex?

Is it important to have the most efficient rules possible? Do you require all weights to be positive and/or all points to be inside the region? Two general resources:

If you are looking for some implementations (in Fortran) then I can recommend this page. There you can find multiple integration rules for different geometries like triangles, simplices, etc. Of course you will also find references there to the original papers.

The classical triangle paper is the one by Dunavant, Symmetrical Gaussian Quadrature Rules, 1985. Where you have rules up to order 20. An implementation can be found here.