Grundmann-Möller n-dimensional Simplex Quadrature
Author eschnett
1 Star
Updated Last
3 Years Ago
Started In
September 2020

Grundmann-Möller n-dimensional Simplex Quadrature

GitHub CI deps version pkgeval


This package calculates integrations points and weights for numerical quadrature (i.e. integrating a function) over an n-dimensional simplex. It supports arbitrary (odd) degrees of accuracy, and supports arbitrary floating-point types.

Sample usage

julia> using GrundmannMoeller

julia> # Obtain quadrature scheme (2 dimensions, degree 5; degree must be odd)

julia> scheme = grundmann_moeller(Float64, Val(2), 5);

julia> length(scheme.weights)

julia> # Apply scheme

julia> vertices = [[0,0], [1,0], [0,1]];

julia> f(x) = x[1]*x[2]
f (generic function with 1 method)

julia> q = integrate(f, scheme, vertices)

julia> q  1/24


Create a quadrature scheme

    grundmann_moeller(::Type{T}, ::Val{D}, degree::Int)
  • T: desired floating point type
  • D: dimension
  • degree: desired polynomial degree of accuracy (must be odd)

Evaluate an integral (apply the scheme)

    integrate(fun, scheme, vertices::AbstractVector)
    integrate(fun, scheme, vertices::AbstractMatrix)
  • fun: integrand, should accept an SVector as argument
  • scheme: quadrature scheme
  • vertices: vertices of the simplex

The vertices need to be passed either as a vector-of-vectors or as a matrix. In the first case, there need to be D+1 points with D coordinates each. In the second case, the matrix needs to have size D×D+1.

Provenance of this package

This package is a Julia translation of one of the algorithms of the Python quadpy package by Nico Schlömer.

The algorithm itself was published by

  • A. Grundmann, H. M. Möller, "Invariant integration formulas for the n-simplex by combinatorial methods", SIAM J. Numer. Anal. 15, 282-290 (1978), DOI 10.1137/0715019.

Related work

See SimplexQuad.jl for a simplex quadrature package that uses a different algorithm.