QuadGK.jl
Documentation:
This package provides support for one-dimensional numerical integration in Julia using adaptive Gauss-Kronrod quadrature. The code was originally part of Base Julia.
The package provides three functions: quadgk, gauss, and kronrod.
quadgk performs the integration, gauss computes Gaussian quadrature points and weights for integrating
over the interval [-1, 1], and kronrod computes Kronrod points, weights, and embedded Gaussian quadrature
weights for integrating over [-1, 1]. Typical usage looks like:
using QuadGK
integral, err = quadgk(x -> exp(-x^2), 0, 1, rtol=1e-8)which computes the integral of exp(–x²) from x=0 to x=1 to a relative tolerance of 10⁻⁸, and returns the approximate integral = 0.746824132812427 and error estimate err = 7.887024366937112e-13 (which is actually smaller than the requested tolerance: convergence was very rapid because the integrand is smooth).
For more information, see the documentation.
Similar packages
The FastGaussQuadrature.jl package provides non-adaptive Gaussian quadrature with a wider variety of weight functions — it is a good choice you need to go to very high orders N, e.g. to integrate rapidly oscillating functions, or use weight functions that incorporate some known singularity in your integrand. QuadGK, on the other hand, keeps the order N of the quadrature rule fixed and improves accuracy by subdividing the integration domain, which can be better if fine resolution is required only in a part of your domain (e.g if your integrand has a sharp peak or singularity somewhere that is not known in advance).
For multidimensional integration, see the HCubature.jl, Cubature.jl, and Cuba.jl packages.