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NIntegration.jl

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This is library intended to provided multidimensional numerical integration routines in pure Julia

Status

For the time being this library can only perform integrals in three dimensions.

TODO

  • Add rules for other dimensions
  • Make sure it works properly with complex valued functions
  • Parallelize
  • Improve the error estimates (the Cuba library and consequently Cuba.jl seem to calculate tighter errors)

Installation

NIntegration.jl should work on Julia 1.0 and later versions and can be installed from a Julia session by running

julia> using Pkg
julia> Pkg.add(PackageSpec(url = "https://github.com/pabloferz/NIntegration.jl.git"))

Usage

Once installed, run

using NIntegration

To integrate a function f(x, y, z) on the hyperrectangle defined by xmin and xmax, just call

nintegrate(
    f::Function, xmin::NTuple{N}, xmax::NTuple{N};
    reltol = 1e-6, abstol = eps(), maxevals = 1000000
)

The above returns a tuple (I, E, n, R) of the calculated integral I, the estimated error E, the number of integrand evaluations n, and a list R of the subregions in which the integration domain was subdivided.

If you need to evaluate multiple functions (f₁, f₂, ...) on the same integration domain, you can evaluate the function f with more "features" and use its subregions list to estimate the integral for the rest of the functions in the list, e.g.

(I, E, n, R) = nintegrate(f, xmin, xmin)
I₁ = nintegrate(f₁, R)

Technical Algorithms and References

The integration algorithm is based on the one decribed in:

  • J. Berntsen, T. O. Espelid, and A. Genz, "An Adaptive Algorithm for the Approximate Calculation of Multiple Integrals," ACM Trans. Math. Soft., 17 (4), 437-451 (1991).

Author

Acknowdlegments

The author expresses his gratitude to Professor Alan Genz for some useful pointers.

This work was financially supported by CONACYT through grant 354884.