Cuba.jl

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Cuba.jl is a library for multidimensional numerical integration with different algorithms in Julia.

This is just a Julia wrapper around the C Cuba library, version 4.2, by Thomas Hahn. All the credits goes to him for the underlying functions, blame me for any problem with the Julia interface. Feel free to report bugs and make suggestions at https://github.com/giordano/Cuba.jl/issues.

All algorithms provided by Cuba library are supported in Cuba.jl:

• vegas (type: Monte Carlo; variance reduction with importance sampling)
• suave (type: Monte Carlo; variance reduction with globally adaptive subdivision + importance sampling)
• divonne (type: Monte Carlo or deterministic; variance reduction with stratified sampling, aided by methods from numerical optimization)
• cuhre (type: deterministic; variance reduction with globally adaptive subdivision)

Integration is performed on the n-dimensional unit hypercube [0, 1]^n. For more details on the algorithms see the manual included in Cuba library and available in deps/usr/share/cuba.pdf after successful installation of Cuba.jl.

Cuba.jl is available on all platforms supported by Julia.

The latest version of Cuba.jl is available for Julia 1.3 and later versions, and can be installed with Julia built-in package manager. In a Julia session, after entering the package manager mode with ], run the command

pkg> update
pkg> add Cuba

Older versions are also available for Julia 0.4-1.2.

After installing the package, run

using Cuba

or put this command into your Julia script.

Cuba.jl provides the following functions to integrate:

vegas(integrand, ndim, ncomp[; keywords...])
suave(integrand, ndim, ncomp[; keywords...])
divonne(integrand, ndim, ncomp[; keywords...])
cuhre(integrand, ndim, ncomp[; keywords...])

These functions wrap the 64-bit integers functions provided by the Cuba library.

The only mandatory argument is:

• function: the name of the function to be integrated

Optional positional arguments are:

• ndim: the number of dimensions of the integration domain. Defaults to 1 in vegas and suave, to 2 in divonne and cuhre. Note: ndim must be at least 2 with the latest two methods.
• ncomp: the number of components of the integrand. Defaults to 1

ndim and ncomp arguments must appear in this order, so you cannot omit ndim but not ncomp. integrand should be a function integrand(x, f) taking two arguments:

• the input vector x of length ndim
• the output vector f of length ncomp, used to set the value of each component of the integrand at point x

Also anonymous functions are allowed as integrand. For those familiar with Cubature.jl package, this is the same syntax used for integrating vector-valued functions.

For example, the integral

∫_0^1 cos(x) dx = sin(1) = 0.8414709848078965

can be computed with one of the following commands

julia> vegas((x, f) -> f[1] = cos(x[1]))
Component:
 1: 0.8414910005259609 ± 5.2708169787733e-5 (prob.: 0.028607201257039333)
Integrand evaluations: 13500
Number of subregions:  0
Note: The desired accuracy was reached

julia> suave((x, f) -> f[1] = cos(x[1]))
Component:
 1: 0.8411523690658836 ± 8.357995611133613e-5 (prob.: 1.0)
Integrand evaluations: 22000
Number of subregions:  22
Note: The desired accuracy was reached

julia> divonne((x, f) -> f[1] = cos(x[1]))
Component:
 1: 0.841468071955942 ± 5.3955070531551656e-5 (prob.: 0.0)
Integrand evaluations: 1686
Number of subregions:  14
Note: The desired accuracy was reached

julia> cuhre((x, f) -> f[1] = cos(x[1]))
Component:
 1: 0.8414709848078966 ± 2.2204460420128823e-16 (prob.: 3.443539937576958e-5)
Integrand evaluations: 195
Number of subregions:  2
Note: The desired accuracy was reached

The integrating functions vegas, suave, divonne, and cuhre return an Integral object whose fields are

integral :: Vector{Float64}
error    :: Vector{Float64}
probl    :: Vector{Float64}
neval    :: Int64
fail     :: Int32
nregions :: Int32

The first three fields are vectors with length ncomp, the last three ones are scalars. The Integral object can also be iterated over like a tuple. In particular, if you assign the output of integration functions to the variable named result, you can access the value of the i-th component of the integral with result[1][i] or result.integral[i] and the associated error with result[2][i] or result.error[i]. The details of other quantities can be found in Cuba manual.

All other arguments listed in Cuba documentation can be passed as optional keywords.

A more detailed manual of Cuba.jl, with many complete examples, is available at https://giordano.github.io/Cuba.jl/stable/.

There are other Julia packages for multidimenensional numerical integration:

• Cubature.jl
• HCubature.jl
• NIntegration.jl

The Cuba.jl package is released under the terms of the MIT "Expat" License. Note that the binary library Cuba is distributed with the GNU Lesser General Public License. The original author of Cuba.jl is Mosè Giordano. If you use this library for your work, please credit Thomas Hahn (citable papers about Cuba library: https://ui.adsabs.harvard.edu/abs/2005CoPhC.168...78H/abstract and https://ui.adsabs.harvard.edu/abs/2015JPhCS.608a2066H/abstract).