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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 invegas
andsuave
, to 2 indivonne
andcuhre
. 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 lengthndim
- the output vector
f
of lengthncomp
, used to set the value of each component of the integrand at pointx
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:
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).