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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)

For more details on the algorithms see the manual included in Cuba library and
available in `deps/cuba-julia/cuba.pdf`

after successful installation
of `Cuba.jl`

.

Integration is performed on the n-dimensional unit hypercube $[0, 1]^n$. If you want to compute an integral over a different set, you have to scale the integrand function in order to have an equivalent integral on $[0, 1]^n$. For example, recall that in one dimension

```
∫_a^b dx f[x] → ∫_0^1 dy f[a + (b - a) y] (b - a)
```

where the final `(b - a)`

is the one-dimensional version of the Jacobian. This
generalizes straightforwardly to more than one dimension.

`Cuba.jl`

is available for GNU/Linux, Mac OS, and Windows (`i686`

and `x86_64`

architectures).

`Cuba.jl`

is available for Julia 0.5 and later versions, and can be installed
with
Julia built-in package manager.
In a Julia session run the commands

```
julia> Pkg.update()
julia> Pkg.add("Cuba")
```

Installation script on GNU/Linux and Mac OS systems will download Cuba Library source code and build the Cuba shared object. In order to accomplish this task a C compiler is needed. Instead, on Windows a prebuilt version of the library is downloaded.

Older versions are also available for Julia 0.4.

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`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: 0.8414910005259612 ± 5.2708169787342156e-5 (prob.: 0.028607201258072673)
Integrand evaluations: 13500
Fail: 0
Number of subregions: 0
julia> suave((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.84115236906584 ± 8.357995609919512e-5 (prob.: 1.0)
Integrand evaluations: 22000
Fail: 0
Number of subregions: 22
julia> divonne((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.841468071955942 ± 5.3955070531551656e-5 (prob.: 0.0)
Integrand evaluations: 1686
Fail: 0
Number of subregions: 14
julia> cuhre((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.8414709848078966 ± 2.2204460420128823e-16 (prob.: 3.443539937576958e-5)
Integrand evaluations: 195
Fail: 0
Number of subregions: 2
```

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.

**Note:** if you used `Cuba.jl`

until version 0.0.4, be aware that the user
interface has been reworked in version 0.0.5 in a backward incompatible way.

A more detailed manual of `Cuba.jl`

, with many complete examples, is available
at http://cubajl.readthedocs.io/. You can also download the latest PDF version
from https://media.readthedocs.org/pdf/cubajl/latest/cubajl.pdf.

Here is an example of a 3-component integral in 3D space (so `ndim=3`

and
`ncomp=3`

) using the integrand function tested in `test/runtests.jl`

:

```
using Cuba
function func(x, f)
f[1] = sin(x[1])*cos(x[2])*exp(x[3])
f[2] = exp(-(x[1]^2 + x[2]^2 + x[3]^2))
f[3] = 1/(1 - x[1]*x[2]*x[3])
end
result = cuhre(func, 3, 3, abstol=1e-12, reltol=1e-10)
println("Results of Cuba:")
for i=1:3; println("Component $i: ", result[1][i], " ± ", result[2][i]); end
println("Exact results:")
println("Component 1: ", (e-1)*(1-cos(1))*sin(1))
println("Component 2: ", (sqrt(pi)*erf(1)/2)^3)
println("Component 3: ", zeta(3))
```

This is the output

```
Results of Cuba:
Component 1: 0.6646696797813739 ± 1.0050367631018485e-13
Component 2: 0.4165383858806454 ± 2.932866749838454e-11
Component 3: 1.2020569031649702 ± 1.1958522385908214e-10
Exact results:
Component 1: 0.6646696797813771
Component 2: 0.41653838588663805
Component 3: 1.2020569031595951
```

`Cuba.jl`

cannot (yet?) take
advantage of parallelization capabilities of Cuba Library. Nonetheless, it has
performances competitive with equivalent native C or Fortran codes based on Cuba
library when `CUBACORES`

environment variable is set to `0`

(i.e.,
multithreading is disabled). The following is the result of running the
benchmark present in `test`

directory on a 64-bit GNU/Linux system running Julia
0.7.0-DEV.363 (commit 6071f1a02e) equipped with an Intel(R) Core(TM) i7-4700MQ
CPU. The C and FORTRAN 77 benchmark codes have been compiled with GCC 6.3.0.

```
$ CUBACORES=0 julia -e 'cd(Pkg.dir("Cuba")); include("test/benchmark.jl")'
INFO: Performance of Cuba.jl:
0.271304 seconds (Vegas)
0.579783 seconds (Suave)
0.329504 seconds (Divonne)
0.238852 seconds (Cuhre)
INFO: Performance of Cuba Library in C:
0.319799 seconds (Vegas)
0.619774 seconds (Suave)
0.340317 seconds (Divonne)
0.266906 seconds (Cuhre)
INFO: Performance of Cuba Library in Fortran:
0.272000 seconds (Vegas)
0.584000 seconds (Suave)
0.308000 seconds (Divonne)
0.232000 seconds (Cuhre)
```

Of course, native C and Fortran codes making use of Cuba Library outperform
`Cuba.jl`

when higher values of `CUBACORES`

are used, for example:

```
$ CUBACORES=1 julia -e 'cd(Pkg.dir("Cuba")); include("test/benchmark.jl")'
INFO: Performance of Cuba.jl:
0.279524 seconds (Vegas)
0.581078 seconds (Suave)
0.327319 seconds (Divonne)
0.241211 seconds (Cuhre)
INFO: Performance of Cuba Library in C:
0.115113 seconds (Vegas)
0.596503 seconds (Suave)
0.152511 seconds (Divonne)
0.085805 seconds (Cuhre)
INFO: Performance of Cuba Library in Fortran:
0.108000 seconds (Vegas)
0.604000 seconds (Suave)
0.160000 seconds (Divonne)
0.092000 seconds (Cuhre)
```

`Cuba.jl`

internally fixes `CUBACORES`

to 0 in order to prevent from forking
`julia`

processes that would only slow down calculations eating up the memory,
without actually taking advantage of concurrency. Furthemore, without this
measure, adding more Julia processes with `addprocs()`

would only make the
program segfault.

Another Julia package for multidimenensional numerical integration is available: Cubature.jl, by Steven G. Johnson.

The Cuba.jl package is licensed under the GNU Lesser General Public License, the same as Cuba library. The original author is Mosè Giordano. If you use this library for your work, please credit Thomas Hahn (citable papers about Cuba library: http://adsabs.harvard.edu/abs/2005CoPhC.168...78H and http://adsabs.harvard.edu/abs/2015JPhCS.608a2066H).

04/03/2016

3 days ago

126 commits