The Cuba library offers a choice of four independent routines for multidimensional numerical integration: Vegas, Suave, Divonne, and Cuhre. They work by very different methods, summarized in the following table:
Routine | Basic integration method | Algorithm type | Variance reduction |
---|---|---|---|
Vegas | Sobol quasi-random sample or Mersenne Twister pseudo-random sample or Ranlux pseudo-random sample |
Monte Carlo Monte Carlo |
importance sampling |
Suave | Sobol quasi-random sample or Mersenne Twister pseudo-random sample or Ranlux pseudo-random sample |
Monte Carlo Monte Carlo |
globally adaptive subdivision + importance sampling |
Divonne | Korobov quasi-random sample or Sobol quasi-random sample or Mersenne Twister pseudo-random sample or Ranlux pseudo-random sample or cubature rules |
Monte Carlo Monte Carlo Monte Carlo deterministic |
stratified sampling, aided by methods from numerical optimization |
Cuhre | cubature rules | deterministic | globally adaptive subdivision |
All four have a C/C++, Fortran, and Mathematica interface and can integrate vector integrands. Their invocation is very similar, so it is easy to substitute one method by another for cross-checking. For further safeguarding, the output is supplemented by a chi-square probability which quantifies the reliability of the error estimate.
The source code compiles with gcc, the GNU C compiler. The C functions can be called from Fortran directly, so there is no need for adapter code. Similarly, linking Fortran code with the library is straightforward and requires no extra tools.
Ready-made MathLink executables (Version 2.1, statically linked as far as possible):
Linux x86 (32-bit):
Mac OS X (universal binaries):
Windows (32-bit):
Vegas is the simplest of the four. It uses importance sampling for variance reduction, but is only in some cases competitive in terms of the number of samples needed to reach a prescribed accuracy. Nevertheless, it has a few improvements over the original algorithm and comes in handy for cross-checking the results of other methods.
Suave is a new algorithm which combines the advantages of two popular methods: importance sampling as done by Vegas and subregion sampling in a manner similar to Miser. By dividing into subregions, Suave manages to a certain extent to get around Vegas' difficulty to adapt its weight function to structures not aligned with the coordinate axes.
Divonne is a further development of the CERNLIB routine D151. Divonne works by stratified sampling, where the partitioning of the integration region is aided by methods from numerical optimization. A number of improvements have been added to this algorithm, the most significant being the possibility to supply knowledge about the integrand. Narrow peaks in particular are difficult to find without sampling very many points, especially in high dimensions. Often the exact or approximate location of such peaks is known from analytic considerations, however, and with such hints the desired accuracy can be reached with far fewer points.
Cuhre employs a cubature rule for subregion estimation in a globally adaptive subdivision scheme. It is hence a deterministic, not a Monte Carlo method. In each iteration, the subregion with the largest error is halved along the axis where the integrand has the largest fourth difference. Cuhre is quite powerful in moderate dimensions, and is usually the only viable method to obtain high precision, say relative accuracies much below 1e-3.
Upward of 75% of all questions regarding Cuba have to do with how to choose bounds different from the unit hypercube in Fortran, C, and C++. The solution is not to choose bounds but to scale the integrand. For the mathematically challenged, the explicit transformation is (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.
For constant integration bounds, this transformation might be implemented in Fortran as
subroutine ScaledIntegrand(ndim, x, ncomp, result) implicit none integer ndim, ncomp double precision x(ndim), result(ncomp) integer maxdim parameter (maxdim = 16) double precision upper(maxdim) common /ubound/ upper double precision lower(maxdim) common /lbound/ lower integer dim, comp double precision range, jacobian, scaledx(maxdim) jacobian = 1 do dim = 1, ndim range = upper(dim) - lower(dim) jacobian = jacobian*range scaledx(dim) = lower(dim) + x(dim)*range enddo call Integrand(ndim, scaledx, ncomp, result) do comp = 1, ncomp result(comp) = result(comp)*jacobian enddo end
This site and the programs offered here are not commercial. Cuba is an open-source package and free of charge. If you want to use Cuba in a commercial application, make sure you understand the GNU Lesser General Public License under which Cuba is distributed. Cuba is being developed at the Max Planck Institute for Physics in Munich.
Last update: 18 Apr 14 Thomas Hahn