Cuba is a library for multidimensional numerical integration. The manual in Cuba is helpful as a reference. Parallelisation is currently not supported.

The algorithms available are

  • Cuhre
  • Divonne
  • Suave
  • Vegas

Make sure you installed everything correctly.

PyCuba Demo Program

Take a look at the demo

PyCuba API

The return value is always a dictionary:

{ 'neval': number of evaluations,
  'fail': 0 or 1,
  'comp': number of components, usually 1,
  'nregions': number of regions used,
  'results': [ # a list of results for each component
       'integral': value of the integral,
       'error':  uncertainty,
       'prob': probability (see manual),

Define your integrand, like so:

def Integrand(ndim, xx, ncomp, ff, userdata):
        # access the current parameters
        x,y,z = [xx[i] for i in range(ndim.contents.value)]
        # compute the result
        result = math.sin(x)*math.cos(y)*math.exp(z)
        # store the result (here only one component)
        ff[0] = result
        return 0

It will be called with xx in the interval from 0 to 1 (so scale to your borders in this function).

The pycuba.demo() in the pycuba source shows how to run all of the algorithms and access the results:

  import math

  def Integrand(ndim, xx, ncomp, ff, userdata):
    x,y,z = [xx[i] for i in range(ndim.contents.value)]
    result = math.sin(x)*math.cos(y)*math.exp(z)
    ff[0] = result
    return 0

  NDIM = 3
  NCOMP = 1

  NNEW = 1000
  NMIN = 2
  FLATNESS = 50.

  KEY1 = 47
  KEY2 = 1
  KEY3 = 1
  BORDER = 0.
  MAXCHISQ = 10.
  NGIVEN = 0
  NEXTRA = 0
  MAXEVAL = 50000

  KEY = 0
  def print_header(name):
    print('-------------------- %s test -------------------' % name)
  def print_results(name, results):
    keys = ['nregions', 'neval', 'fail']
    keys = list(filter(results.has_key, keys))
    text = ["%s %d" % (k, results[k]) for k in keys]
    print("%s RESULT:\t" % name.upper() + "\t".join(text))
    for comp in results['results']:
      print("%s RESULT:\t" % name.upper() + \
	"%(integral).8f +- %(error).8f\tp = %(prob).3f\n" % comp)
  from os import environ as env
  verbose = 2
  if 'CUBAVERBOSE' in env:
    verbose = int(env['CUBAVERBOSE'])
  print_results('Vegas', Vegas(Integrand, NDIM, verbose=2))
  print_results('Suave', Suave(Integrand, NDIM, NNEW, FLATNESS, verbose=2 | 4))
  print_results('Divonne', Divonne(Integrand, NDIM, 
    mineval=MINEVAL, maxeval=MAXEVAL,
    key1=KEY1, key2=KEY2, key3=KEY3, maxpass=MAXPASS,
    border=BORDER, maxchisq=MAXCHISQ, mindeviation=MINDEVIATION,
    ldxgiven=LDXGIVEN, verbose=2))
  print_results('Cuhre', Cuhre(Integrand, NDIM, key=KEY, verbose=2 | 4))

API documentation

pycuba.Cuhre(integrand, ndim, key=0, mineval=0, maxeval=50000, ncomp=1, userdata=<pycuba.LP_c_int object>, seed=None, epsrel=0.001, epsabs=1e-12, verbose=0, statefile=<pycuba.LP_c_int object>, nvec=1)[source]
key chooses the basic integration rule:
key = 7, 9, 11, 13 selects the cubature rule of degree key. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions. For other values, the default rule is taken, which is the degree-13 rule in 2 dimensions, the degree-11 rule in 3 dimensions, and the degree-9 rule otherwise.
pycuba.Divonne(integrand, ndim, key1, key2, key3, maxpass, border, maxchisq, mindeviation, mineval=0, maxeval=50000, ncomp=1, ldxgiven=None, xgiven=None, nextra=0, peakfinder=None, userdata=<pycuba.LP_c_int object>, seed=None, epsrel=0.001, epsabs=1e-12, verbose=0, statefile=<pycuba.LP_c_int object>, nvec=1)[source]
key1: determines sampling in the partitioning phase:
key1 = 7, 9, 11, 13 selects the cubature rule of degree key1. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions. For other values of key1, a quasi-random sample of n1 = |key1| points is used, where the sign of key1 determines the type of sample, - key1 > 0, use a Korobov quasi-random sample, - key1 < 0, use a “standard” sample (a Sobol quasi-random sample if seed = 0, otherwise a pseudo-random sample).
key2: determines sampling in the final integration phase:
key2 = 7, 9, 11, 13 selects the cubature rule of degree key2. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions. For other values of key2, a quasi-random sample is used, where the sign of key2 determines the type of sample, - key2 > 0, use a Korobov quasi-random sample, - key2 < 0, use a ‘standard’ sample (see description of key1 above), and n2 = |key2| determines the number of points, - n2 >= 40, sample n2 points, - n2 < 40, sample n2 need points, where nneed is the number of points needed to reach the prescribed accuracy, as estimated by Divonne from the results of the partitioning phase.
key3 sets the strategy for the refinement phase:
key3 = 0, do not treat the subregion any further. key3 = 1, split the subregion up once more. Otherwise, the subregion is sampled a third time with key3 specifying the sampling parameters exactly as key2 above.
maxpass: controls the thoroughness of the partitioning phase: The
partitioning phase terminates when the estimated total number of integrand evalu- ations (partitioning plus final integration) does not decrease for maxpass successive iterations. A decrease in points generally indicates that Divonne discovered new structures of the integrand and was able to find a more effective partitioning. maxpass can be understood as the number of ‘safety’ iterations that are performed before the par- tition is accepted as final and counting consequently restarts at zero whenever new structures are found.
border: the width of the border of the integration region.
Points falling into this border region will not be sampled directly, but will be extrap- olated from two samples from the interior. Use a non-zero border if the integrand subroutine cannot produce values directly on the integration boundary.
maxchisq: the maximum chisquare value a single subregion is al-
lowed to have in the final integration phase. Regions which fail this chisquare test and whose sample averages differ by more than mindeviation move on to the refinement phase.
mindeviation: a bound, given as the fraction of the re-
quested error of the entire integral, which determines whether it is worthwhile fur- ther examining a region that failed the chisquare test. Only if the two sampling averages obtained for the region differ by more than this bound is the region further treated.
ldxgiven: the leading dimension of xgiven, i.e. the offset between one
point and the next in memory.
xgiven(ldxgiven,ngiven): a list of points where the inte-
grand might have peaks. Divonne will consider these points when partitioning the integration region. The idea here is to help the integrator find the extrema of the in- tegrand in the presence of very narrow peaks. Even if only the approximate location of such peaks is known, this can considerably speed up convergence.
nextra: the maximum number of extra points the peak-finder subrou-
tine will return. If nextra is zero, peakfinder is not called and an arbitrary object may be passed in its place, e.g. just 0.
peakfinder: the peak-finder subroutine. This subroutine is called

whenever a region is up for subdivision and is supposed to point out possible peaks lying in the region, thus acting as the dynamic counterpart of the static list of points supplied in xgiven. It is expected to be declared as def peakfinder(ndim, b, n, x):

The bounds of the subregion are passed in the array b, where b(1,d ) is the lower and b(2,d ) the upper bound in dimension d . On entry, n specifies the maximum number of points that may be written to x. On exit, n must contain the actual number of points in x.

Divonne actually passes the integrand one more argument, i.e. the integrand subroutine is really declared as

def integrand(ndim, x, ncomp, f, phase):

The fifth argument, phase, indicates the integration phase:
0, sampling of the points in xgiven, 1, partitioning phase, 2, final integration phase, 3, refinement phase.

This information might be useful if the integrand takes long to compute and a sufficiently accurate approximation of the integrand is available. The actual value of the integral is only of minor importance in the partitioning phase, which is instead much more dependent on the peak structure of the integrand to find an appropriate tessellation. An approximation which reproduces the peak structure while leaving out the fine details might hence be a perfectly viable and much faster substitute when phase < 2.

In all other instances, phase can be ignored and it is entirely admissible to define the integrand with only five arguments.

pycuba.Suave(integrand, ndim, nnew=1000, nmin=2, flatness=50.0, userdata=<pycuba.LP_c_int object>, epsrel=0.001, epsabs=1e-12, verbose=0, ncomp=1, seed=None, mineval=0, maxeval=50000, statefile=<pycuba.LP_c_int object>, nvec=1)[source]

nnew: the number of new integrand evaluations in each subdivision.

nmin: the minimum number of samples a former pass must contribute to a subregion to be considered in that region’s compound integral value. Increasing nmin may reduce jumps in the chi^2 value.

flatness: the parameter p in Eq. (1), i.e. the type of norm used to compute the fluctuation of a sample. This determines how prominently ‘out- liers,’ i.e. individual samples with a large fluctuation, figure in the total fluctuation, which in turn determines how a region is split up. As suggested by its name, flatness should be chosen large for ‘flat’ integrands and small for ‘volatile’ integrands with high peaks. Note that since flatness appears in the exponent, one should not use too large values (say, no more than a few hundred) lest terms be truncated internally to prevent overflow.

pycuba.Vegas(integrand, ndim, userdata=<pycuba.LP_c_int object>, epsrel=0.001, epsabs=1e-12, verbose=0, ncomp=1, seed=None, mineval=0, maxeval=50000, nstart=1000, nincrease=500, nbatch=1000, gridno=0, statefile=<pycuba.LP_c_int object>, nvec=1)[source]
nstart: the number of integrand evaluations per iteration to start
nincrease: the increase in the number of integrand evaluations per
nbatch: the batch size for sampling.
Vegas samples points not all at once, but in batches of size nbatch, to avoid exces- sive memory consumption. 1000 is a reasonable value, though it should not affect performance too much.
gridno: the slot in the internal grid table.
It may accelerate convergence to keep the grid accumulated during one integration for the next one, if the integrands are reasonably similar to each other. Vegas maintains an internal table with space for ten grids for this purpose. The slot in this grid is specified by gridno. If a grid number between 1 and 10 is selected, the grid is not discarded at the end of the integration, but stored in the respective slot of the table for a future invocation. The grid is only re-used if the dimension of the subsequent integration is the same as the one it originates from. In repeated invocations it may become necessary to flush a slot in memory, in which case the negative of the grid number should be set.
statefile: a filename for storing the internal state.
Vegas can store its entire internal state (i.e. all the information to resume an inter- rupted integration) in an external file. The state file is updated after every iteration. If, on a subsequent invocation, Vegas finds a file of the specified name, it loads the internal state and continues from the point it left off. Needless to say, using an ex- isting state file with a different integrand generally leads to wrong results. Once the integration finishes successfully, i.e. the prescribed accuracy is attained, the state file is removed. This feature is useful mainly to define ‘check-points’ in long-running integrations from which the calculation can be restarted.

Vegas actually passes the integrand two more arguments, i.e. the integrand subroutine is really declared as

def integrand(ndim, x, ncomp, f, userdata, weight, iter):

where weight contains the weight of the point being sampled and iter the current iteration number. These extra arguments can safely be ignored, i.e. the integrand may be (and usually is) defined with just five (or even four, if userdata is also ignored) arguments.