vegas {cubature}  R Documentation 
Implement a Monte Carlo algorithm for multidimensional numerical integration. This algorithm uses importance sampling as a variancereduction technique. Vegas iteratively builds up a piecewise constant weight function, represented on a rectangular grid. Each iteration consists of a sampling step followed by a refinement of the grid.
vegas(
f,
nComp = 1L,
lowerLimit,
upperLimit,
...,
relTol = 1e05,
absTol = 1e12,
minEval = 0L,
maxEval = 10^6,
flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, load_state = 0L,
level = 0L),
rngSeed = 12345L,
nVec = 1L,
nStart = 1000L,
nIncrease = 500L,
nBatch = 1000L,
gridNo = 0L,
stateFile = NULL
)
f 
The function (integrand) to be integrated as in

nComp 
The number of components of f, default 1, bears no relation to the dimension of the hypercube over which integration is performed. 
lowerLimit 
The lower limit of integration, a vector for hypercubes. 
upperLimit 
The upper limit of integration, a vector for hypercubes. 
... 
All other arguments passed to the function f. 
relTol 
The maximum tolerance, default 1e5. 
absTol 
the absolute tolerance, default 1e12. 
minEval 
the minimum number of function evaluations required 
maxEval 
The maximum number of function evaluations needed, default 10^6. Note that the actual number of function evaluations performed is only approximately guaranteed not to exceed this number. 
flags 
flags governing the integration. The list here is exhaustive to keep the documentation and invocation uniform, but not all flags may be used for a particular method as noted below. List components:

rngSeed 
seed, default 0, for the random number
generator. Note the articulation with 
nVec 
the number of vectorization points, default 1, but can be set to an integer > 1 for vectorization, for example, 1024 and the function f above needs to handle the vector of points appropriately. See vignette examples. 
nStart 
the number of integrand evaluations per iteration to start with. 
nIncrease 
the increase in the number of integrand evaluations per iteration. The jth iteration evaluates the integrand at nStart+(j1)*nincrease points. 
nBatch 
Vegas samples points not all at once, but in batches
of a predetermined size, to avoid excessive memory
consumption. 
gridNo 
an integer. Vegas 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. If 
stateFile 
the name of an external file. Vegas can store its entire internal state (i.e. all the information to resume an interrupted 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 existing 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 ‘checkpoints’ in longrunning integrations from which the calculation can be restarted. 
See details in the documentation.
A list with components:
the actual number of subregions needed
the actual number of integrand evaluations needed
if zero, the desired accuracy was reached, if 1, dimension out of range, if 1, the accuracy goal was not met within the allowed maximum number of integrand evaluations.
vector of length nComp
; the integral of
integrand
over the hypercube
vector of
length nComp
; the presumed absolute error of
integral
vector of length nComp
;
the \chi^2
probability (not the
\chi^2
value itself!) that error
is not a
reliable estimate of the true integration error.
G. P. Lepage (1978) A new algorithm for adaptive multidimensional integration. J. Comput. Phys., 27, 192210.
G. P. Lepage (1980) VEGAS  An adaptive multidimensional integration program. Research Report CLNS80/447. Cornell University, Ithaca, N.Y.
T. Hahn (2005) CUBAa library for multidimensional numerical integration. Computer Physics Communications, 168, 7895.
integrand < function(arg, weight) {
x < arg[1]
y < arg[2]
z < arg[3]
ff < sin(x)*cos(y)*exp(z);
return(ff)
} # end integrand
vegas(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3),
relTol=1e3, absTol=1e12,
flags=list(verbose=2, final=0))