Implement a Monte Carlo algorithm for multidimensional numerical integration. This algorithm uses importance sampling as a variance-reduction 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 = 1e-05,
absTol = 1e-12,
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
)The function (integrand) to be integrated as in
cuhre(). Optionally, the function can take two
additional arguments in addition to the variable being
integrated: - cuba_weight which is the weight of the
point being sampled, - cuba_iter the current iteration
number. The function author may choose to use these in any
appropriate way or ignore them altogether.
The number of components of f, default 1, bears no relation to the dimension of the hypercube over which integration is performed.
The lower limit of integration, a vector for hypercubes.
The upper limit of integration, a vector for hypercubes.
All other arguments passed to the function f.
The maximum tolerance, default 1e-5.
the absolute tolerance, default 1e-12.
the minimum number of function evaluations required
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 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:
encodes the verbosity level, from 0 (default) to 3. Level 0 does not print any output, level 1 prints reasonable information on the progress of the integration, level 2 also echoes the input parameters, and level 3 further prints the subregion results.
when 0, all sets of samples collected on a subregion during the various iterations or phases contribute to the final result. When 1, only the last (largest) set of samples is used in the final result.
Applies to Suave and Vegas only. When 0, apply additional smoothing to the importance function, this moderately improves convergence for many integrands. When 1, use the importance function without smoothing, this should be chosen if the integrand has sharp edges.
when nonzero, retain state file if argument stateFile is non-null, else delete stateFile if specified.
Applies to Vegas only. Reset the integrator state even if a state file is present, i.e. keep only the grid. Together with keep_state this allows a grid adapted by one integration to be used for another integrand.
applies only to Divonne, Suave and Vegas. When 0, Mersenne Twister random numbers are used. When nonzero Ranlux random numbers are used, except when rngSeed is zero which forces use of Sobol quasi-random numbers. Ranlux implements Marsaglia and Zaman's 24-bit RCARRY algorithm with generation period \(p\), i.e. for every 24 generated numbers used, another \(p-24\) are skipped. The luxury level for the Ranlux generator may be encoded in level as follows:
gives very long period, passes the gap test but fails spectral test
passes all known tests, but theoretically still defective
any theoretically possible correlations have very small chance of being observed
highest possible luxury, all 24 bits chaotic
default to 3, values above 24 directly specify the period p. Note that Ranlux's original level 0, (mis)used for selecting Mersenne Twister in Cuba, is equivalent to level = 24
seed, default 0, for the random number
generator. Note the articulation with level settings for
flag
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.
the number of integrand evaluations per iteration to start with.
the increase in the number of integrand evaluations per iteration. The j-th iteration evaluates the integrand at nStart+(j-1)*nincrease points.
Vegas samples points not all at once, but in batches
of a predetermined size, to avoid excessive memory
consumption. nbatch is the number of points sampled in
each batch. Tuning this number should usually not be necessary
as performance is affected significantly only as far as the
batch of samples fits into the CPU cache.
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 gridno is a number between 1 and 10, 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 this case the negative of the grid number
should be set. Vegas will then start with a new grid and also
restore the grid number to its positive value, such that at the
end of the integration the grid is again stored in the
indicated slot.
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 ‘check-points’ in long-running integrations from which the calculation can be restarted.
A list with components:
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.
See details in the documentation.
G. P. Lepage (1978) A new algorithm for adaptive multidimensional integration. J. Comput. Phys., 27, 192-210.
G. P. Lepage (1980) VEGAS - An adaptive multi-dimensional integration program. Research Report CLNS-80/447. Cornell University, Ithaca, N.-Y.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
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=1e-3, absTol=1e-12,
flags=list(verbose=2, final=0))
#> $integral
#> [1] 0.6652311
#>
#> $error
#> [1] 0.0006170268
#>
#> $neval
#> [1] 13500
#>
#> $prob
#> [1] 0.6053001
#>
#> $returnCode
#> [1] 0
#>