Suave uses vegas()-like importance sampling combined with a
globally adaptive subdivision strategy: Until the requested accuracy is
reached, the region with the largest error at the time is bisected in the
dimension in which the fluctuations of the integrand are reduced most. The
number of new samples in each half is prorated for the fluctuation in that
half.
suave(
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, level = 0L),
rngSeed = 0L,
nVec = 1L,
nNew = 1000L,
nMin = 50L,
flatness = 50,
stateFile = NULL
)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.
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 new integrand evaluations in each subdivision.
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.
the parameter p, or the type of norm used to compute the fluctuation of a sample. This determines how prominently "outliers," 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.
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.
See details in the documentation.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
cuhre(), divonne(), vegas()
integrand <- function(arg) {
x <- arg[1]
y <- arg[2]
z <- arg[3]
ff <- sin(x)*cos(y)*exp(z);
return(ff)
} # end integrand
suave(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3),
relTol=1e-3, absTol=1e-12,
flags=list(verbose=2, final=0))
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