Divonne works by stratified sampling, where the partioning of the integration region is aided by methods from numerical optimization.
divonne(
f,
nComp = 1L,
lowerLimit,
upperLimit,
...,
relTol = 1e-05,
absTol = 1e-12,
minEval = 0L,
maxEval = 10^6,
flags = list(verbose = 0L, final = 1L, keep_state = 0L, level = 0L),
rngSeed = 0L,
nVec = 1L,
key1 = 47L,
key2 = 1L,
key3 = 1L,
maxPass = 5L,
border = 0,
maxChisq = 10,
minDeviation = 0.25,
xGiven = NULL,
nExtra = 0L,
peakFinder = NULL,
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 an
additional argument in addition to the variable being
integrated: - cuba_phase - indicating the integration phase:
sampling of the points in xgiven
partitioning phase
final integration phase
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
cuba_phase < 2. In all other instances, phase can be
ignored and it is entirely admissible to define the integrand
without it.
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.
integer that 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 \(n=|key1|\) points is used, where
the sign of key1 determines the type of sample,
key1 = 0, use the default rule. key1 > 0, use a
Korobov quasi-random sample, key1 < 0, use a Sobol
quasi-random sample if flags$seed is zero, otherwise a
“standard” sample (Mersenne Twister) pseudo-random
sample
integer that determines sampling in the final
integration phase: same as key1, but here
\(n=|key2|\) determines the number of
points, \(n > 39\), sample \(n\) points,
\(n < 40\), sample \(n\)
nneed 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.
integer that 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.
integer that controls the thoroughness of the
partitioning phase: The partitioning phase terminates when the
estimated total number of integrand evaluations (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 partition is accepted as final and
counting consequently restarts at zero whenever new structures
are found.
the relative width of the border of the integration
region. Points falling into the border region will not be
sampled directly, but will be extrapolated from two samples
from the interior. Use a non-zero border if the
integrand subroutine cannot produce values directly on the
integration boundary. The relative width of the border is
identical in all the dimensions. For example, set
border=0.1 for a border of width equal to 10\
width of the integration region.
the maximum \(\chi^2\) value a single
subregion is allowed to have in the final integration
phase. Regions which fail this \(\chi^2\) test and
whose sample averages differ by more than min.deviation
move on to the refinement phase.
a bound, given as the fraction of the requested error of the entire integral, which determines whether it is worthwhile further examining a region that failed the \(\chi^2\) test. Only if the two sampling averages obtained for the region differ by more than this bound is the region further treated.
a matrix (nDim, nGiven). A list of
nGiven points where the integrand 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 integrand in the presence of very
narrow peaks. Even if only the approximate location of such
peaks is known, this can considerably speed up convergence.
the maximum number of extra points the peak-finder
subroutine will return. If nextra is zero,
peakfinder is not called and an arbitrary object may be
passed in its place, e.g. just 0.
the peak-finder subroutine. This R function 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
peakFinder <- function(bounds, nMax) where bounds
is a matrix of dimension (2, nDim) which contains the
lower (row 1) and upper (row 2) bounds of the subregion. The
returned value should be a matrix (nX, nDim) where
nX is the actual number of points (should be less or
equal to nMax).
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.
Divonne uses stratified sampling for variance reduction, that is, it partitions the integration region such that all subregions have an approximately equal value of a quantity called the spread (volume times half-range).
See details in the documentation.
J. H. Friedman, M. H. Wright (1981) A nested partitioning procedure for numerical multiple integration. ACM Trans. Math. Software, 7(1), 76-92.
J. H. Friedman, M. H. Wright (1981) User's guide for DIVONNE. SLAC Report CGTM-193-REV, CGTM-193, Stanford University.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
cuhre(), suave(), vegas()
integrand <- function(arg, phase) {
x <- arg[1]
y <- arg[2]
z <- arg[3]
ff <- sin(x)*cos(y)*exp(z);
return(ff)
}
divonne(integrand, relTol=1e-3, absTol=1e-12, lowerLimit = rep(0, 3), upperLimit = rep(1, 3),
flags=list(verbose = 2), key1= 47)
# Example with a peak-finder function
nDim <- 3L
peakf <- function(bounds, nMax) {
# print(bounds) # matrix (ndim,2)
x <- matrix(0, ncol = nMax, nrow = nDim)
pas <- 1 / (nMax - 1)
# 1ier point
x[, 1] <- rep(0, nDim)
# Les autres points
for (i in 2L:nMax) {
x[, i] <- x[, (i - 1)] + pas
}
x
} #end peakf
divonne(integrand, relTol=1e-3, absTol=1e-12,
lowerLimit = rep(0, 3), upperLimit = rep(1, 3),
flags=list(verbose = 2), peakFinder = peakf, nExtra = 4L)
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