Description Usage Arguments Details Value References See Also Examples
Divonne works by stratified sampling, where the partioning of the integration region is aided by methods from numerical optimization.
1 2 3 4 5 6 7 8 9  divonne(ndim, ncomp, integrand, ...,
lower=rep(0,ndim), upper=rep(1,ndim),
rel.tol= 0.001, abs.tol = 0,
flags=list(verbose=1, final=1, pseudo.random=0, mersenne.seed=NULL),
min.eval=0, max.eval=50000,
key1=47, key2=1, key3=1,
max.pass=5, border=0, max.chisq=10,
min.deviation=0.25,
xgiven=NULL, nextra=0, peakfinder=NULL)

ndim 
same as 
ncomp 
same as 
integrand 
same as
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 integrand 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 In all other instances, 
... 
same as 
lower 
same as 
upper 
same as 
rel.tol 
same as 
abs.tol 
same as 
flags 
same as

min.eval 
same as 
max.eval 
same as 
key1 
integer that determines sampling in the partitioning phase:

key2 
integer that determines sampling in
the final integration phase:
same as 
key3 
integer that sets the strategy for the refinement phase:
Otherwise, the subregion is sampled a third time with 
max.pass 
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 A decrease in points generally indicates that Divonne discovered new structures of
the integrand and was able to find a more effective partitioning.

border 
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 nonzero 
max.chisq 
the maximum Chi2 value a single subregion is
allowed to have in the final integration phase. Regions which fail this Chi2 test and whose
sample averages differ by more than 
min.deviation 
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 Chi2 test. Only if the two sampling averages obtained for the region differ by more than this bound is the region further treated. 
xgiven 
a matrix ( 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. 
nextra 
the maximum number of extra points the peakfinder
subroutine will return. If 
peakfinder 
the peakfinder 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
where The returned value should be a matrix ( 
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 halfrange).
See details in the documentation.
Idem as cuhre
.
Here ifail
may be >1
when
the accuracy goal was not met within the allowed maximum number of
integrand evaluations. Divonne
can estimate the number of points by which
maxeval
needs to be increased to
reach the desired accuracy and returns this value.
J. H. Friedman, M. H. Wright (1981) A nested partitioning procedure for numerical multiple integration. ACM Trans. Math. Software, 7(1), 7692.
J. H. Friedman, M. H. Wright (1981) User's guide for DIVONNE. SLAC Report CGTM193REV, CGTM193, Stanford University.
T. Hahn (2005) CUBAa library for multidimensional numerical integration. Computer Physics Communications, 168, 7895.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  NDIM < 3
NCOMP < 1
integrand < function(arg, phase) {
x < arg[1]
y < arg[2]
z < arg[3]
ff < sin(x)*cos(y)*exp(z);
return(ff)
}
divonne(NDIM, NCOMP, integrand, rel.tol=1e3, abs.tol=1e12,
flags=list(verbose=2), key1= 47)
# Example with a peakfinder function
NMAX < 4
peakf < function(bounds) {
# print(bounds) # matrix (ndim,2)
x < matrix(0, ncol=NMAX, nrow=NDIM)
pas < 1/(NMAX1)
# 1ier point
x[,1] < rep(0, NDIM)
# Les autres points
for (i in 2:NMAX) {
x[,i] < x[,(i1)] + pas
}
return(x)
} #end peakf
divonne(NDIM, NCOMP, integrand,
flags=list(verbose=0) ,
peakfinder=peakf, nextra=NMAX)

Divonne input parameters:
ndim 3
ncomp 1
rel.tol 0.001
abs.tol 1e12
pseudo.random 0
final 0
verbose 2
min.eval 0
max.eval 50000
key1 47
key2 1
key3 1
max.pass 5
border 0
max.chisq 10
min.deviation 0.25
ngiven 0
nextra 0
Partitioning phase:
Iteration 1 (pass 0): 8 regions
836 integrand evaluations so far,
406 in optimizing regions,
70 in finding cuts
[1] 0.665011 + 0.00470198
Iteration 2 (pass 0): 9 regions
966 integrand evaluations so far,
478 in optimizing regions,
80 in finding cuts
[1] 0.664964 + 0.00429467
Iteration 3 (pass 1): 10 regions
1096 integrand evaluations so far,
550 in optimizing regions,
90 in finding cuts
[1] 0.664949 + 0.00388393
Iteration 4 (pass 2): 11 regions
1194 integrand evaluations so far,
590 in optimizing regions,
100 in finding cuts
[1] 0.664887 + 0.0035842
Iteration 5 (pass 3): 12 regions
1308 integrand evaluations so far,
646 in optimizing regions,
110 in finding cuts
[1] 0.664853 + 0.0033385
Iteration 6 (pass 4): 13 regions
1438 integrand evaluations so far,
718 in optimizing regions,
120 in finding cuts
[1] 0.664825 + 0.0031276
Iteration 7 (pass 5): 14 regions
1568 integrand evaluations so far,
790 in optimizing regions,
130 in finding cuts
[1] 0.664816 + 0.00292074
Main integration on 14 regions with 211 samples per region.integral: 0.6646195 (+0.00064)
nregions: 14; number of evaluations: 3052; probability: 1.110223e16
integral: 0.6646195 (+0.00064)
nregions: 14; number of evaluations: 3136; probability: 1.110223e16
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