Integration by a Deterministic Iterative Adaptive Algorithm

Implement a deterministic algorithm for multidimensional numerical integration. Its algorithm uses one of several cubature rules in a globally adaptive subdivision scheme. The subdivision algorithm is similar to suave().

Usage

cuhre(
  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),
  key = 0L,
  nVec = 1L,
  stateFile = NULL
)

Arguments

f

The function (integrand) to be integrated. For cuhre, it can be something as simple as a function of a single argument, say x.

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 1e-5.

absTol

the absolute tolerance, default 1e-12.

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:

verbose

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.

final

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.

smooth

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.

keep_state

when nonzero, retain state file if argument stateFile is non-null, else delete stateFile if specified.

load_state

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.

level

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:

Level 1 (p = 48)

gives very long period, passes the gap test but fails spectral test

Level 2 (p = 97)

passes all known tests, but theoretically still defective

Level 3 (p = 223)

any theoretically possible correlations have very small chance of being observed

Level 4 (p = 389)

highest possible luxury, all 24 bits chaotic

Levels 5-23

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

key

the quadrature rule key: key = 7, 9, 11, 13 selects the cubature rule of degree key. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions. For other values, including the default 0, the rule is the degree-13 rule in 2 dimensions, the degree-11 rule in 3 dimensions, and the degree-9 rule otherwise.

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.

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 ‘check-points’ in long-running integrations from which the calculation can be restarted.

Value

A list with components:

neval

the actual number of integrand evaluations needed

returnCode

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.

integral

vector of length nComp; the integral of integrand over the hypercube

error

vector of length nComp; the presumed absolute error of integral

prob

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.

Details

See details in the documentation.

References

J. Berntsen, T. O. Espelid (1991) An adaptive algorithm for the approximate calculation of multiple integrals. ACM Transactions on Mathematical Software, 17(4), 437-451.

T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95. See https://feynarts.de/cuba/

See also

vegas(), suave(), divonne()

Examples

integrand <- function(arg) {
  x <- arg[1]
  y <- arg[2]
  z <- arg[3]
  ff <- sin(x)*cos(y)*exp(z);
return(ff)
} # End integrand

NDIM <- 3
NCOMP <- 1
cuhre(f = integrand,
      lowerLimit = rep(0, NDIM),
      upperLimit = rep(1, NDIM),
      relTol = 1e-3, absTol= 1e-12,
      flags = list(verbose = 2, final = 0))
#> $integral
#> [1] 0.6646697
#> 
#> $error
#> [1] 3.33018e-11
#> 
#> $nregions
#> [1] 2
#> 
#> $neval
#> [1] 381
#> 
#> $prob
#> [1] 0
#> 
#> $returnCode
#> [1] 0
#>