Unified Cubature Integration Interface

Integrate a function within specified limits using method specified. Further arguments specific to method as well as other arguments to f may be passed. For defaults used in each method, see help on the method or default_args().

Usage

cubintegrate(
  f,
  lower,
  upper,
  fDim = 1,
  method = c("hcubature", "pcubature", "cuhre", "divonne", "suave", "vegas"),
  relTol = 1e-05,
  absTol = 1e-12,
  maxEval = 10^6,
  nVec = 1L,
  ...
)

Arguments

f: The function (integrand) to be integrated. Can be vectorized version, but the additional arguments ... must indicate via either vectorInterface = TRUE for hcubature and pcubature, or a value for nVec. See details on each method.
lower: The lower limit of integration, a vector for hypercubes.
upper: The upper limit of integration, a vector for hypercubes.
fDim: The number of components of f, default 1, bears no relation to the dimension of the hypercube over which integration is performed.
method: the method to use should be one of "hcubature", "pcubature", "cuhre", "divonne", "suave" or "vegas".
relTol: The maximum tolerance, default 1e-5.
absTol: the absolute tolerance, default 1e-12.
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.
nVec: the number of vectorization points for Cuba C library, default 1, but can be set to an integer > 1 for vectorization, for example, 1024. The function f above needs to handle the vector of points appropriately; see vignette examples. Unlike Cuba, the cubature C library manages the number of points on its own and can vary between calls. Therefore, any value for nVec greater than one implies vectorization for a cubature method.
...: All other arguments which may include integration method specific parameters and those for f. Unrecognized parameters for integration method are presumed to be intended for f and so processed.

Value

The returned value is a list of items:

integral: the value of the integral
error: the estimated absolute error
neval: the number of times the function was evaluated
returnCode: the actual integer return code of the C routine; a non-zero value usually indicates problems; further interpretation depends on method
nregions: forcCuba routines, the actual number of subregions needed
prob: the \(\chi^2\)-probability (not the \(\chi^2\)-value itself!) that error is not a reliable estimate of the true integration error.

Examples

I.1d <- function(x) {
  sin(4*x) *
    x * ((x * ( x * (x*x-4) + 1) - 1))
}
I.1d_v <- function(x) {
   matrix(apply(x, 2, function(z)
       sin(4 * z) *
       z * ((z * ( z * (z * z - 4) + 1) - 1))),
       ncol = ncol(x))
}
cubintegrate(f = I.1d, lower = -2, upper = 2, method = "pcubature")
#> $integral
#> [1] 1.635644
#> 
#> $error
#> [1] 1.332268e-15
#> 
#> $neval
#> [1] 65
#> 
#> $returnCode
#> [1] 0
#> 
cubintegrate(f = I.1d, lower = -2, upper = 2, method = "cuhre", flags=list(verbose = 2))
#> Cuhre input parameters:
#>   ndim 1
#>   ncomp 1
#>   nvec 1
#>   epsrel 1e-05
#>   epsabs 1e-12
#>   flags 6
#>   mineval 0
#>   maxeval 1000000
#>   key 0
#>   statefile "(null)"
#> 
#> Iteration 1:  11 integrand evaluations so far
#> [1] 1.79496 +- 36.4275  	chisq 0 (0 df)
#> 
#> Iteration 2:  33 integrand evaluations so far
#> [1] 1.60939 +- 15.0794  	chisq 2.21553e-05 (1 df)
#> 
#> Iteration 3:  55 integrand evaluations so far
#> [1] 1.62346 +- 7.01714  	chisq 2.28586e-05 (2 df)
#> 
#> Iteration 4:  77 integrand evaluations so far
#> [1] 1.63569 +- 0.145134  	chisq 2.5197e-05 (3 df)
#> 
#> Iteration 5:  99 integrand evaluations so far
#> [1] 1.63568 +- 0.0643672  	chisq 2.51973e-05 (4 df)
#> 
#> Iteration 6:  121 integrand evaluations so far
#> [1] 1.63568 +- 0.0123924  	chisq 2.52128e-05 (5 df)
#> 
#> Iteration 7:  143 integrand evaluations so far
#> [1] 1.63566 +- 0.00587027  	chisq 2.67284e-05 (6 df)
#> 
#> Iteration 8:  165 integrand evaluations so far
#> [1] 1.63564 +- 5.32936e-05  	chisq 3.85791e-05 (7 df)
#> 
#> Iteration 9:  187 integrand evaluations so far
#> [1] 1.63564 +- 3.85904e-05  	chisq 3.8594e-05 (8 df)
#> 
#> Iteration 10:  209 integrand evaluations so far
#> [1] 1.63564 +- 2.5705e-05  	chisq 3.86238e-05 (9 df)
#> 
#> Iteration 11:  231 integrand evaluations so far
#> [1] 1.63564 +- 2.02465e-05  	chisq 3.86238e-05 (10 df)
#> 
#> Iteration 12:  253 integrand evaluations so far
#> [1] 1.63564 +- 1.51801e-05  	chisq 3.8147e-05 (11 df)
#> $integral
#> [1] 1.635644
#> 
#> $error
#> [1] 1.518009e-05
#> 
#> $nregions
#> [1] 12
#> 
#> $neval
#> [1] 253
#> 
#> $prob
#> [1] 0
#> 
#> $returnCode
#> [1] 0
#> 
cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "hcubature", nVec = 2L)
#> $integral
#> [1] 1.635644
#> 
#> $error
#> [1] 4.024021e-09
#> 
#> $neval
#> [1] 105
#> 
#> $returnCode
#> [1] 0
#> 
cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "cuhre", nVec = 128L)
#> $integral
#> [1] 1.635644
#> 
#> $error
#> [1] 1.518009e-05
#> 
#> $nregions
#> [1] 12
#> 
#> $neval
#> [1] 253
#> 
#> $prob
#> [1] 0
#> 
#> $returnCode
#> [1] 0
#>

Usage

Arguments

Value

See also

Examples