Triple integral in R (how to specifying the domain)

Question

I would like to compute the triple integral of a function of three variables f(x,y,z) in R. I'm using the package cubature and the function adaptIntegrate(). The integrand is equal to 1 only in a certain domain (x<y<z, 0 otherwise) which I don't know how to specify. I'm trying 2 different implementations of the function, but none of them work:

#First implementation
fxyz <- function(w) {
x <- w[1]
y <- w[2]
z <- w[3]
x*y*z*(x < y)&(y < z)
}

#Second implementation
fxyz <- function(w) {
x <- w[1]
y <- w[2]
z <- w[3]
if(x<y&y<z)
    out<-1
else
    out<-0
out
}

#Computation of integral
library(cubature)
lower <- rep(0,3)
upper <- rep(1, 3)
adaptIntegrate(f=fxyz, lowerLimit=lower, upperLimit=upper, fDim = 3)

Any idea on how to specify the domain correctly?

Answer 1

In your first function the return value is wrong. It should be as.numeric(x<=y)*as.numeric(y<=z). In your second function you should also use <= instead of <, otherwise `adapIntegrate won't work correctly. You also need to specify a maximum number of evaluations. Try this

library(cubature)
lower <- rep(0,3)
upper <- rep(1,3)

# First implementation (modified)
fxyz <- function(w) {
    x <- w[1]
    y <- w[2]
    z <- w[3]
    as.numeric(x <= y)*as.numeric(y <= z)
}

adaptIntegrate(f=fxyz,lowerLimit=lower,upperLimit=upper,doChecking=TRUE,
          maxEval=2000000,absError=10e-5,tol=1e-5)
#$integral
#[1] 0.1664146
#$error
#[1] 0.0001851699
#$functionEvaluations
#[1] 2000031
#$returnCode
#[1] 0