cubature (version 1.0)

adaptIntegrate: Adaptive multivariate integration over hypercubes

Description

The function performs adaptive multidimensional integration (cubature) of (possibly) vector-valued integrands over hypercubes.

Usage

adaptIntegrate(f, lowerLimit, upperLimit, tol = 1e-05, fDim = 1, maxEval = 0, absError=0)

Arguments

f

The function (integrand) to be integrated

lowerLimit

The lower limit of integration, a vector for hypercubes

upperLimit

The upper limit of integration, a vector for hypercubes

tol

The maximum tolerance, default 1e-5.

fDim

The dimension of the integrand, default 1, bears no relation to the dimension of the hypercube

maxEval

The maximum number of function evaluations needed, default 0 implying no limit

absError

The maximum absolute error tolerated

Value

• The returned value is a list of three items:
• integral
  the value of the integral
• error
  the estimated relative error
• functionEvaluations
  the number of times the function was evaluated
• returnCode
  the actual integer return code of the C routine

Details

The function merely calls Johnson's C code and returns the results. The original C code by Johnson was modified for use with R memory allocation functions and a helper function does the callback.

One can specify a maximum number of function evaluations (default is 0 for no limit). Otherwise, the integration stops when the estimated error is less than the absolute error requested, or when the estimated error is less than tol times the integral, in absolute value.

References

See http://ab-initio.mit.edu/wiki/index.php/Cubature.

Examples

## Test function 0
## Compare with original cubature result of
## ./cubature_test 2 1e-4 0 0
## 2-dim integral, tolerance = 0.0001
## integrand 0: integral = 0.708073, est err = 1.70943e-05, true err = 7.69005e-09
## #evals = 17

testFn0 <- function(x) {
  prod(cos(x))
}

adaptIntegrate(testFn0, rep(0,2), rep(1,2), tol=1e-4)

M_2_SQRTPI <- 2/sqrt(pi)  

## Test function 1
## Compare with original cubature result of
## ./cubature_test 3 1e-4 1 0
## 3-dim integral, tolerance = 0.0001
## integrand 1: integral = 1.00001, est err = 9.67798e-05, true err = 9.76919e-06
## #evals = 5115

testFn1 <- function(x) {
  scale = 1.0
  val = 0
  dim = length(x)
  val = sum (((1-x) / x)^2)
  scale = prod(M_2_SQRTPI/x^2)
  exp(-val) * scale
}

adaptIntegrate(testFn1, rep(0, 3), rep(1, 3), tol=1e-4)

##
## Test function 2
## Compare with original cubature result of
## ./cubature_test 2 1e-4 2 0
## 2-dim integral, tolerance = 0.0001
## integrand 2: integral = 0.19728, est err = 1.97278e-05, true err = 4.58288e-05
## #evals = 166175
##

testFn2 <- function(x) {
  ## discontinuous objective: volume of hypersphere
  radius = 0.5012414
  ifelse(sum(x*x) < radius*radius, 1, 0)
}

adaptIntegrate(testFn2, rep(0, 2), rep(1, 2), tol=1e-4)

##
## Test function 3
## Compare with original cubature result of
## ./cubature_test 3 1e-4 3 0
## 3-dim integral, tolerance = 0.0001
## integrand 3: integral = 1, est err = 0, true err = 2.22045e-16
## #evals = 33

testFn3 <- function(x) {
  prod(2*x)
}

adaptIntegrate(testFn3, rep(0,3), rep(1,3), tol=1e-4)

##
## Test function 4 (Gaussian centered at 1/2)
## Compare with original cubature result of
## ./cubature_test 2 1e-4 4 0
## 2-dim integral, tolerance = 0.0001
## integrand 4: integral = 1, est err = 9.84399e-05, true err = 2.78894e-06
## #evals = 1853

testFn4 <- function(x) {
  a = 0.1
  s = sum((x-0.5)^2)
  (M_2_SQRTPI / (2. * a))^length(x) * exp (-s / (a * a))
}
  
adaptIntegrate(testFn4, rep(0,2), rep(1,2), tol=1e-4)


##
## Test function 5 (double Gaussian)
## Compare with original cubature result of
## ./cubature_test 3 1e-4 5 0
## 3-dim integral, tolerance = 0.0001
## integrand 5: integral = 0.999994, est err = 9.98015e-05, true err = 6.33407e-06
## #evals = 59631

testFn5 <- function(x) {
  a = 0.1
  s1 = sum((x-1/3)^2)
  s2 = sum((x-2/3)^2)
  0.5 * (M_2_SQRTPI / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
}
  
adaptIntegrate(testFn5, rep(0,3), rep(1,3), tol=1e-4)

##
## Test function 6 (Tsuda's example)
## Compare with original cubature result of
## ./cubature_test 4 1e-4 6 0
## 4-dim integral, tolerance = 0.0001
## integrand 6: integral = 0.999998, est err = 9.99685e-05, true err = 1.5717e-06
## #evals = 18753

testFn6 <- function(x) {
  a = (1+sqrt(10.0))/9.0
  prod(a/(a+1)*((a+1)/(a+x))^2)
}
  
adaptIntegrate(testFn6, rep(0,4), rep(1,4), tol=1e-4)


##
## Test function 7
##   test integrand from W. J. Morokoff and R. E. Caflisch, "Quasi=
##   Monte Carlo integration," J. Comput. Phys 122, 218-230 (1995).
##   Designed for integration on [0,1]^dim, integral = 1. */
## Compare with original cubature result of
## ./cubature_test 3 1e-4 7 0
## 3-dim integral, tolerance = 0.0001
## integrand 7: integral = 1.00001, est err = 9.96657e-05, true err = 1.15994e-05
## #evals = 7887

testFn7 <- function(x) {
  n <- length(x)
  p <- 1/n
  (1+p)^n * prod(x^p)
}
  
adaptIntegrate(testFn7, rep(0,3), rep(1,3), tol=1e-4)


## Example from web page
## http://ab-initio.mit.edu/wiki/index.php/Cubature
##
## f(x) = exp(-0.5(euclidean_norm(x)^2)) over the three-dimensional
## hyperbcube [-2, 2]^3
## Compare with original cubature result
testFnWeb <-  function(x) {
  exp(-0.5*sum(x^2))
}

adaptIntegrate(testFnWeb, rep(-2,3), rep(2,3), tol=1e-4)

## Test function I.1d from
## Numerical integration using Wang-Landau sampling
## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin
## Computer Physics Communications, 2007, 524-529
## Compare with exact answer: 1.63564436296
##
I.1d <- function(x) {
  sin(4*x) *
    x * ((x * ( x * (x*x-4) + 1) - 1))
}

adaptIntegrate(I.1d, -2, 2, tol=1e-7)

## Test function I.2d from
## Numerical integration using Wang-Landau sampling
## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin
## Computer Physics Communications, 2007, 524-529
## Compare with exact answer: -0.01797992646
##
I.2d <- function(x) {
  x1 = x[1]
  x2 = x[2]
  sin(4*x1+1) * cos(4*x2) * x1 * (x1*(x1*x1)^2 - x2*(x2*x2 - x1) +2)
}

adaptIntegrate(I.2d, rep(-1, 2), rep(1, 2), maxEval=10000)