Adaptive multivariate integration over hypercubes (hcubature and pcubature)

The function performs adaptive multidimensional integration (cubature) of (possibly) vector-valued integrands over hypercubes. The function includes a vector interface where the integrand may be evaluated at several hundred points in a single call.

Usage

hcubature(
  f,
  lowerLimit,
  upperLimit,
  ...,
  tol = 1e-05,
  fDim = 1,
  maxEval = 0,
  absError = .Machine$double.eps * 10^2/2,
  doChecking = FALSE,
  vectorInterface = FALSE,
  norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")
)

pcubature(
  f,
  lowerLimit,
  upperLimit,
  ...,
  tol = 1e-05,
  fDim = 1,
  maxEval = 0,
  absError = .Machine$double.eps * 10^2,
  doChecking = FALSE,
  vectorInterface = FALSE,
  norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")
)

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
...: All other arguments passed to the function f
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. Note that the actual number of function evaluations performed is only approximately guaranteed not to exceed this number.
absError: The maximum absolute error tolerated, default .Machine$double_eps * 10^2.
doChecking: As of version 2.0, this flag is ignored and will be dropped in forthcoming versions
vectorInterface: A flag that indicates whether to use the vector interface and is by default FALSE. See details below
norm: For vector-valued integrands, norm specifies the norm that is used to measure the error and determine convergence properties. See below.

Value

The returned value is a list of four items:

integral: the value of the integral
error: the estimated absolute 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.

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, or the maximum number of iterations is reached (see parameter info below), whichever is earlier.

For compatibility with earlier versions, the adaptIntegrate function is an alias for the underlying hcubature function which uses h-adaptive integration. Otherwise, the calling conventions are the same.

We highly recommend referring to the vignette to achieve the best results!

The hcubature function is the h-adaptive version that recursively partitions the integration domain into smaller subdomains, applying the same integration rule to each, until convergence is achieved.

The p-adaptive version, pcubature, repeatedly doubles the degree of the quadrature rules until convergence is achieved, and is based on a tensor product of Clenshaw-Curtis quadrature rules. This algorithm is often superior to h-adaptive integration for smooth integrands in a few (<=3) dimensions, but is a poor choice in higher dimensions or for non-smooth integrands. Compare with hcubature which also takes the same arguments.

The vector interface requires the integrand to take a matrix as its argument. The return value should also be a matrix. The number of points at which the integrand may be evaluated is not under user control: the integration routine takes care of that and this number may run to several hundreds. We strongly advise vectorization; see vignette.

The norm argument is irrelevant for scalar integrands and is ignored. Given vectors $v$ and $e$ of estimated integrals and errors therein, respectively, the norm argument takes on one of the following values:

INDIVIDUAL: Convergence is achieved only when each integrand (each component of $v$ and $e$) individually satisfies the requested error tolerances
L1, L2, LINF: The absolute error is measured as $|e|$ and the relative error as $|e|/|v|$, where $|...|$ is the $L_1$, $L_2$, or $L_{\infty}$ norm, respectively
PAIRED: Like INDIVIDUAL, except that the integrands are grouped into consecutive pairs, with the error tolerance applied in an $L_2$ sense to each pair. This option is mainly useful for integrating vectors of complex numbers, where each consecutive pair of real integrands is the real and imaginary parts of a single complex integrand, and the concern is only the error in the complex plane rather than the error in the real and imaginary parts separately

Author

Balasubramanian Narasimhan

Examples


if (FALSE) { # \dontrun{
## 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))
}

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

pcubature(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) {
  val <- sum (((1-x) / x)^2)
  scale <- prod(M_2_SQRTPI/x^2)
  exp(-val) * scale
}

hcubature(testFn1, rep(0, 3), rep(1, 3), tol=1e-4)
pcubature(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 = 1.97261e-05, true err = 4.58316e-05
## #evals = 166141

testFn2 <- function(x) {
  ## discontinuous objective: volume of hypersphere
  radius <- as.double(0.50124145262344534123412)
  ifelse(sum(x*x) < radius*radius, 1, 0)
}

hcubature(testFn2, rep(0, 2), rep(1, 2), tol=1e-4)
pcubature(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)
}

hcubature(testFn3, rep(0,3), rep(1,3), tol=1e-4)
pcubature(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))
}

hcubature(testFn4, rep(0,2), rep(1,2), tol=1e-4)
pcubature(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)))
}

hcubature(testFn5, rep(0,3), rep(1,3), tol=1e-4)
pcubature(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)
}

hcubature(testFn6, rep(0,4), rep(1,4), tol=1e-4)
pcubature(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)
}

hcubature(testFn7, rep(0,3), rep(1,3), tol=1e-4)
pcubature(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))
}

hcubature(testFnWeb, rep(-2,3), rep(2,3), tol=1e-4)
pcubature(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))
}

hcubature(I.1d, -2, 2, tol=1e-7)
pcubature(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)
}

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

##
## Example of multivariate normal integration borrowed from
## package mvtnorm (on CRAN) to check ... argument
## Compare with output of
## pmvnorm(lower=rep(-0.5, m), upper=c(1,4,2), mean=rep(0, m), corr=sigma, alg=Miwa())
##     0.3341125.  Blazing quick as well!  Ours is, not unexpectedly, much slower.
##
dmvnorm <- function (x, mean, sigma, log = FALSE) {
    if (is.vector(x)) {
        x <- matrix(x, ncol = length(x))
    }
    if (missing(mean)) {
        mean <- rep(0, length = ncol(x))
    }
    if (missing(sigma)) {
        sigma <- diag(ncol(x))
    }
    if (NCOL(x) != NCOL(sigma)) {
        stop("x and sigma have non-conforming size")
    }
    if (!isSymmetric(sigma, tol = sqrt(.Machine$double.eps),
        check.attributes = FALSE)) {
        stop("sigma must be a symmetric matrix")
    }
    if (length(mean) != NROW(sigma)) {
        stop("mean and sigma have non-conforming size")
    }
    distval <- mahalanobis(x, center = mean, cov = sigma)
    logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values))
    logretval <- -(ncol(x) * log(2 * pi) + logdet + distval)/2
    if (log)
        return(logretval)
    exp(logretval)
}

m <- 3
sigma <- diag(3)
sigma[2,1] <- sigma[1, 2] <- 3/5 ; sigma[3,1] <- sigma[1, 3] <- 1/3
sigma[3,2] <- sigma[2, 3] <- 11/15
hcubature(dmvnorm, lower=rep(-0.5, m), upper=c(1,4,2),
                        mean=rep(0, m), sigma=sigma, log=FALSE,
               maxEval=10000)
pcubature(dmvnorm, lower=rep(-0.5, m), upper=c(1,4,2),
                        mean=rep(0, m), sigma=sigma, log=FALSE,
               maxEval=10000)
} # }