Title: | Adaptive Multivariate Integration over Hypercubes |
---|---|
Description: | R wrappers around the cubature C library of Steven G. Johnson for adaptive multivariate integration over hypercubes and the Cuba C library of Thomas Hahn for deterministic and Monte Carlo integration. Scalar and vector interfaces for cubature and Cuba routines are provided; the vector interfaces are highly recommended as demonstrated in the package vignette. |
Authors: | Balasubramanian Narasimhan [aut, cre], Manuel Koller [ctb], Steven G. Johnson [aut], Thomas Hahn [aut], Annie Bouvier [aut], Kiên Kiêu [aut], Simen Gaure [ctb] |
Maintainer: | Balasubramanian Narasimhan <[email protected]> |
License: | GPL-3 |
Version: | 2.1.1 |
Built: | 2024-12-12 07:08:20 UTC |
Source: | CRAN |
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()
.
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, ... )
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, ... )
f |
The function (integrand) to be integrated. Can be
vectorized version, but the additional arguments |
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. |
The returned value is a list of items:
the value of the integral
the estimated absolute error
the number of times the function was evaluated
the actual integer return code of the C routine; a non-zero value usually indicates problems; further interpretation depends on method
forcCuba routines, the actual number of subregions needed
the -probability (not the
-value itself!) that
error
is not a reliable estimate of the true integration error.
default_args()
, hcubature()
,
pcubature()
, cuhre()
,
vegas()
, suave()
, divonne()
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") cubintegrate(f = I.1d, lower = -2, upper = 2, method = "cuhre", flags=list(verbose = 2)) cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "hcubature", nVec = 2L) cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "cuhre", nVec = 128L)
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") cubintegrate(f = I.1d, lower = -2, upper = 2, method = "cuhre", flags=list(verbose = 2)) cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "hcubature", nVec = 2L) cubintegrate(f = I.1d_v, lower = -2, upper = 2, method = "cuhre", nVec = 128L)
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()
.
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 )
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 )
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:
|
key |
the quadrature rule key: |
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. |
See details in the documentation.
A list with components:
the actual number of subregions needed
the actual number of integrand evaluations needed
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.
vector of length nComp
; the integral of
integrand
over the hypercube
vector of
length nComp
; the presumed absolute error of
integral
vector of length nComp
; the
-probability (not the
-value itself!) that
error
is not a
reliable estimate of the true integration error.
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/
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))
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))
Since each method has a different set of parameters, this function returns the default values of all parameters that can be modified and passed to integration routines.
default_args()
default_args()
a named list of parameters for each method.
default_args()
default_args()
Divonne works by stratified sampling, where the partioning of the integration region is aided by methods from numerical optimization.
divonne( 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), rngSeed = 0L, nVec = 1L, key1 = 47L, key2 = 1L, key3 = 1L, maxPass = 5L, border = 0, maxChisq = 10, minDeviation = 0.25, xGiven = NULL, nExtra = 0L, peakFinder = NULL, stateFile = NULL )
divonne( 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), rngSeed = 0L, nVec = 1L, key1 = 47L, key2 = 1L, key3 = 1L, maxPass = 5L, border = 0, maxChisq = 10, minDeviation = 0.25, xGiven = NULL, nExtra = 0L, peakFinder = NULL, stateFile = NULL )
f |
The function (integrand) to be integrated as in
This information might be useful if
the integrand takes long to compute and a sufficiently accurate
approximation of the integrand is available. The actual value
of the integral is only of minor importance in the partitioning
phase, which is instead much more dependent on the peak
structure of the integrand to find an appropriate
tessellation. An approximation which reproduces the peak
structure while leaving out the fine details might hence be a
perfectly viable and much faster substitute when
|
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:
|
rngSeed |
seed, default 0, for the random number
generator. Note the articulation with |
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. |
key1 |
integer that determines sampling in the partitioning
phase: |
key2 |
integer that determines sampling in the final
integration phase: same as |
key3 |
integer that sets the strategy for the refinement
phase: |
maxPass |
integer that controls the thoroughness of the
partitioning phase: The partitioning phase terminates when the
estimated total number of integrand evaluations (partitioning
plus final integration) does not decrease for |
border |
the relative width of the border of the integration
region. Points falling into the border region will not be
sampled directly, but will be extrapolated from two samples
from the interior. Use a non-zero |
maxChisq |
the maximum |
minDeviation |
a bound, given as the fraction of the requested
error of the entire integral, which determines whether it is
worthwhile further examining a region that failed the
|
xGiven |
a matrix ( |
nExtra |
the maximum number of extra points the peak-finder
subroutine will return. If |
peakFinder |
the peak-finder subroutine. This R function is
called whenever a region is up for subdivision and is supposed
to point out possible peaks lying in the region, thus acting as
the dynamic counterpart of the static list of points supplied
in |
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. |
Divonne uses stratified sampling for variance reduction, that is, it partitions the integration region such that all subregions have an approximately equal value of a quantity called the spread (volume times half-range).
See details in the documentation.
A list with components:
the actual number of subregions needed
the actual number of integrand evaluations needed
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.
vector of length nComp
; the integral of
integrand
over the hypercube
vector of
length nComp
; the presumed absolute error of
integral
vector of length nComp
;
the -probability (not the
-value itself!) that
error
is not a
reliable estimate of the true integration error.
J. H. Friedman, M. H. Wright (1981) A nested partitioning procedure for numerical multiple integration. ACM Trans. Math. Software, 7(1), 76-92.
J. H. Friedman, M. H. Wright (1981) User's guide for DIVONNE. SLAC Report CGTM-193-REV, CGTM-193, Stanford University.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
integrand <- function(arg, phase) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } divonne(integrand, relTol=1e-3, absTol=1e-12, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), flags=list(verbose = 2), key1= 47) # Example with a peak-finder function nDim <- 3L peakf <- function(bounds, nMax) { # print(bounds) # matrix (ndim,2) x <- matrix(0, ncol = nMax, nrow = nDim) pas <- 1 / (nMax - 1) # 1ier point x[, 1] <- rep(0, nDim) # Les autres points for (i in 2L:nMax) { x[, i] <- x[, (i - 1)] + pas } x } #end peakf divonne(integrand, relTol=1e-3, absTol=1e-12, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), flags=list(verbose = 2), peakFinder = peakf, nExtra = 4L)
integrand <- function(arg, phase) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } divonne(integrand, relTol=1e-3, absTol=1e-12, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), flags=list(verbose = 2), key1= 47) # Example with a peak-finder function nDim <- 3L peakf <- function(bounds, nMax) { # print(bounds) # matrix (ndim,2) x <- matrix(0, ncol = nMax, nrow = nDim) pas <- 1 / (nMax - 1) # 1ier point x[, 1] <- rep(0, nDim) # Les autres points for (i in 2L:nMax) { x[, i] <- x[, (i - 1)] + pas } x } #end peakf divonne(integrand, relTol=1e-3, absTol=1e-12, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), flags=list(verbose = 2), peakFinder = peakf, nExtra = 4L)
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.
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") )
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") )
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 |
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, |
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 and
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 and
) individually satisfies the requested
error tolerances
L1
, L2
, LINF
The absolute error is measured as
and the relative error as
, where
is the
,
, or
norm, respectively
PAIRED
Like INDIVIDUAL
, except that the integrands are
grouped into consecutive pairs, with the error tolerance applied in an
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
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 |
Balasubramanian Narasimhan
## Not run: ## 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) ## End(Not run)
## Not run: ## 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) ## End(Not run)
Suave uses vegas()
-like importance sampling combined with a
globally adaptive subdivision strategy: Until the requested accuracy is
reached, the region with the largest error at the time is bisected in the
dimension in which the fluctuations of the integrand are reduced most. The
number of new samples in each half is prorated for the fluctuation in that
half.
suave( f, nComp = 1L, lowerLimit, upperLimit, ..., relTol = 1e-05, absTol = 1e-12, minEval = 0L, maxEval = 10^6, flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, level = 0L), rngSeed = 0L, nVec = 1L, nNew = 1000L, nMin = 50L, flatness = 50, stateFile = NULL )
suave( f, nComp = 1L, lowerLimit, upperLimit, ..., relTol = 1e-05, absTol = 1e-12, minEval = 0L, maxEval = 10^6, flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, level = 0L), rngSeed = 0L, nVec = 1L, nNew = 1000L, nMin = 50L, flatness = 50, stateFile = NULL )
f |
The function (integrand) to be integrated as in
|
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:
|
rngSeed |
seed, default 0, for the random number
generator. Note the articulation with |
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. |
nNew |
the number of new integrand evaluations in each subdivision. |
nMin |
the minimum number of samples a former pass must
contribute to a subregion to be considered in that region's
compound integral value. Increasing nmin may reduce jumps in
the |
flatness |
the parameter p, or the type of norm used to compute the fluctuation of a sample. This determines how prominently "outliers," i.e. individual samples with a large fluctuation, figure in the total fluctuation, which in turn determines how a region is split up. As suggested by its name, flatness should be chosen large for "flat" integrands and small for "volatile" integrands with high peaks. Note that since flatness appears in the exponent, one should not use too large values (say, no more than a few hundred) lest terms be truncated internally to prevent overflow. |
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. |
See details in the documentation.
A list with components:
the actual number of subregions needed
the actual number of integrand evaluations needed
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.
vector of length nComp
; the integral of
integrand
over the hypercube
vector of
length nComp
; the presumed absolute error of
integral
vector of length nComp
;
the -probability (not the
-value itself!) that
error
is not a
reliable estimate of the true integration error.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
integrand <- function(arg) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } # end integrand suave(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), relTol=1e-3, absTol=1e-12, flags=list(verbose=2, final=0))
integrand <- function(arg) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } # end integrand suave(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), relTol=1e-3, absTol=1e-12, flags=list(verbose=2, final=0))
Implement a Monte Carlo algorithm for multidimensional numerical integration. This algorithm uses importance sampling as a variance-reduction technique. Vegas iteratively builds up a piecewise constant weight function, represented on a rectangular grid. Each iteration consists of a sampling step followed by a refinement of the grid.
vegas( f, nComp = 1L, lowerLimit, upperLimit, ..., relTol = 1e-05, absTol = 1e-12, minEval = 0L, maxEval = 10^6, flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, load_state = 0L, level = 0L), rngSeed = 12345L, nVec = 1L, nStart = 1000L, nIncrease = 500L, nBatch = 1000L, gridNo = 0L, stateFile = NULL )
vegas( f, nComp = 1L, lowerLimit, upperLimit, ..., relTol = 1e-05, absTol = 1e-12, minEval = 0L, maxEval = 10^6, flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, load_state = 0L, level = 0L), rngSeed = 12345L, nVec = 1L, nStart = 1000L, nIncrease = 500L, nBatch = 1000L, gridNo = 0L, stateFile = NULL )
f |
The function (integrand) to be integrated as in
|
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:
|
rngSeed |
seed, default 0, for the random number
generator. Note the articulation with |
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. |
nStart |
the number of integrand evaluations per iteration to start with. |
nIncrease |
the increase in the number of integrand evaluations per iteration. The j-th iteration evaluates the integrand at nStart+(j-1)*nincrease points. |
nBatch |
Vegas samples points not all at once, but in batches
of a predetermined size, to avoid excessive memory
consumption. |
gridNo |
an integer. Vegas may accelerate convergence to keep
the grid accumulated during one integration for the next one,
if the integrands are reasonably similar to each other. Vegas
maintains an internal table with space for ten grids for this
purpose. If |
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. |
See details in the documentation.
A list with components:
the actual number of subregions needed
the actual number of integrand evaluations needed
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.
vector of length nComp
; the integral of
integrand
over the hypercube
vector of
length nComp
; the presumed absolute error of
integral
vector of length nComp
;
the -probability (not the
-value itself!) that
error
is not a
reliable estimate of the true integration error.
G. P. Lepage (1978) A new algorithm for adaptive multidimensional integration. J. Comput. Phys., 27, 192-210.
G. P. Lepage (1980) VEGAS - An adaptive multi-dimensional integration program. Research Report CLNS-80/447. Cornell University, Ithaca, N.-Y.
T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.
integrand <- function(arg, weight) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } # end integrand vegas(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), relTol=1e-3, absTol=1e-12, flags=list(verbose=2, final=0))
integrand <- function(arg, weight) { x <- arg[1] y <- arg[2] z <- arg[3] ff <- sin(x)*cos(y)*exp(z); return(ff) } # end integrand vegas(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), relTol=1e-3, absTol=1e-12, flags=list(verbose=2, final=0))