cubature/0000755000176200001440000000000013133704177012072 5ustar liggesuserscubature/inst/0000755000176200001440000000000013133476177013055 5ustar liggesuserscubature/inst/doc/0000755000176200001440000000000013133476177013622 5ustar liggesuserscubature/inst/doc/cubature.Rmd0000644000176200001440000003360313133475417016101 0ustar liggesusers--- title: "Cubature Vectorization Results" author: "Balasubramanian Narasimhan" date: '`r Sys.Date()`' output: html_document: fig_caption: yes theme: cerulean toc: yes toc_depth: 2 vignette: > %\VignetteIndexEntry{Cubature Vectorization Efficiency} %\VignetteEngine{knitr::rmarkdown} \usepackage[utf8]{inputenc} --- ```{r echo=FALSE} knitr::opts_chunk$set( message = FALSE, warning = FALSE, error = FALSE, tidy = FALSE, cache = FALSE ) ``` ## Introduction Starting with version 1.2, `cubature` now uses `Rcpp`. Also, version 1.3 uses the newer version (1.0.2) of Steven G. Johnson's `hcubature` and `pcubature` routines, including the vectorized interface. Per the documentation, use of `pcubature` is advisable only for smooth integrands in dimesions up to three at most. In fact, the `pcubature` routines perform significantly worse than the vectorized `hcubature` in inappropriate cases. So when in doubt, you are better off using `hcubature`. The main point of this note is to examine the difference vectorization makes. My recommendations are below in the summary section. ## A Timing Harness Our harness will provide timing results for `hcubature`, `pcubature` (where appropriate) and `R2Cuba` calls. We begin by creating a harness for these calls. ```{r} loadedSuggested <- c(benchr = FALSE, R2Cuba = FALSE) if (requireNamespace("benchr", quietly = TRUE)) { loadedSuggested["benchr"] <- TRUE } if (requireNamespace("R2Cuba", quietly = TRUE)) { loadedSuggested["R2Cuba"] <- TRUE } library(cubature) harness <- function(which = NULL, f, fv, lowerLimit, upperLimit, tol = 1e-3, times = 20, ...) { fns <- c(hc = "Non-vectorized Hcubature", hc.v = "Vectorized Hcubature", pc = "Non-vectorized Pcubature", pc.v = "Vectorized Pcubature") if (loadedSuggested["R2Cuba"]) { fns <- c(fns, cc = "R2Cuba::cuhre") cc <- function() R2Cuba::cuhre(ndim = ndim, ncomp = 1, integrand = f, lower = lowerLimit, upper = upperLimit, flags = list(verbose = 0, final = 1), rel.tol = tol, max.eval = 10^6, ...) } hc <- function() cubature::hcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) hc.v <- function() cubature::hcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) pc <- function() cubature::pcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) pc.v <- function() cubature::pcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) ndim = length(lowerLimit) if (is.null(which)) { fnIndices <- seq_along(fns) } else { fnIndices <- na.omit(match(which, names(fns))) } fnList <- lapply(names(fns)[fnIndices], function(x) call(x)) if (loadedSuggested["benchr"]) { argList <- c(fnList, times = times, progress = FALSE) result <- do.call(benchr::benchmark, args = argList) d <- summary(result)[seq_along(fnIndices), ] d$expr <- fns[fnIndices] d } else { d <- data.frame(expr = names(fns)[fnIndices], timing = NA) } } ``` We reel off the timing runs. ## Example 1. ```{r} func <- function(x) sin(x[1]) * cos(x[2]) * exp(x[3]) func.v <- function(x) { matrix(apply(x, 2, function(z) sin(z[1]) * cos(z[2]) * exp(z[3])), ncol = ncol(x)) } d <- harness(f = func, fv = func.v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = 1e-5, times = 100) knitr::kable(d, digits = 3, row.names = FALSE) ``` ## Multivariate Normal Using `cubature`, we evaluate $$ \int_R\phi(x)dx $$ where $\phi(x)$ is the three-dimensional multivariate normal density with mean 0, and variance $$ \Sigma = \left(\begin{array}{rrr} 1 &\frac{3}{5} &\frac{1}{3}\\ \frac{3}{5} &1 &\frac{11}{15}\\ \frac{1}{3} &\frac{11}{15} & 1 \end{array} \right) $$ and $R$ is $[-\frac{1}{2}, 1] \times [-\frac{1}{2}, 4] \times [-\frac{1}{2}, 2].$ We construct a scalar function (`my_dmvnorm`) and a vector analog (`my_dmvnorm_v`). First the functions. ```{r} 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 logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values)) my_dmvnorm <- function (x, mean, sigma, logdet) { x <- matrix(x, ncol = length(x)) distval <- stats::mahalanobis(x, center = mean, cov = sigma) exp(-(3 * log(2 * pi) + logdet + distval)/2) } my_dmvnorm_v <- function (x, mean, sigma, logdet) { distval <- stats::mahalanobis(t(x), center = mean, cov = sigma) exp(matrix(-(3 * log(2 * pi) + logdet + distval)/2, ncol = ncol(x))) } ``` Now the timing. ```{r} d <- harness(f = my_dmvnorm, fv = my_dmvnorm_v, lowerLimit = rep(-0.5, 3), upperLimit = c(1, 4, 2), tol = 1e-5, times = 10, mean = rep(0, m), sigma = sigma, logdet = logdet) knitr::kable(d, digits = 3) ``` The effect of vectorization is huge. So it makes sense for users to vectorize the integrands as much as possible for efficiency. Furthermore, for this particular example, we expect `mvtnorm::pmvnorm` to do pretty well since it is specialized for the multivariate normal. The good news is that the vectorized versions of `hcubature` and `pcubature` are quite competitive if you compare the table above to the one below. ```{r} library(mvtnorm) g1 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = Miwa(), abseps = 1e-5, releps = 1e-5) g2 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = GenzBretz(), abseps = 1e-5, releps = 1e-5) g3 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = TVPACK(), abseps = 1e-5, releps = 1e-5) knitr::kable(summary(benchr::benchmark(g1(), g2(), g3(), times = 20, progress = FALSE)), digits = 3, row.names = FALSE) ``` ## Product of cosines ```{r} testFn0 <- function(x) prod(cos(x)) testFn0_v <- function(x) matrix(apply(x, 2, function(z) prod(cos(z))), ncol = ncol(x)) d <- harness(f = testFn0, fv = testFn0_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 1000) knitr::kable(d, digits = 3) ``` ## Gaussian function ```{r} testFn1 <- function(x) { val <- sum(((1 - x) / x)^2) scale <- prod((2 / sqrt(pi)) / x^2) exp(-val) * scale } testFn1_v <- function(x) { val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x)) scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x)) exp(-val) * scale } d <- harness(f = testFn1, fv = testFn1_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 10) knitr::kable(d, digits = 3) ``` ## Discontinuous function ```{r} testFn2 <- function(x) { radius <- 0.50124145262344534123412 ifelse(sum(x * x) < radius * radius, 1, 0) } testFn2_v <- function(x) { radius <- 0.50124145262344534123412 matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x)) } d <- harness(which = c("hc", "hc.v", "cc"), f = testFn2, fv = testFn2_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 10) knitr::kable(d, digits = 3) ``` ## A Simple Polynomial (product of coordinates) ```{r} testFn3 <- function(x) prod(2 * x) testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x)) d <- harness(f = testFn3, fv = testFn3_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 50) knitr::kable(d, digits = 3) ``` ## Gaussian centered at \frac{1}{2} ```{r} testFn4 <- function(x) { a <- 0.1 s <- sum((x - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a)) } testFn4_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s <- sum((z - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a)) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn4, fv = testFn4_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ``` ## Double Gaussian ```{r} testFn5 <- function(x) { a <- 0.1 s1 <- sum((x - 1 / 3)^2) s2 <- sum((x - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) } testFn5_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s1 <- sum((z - 1 / 3)^2) s2 <- sum((z - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn5, fv = testFn5_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ``` ## Tsuda's Example ```{r} testFn6 <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 prod( a / (a + 1) * ((a + 1) / (a + x))^2) } testFn6_v <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2)) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn6, fv = testFn6_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ``` ## Morokoff & Calflish Example ```{r} testFn7 <- function(x) { n <- length(x) p <- 1/n (1 + p)^n * prod(x^p) } testFn7_v <- function(x) { matrix(apply(x, 2, function(z) { n <- length(z) p <- 1/n (1 + p)^n * prod(z^p) }), ncol = ncol(x)) } d <- harness(f = testFn7, fv = testFn7_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ``` ## Wang-Landau Sampling 1d, 2d Examples ```{r} 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)) } d <- harness(f = I.1d, fv = I.1d_v, lowerLimit = -2, upperLimit = 2, times = 100) knitr::kable(d, digits = 3) ``` ```{r} 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) } I.2d_v <- function(x) { matrix(apply(x, 2, function(z) { x1 <- z[1]; x2 <- z[2] sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2) }), ncol = ncol(x)) } d <- harness(f = I.2d, fv = I.2d_v, lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), times = 100) knitr::kable(d, digits = 3) ``` ## An implementation note About the only real modification we have made to the underlying `cubature-1.0.2` library is that we use `M = 16` rather than the default `M = 19` suggested by the original author for `pcubature`. This allows us to comply with CRAN package size limits and seems to work reasonably well for the above tests. Future versions will allow for such customization on demand. ### Apropos the `Cuba` library The package [`R2Cuba`](https://cran.r-project.org/package=R2Cuba) provides a suite of cubature and other useful Monte Carlo integration routines linked against version 1.6 of the C library. The authors of `R2Cuba` have obviously put a lot of work has into it since it uses C-style R API. This approach also means that it is harder to keep the R package in sync with new versions of the underlying C library. In fact, the Cuba C library has marched on now to version 4.2. In a matter of a couple of hours, I was able to link the latest version (4.2) of the [Cuba libraries](http://www.feynarts.de/cuba/) with R using Rcpp; you can see it on the `Cuba` branch of my [Github repo](https://github.com/bnaras/cubature). This branch package builds and installs in R on my Mac and Ubuntu machines and gives correct answers at least for `cuhre`. The 4.2 version of the Cuba library also has vectorized versions of the routines that can be gainfully exploited (not implemented in the branch). As of this writing, I have also not yet carefully considered the issue of parallel execution (via `fork()`) which might be problematic in the Windows version. In addition, my timing benchmarks showed very disappointing results. For the above reasons, I decided not to bother with Cuba for now, but if there is enough interest, I might consider rolling `Cuba-4.2+` into this `cubature` package in the future. ## Summary The following is therefore my recommendation. 1. _Vectorize_ your function. The time spent in so doing pays back enormously. This is easy to do and the examples above show how. 2. Vectorized `hcubature` seems to be a good starting point. 3. For smooth integrands in low dimensions ($\leq 3$), `pcubature` might be worth trying out. Experiment before using in a production package. cubature/inst/doc/cubature.html0000644000176200001440000266427313133476177016346 0ustar liggesusers Cubature Vectorization Results

Introduction

Starting with version 1.2, cubature now uses Rcpp. Also, version 1.3 uses the newer version (1.0.2) of Steven G. Johnson’s hcubature and pcubature routines, including the vectorized interface.

Per the documentation, use of pcubature is advisable only for smooth integrands in dimesions up to three at most. In fact, the pcubature routines perform significantly worse than the vectorized hcubature in inappropriate cases. So when in doubt, you are better off using hcubature.

The main point of this note is to examine the difference vectorization makes. My recommendations are below in the summary section.

A Timing Harness

Our harness will provide timing results for hcubature, pcubature (where appropriate) and R2Cuba calls. We begin by creating a harness for these calls.

loadedSuggested  <- c(benchr = FALSE, R2Cuba = FALSE)
if (requireNamespace("benchr", quietly = TRUE)) {
    loadedSuggested["benchr"] <- TRUE
}
if (requireNamespace("R2Cuba", quietly = TRUE)) {
    loadedSuggested["R2Cuba"] <- TRUE
}

library(cubature)

harness <- function(which = NULL,
                    f, fv, lowerLimit, upperLimit, tol = 1e-3, times = 20, ...) {

    fns <- c(hc = "Non-vectorized Hcubature",
             hc.v = "Vectorized Hcubature",
             pc = "Non-vectorized Pcubature",
             pc.v = "Vectorized Pcubature")

    if (loadedSuggested["R2Cuba"]) {
        fns <- c(fns, cc = "R2Cuba::cuhre")
        cc <- function() R2Cuba::cuhre(ndim = ndim, ncomp = 1, integrand = f,
                                       lower = lowerLimit, upper = upperLimit,
                                       flags = list(verbose = 0, final = 1),
                                       rel.tol = tol,
                                       max.eval = 10^6,
                                       ...)
    }

    hc <- function() cubature::hcubature(f = f,
                                         lowerLimit = lowerLimit,
                                         upperLimit = upperLimit,
                                         tol = tol,
                                         ...)

    hc.v <- function() cubature::hcubature(f = fv,
                                           lowerLimit = lowerLimit,
                                           upperLimit = upperLimit,
                                           tol = tol,
                                           vectorInterface = TRUE,
                                           ...)

    pc <- function() cubature::pcubature(f = f,
                                         lowerLimit = lowerLimit,
                                         upperLimit = upperLimit,
                                         tol = tol,
                                         ...)

    pc.v <- function() cubature::pcubature(f = fv,
                                           lowerLimit = lowerLimit,
                                           upperLimit = upperLimit,
                                           tol = tol,
                                           vectorInterface = TRUE,
                                           ...)

    ndim = length(lowerLimit)

    if (is.null(which)) {
        fnIndices <- seq_along(fns)
    } else {
        fnIndices <- na.omit(match(which, names(fns)))
    }
    fnList <- lapply(names(fns)[fnIndices], function(x) call(x))

    if (loadedSuggested["benchr"]) {
        argList <- c(fnList, times = times, progress = FALSE)
        result <- do.call(benchr::benchmark, args = argList)
        d <- summary(result)[seq_along(fnIndices), ]
        d$expr <- fns[fnIndices]
        d
    } else {
        d <- data.frame(expr = names(fns)[fnIndices], timing = NA)
    }
}

We reel off the timing runs.

Example 1.

func <- function(x) sin(x[1]) * cos(x[2]) * exp(x[3])
func.v <- function(x) {
    matrix(apply(x, 2, function(z) sin(z[1]) * cos(z[2]) * exp(z[3])), ncol = ncol(x))
}

d <- harness(f = func, fv = func.v,
             lowerLimit = rep(0, 3),
             upperLimit = rep(1, 3),
             tol = 1e-5,
             times = 100)
knitr::kable(d, digits = 3, row.names = FALSE)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 100 0.002 0.003 0.003 0.003 0.003 0.005 0.283 6.17
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.046 1.00
Non-vectorized Pcubature 100 0.008 0.008 0.009 0.009 0.009 0.043 0.928 20.00
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.127 2.86

Multivariate Normal

Using cubature, we evaluate \[ \int_R\phi(x)dx \] where \(\phi(x)\) is the three-dimensional multivariate normal density with mean 0, and variance \[ \Sigma = \left(\begin{array}{rrr} 1 &\frac{3}{5} &\frac{1}{3}\\ \frac{3}{5} &1 &\frac{11}{15}\\ \frac{1}{3} &\frac{11}{15} & 1 \end{array} \right) \] and \(R\) is \([-\frac{1}{2}, 1] \times [-\frac{1}{2}, 4] \times [-\frac{1}{2}, 2].\)

We construct a scalar function (my_dmvnorm) and a vector analog (my_dmvnorm_v). First the functions.

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
logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values))
my_dmvnorm <- function (x, mean, sigma, logdet) {
    x <- matrix(x, ncol = length(x))
    distval <- stats::mahalanobis(x, center = mean, cov = sigma)
    exp(-(3 * log(2 * pi) + logdet + distval)/2)
}

my_dmvnorm_v <- function (x, mean, sigma, logdet) {
    distval <- stats::mahalanobis(t(x), center = mean, cov = sigma)
    exp(matrix(-(3 * log(2 * pi) + logdet + distval)/2, ncol = ncol(x)))
}

Now the timing.

d <- harness(f = my_dmvnorm, fv = my_dmvnorm_v,
             lowerLimit = rep(-0.5, 3),
             upperLimit = c(1, 4, 2),
             tol = 1e-5,
             times = 10,
             mean = rep(0, m), sigma = sigma, logdet = logdet)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.924 1.011 1.075 1.078 1.114 1.329 10.781 594.0
Vectorized Hcubature 10 0.002 0.003 0.003 0.003 0.003 0.004 0.029 1.6
Non-vectorized Pcubature 10 0.398 0.475 0.487 0.497 0.521 0.590 4.969 270.0
Vectorized Pcubature 10 0.001 0.001 0.002 0.002 0.002 0.003 0.018 1.0

The effect of vectorization is huge. So it makes sense for users to vectorize the integrands as much as possible for efficiency.

Furthermore, for this particular example, we expect mvtnorm::pmvnorm to do pretty well since it is specialized for the multivariate normal. The good news is that the vectorized versions of hcubature and pcubature are quite competitive if you compare the table above to the one below.

library(mvtnorm)
g1 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
                                  upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
                                  alg = Miwa(), abseps = 1e-5, releps = 1e-5)
g2 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
                                  upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
                                  alg = GenzBretz(), abseps = 1e-5, releps = 1e-5)
g3 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m),
                                  upper = c(1, 4, 2), mean = rep(0, m), corr = sigma,
                                  alg = TVPACK(), abseps = 1e-5, releps = 1e-5)

knitr::kable(summary(benchr::benchmark(g1(), g2(), g3(), times = 20, progress = FALSE)),
             digits = 3, row.names = FALSE)
expr n.eval min lw.qu median mean up.qu max total relative
g1() 20 0.001 0.003 0.003 0.003 0.003 0.006 0.057 1.03
g2() 20 0.001 0.002 0.003 0.003 0.003 0.005 0.052 1.01
g3() 20 0.001 0.002 0.003 0.003 0.003 0.004 0.052 1.00

Product of cosines

testFn0 <- function(x) prod(cos(x))
testFn0_v <- function(x) matrix(apply(x, 2, function(z) prod(cos(z))), ncol = ncol(x))

d <- harness(f = testFn0, fv = testFn0_v,
             lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 1000)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 1000 0 0 0 0 0 0.001 0.217 2.84
Vectorized Hcubature 1000 0 0 0 0 0 0.001 0.076 1.00
Non-vectorized Pcubature 1000 0 0 0 0 0 0.003 0.307 4.07
Vectorized Pcubature 1000 0 0 0 0 0 0.001 0.146 1.93

Gaussian function

testFn1 <- function(x) {
    val <- sum(((1 - x) / x)^2)
    scale <- prod((2 / sqrt(pi)) / x^2)
    exp(-val) * scale
}

testFn1_v <- function(x) {
    val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x))
    scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x))
    exp(-val) * scale
}

d <- harness(f = testFn1, fv = testFn1_v,
             lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 10)

knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.015 0.016 0.018 0.020 0.019 0.047 0.205 95.70
Vectorized Hcubature 10 0.004 0.004 0.004 0.004 0.004 0.005 0.042 22.10
Non-vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.004 2.12
Vectorized Pcubature 10 0.000 0.000 0.000 0.000 0.000 0.000 0.002 1.00

Discontinuous function

testFn2 <- function(x) {
    radius <- 0.50124145262344534123412
    ifelse(sum(x * x) < radius * radius, 1, 0)
}

testFn2_v <- function(x) {
    radius <- 0.50124145262344534123412
    matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x))
}

d <- harness(which = c("hc", "hc.v", "cc"),
             f = testFn2, fv = testFn2_v,
             lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 10)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 10 0.202 0.248 0.251 0.246 0.251 0.258 2.458 4.96
Vectorized Hcubature 10 0.043 0.050 0.051 0.050 0.051 0.057 0.503 1.00

A Simple Polynomial (product of coordinates)

testFn3 <- function(x) prod(2 * x)
testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x))

d <- harness(f = testFn3, fv = testFn3_v,
             lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 50)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 50 0 0 0 0 0 0.001 0.024 4.40
Vectorized Hcubature 50 0 0 0 0 0 0.000 0.006 1.07
Non-vectorized Pcubature 50 0 0 0 0 0 0.000 0.019 3.62
Vectorized Pcubature 50 0 0 0 0 0 0.000 0.005 1.00

Gaussian centered at 

testFn4 <- function(x) {
    a <- 0.1
    s <- sum((x - 0.5)^2)
    ((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a))
}

testFn4_v <- function(x) {
    a <- 0.1
    r <- apply(x, 2, function(z) {
        s <- sum((z - 0.5)^2)
        ((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a))
    })
    matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn4, fv = testFn4_v,
             lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.007 0.008 0.008 0.008 0.008 0.013 0.167 5.92
Vectorized Hcubature 20 0.001 0.001 0.001 0.001 0.001 0.002 0.028 1.00
Non-vectorized Pcubature 20 0.011 0.012 0.012 0.014 0.013 0.049 0.286 8.75
Vectorized Pcubature 20 0.002 0.002 0.002 0.002 0.002 0.003 0.045 1.49

Double Gaussian

testFn5 <- function(x) {
    a <- 0.1
    s1 <- sum((x - 1 / 3)^2)
    s2 <- sum((x - 2 / 3)^2)
    0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
}
testFn5_v <- function(x) {
    a <- 0.1
    r <- apply(x, 2, function(z) {
        s1 <- sum((z - 1 / 3)^2)
        s2 <- sum((z - 2 / 3)^2)
        0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
    })
    matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn5, fv = testFn5_v,
             lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.017 0.018 0.019 0.020 0.021 0.023 0.392 6.84
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.005 0.081 1.46
Non-vectorized Pcubature 20 0.012 0.013 0.013 0.014 0.015 0.016 0.275 4.89
Vectorized Pcubature 20 0.002 0.003 0.003 0.003 0.003 0.004 0.057 1.00

Tsuda’s Example

testFn6 <- function(x) {
    a <- (1 + sqrt(10.0)) / 9.0
    prod( a / (a + 1) * ((a + 1) / (a + x))^2)
}

testFn6_v <- function(x) {
    a <- (1 + sqrt(10.0)) / 9.0
    r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2))
    matrix(r, ncol = ncol(x))
}

d <- harness(f = testFn6, fv = testFn6_v,
             lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.010 0.011 0.011 0.011 0.012 0.013 0.225 6.21
Vectorized Hcubature 20 0.002 0.002 0.002 0.002 0.002 0.006 0.041 1.00
Non-vectorized Pcubature 20 0.053 0.055 0.055 0.060 0.060 0.092 1.192 31.80
Vectorized Pcubature 20 0.007 0.008 0.008 0.008 0.009 0.011 0.167 4.52

Morokoff & Calflish Example

testFn7 <- function(x) {
    n <- length(x)
    p <- 1/n
    (1 + p)^n * prod(x^p)
}
testFn7_v <- function(x) {
    matrix(apply(x, 2, function(z) {
        n <- length(z)
        p <- 1/n
        (1 + p)^n * prod(z^p)
    }), ncol = ncol(x))
}

d <- harness(f = testFn7, fv = testFn7_v,
             lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 20 0.022 0.022 0.024 0.025 0.026 0.030 0.497 6.89
Vectorized Hcubature 20 0.003 0.003 0.004 0.004 0.004 0.006 0.079 1.00
Non-vectorized Pcubature 20 0.053 0.055 0.057 0.060 0.065 0.074 1.198 16.10
Vectorized Pcubature 20 0.007 0.008 0.009 0.009 0.010 0.015 0.187 2.48

Wang-Landau Sampling 1d, 2d 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))
}
d <- harness(f = I.1d, fv = I.1d_v,
             lowerLimit = -2, upperLimit = 2, times = 100)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.126 5.54
Vectorized Hcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.027 1.13
Non-vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.042 1.85
Vectorized Pcubature 100 0.000 0.000 0.000 0.000 0.000 0.001 0.023 1.00 
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)
}
I.2d_v <- function(x) {
    matrix(apply(x, 2,
                 function(z) {
                     x1 <- z[1]; x2 <- z[2]
                     sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2)
                 }),
           ncol = ncol(x))
}
d <- harness(f = I.2d, fv = I.2d_v,
             lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), times = 100)
knitr::kable(d, digits = 3)
expr n.eval min lw.qu median mean up.qu max total relative
Non-vectorized Hcubature 100 0.040 0.044 0.046 0.047 0.048 0.079 4.707 56.30
Vectorized Hcubature 100 0.005 0.006 0.006 0.006 0.007 0.009 0.644 7.70
Non-vectorized Pcubature 100 0.003 0.004 0.004 0.004 0.004 0.005 0.399 4.75
Vectorized Pcubature 100 0.001 0.001 0.001 0.001 0.001 0.002 0.087 1.00

An implementation note

About the only real modification we have made to the underlying cubature-1.0.2 library is that we use M = 16 rather than the default M = 19 suggested by the original author for pcubature. This allows us to comply with CRAN package size limits and seems to work reasonably well for the above tests. Future versions will allow for such customization on demand.

Apropos the Cuba library

The package R2Cuba provides a suite of cubature and other useful Monte Carlo integration routines linked against version 1.6 of the C library. The authors of R2Cuba have obviously put a lot of work has into it since it uses C-style R API. This approach also means that it is harder to keep the R package in sync with new versions of the underlying C library. In fact, the Cuba C library has marched on now to version 4.2.

In a matter of a couple of hours, I was able to link the latest version (4.2) of the Cuba libraries with R using Rcpp; you can see it on the Cuba branch of my Github repo. This branch package builds and installs in R on my Mac and Ubuntu machines and gives correct answers at least for cuhre. The 4.2 version of the Cuba library also has vectorized versions of the routines that can be gainfully exploited (not implemented in the branch). As of this writing, I have also not yet carefully considered the issue of parallel execution (via fork()) which might be problematic in the Windows version. In addition, my timing benchmarks showed very disappointing results.

For the above reasons, I decided not to bother with Cuba for now, but if there is enough interest, I might consider rolling Cuba-4.2+ into this cubature package in the future.

Summary

The following is therefore my recommendation.

  1. Vectorize your function. The time spent in so doing pays back enormously. This is easy to do and the examples above show how.

  2. Vectorized hcubature seems to be a good starting point.

  3. For smooth integrands in low dimensions (\(\leq 3\)), pcubature might be worth trying out. Experiment before using in a production package.

cubature/inst/doc/cubature.R0000644000176200001440000002453613133476176015570 0ustar liggesusers## ----echo=FALSE---------------------------------------------------------- knitr::opts_chunk$set( message = FALSE, warning = FALSE, error = FALSE, tidy = FALSE, cache = FALSE ) ## ------------------------------------------------------------------------ loadedSuggested <- c(benchr = FALSE, R2Cuba = FALSE) if (requireNamespace("benchr", quietly = TRUE)) { loadedSuggested["benchr"] <- TRUE } if (requireNamespace("R2Cuba", quietly = TRUE)) { loadedSuggested["R2Cuba"] <- TRUE } library(cubature) harness <- function(which = NULL, f, fv, lowerLimit, upperLimit, tol = 1e-3, times = 20, ...) { fns <- c(hc = "Non-vectorized Hcubature", hc.v = "Vectorized Hcubature", pc = "Non-vectorized Pcubature", pc.v = "Vectorized Pcubature") if (loadedSuggested["R2Cuba"]) { fns <- c(fns, cc = "R2Cuba::cuhre") cc <- function() R2Cuba::cuhre(ndim = ndim, ncomp = 1, integrand = f, lower = lowerLimit, upper = upperLimit, flags = list(verbose = 0, final = 1), rel.tol = tol, max.eval = 10^6, ...) } hc <- function() cubature::hcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) hc.v <- function() cubature::hcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) pc <- function() cubature::pcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) pc.v <- function() cubature::pcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) ndim = length(lowerLimit) if (is.null(which)) { fnIndices <- seq_along(fns) } else { fnIndices <- na.omit(match(which, names(fns))) } fnList <- lapply(names(fns)[fnIndices], function(x) call(x)) if (loadedSuggested["benchr"]) { argList <- c(fnList, times = times, progress = FALSE) result <- do.call(benchr::benchmark, args = argList) d <- summary(result)[seq_along(fnIndices), ] d$expr <- fns[fnIndices] d } else { d <- data.frame(expr = names(fns)[fnIndices], timing = NA) } } ## ------------------------------------------------------------------------ func <- function(x) sin(x[1]) * cos(x[2]) * exp(x[3]) func.v <- function(x) { matrix(apply(x, 2, function(z) sin(z[1]) * cos(z[2]) * exp(z[3])), ncol = ncol(x)) } d <- harness(f = func, fv = func.v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = 1e-5, times = 100) knitr::kable(d, digits = 3, row.names = FALSE) ## ------------------------------------------------------------------------ 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 logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values)) my_dmvnorm <- function (x, mean, sigma, logdet) { x <- matrix(x, ncol = length(x)) distval <- stats::mahalanobis(x, center = mean, cov = sigma) exp(-(3 * log(2 * pi) + logdet + distval)/2) } my_dmvnorm_v <- function (x, mean, sigma, logdet) { distval <- stats::mahalanobis(t(x), center = mean, cov = sigma) exp(matrix(-(3 * log(2 * pi) + logdet + distval)/2, ncol = ncol(x))) } ## ------------------------------------------------------------------------ d <- harness(f = my_dmvnorm, fv = my_dmvnorm_v, lowerLimit = rep(-0.5, 3), upperLimit = c(1, 4, 2), tol = 1e-5, times = 10, mean = rep(0, m), sigma = sigma, logdet = logdet) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ library(mvtnorm) g1 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = Miwa(), abseps = 1e-5, releps = 1e-5) g2 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = GenzBretz(), abseps = 1e-5, releps = 1e-5) g3 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = TVPACK(), abseps = 1e-5, releps = 1e-5) knitr::kable(summary(benchr::benchmark(g1(), g2(), g3(), times = 20, progress = FALSE)), digits = 3, row.names = FALSE) ## ------------------------------------------------------------------------ testFn0 <- function(x) prod(cos(x)) testFn0_v <- function(x) matrix(apply(x, 2, function(z) prod(cos(z))), ncol = ncol(x)) d <- harness(f = testFn0, fv = testFn0_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 1000) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn1 <- function(x) { val <- sum(((1 - x) / x)^2) scale <- prod((2 / sqrt(pi)) / x^2) exp(-val) * scale } testFn1_v <- function(x) { val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x)) scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x)) exp(-val) * scale } d <- harness(f = testFn1, fv = testFn1_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 10) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn2 <- function(x) { radius <- 0.50124145262344534123412 ifelse(sum(x * x) < radius * radius, 1, 0) } testFn2_v <- function(x) { radius <- 0.50124145262344534123412 matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x)) } d <- harness(which = c("hc", "hc.v", "cc"), f = testFn2, fv = testFn2_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 10) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn3 <- function(x) prod(2 * x) testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x)) d <- harness(f = testFn3, fv = testFn3_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 50) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn4 <- function(x) { a <- 0.1 s <- sum((x - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a)) } testFn4_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s <- sum((z - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a)) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn4, fv = testFn4_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn5 <- function(x) { a <- 0.1 s1 <- sum((x - 1 / 3)^2) s2 <- sum((x - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) } testFn5_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s1 <- sum((z - 1 / 3)^2) s2 <- sum((z - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn5, fv = testFn5_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn6 <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 prod( a / (a + 1) * ((a + 1) / (a + x))^2) } testFn6_v <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2)) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn6, fv = testFn6_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ testFn7 <- function(x) { n <- length(x) p <- 1/n (1 + p)^n * prod(x^p) } testFn7_v <- function(x) { matrix(apply(x, 2, function(z) { n <- length(z) p <- 1/n (1 + p)^n * prod(z^p) }), ncol = ncol(x)) } d <- harness(f = testFn7, fv = testFn7_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ 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)) } d <- harness(f = I.1d, fv = I.1d_v, lowerLimit = -2, upperLimit = 2, times = 100) knitr::kable(d, digits = 3) ## ------------------------------------------------------------------------ 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) } I.2d_v <- function(x) { matrix(apply(x, 2, function(z) { x1 <- z[1]; x2 <- z[2] sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2) }), ncol = ncol(x)) } d <- harness(f = I.2d, fv = I.2d_v, lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), times = 100) knitr::kable(d, digits = 3) cubature/inst/include/0000755000176200001440000000000013133443063014464 5ustar liggesuserscubature/inst/include/exp_cubature.h0000644000176200001440000000754013133443055017332 0ustar liggesusers#ifndef __exp_cubature_h__ #define __exp_cubature_h__ #include #include #include "exp_cubature_typedefs.h" #ifdef __cplusplus extern "C" { #endif /* adapative integration by partitioning the integration domain ("h-adaptive") and using the same fixed-degree quadrature in each subdomain, recursively, until convergence is achieved. */ int hcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { typedef int (*Fun)(unsigned, integrand, void*, unsigned, const double*, const double*, size_t, double, double, error_norm, double*, double*); static Fun fun = NULL; if (fun == NULL) { Rf_eval(Rf_lang2(Rf_install("loadNamespace"), Rf_ScalarString(Rf_mkChar("cubature")) ), R_GlobalEnv); fun = (Fun) R_GetCCallable("cubature", "hcubature"); } return fun(fdim,f,fdata,dim,xmin,xmax,maxEval,reqAbsError,reqRelError,norm,val,err); } /* as hcubature, but vectorized integrand */ int hcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { typedef int (*Fun)(unsigned, integrand_v, void*, unsigned, const double*, const double*, size_t, double, double, error_norm, double*, double*); static Fun fun = NULL; if (fun == NULL) { Rf_eval(Rf_lang2(Rf_install("loadNamespace"), Rf_ScalarString(Rf_mkChar("cubature")) ), R_GlobalEnv); fun = (Fun) R_GetCCallable("cubature", "hcubature_v"); } return fun(fdim,f,fdata,dim,xmin,xmax,maxEval,reqAbsError,reqRelError,norm,val,err); } /* adaptive integration by increasing the degree of (tensor-product Clenshaw-Curtis) quadrature rules ("p-adaptive"), rather than subdividing the domain ("h-adaptive"). Possibly better for smooth integrands in low dimensions. */ int pcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { typedef int (*Fun)(unsigned, integrand, void*, unsigned, const double*, const double*, size_t, double, double, error_norm, double*, double*); static Fun fun = NULL; if (fun == NULL) { Rf_eval(Rf_lang2(Rf_install("loadNamespace"), Rf_ScalarString(Rf_mkChar("cubature")) ), R_GlobalEnv); fun = (Fun) R_GetCCallable("cubature", "pcubature"); } return fun(fdim,f,fdata,dim,xmin,xmax,maxEval,reqAbsError,reqRelError,norm,val,err); } /* as pcubature, but vectorized integrand */ int pcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { typedef int (*Fun)(unsigned, integrand_v, void*, unsigned, const double*, const double*, size_t, double, double, error_norm, double*, double*); static Fun fun = NULL; if (fun == NULL) { Rf_eval(Rf_lang2(Rf_install("loadNamespace"), Rf_ScalarString(Rf_mkChar("cubature")) ), R_GlobalEnv); fun = (Fun) R_GetCCallable("cubature", "pcubature_v"); } return fun(fdim,f,fdata,dim,xmin,xmax,maxEval,reqAbsError,reqRelError,norm,val,err); } #ifdef __cplusplus } #endif #endif cubature/inst/include/exp_cubature_typedefs.h0000644000176200001440000000352213133433710021226 0ustar liggesusers#ifndef __exp_cubature_typedefs_h__ #define __exp_cubature_typedefs_h__ /* separate h file for the types and the error_norm enum, to avoid duplicate symbol error. */ #ifdef __cplusplus extern "C" { #endif /* a vector integrand - evaluates the function at the given point x (an array of length ndim) and returns the result in fval (an array of length fdim). The void* parameter is there in case you have to pass any additional data through to your function (it corresponds to the fdata parameter you pass to cubature). Return 0 on success or nonzero to terminate the integration. */ typedef int (*integrand) (unsigned ndim, const double *x, void *, unsigned fdim, double *fval); /* a vector integrand of a vector of npt points: x[i*ndim + j] is the j-th coordinate of the i-th point, and the k-th function evaluation for the i-th point is returned in fval[i*fdim + k]. Return 0 on success or nonzero to terminate the integration. */ typedef int (*integrand_v) (unsigned ndim, size_t npt, const double *x, void *, unsigned fdim, double *fval); /* Different ways of measuring the absolute and relative error when we have multiple integrands, given a vector e of error estimates in the individual components of a vector v of integrands. These are all equivalent when there is only a single integrand. */ typedef enum { ERROR_INDIVIDUAL = 0, /* individual relerr criteria in each component */ ERROR_PAIRED, /* paired L2 norms of errors in each component, mainly for integrating vectors of complex numbers */ ERROR_L2, /* abserr is L_2 norm |e|, and relerr is |e|/|v| */ ERROR_L1, /* abserr is L_1 norm |e|, and relerr is |e|/|v| */ ERROR_LINF /* abserr is L_\infty norm |e|, and relerr is |e|/|v| */ } error_norm; #ifdef __cplusplus } #endif #endif cubature/inst/include/cubature.h0000644000176200001440000001233113133172427016452 0ustar liggesusers/* Adaptive multidimensional integration of a vector of integrands. * * Copyright (c) 2005-2013 Steven G. Johnson * * Portions (see comments) based on HIntLib (also distributed under * the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. * (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) * * Portions (see comments) based on GNU GSL (also distributed under * the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. * (http://www.gnu.org/software/gsl/) * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * */ #ifndef CUBATURE_H #define CUBATURE_H #include /* for size_t */ #ifdef __cplusplus extern "C" { #endif /* __cplusplus */ /* USAGE: Call hcubature or pcubature with your function as described in the README file. */ /* a vector integrand - evaluates the function at the given point x (an array of length ndim) and returns the result in fval (an array of length fdim). The void* parameter is there in case you have to pass any additional data through to your function (it corresponds to the fdata parameter you pass to cubature). Return 0 on success or nonzero to terminate the integration. */ typedef int (*integrand) (unsigned ndim, const double *x, void *, unsigned fdim, double *fval); /* a vector integrand of a vector of npt points: x[i*ndim + j] is the j-th coordinate of the i-th point, and the k-th function evaluation for the i-th point is returned in fval[i*fdim + k]. Return 0 on success or nonzero to terminate the integration. */ typedef int (*integrand_v) (unsigned ndim, size_t npt, const double *x, void *, unsigned fdim, double *fval); /* Different ways of measuring the absolute and relative error when we have multiple integrands, given a vector e of error estimates in the individual components of a vector v of integrands. These are all equivalent when there is only a single integrand. */ typedef enum { ERROR_INDIVIDUAL = 0, /* individual relerr criteria in each component */ ERROR_PAIRED, /* paired L2 norms of errors in each component, mainly for integrating vectors of complex numbers */ ERROR_L2, /* abserr is L_2 norm |e|, and relerr is |e|/|v| */ ERROR_L1, /* abserr is L_1 norm |e|, and relerr is |e|/|v| */ ERROR_LINF /* abserr is L_\infty norm |e|, and relerr is |e|/|v| */ } error_norm; /* Integrate the function f from xmin[dim] to xmax[dim], with at most maxEval function evaluations (0 for no limit), until the given absolute or relative error is achieved. val returns the integral, and err returns the estimate for the absolute error in val; both of these are arrays of length fdim, the dimension of the vector integrand f(x). The return value of the function is 0 on success and non-zero if there was an error. */ /* adapative integration by partitioning the integration domain ("h-adaptive") and using the same fixed-degree quadrature in each subdomain, recursively, until convergence is achieved. */ int hcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); /* as hcubature, but vectorized integrand */ int hcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); /* adaptive integration by increasing the degree of (tensor-product Clenshaw-Curtis) quadrature rules ("p-adaptive"), rather than subdividing the domain ("h-adaptive"). Possibly better for smooth integrands in low dimensions. */ int pcubature_v_buf(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, unsigned *m, double **buf, size_t *nbuf, size_t max_nbuf, double *val, double *err); int pcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); int pcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); #ifdef __cplusplus } /* extern "C" */ #endif /* __cplusplus */ #endif /* CUBATURE_H */ cubature/tests/0000755000176200001440000000000013133172427013231 5ustar liggesuserscubature/tests/testthat.R0000644000176200001440000000007413133172427015215 0ustar liggesuserslibrary(testthat) library(cubature) test_check("cubature") cubature/tests/testthat/0000755000176200001440000000000013133704177015074 5ustar liggesuserscubature/tests/testthat/test_integration.R0000644000176200001440000005212213133172427020600 0ustar liggesuserslibrary(cubature) context("Test known integration results") test_that("Test a product of Cosine functions", { expected <- 0.708073 tol <- 1e-4 ## Product of cosines testFn0 <- function(x) prod(cos(x)) result <- cubature::adaptIntegrate(f = testFn0, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn0_v <- function(x) { r <- apply(x, 2, function(z) prod(cos(z))) matrix(r, ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn0_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Vector mode Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached in vector mode") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached in vector mode") }) test_that("Test a Gaussian integral remapped to [0, infinity] limits", { expected <- 1.00001 tol <- 1e-4 ## Gaussian function testFn1 <- function(x) { val <- sum(((1 - x) / x)^2) scale <- prod((2 / sqrt(pi)) / x^2) exp(-val) * scale } result <- cubature::adaptIntegrate(f = testFn1, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn1_v <- function(x) { val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x)) scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x)) exp(-val) * scale } result <- cubature::adaptIntegrate(f = testFn1_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Vector mode Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached in vector mode") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached in vector mode") }) test_that("Test volume of a hypersphere (integrating a discountinuous function)", { expected <- 0.19728 tol <- 1e-4 ## discontinuous objective: volume of hypersphere testFn2 <- function(x) { radius <- 0.50124145262344534123412 ifelse(sum(x * x) < radius * radius, 1, 0) } result <- cubature::adaptIntegrate(f = testFn2, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn2_v <- function(x) { radius <- 0.50124145262344534123412 matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn2_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Vector mode Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached in vector mode") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached in vector mode") }) test_that("Test a simple polynomial (product of coordinates)", { expected <- 1 tol <- 1e-4 ## product of coordinates testFn3 <- function(x) prod(2 * x) result <- cubature::adaptIntegrate(f = testFn3, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x)) result <- cubature::adaptIntegrate(f = testFn3_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Vector mode Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached in vector mode") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached in vector mode") }) test_that("Test Gaussian centered at 1/2", { expected <- 1 tol <- 1e-4 ## guassian centered at 1/2 testFn4 <- function(x) { a <- 0.1 s <- sum((x - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a)) } result <- cubature::adaptIntegrate(f = testFn4, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn4_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s <- sum((z - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a)) }) matrix(r, ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn4_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test double Gaussian", { expected <- 0.999994 tol <- 1e-4 ## double guassian testFn5 <- function(x) { a = 0.1 s1 = sum((x - 1 / 3)^2) s2 = sum((x - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) } result <- cubature::adaptIntegrate(f = testFn5, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn5_v <- function(x) { a = 0.1 r <- apply(x, 2, function(z) { s1 <- sum((z - 1 / 3)^2) s2 <- sum((z - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) }) matrix(r, ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn5_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test Tsuda's example", { expected <- 0.999998 tol <- 1e-4 ## Tsuda's example testFn6 <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 prod( a / (a + 1) * ((a + 1) / (a + x))^2) } result <- cubature::adaptIntegrate(f = testFn6, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn6_v <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2)) matrix(r, ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn6_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test Morokoff & Calflish example", { expected <- 1.00001 tol <- 1e-4 ## Morokoff & Calflish function testFn7 <- function(x) { n <- length(x) p <- 1/n (1 + p)^n * prod(x^p) } result <- cubature::adaptIntegrate(f = testFn7, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") testFn7_v <- function(x) { matrix(apply(x, 2, function(z) { n <- length(z) p <- 1/n (1 + p)^n * prod(z^p) }), ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = testFn7_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test cubature web example", { expected <- 13.69609 tol <- 1e-4 result <- cubature::adaptIntegrate(f = function(x) exp(-0.5 * sum(x^2)), lowerLimit = rep(-2, 3), upperLimit = rep(2, 3), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") result <- cubature::adaptIntegrate(f = function(x) matrix( apply(x, 2, function(z) exp(-0.5 * sum(z^2))), ncol = ncol(x)), lowerLimit = rep(-2, 3), upperLimit = rep(2, 3), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test Wang-Landau sampling 1d example", { expected <- 1.63564436296 tol <- 1e-5 ## Numerical integration using Wang-Landau sampling ## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin ## Computer Physics Communications, 2007, 524-529 ## Compare with exact answer: 1.63564436296 ## I.1d <- function(x) { sin(4 * x) * x * ((x * ( x * (x * x - 4) + 1) - 1)) } result <- cubature::adaptIntegrate(f = I.1d, lowerLimit = -2, upperLimit = 2, tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") I.1d_v <- function(x) { matrix(apply(x, 2, function(z) sin(4 * z) * z * ((z * ( z * (z * z - 4) + 1) - 1))), ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = I.1d_v, lowerLimit = -2, upperLimit = 2, tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test Wang-Landau sampling 2d example", { expected <- -0.01797992646 tol <- 1e-5 ## Numerical integration using Wang-Landau sampling ## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin ## Computer Physics Communications, 2007, 524-529 ## Compare with exact answer: -0.01797992646 ## I.2d <- function(x) { x1 <- x[1]; x2 <- x[2] sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2) } result <- cubature::adaptIntegrate(f = I.2d, lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), tol = tol) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") I.2d_v <- function(x) { matrix(apply(x, 2, function(z) { x1 <- z[1]; x2 <- z[2] sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2) }), ncol = ncol(x)) } result <- cubature::adaptIntegrate(f = I.2d_v, lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), tol = tol, vectorInterface = TRUE) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") }) test_that("Test Multivariate Normal", { expected <- 0.3341125 tol <- 1e-5 dmvnorm <- function (x, mean, sigma) { x <- matrix(x, ncol = length(x)) distval <- stats::mahalanobis(x, center = mean, cov = sigma) logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values)) logretval <- -(ncol(x) * log(2 * pi) + logdet + distval)/2 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 result <- cubature::adaptIntegrate(f = dmvnorm, lowerLimit = rep(-0.5, 3), upperLimit = c(1, 4, 2), tol = tol, mean=rep(0, m), sigma=sigma) testthat::expect_equal(0, result$returnCode, info = "Integration unsuccessful!") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = result$integral, info = "Relative error not reached") testthat::expect_equal(expected, result$integral, tolerance = tol, scale = 1, info = "Absolute error not reached") dmvnorm_v <- function (x, mean, sigma) { x <- t(x) distval <- stats::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 logretval <- matrix(logretval, ncol = nrow(x)) exp(logretval) } result <- cubature::adaptIntegrate(f = dmvnorm_v, lowerLimit = rep(-0.5, 3), upperLimit = c(1, 4, 2), tol = tol, vectorInterface = TRUE, mean=rep(0, m), sigma=sigma) }) cubature/src/0000755000176200001440000000000013133476177012667 5ustar liggesuserscubature/src/Makevars0000644000176200001440000000125513133476200014351 0ustar liggesusersPKG_CFLAGS=-DR_PACKAGE=1 -DTEST_INTEGRATOR=0 -I./cubature-1.0.2 -I../inst/include PKG_CPPFLAGS=-DR_PACKAGE=1 -DTEST_INTEGRATOR=0 -I./cubature-1.0.2 -I../inst/include PKG_LIBS=-Lcubature-1.0.2 -lcubature $(SHLIB): Rcpp-cubature.o RcppExports.o rcubature.o Rcpp-cubature.o: cubature.ts RcppExports.o: cubature.ts rcubature.o: cubature.ts cubature.ts: (cd cubature-1.0.2 && \ ($(MAKE) libcubature.a CC="$(CC)" CFLAGS="$(CFLAGS) -I../../inst/include $(CPICFLAGS) -DR_PACKAGE=1" AR="$(AR)" RANLIB="$(RANLIB)") && \ touch $@) clean: rm -f Rcpp-cubature.o RcppExports.o rcubature.o cubature.so rm -f cubature-1.0.2/*.o rm -f cubature-1.0.2/libcubature.a cubature-1.0.2/cubature.ts cubature/src/cubature_init.c0000644000176200001440000000234513133476200015657 0ustar liggesusers#include #include #include #include /* .Call entry points */ extern SEXP _cubature_doHCubature(SEXP fDimSEXP, SEXP fSEXP, SEXP xLLSEXP, SEXP xULSEXP, SEXP maxEvalSEXP, SEXP absErrSEXP, SEXP tolSEXP, SEXP vectorInterfaceSEXP, SEXP normSEXP); extern SEXP _cubature_doPCubature(SEXP fDimSEXP, SEXP fSEXP, SEXP xLLSEXP, SEXP xULSEXP, SEXP maxEvalSEXP, SEXP absErrSEXP, SEXP tolSEXP, SEXP vectorInterfaceSEXP, SEXP normSEXP); static const R_CallMethodDef CallEntries[] = { {"_cubature_doHCubature", (DL_FUNC) &_cubature_doHCubature, 9}, {"_cubature_doPCubature", (DL_FUNC) &_cubature_doPCubature, 9}, {NULL, NULL, 0} }; void R_init_cubature(DllInfo *dll) { R_registerRoutines(dll, NULL, CallEntries, NULL, NULL); R_useDynamicSymbols(dll, FALSE); R_RegisterCCallable("cubature", "adapt_integrate", (DL_FUNC) hcubature); R_RegisterCCallable("cubature", "adapt_integrate_v", (DL_FUNC) hcubature_v); R_RegisterCCallable("cubature", "hcubature", (DL_FUNC) hcubature); R_RegisterCCallable("cubature", "hcubature_v", (DL_FUNC) hcubature_v); R_RegisterCCallable("cubature", "pcubature", (DL_FUNC) pcubature); R_RegisterCCallable("cubature", "pcubature_v", (DL_FUNC) pcubature_v); } cubature/src/cubature-1.0.2/0000755000176200001440000000000013133476177015135 5ustar liggesuserscubature/src/cubature-1.0.2/hcubature.c0000644000176200001440000010220013133476200017241 0ustar liggesusers/* Adaptive multidimensional integration of a vector of integrands. * * Copyright (c) 2005-2013 Steven G. Johnson * * Portions (see comments) based on HIntLib (also distributed under * the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. * (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) * * Portions (see comments) based on GNU GSL (also distributed under * the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. * (http://www.gnu.org/software/gsl/) * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * */ #include #include #include #include #include #include /* Adaptive multidimensional integration on hypercubes (or, really, hyper-rectangles) using cubature rules. A cubature rule takes a function and a hypercube and evaluates the function at a small number of points, returning an estimate of the integral as well as an estimate of the error, and also a suggested dimension of the hypercube to subdivide. Given such a rule, the adaptive integration is simple: 1) Evaluate the cubature rule on the hypercube(s). Stop if converged. 2) Pick the hypercube with the largest estimated error, and divide it in two along the suggested dimension. 3) Goto (1). The basic algorithm is based on the adaptive cubature described in A. C. Genz and A. A. Malik, "An adaptive algorithm for numeric integration over an N-dimensional rectangular region," J. Comput. Appl. Math. 6 (4), 295-302 (1980). and subsequently extended to integrating a vector of integrands in J. Berntsen, T. O. Espelid, and A. Genz, "An adaptive algorithm for the approximate calculation of multiple integrals," ACM Trans. Math. Soft. 17 (4), 437-451 (1991). Note, however, that we do not use any of code from the above authors (in part because their code is Fortran 77, but mostly because it is under the restrictive ACM copyright license). I did make use of some GPL code from Rudolf Schuerer's HIntLib and from the GNU Scientific Library as listed in the copyright notice above, on the other hand. I am also grateful to Dmitry Turbiner , who implemented an initial prototype of the "vectorized" functionality for evaluating multiple points in a single call (as opposed to multiple functions in a single call). (Although Dmitry implemented a working version, I ended up re-implementing this feature from scratch as part of a larger code-cleanup, and in order to have a single code path for the vectorized and non-vectorized APIs. I subsequently implemented the algorithm by Gladwell to extract even more parallelism by evalutating many hypercubes at once.) TODO: * Putting these routines into the GNU GSL library would be nice. * A Python interface would be nice. (Also a Matlab interface, a GNU Octave interface, ...) * For high-dimensional integrals, it would be nice to implement a sparse-grid cubature scheme using Clenshaw-Curtis quadrature. Currently, for dimensions > 7 or so, quasi Monte Carlo methods win. * Berntsen et. al also describe a "two-level" error estimation scheme that they claim makes the algorithm more robust. It might be nice to implement this, at least as an option (although I seem to remember trying it once and it made the number of evaluations substantially worse for my test integrands). */ /* USAGE: Call cubature with your function as described in cubature.h. To compile a test program, compile cubature.c with -DTEST_INTEGRATOR as described at the end. */ #include /* error return codes */ #define SUCCESS 0 #define FAILURE 1 /***************************************************************************/ /* Basic datatypes */ typedef struct { double val, err; } esterr; static double errMax(unsigned fdim, const esterr *ee) { double errmax = 0; unsigned k; for (k = 0; k < fdim; ++k) if (ee[k].err > errmax) errmax = ee[k].err; return errmax; } typedef struct { unsigned dim; double *data; /* length 2*dim = center followed by half-widths */ double vol; /* cache volume = product of widths */ } hypercube; static double compute_vol(const hypercube *h) { unsigned i; double vol = 1; for (i = 0; i < h->dim; ++i) vol *= 2 * h->data[i + h->dim]; return vol; } static hypercube make_hypercube(unsigned dim, const double *center, const double *halfwidth) { unsigned i; hypercube h; h.dim = dim; h.data = (double *) malloc(sizeof(double) * dim * 2); h.vol = 0; if (h.data) { for (i = 0; i < dim; ++i) { h.data[i] = center[i]; h.data[i + dim] = halfwidth[i]; } h.vol = compute_vol(&h); } return h; } static hypercube make_hypercube_range(unsigned dim, const double *xmin, const double *xmax) { hypercube h = make_hypercube(dim, xmin, xmax); unsigned i; if (h.data) { for (i = 0; i < dim; ++i) { h.data[i] = 0.5 * (xmin[i] + xmax[i]); h.data[i + dim] = 0.5 * (xmax[i] - xmin[i]); } h.vol = compute_vol(&h); } return h; } static void destroy_hypercube(hypercube *h) { free(h->data); h->dim = 0; } typedef struct { hypercube h; unsigned splitDim; unsigned fdim; /* dimensionality of vector integrand */ esterr *ee; /* array of length fdim */ double errmax; /* max ee[k].err */ } region; static region make_region(const hypercube *h, unsigned fdim) { region R; R.h = make_hypercube(h->dim, h->data, h->data + h->dim); R.splitDim = 0; R.fdim = fdim; R.ee = R.h.data ? (esterr *) malloc(sizeof(esterr) * fdim) : NULL; R.errmax = HUGE_VAL; return R; } static void destroy_region(region *R) { destroy_hypercube(&R->h); free(R->ee); R->ee = 0; } static int cut_region(region *R, region *R2) { unsigned d = R->splitDim, dim = R->h.dim; *R2 = *R; R->h.data[d + dim] *= 0.5; R->h.vol *= 0.5; R2->h = make_hypercube(dim, R->h.data, R->h.data + dim); if (!R2->h.data) return FAILURE; R->h.data[d] -= R->h.data[d + dim]; R2->h.data[d] += R->h.data[d + dim]; R2->ee = (esterr *) malloc(sizeof(esterr) * R2->fdim); return R2->ee == NULL; } struct rule_s; /* forward declaration */ typedef int (*evalError_func)(struct rule_s *r, unsigned fdim, integrand_v f, void *fdata, unsigned nR, region *R); typedef void (*destroy_func)(struct rule_s *r); typedef struct rule_s { unsigned dim, fdim; /* the dimensionality & number of functions */ unsigned num_points; /* number of evaluation points */ unsigned num_regions; /* max number of regions evaluated at once */ double *pts; /* points to eval: num_regions * num_points * dim */ double *vals; /* num_regions * num_points * fdim */ evalError_func evalError; destroy_func destroy; } rule; static void destroy_rule(rule *r) { if (r) { if (r->destroy) r->destroy(r); free(r->pts); free(r); } } static int alloc_rule_pts(rule *r, unsigned num_regions) { if (num_regions > r->num_regions) { free(r->pts); r->pts = r->vals = NULL; r->num_regions = 0; num_regions *= 2; /* allocate extra so that repeatedly calling alloc_rule_pts with growing num_regions only needs a logarithmic number of allocations */ r->pts = (double *) malloc(sizeof(double) * (num_regions * r->num_points * (r->dim + r->fdim))); if (r->fdim + r->dim > 0 && !r->pts) return FAILURE; r->vals = r->pts + num_regions * r->num_points * r->dim; r->num_regions = num_regions; } return SUCCESS; } static rule *make_rule(size_t sz, /* >= sizeof(rule) */ unsigned dim, unsigned fdim, unsigned num_points, evalError_func evalError, destroy_func destroy) { rule *r; if (sz < sizeof(rule)) return NULL; r = (rule *) malloc(sz); if (!r) return NULL; r->pts = r->vals = NULL; r->num_regions = 0; r->dim = dim; r->fdim = fdim; r->num_points = num_points; r->evalError = evalError; r->destroy = destroy; return r; } /* note: all regions must have same fdim */ static int eval_regions(unsigned nR, region *R, integrand_v f, void *fdata, rule *r) { unsigned iR; if (nR == 0) return SUCCESS; /* nothing to evaluate */ if (r->evalError(r, R->fdim, f, fdata, nR, R)) return FAILURE; for (iR = 0; iR < nR; ++iR) R[iR].errmax = errMax(R->fdim, R[iR].ee); return SUCCESS; } /***************************************************************************/ /* Functions to loop over points in a hypercube. */ /* Based on orbitrule.cpp in HIntLib-0.0.10 */ /* ls0 returns the least-significant 0 bit of n (e.g. it returns 0 if the LSB is 0, it returns 1 if the 2 LSBs are 01, etcetera). */ static unsigned ls0(unsigned n) { #if defined(__GNUC__) && \ ((__GNUC__ == 3 && __GNUC_MINOR__ >= 4) || __GNUC__ > 3) return __builtin_ctz(~n); /* gcc builtin for version >= 3.4 */ #else const unsigned bits[256] = { 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 8, }; unsigned bit = 0; while ((n & 0xff) == 0xff) { n >>= 8; bit += 8; } return bit + bits[n & 0xff]; #endif } /** * Evaluate the integration points for all 2^n points (+/-r,...+/-r) * * A Gray-code ordering is used to minimize the number of coordinate updates * in p, although this doesn't matter as much now that we are saving all pts. */ static void evalR_Rfs(double *pts, unsigned dim, double *p, const double *c, const double *r) { unsigned i; unsigned signs = 0; /* 0/1 bit = +/- for corresponding element of r[] */ /* We start with the point where r is ADDed in every coordinate (this implies signs=0). */ for (i = 0; i < dim; ++i) p[i] = c[i] + r[i]; /* Loop through the points in Gray-code ordering */ for (i = 0;; ++i) { unsigned mask, d; memcpy(pts, p, sizeof(double) * dim); pts += dim; d = ls0(i); /* which coordinate to flip */ if (d >= dim) break; /* flip the d-th bit and add/subtract r[d] */ mask = 1U << d; signs ^= mask; p[d] = (signs & mask) ? c[d] - r[d] : c[d] + r[d]; } } static void evalRR0_0fs(double *pts, unsigned dim, double *p, const double *c, const double *r) { unsigned i, j; for (i = 0; i < dim - 1; ++i) { p[i] = c[i] - r[i]; for (j = i + 1; j < dim; ++j) { p[j] = c[j] - r[j]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i] + r[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[j] = c[j] + r[j]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i] - r[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[j] = c[j]; /* Done with j -> Restore p[j] */ } p[i] = c[i]; /* Done with i -> Restore p[i] */ } } static void evalR0_0fs4d(double *pts, unsigned dim, double *p, const double *c, const double *r1, const double *r2) { unsigned i; memcpy(pts, p, sizeof(double) * dim); pts += dim; for (i = 0; i < dim; i++) { p[i] = c[i] - r1[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i] + r1[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i] - r2[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i] + r2[i]; memcpy(pts, p, sizeof(double) * dim); pts += dim; p[i] = c[i]; } } #define num0_0(dim) (1U) #define numR0_0fs(dim) (2 * (dim)) #define numRR0_0fs(dim) (2 * (dim) * (dim-1)) #define numR_Rfs(dim) (1U << (dim)) /***************************************************************************/ /* Based on rule75genzmalik.cpp in HIntLib-0.0.10: An embedded cubature rule of degree 7 (embedded rule degree 5) due to A. C. Genz and A. A. Malik. See: A. C. Genz and A. A. Malik, "An imbedded [sic] family of fully symmetric numerical integration rules," SIAM J. Numer. Anal. 20 (3), 580-588 (1983). */ typedef struct { rule parent; /* temporary arrays of length dim */ double *widthLambda, *widthLambda2, *p; /* dimension-dependent constants */ double weight1, weight3, weight5; double weightE1, weightE3; } rule75genzmalik; #define real(x) ((double)(x)) #define to_int(n) ((int)(n)) static int isqr(int x) { return x * x; } static void destroy_rule75genzmalik(rule *r_) { rule75genzmalik *r = (rule75genzmalik *) r_; free(r->p); } static int rule75genzmalik_evalError(rule *r_, unsigned fdim, integrand_v f, void *fdata, unsigned nR, region *R) { /* lambda2 = sqrt(9/70), lambda4 = sqrt(9/10), lambda5 = sqrt(9/19) */ const double lambda2 = 0.3585685828003180919906451539079374954541; const double lambda4 = 0.9486832980505137995996680633298155601160; const double lambda5 = 0.6882472016116852977216287342936235251269; const double weight2 = 980. / 6561.; const double weight4 = 200. / 19683.; const double weightE2 = 245. / 486.; const double weightE4 = 25. / 729.; const double ratio = (lambda2 * lambda2) / (lambda4 * lambda4); rule75genzmalik *r = (rule75genzmalik *) r_; unsigned i, j, iR, dim = r_->dim; size_t npts = 0; double *diff, *pts, *vals; if (alloc_rule_pts(r_, nR)) return FAILURE; pts = r_->pts; vals = r_->vals; for (iR = 0; iR < nR; ++iR) { const double *center = R[iR].h.data; const double *halfwidth = R[iR].h.data + dim; for (i = 0; i < dim; ++i) r->p[i] = center[i]; for (i = 0; i < dim; ++i) r->widthLambda2[i] = halfwidth[i] * lambda2; for (i = 0; i < dim; ++i) r->widthLambda[i] = halfwidth[i] * lambda4; /* Evaluate points in the center, in (lambda2,0,...,0) and (lambda3=lambda4, 0,...,0). */ evalR0_0fs4d(pts + npts*dim, dim, r->p, center, r->widthLambda2, r->widthLambda); npts += num0_0(dim) + 2 * numR0_0fs(dim); /* Calculate points for (lambda4, lambda4, 0, ...,0) */ evalRR0_0fs(pts + npts*dim, dim, r->p, center, r->widthLambda); npts += numRR0_0fs(dim); /* Calculate points for (lambda5, lambda5, ..., lambda5) */ for (i = 0; i < dim; ++i) r->widthLambda[i] = halfwidth[i] * lambda5; evalR_Rfs(pts + npts*dim, dim, r->p, center, r->widthLambda); npts += numR_Rfs(dim); } /* Evaluate the integrand function(s) at all the points */ if (f(dim, npts, pts, fdata, fdim, vals)) return FAILURE; /* we are done with the points, and so we can re-use the pts array to store the maximum difference diff[i] in each dimension for each hypercube */ diff = pts; for (i = 0; i < dim * nR; ++i) diff[i] = 0; for (j = 0; j < fdim; ++j) { const double *v = vals + j; # define VALS(i) v[fdim*(i)] for (iR = 0; iR < nR; ++iR) { double result, res5th; double val0, sum2=0, sum3=0, sum4=0, sum5=0; unsigned k, k0 = 0; /* accumulate j-th function values into j-th integrals NOTE: this relies on the ordering of the eval functions above, as well as on the internal structure of the evalR0_0fs4d function */ val0 = VALS(0); /* central point */ k0 += 1; for (k = 0; k < dim; ++k) { double v0 = VALS(k0 + 4*k); double v1 = VALS((k0 + 4*k) + 1); double v2 = VALS((k0 + 4*k) + 2); double v3 = VALS((k0 + 4*k) + 3); sum2 += v0 + v1; sum3 += v2 + v3; diff[iR * dim + k] += fabs(v0 + v1 - 2*val0 - ratio * (v2 + v3 - 2*val0)); } k0 += 4*k; for (k = 0; k < numRR0_0fs(dim); ++k) sum4 += VALS(k0 + k); k0 += k; for (k = 0; k < numR_Rfs(dim); ++k) sum5 += VALS(k0 + k); /* Calculate fifth and seventh order results */ result = R[iR].h.vol * (r->weight1 * val0 + weight2 * sum2 + r->weight3 * sum3 + weight4 * sum4 + r->weight5 * sum5); res5th = R[iR].h.vol * (r->weightE1 * val0 + weightE2 * sum2 + r->weightE3 * sum3 + weightE4 * sum4); R[iR].ee[j].val = result; R[iR].ee[j].err = fabs(res5th - result); v += r_->num_points * fdim; } # undef VALS } /* figure out dimension to split: */ for (iR = 0; iR < nR; ++iR) { double maxdiff = 0; unsigned dimDiffMax = 0; for (i = 0; i < dim; ++i) if (diff[iR*dim + i] > maxdiff) { maxdiff = diff[iR*dim + i]; dimDiffMax = i; } R[iR].splitDim = dimDiffMax; } return SUCCESS; } static rule *make_rule75genzmalik(unsigned dim, unsigned fdim) { rule75genzmalik *r; if (dim < 2) return NULL; /* this rule does not support 1d integrals */ /* Because of the use of a bit-field in evalR_Rfs, we are limited to be < 32 dimensions (or however many bits are in unsigned). This is not a practical limitation...long before you reach 32 dimensions, the Genz-Malik cubature becomes excruciatingly slow and is superseded by other methods (e.g. Monte-Carlo). */ if (dim >= sizeof(unsigned) * 8) return NULL; r = (rule75genzmalik *) make_rule(sizeof(rule75genzmalik), dim, fdim, num0_0(dim) + 2 * numR0_0fs(dim) + numRR0_0fs(dim) + numR_Rfs(dim), rule75genzmalik_evalError, destroy_rule75genzmalik); if (!r) return NULL; r->weight1 = (real(12824 - 9120 * to_int(dim) + 400 * isqr(to_int(dim))) / real(19683)); r->weight3 = real(1820 - 400 * to_int(dim)) / real(19683); r->weight5 = real(6859) / real(19683) / real(1U << dim); r->weightE1 = (real(729 - 950 * to_int(dim) + 50 * isqr(to_int(dim))) / real(729)); r->weightE3 = real(265 - 100 * to_int(dim)) / real(1458); r->p = (double *) malloc(sizeof(double) * dim * 3); if (!r->p) { destroy_rule((rule *) r); return NULL; } r->widthLambda = r->p + dim; r->widthLambda2 = r->p + 2 * dim; return (rule *) r; } /***************************************************************************/ /* 1d 15-point Gaussian quadrature rule, based on qk15.c and qk.c in GNU GSL (which in turn is based on QUADPACK). */ static int rule15gauss_evalError(rule *r, unsigned fdim, integrand_v f, void *fdata, unsigned nR, region *R) { /* Gauss quadrature weights and kronrod quadrature abscissae and weights as evaluated with 80 decimal digit arithmetic by L. W. Fullerton, Bell Labs, Nov. 1981. */ const unsigned n = 8; const double xgk[8] = { /* abscissae of the 15-point kronrod rule */ 0.991455371120812639206854697526329, 0.949107912342758524526189684047851, 0.864864423359769072789712788640926, 0.741531185599394439863864773280788, 0.586087235467691130294144838258730, 0.405845151377397166906606412076961, 0.207784955007898467600689403773245, 0.000000000000000000000000000000000 /* xgk[1], xgk[3], ... abscissae of the 7-point gauss rule. xgk[0], xgk[2], ... to optimally extend the 7-point gauss rule */ }; static const double wg[4] = { /* weights of the 7-point gauss rule */ 0.129484966168869693270611432679082, 0.279705391489276667901467771423780, 0.381830050505118944950369775488975, 0.417959183673469387755102040816327 }; static const double wgk[8] = { /* weights of the 15-point kronrod rule */ 0.022935322010529224963732008058970, 0.063092092629978553290700663189204, 0.104790010322250183839876322541518, 0.140653259715525918745189590510238, 0.169004726639267902826583426598550, 0.190350578064785409913256402421014, 0.204432940075298892414161999234649, 0.209482141084727828012999174891714 }; unsigned j, k, iR; size_t npts = 0; double *pts, *vals; if (alloc_rule_pts(r, nR)) return FAILURE; pts = r->pts; vals = r->vals; for (iR = 0; iR < nR; ++iR) { const double center = R[iR].h.data[0]; const double halfwidth = R[iR].h.data[1]; pts[npts++] = center; for (j = 0; j < (n - 1) / 2; ++j) { int j2 = 2*j + 1; double w = halfwidth * xgk[j2]; pts[npts++] = center - w; pts[npts++] = center + w; } for (j = 0; j < n/2; ++j) { int j2 = 2*j; double w = halfwidth * xgk[j2]; pts[npts++] = center - w; pts[npts++] = center + w; } R[iR].splitDim = 0; /* no choice but to divide 0th dimension */ } if (f(1, npts, pts, fdata, fdim, vals)) return FAILURE; for (k = 0; k < fdim; ++k) { const double *vk = vals + k; for (iR = 0; iR < nR; ++iR) { const double halfwidth = R[iR].h.data[1]; double result_gauss = vk[0] * wg[n/2 - 1]; double result_kronrod = vk[0] * wgk[n - 1]; double result_abs = fabs(result_kronrod); double result_asc, mean, err; /* accumulate integrals */ npts = 1; for (j = 0; j < (n - 1) / 2; ++j) { int j2 = 2*j + 1; double v = vk[fdim*npts] + vk[fdim*npts+fdim]; result_gauss += wg[j] * v; result_kronrod += wgk[j2] * v; result_abs += wgk[j2] * (fabs(vk[fdim*npts]) + fabs(vk[fdim*npts+fdim])); npts += 2; } for (j = 0; j < n/2; ++j) { int j2 = 2*j; result_kronrod += wgk[j2] * (vk[fdim*npts] + vk[fdim*npts+fdim]); result_abs += wgk[j2] * (fabs(vk[fdim*npts]) + fabs(vk[fdim*npts+fdim])); npts += 2; } /* integration result */ R[iR].ee[k].val = result_kronrod * halfwidth; /* error estimate (from GSL, probably dates back to QUADPACK ... not completely clear to me why we don't just use fabs(result_kronrod - result_gauss) * halfwidth */ mean = result_kronrod * 0.5; result_asc = wgk[n - 1] * fabs(vk[0] - mean); npts = 1; for (j = 0; j < (n - 1) / 2; ++j) { int j2 = 2*j + 1; result_asc += wgk[j2] * (fabs(vk[fdim*npts]-mean) + fabs(vk[fdim*npts+fdim]-mean)); npts += 2; } for (j = 0; j < n/2; ++j) { int j2 = 2*j; result_asc += wgk[j2] * (fabs(vk[fdim*npts]-mean) + fabs(vk[fdim*npts+fdim]-mean)); npts += 2; } err = fabs(result_kronrod - result_gauss) * halfwidth; result_abs *= halfwidth; result_asc *= halfwidth; if (result_asc != 0 && err != 0) { double scale = pow((200 * err / result_asc), 1.5); err = (scale < 1) ? result_asc * scale : result_asc; } if (result_abs > DBL_MIN / (50 * DBL_EPSILON)) { double min_err = 50 * DBL_EPSILON * result_abs; if (min_err > err) err = min_err; } R[iR].ee[k].err = err; /* increment vk to point to next batch of results */ vk += 15*fdim; } } return SUCCESS; } static rule *make_rule15gauss(unsigned dim, unsigned fdim) { if (dim != 1) return NULL; /* this rule is only for 1d integrals */ return make_rule(sizeof(rule), dim, fdim, 15, rule15gauss_evalError, 0); } /***************************************************************************/ /* binary heap implementation (ala _Introduction to Algorithms_ by Cormen, Leiserson, and Rivest), for use as a priority queue of regions to integrate. */ typedef region heap_item; #define KEY(hi) ((hi).errmax) typedef struct { size_t n, nalloc; heap_item *items; unsigned fdim; esterr *ee; /* array of length fdim of the total integrand & error */ } heap; static void heap_resize(heap *h, size_t nalloc) { h->nalloc = nalloc; if (nalloc) h->items = (heap_item *) realloc(h->items, sizeof(heap_item)*nalloc); else { /* BSD realloc does not free for a zero-sized reallocation */ free(h->items); h->items = NULL; } } static heap heap_alloc(size_t nalloc, unsigned fdim) { heap h; unsigned i; h.n = 0; h.nalloc = 0; h.items = 0; h.fdim = fdim; h.ee = (esterr *) malloc(sizeof(esterr) * fdim); if (h.ee) { for (i = 0; i < fdim; ++i) h.ee[i].val = h.ee[i].err = 0; heap_resize(&h, nalloc); } return h; } /* note that heap_free does not deallocate anything referenced by the items */ static void heap_free(heap *h) { h->n = 0; heap_resize(h, 0); h->fdim = 0; free(h->ee); } static int heap_push(heap *h, heap_item hi) { int insert; unsigned i, fdim = h->fdim; for (i = 0; i < fdim; ++i) { h->ee[i].val += hi.ee[i].val; h->ee[i].err += hi.ee[i].err; } insert = h->n; if (++(h->n) > h->nalloc) { heap_resize(h, h->n * 2); if (!h->items) return FAILURE; } while (insert) { int parent = (insert - 1) / 2; if (KEY(hi) <= KEY(h->items[parent])) break; h->items[insert] = h->items[parent]; insert = parent; } h->items[insert] = hi; return SUCCESS; } static int heap_push_many(heap *h, size_t ni, heap_item *hi) { size_t i; for (i = 0; i < ni; ++i) if (heap_push(h, hi[i])) return FAILURE; return SUCCESS; } static heap_item heap_pop(heap *h) { heap_item ret; int i, n, child; if (!(h->n)) { fprintf(stderr, "attempted to pop an empty heap\n"); exit(EXIT_FAILURE); } ret = h->items[0]; h->items[i = 0] = h->items[n = --(h->n)]; while ((child = i * 2 + 1) < n) { int largest; heap_item swap; if (KEY(h->items[child]) <= KEY(h->items[i])) largest = i; else largest = child; if (++child < n && KEY(h->items[largest]) < KEY(h->items[child])) largest = child; if (largest == i) break; swap = h->items[i]; h->items[i] = h->items[largest]; h->items[i = largest] = swap; } { unsigned i, fdim = h->fdim; for (i = 0; i < fdim; ++i) { h->ee[i].val -= ret.ee[i].val; h->ee[i].err -= ret.ee[i].err; } } return ret; } /***************************************************************************/ static int converged(unsigned fdim, const esterr *ee, double reqAbsError, double reqRelError, error_norm norm) #define ERR(j) ee[j].err #define VAL(j) ee[j].val #include "converged.h" /***************************************************************************/ /* adaptive integration, analogous to adaptintegrator.cpp in HIntLib */ static int rulecubature(rule *r, unsigned fdim, integrand_v f, void *fdata, const hypercube *h, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err, int parallel) { size_t numEval = 0; heap regions; unsigned i, j; region *R = NULL; /* array of regions to evaluate */ size_t nR_alloc = 0; esterr *ee = NULL; if (fdim <= 1) norm = ERROR_INDIVIDUAL; /* norm is irrelevant */ #ifndef R_PACKAGE if (norm < 0 || norm > ERROR_LINF) return FAILURE; /* invalid norm */ #endif regions = heap_alloc(1, fdim); if (!regions.ee || !regions.items) goto bad; ee = (esterr *) malloc(sizeof(esterr) * fdim); if (!ee) goto bad; nR_alloc = 2; R = (region *) malloc(sizeof(region) * nR_alloc); if (!R) goto bad; R[0] = make_region(h, fdim); if (!R[0].ee || eval_regions(1, R, f, fdata, r) || heap_push(®ions, R[0])) goto bad; numEval += r->num_points; while (numEval < maxEval || !maxEval) { if (converged(fdim, regions.ee, reqAbsError, reqRelError, norm)) break; if (parallel) { /* maximize potential parallelism */ /* adapted from I. Gladwell, "Vectorization of one dimensional quadrature codes," pp. 230--238 in _Numerical Integration. Recent Developments, Software and Applications_, G. Fairweather and P. M. Keast, eds., NATO ASI Series C203, Dordrecht (1987), as described in J. M. Bull and T. L. Freeman, "Parallel Globally Adaptive Algorithms for Multi-dimensional Integration," http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.6638 (1994). Basically, this evaluates in one shot all regions that *must* be evaluated in order to reduce the error to the requested bound: the minimum set of largest-error regions whose errors push the total error over the bound. [Note: Bull and Freeman claim that the Gladwell approach is intrinsically inefficent because it "requires sorting", and propose an alternative algorithm that "only" requires three passes over the entire set of regions. Apparently, they didn't realize that one could use a heap data structure, in which case the time to pop K biggest-error regions out of N is only O(K log N), much better than the O(N) cost of the Bull and Freeman algorithm if K << N, and it is also much simpler.] */ size_t nR = 0; for (j = 0; j < fdim; ++j) ee[j] = regions.ee[j]; do { if (nR + 2 > nR_alloc) { nR_alloc = (nR + 2) * 2; R = (region *) realloc(R, nR_alloc * sizeof(region)); if (!R) goto bad; } R[nR] = heap_pop(®ions); for (j = 0; j < fdim; ++j) ee[j].err -= R[nR].ee[j].err; if (cut_region(R+nR, R+nR+1)) goto bad; numEval += r->num_points * 2; nR += 2; if (converged(fdim, ee, reqAbsError, reqRelError, norm)) break; /* other regions have small errs */ } while (regions.n > 0 && (numEval < maxEval || !maxEval)); if (eval_regions(nR, R, f, fdata, r) || heap_push_many(®ions, nR, R)) goto bad; } else { /* minimize number of function evaluations */ R[0] = heap_pop(®ions); /* get worst region */ if (cut_region(R, R+1) || eval_regions(2, R, f, fdata, r) || heap_push_many(®ions, 2, R)) goto bad; numEval += r->num_points * 2; } } /* re-sum integral and errors */ for (j = 0; j < fdim; ++j) val[j] = err[j] = 0; for (i = 0; i < regions.n; ++i) { for (j = 0; j < fdim; ++j) { val[j] += regions.items[i].ee[j].val; err[j] += regions.items[i].ee[j].err; } destroy_region(®ions.items[i]); } /* printf("regions.nalloc = %d\n", regions.nalloc); */ free(ee); heap_free(®ions); free(R); return SUCCESS; bad: free(ee); heap_free(®ions); free(R); return FAILURE; } static int cubature(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err, int parallel) { rule *r; hypercube h; int status; unsigned i; if (fdim == 0) /* nothing to do */ return SUCCESS; if (dim == 0) { /* trivial integration */ if (f(0, 1, xmin, fdata, fdim, val)) return FAILURE; for (i = 0; i < fdim; ++i) err[i] = 0; return SUCCESS; } r = dim == 1 ? make_rule15gauss(dim, fdim) : make_rule75genzmalik(dim, fdim); if (!r) { for (i = 0; i < fdim; ++i) { val[i] = 0; err[i] = HUGE_VAL; } return FAILURE; } h = make_hypercube_range(dim, xmin, xmax); status = !h.data ? FAILURE : rulecubature(r, fdim, f, fdata, &h, maxEval, reqAbsError, reqRelError, norm, val, err, parallel); destroy_hypercube(&h); destroy_rule(r); return status; } int hcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { return cubature(fdim, f, fdata, dim, xmin, xmax, maxEval, reqAbsError, reqRelError, norm, val, err, 1); } #include "vwrapper.h" int hcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { int ret; fv_data d; if (fdim == 0) return SUCCESS; /* nothing to do */ d.f = f; d.fdata = fdata; ret = cubature(fdim, fv, &d, dim, xmin, xmax, maxEval, reqAbsError, reqRelError, norm, val, err, 0); return ret; } /***************************************************************************/ cubature/src/cubature-1.0.2/Makefile0000644000176200001440000000034213133476200016557 0ustar liggesusers# NOT PART OF cubature-1.0.2 # # Build the library. # libcubature.a: hcubature.o pcubature.o ${AR} cr libcubature.a hcubature.o pcubature.o # # To clean things up. # clean: rm -f *.o rm -f libcubature.a cubature.ts cubature/src/cubature-1.0.2/COPYING0000644000176200001440000010451313133172427016163 0ustar liggesusers GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. 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Copyright (C) This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: Copyright (C) This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an "about box". You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see . The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read . cubature/src/cubature-1.0.2/NEWS0000644000176200001440000000120713133172427015623 0ustar liggesusersVersion 1.0.2 * Fix memory leak in hcubature on BSD (and MacOS) systems. Thanks to Nicolas Tessore (@ntessore) for the bug report. Version 1.0.1 * cubature.h header now includes , to make sure size_t is defined Version 1.0 * Many API changes compared to pre-1.0 versions: - rename adapt_integrate -> hcubature - integrand now returns int to signal errors - error_norm argument for vector-valued integrands - maxeval and npt args are now size_t, not int * New pcubature routines for p-adaptive (Clenshaw-Curtis) integration. * Split test.c and some #include stuff from hcubature.c (to share with pcubature.c). cubature/src/cubature-1.0.2/clencurt_gen.c0000644000176200001440000001354413133476200017743 0ustar liggesusers/* * Copyright (c) 2005-2013 Steven G. Johnson * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * */ /* This stand-alone program, which should be compiled and linked against FFTW (www.fftw.org) version 3 or later, is used to generate the clencurt.h file for pcubature.c. You only need to run it if you want to do p-adaptive cubature with more than 8193 points per dimension. See the README file for more information. */ #include #include #include extern long double cosl(long double x); /* Program to generate tables of precomputed points and weights for Clenshaw-Curtis quadrature on an interval [-1, 1] with 3, 5, 9, ..., 2^(m+1)+1, ..., 2^(M+1)+1 points up to some given M. Because the quadrature rules are mirror-symmetric, we only need to store 2^m+1 weights for each rule. Furthermore, the rules are nested, so we only need to store the points for the M rule and the points for the other rules are a subset. We store the points and weights in a permuted order P corresponding to a usage where we first evaluate m=0, then m=1, etc. until it is converged. In particular, for the m rule (2^m+1 weights w[j], j=0,1,...,2^m), the corresponding points are x[j] = +/- cos(pi * j / 2^(m+1)) (Note that for x[2^m] = 0; this point must be specially handled so that it is not counted twice.) So, we precompute an array clencurt_x of length 2^M storing clencurt_x[j] = cos(pi * P_M(j) / 2^(M+1)) for j = 0,1,...,2^M-1. Then, for a given rule m, we use x[P_m(j)] = clencurt_x[j] for j = 0,1,...,2^m-1 and x=0 for j = 2^m. P_m is the permutation P_0(j) = j P_m(j) = 2 * P_{m-1}(j) if j < 2^(m-1) 2 * (j - 2^(m-1)) + 1 otherwise The 2^m+1 weights w are stored for m=0,1,..,M in the same order in an array clencurt_w of length M+2^(M+1), in order of m. So, the weights for a given m start at clencurt_w + (m + 2^m - 1), in the same order as clencurt_x except that it starts with the weight for x=0. */ static int P(int m, int j) { if (m == 0) return j; else if (j < (1<<(m-1))) return 2 * P(m-1,j); else return 2 * (j - (1<<(m-1))) + 1; } /***************************************************************************/ /* The general principle is this: in Fejer and Clenshaw-Curtis quadrature, we take our function f(x) evaluated at f(cos(theta)), discretize it in theta to a vector f of points, compute the DCT, then multiply by some coefficients c (the integrals of cos(theta) * sin(theta)). In matrix form, given the DCT matrix D, this is: c' * D * f = (D' * c)' * f = w' * f where w is the vector of weights for each function value. It is obviously much nicer to precompute w if we are going to be integrating many functions. Since w = D' * c, and the transpose D' of a DCT is another DCT (e.g. the transpose of DCT-I is a DCT-I, modulo some rescaling in FFTW's normalization), to compute w is just another DCT. There is an additional wrinkle, because the odd-frequency DCT components of f integrate to zero, so every other entry in c is zero. We can take advantage of this in computing w, because we can essentially do a radix-2 step in the DCT where one of the two subtransforms is zero. Therefore, for 2n+1 inputs, we only need to do a DCT of size n+1, and the weights w are a nice symmetric function. The weights are for integration of functions on (-1,1). */ void clencurt_weights(int n, long double *w) { fftwl_plan p; int j; long double scale = 1.0 / n; p = fftwl_plan_r2r_1d(n+1, w, w, FFTW_REDFT00, FFTW_ESTIMATE); for (j = 0; j <= n; ++j) w[j] = scale / (1 - 4*j*j); fftwl_execute(p); w[0] *= 0.5; fftwl_destroy_plan(p); } /***************************************************************************/ #define KPI 3.1415926535897932384626433832795028841971L int main(int argc, char **argv) { int M = argc > 1 ? atoi(argv[1]) : 11; long double *w; int j, m; long double k; if (argc > 2 || M < 0) { fprintf(stderr, "clencurt_gen usage: clencurt_gen [M]\n"); return EXIT_FAILURE; } w = (long double *) fftwl_malloc(sizeof(long double) * ((1<<(M+2)) - 2)); if (!w) { fprintf(stderr, "clencurt_gen: out of memory\n"); return EXIT_FAILURE; } printf("/* AUTOMATICALLY GENERATED -- DO NOT EDIT */\n\n"); printf("static const int clencurt_M = %d;\n\n", M); printf("static const double clencurt_x[%d] = { /* length 2^M */\n", 1<` rather than `"cubature.h"`. cubature/src/cubature-1.0.2/converged.h0000644000176200001440000000453713133476200017256 0ustar liggesusers/* Body of convergence test, shared between hcubature.c and pcubature.c. We use an #include file because the two routines use somewhat different data structures, and define macros ERR(j) and VAL(j) to get the error and value estimates, respectively, for integrand j. */ { unsigned j; # define SQR(x) ((x) * (x)) switch (norm) { case ERROR_INDIVIDUAL: for (j = 0; j < fdim; ++j) if (ERR(j) > reqAbsError && ERR(j) > fabs(VAL(j))*reqRelError) return 0; return 1; case ERROR_PAIRED: for (j = 0; j+1 < fdim; j += 2) { double maxerr, serr, err, maxval, sval, val; /* scale to avoid overflow/underflow */ maxerr = ERR(j) > ERR(j+1) ? ERR(j) : ERR(j+1); maxval = VAL(j) > VAL(j+1) ? VAL(j) : VAL(j+1); serr = maxerr > 0 ? 1/maxerr : 1; sval = maxval > 0 ? 1/maxval : 1; err = sqrt(SQR(ERR(j)*serr) + SQR(ERR(j+1)*serr)) * maxerr; val = sqrt(SQR(VAL(j)*sval) + SQR(VAL(j+1)*sval)) * maxval; if (err > reqAbsError && err > val*reqRelError) return 0; } if (j < fdim) /* fdim is odd, do last dimension individually */ if (ERR(j) > reqAbsError && ERR(j) > fabs(VAL(j))*reqRelError) return 0; return 1; case ERROR_L1: { double err = 0, val = 0; for (j = 0; j < fdim; ++j) { err += ERR(j); val += fabs(VAL(j)); } return err <= reqAbsError || err <= val*reqRelError; } case ERROR_LINF: { double err = 0, val = 0; for (j = 0; j < fdim; ++j) { double absval = fabs(VAL(j)); if (ERR(j) > err) err = ERR(j); if (absval > val) val = absval; } return err <= reqAbsError || err <= val*reqRelError; } case ERROR_L2: { double maxerr = 0, maxval = 0, serr, sval, err = 0, val = 0; /* scale values by 1/max to avoid overflow/underflow */ for (j = 0; j < fdim; ++j) { double absval = fabs(VAL(j)); if (ERR(j) > maxerr) maxerr = ERR(j); if (absval > maxval) maxval = absval; } serr = maxerr > 0 ? 1/maxerr : 1; sval = maxval > 0 ? 1/maxval : 1; for (j = 0; j < fdim; ++j) { err += SQR(ERR(j) * serr); val += SQR(fabs(VAL(j)) * sval); } err = sqrt(err) * maxerr; val = sqrt(val) * maxval; return err <= reqAbsError || err <= val*reqRelError; } } return 1; /* unreachable */ } cubature/src/cubature-1.0.2/README0000644000176200001440000002771013133172427016013 0ustar liggesusers== Cubature Package == Multidimensional integration ("cubature") code by Steven G. Johnson , based in part on code from the GNU Scientific Library (GSL) by Brian Gough and others and from the HIntLib numeric-integration library by Rudolf Schuerer. Adaptive integration of either one integrand or a vector of integrands is supported. Both h-adaptive integration (partitioning the integration domain) and p-adaptive integration (increasing the order of the integration rule) are supported (the latter being often better for smooth functions in low dimensions). === Download and Copyright === The latest version can be downloaded from: http://ab-initio.mit.edu/cubature/ Copyright (c) 2005-2013 Steven G. Johnson This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Portions (see comments) based on HIntLib (also distributed under the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) Portions (see comments) based on GNU GSL (also distributed under the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. (http://www.gnu.org/software/gsl/) === h-adaptive cubature === The basic algorithm is based on the adaptive cubature described in A. C. Genz and A. A. Malik, "An adaptive algorithm for numeric integration over an N-dimensional rectangular region," J. Comput. Appl. Math. 6 (4), 295-302 (1980). and subsequently extended to integrating a vector of integrands in J. Berntsen, T. O. Espelid, and A. Genz, "An adaptive algorithm for the approximate calculation of multiple integrals," ACM Trans. Math. Soft. 17 (4), 437-451 (1991). Note, however, that we do not use any of code from the above authors (in part because their code is Fortran 77, but mostly because it is under the restrictive ACM copyright license). I did make use of some GPL code from Rudolf Schuerer's HIntLib and from the GNU Scientific Library as noted above, however. I am also grateful to Dmitry Turbiner , who implemented an initial prototype of the "vectorized" functionality for evaluating multiple points in a single call (as opposed to multiple functions in a single call). (Although Dmitry implemented a working version, I ended up re-implementing this feature from scratch as part of a larger code-cleanup, and in order to have a single code path for the vectorized and non-vectorized APIs.) I subsequently extended the "vectorized" interface to use an algorithm by Gladwell to increase the potential parallelism: I. Gladwell, "Vectorization of one dimensional quadrature codes," pp. 230--238 in _Numerical Integration. Recent Developments, Software and Applications_, G. Fairweather and P. M. Keast, eds., NATO ASI Series C203, Dordrecht (1987). as described in: J. M. Bull and T. L. Freeman, "Parallel globally adaptive algorithms for multi-dimensional integration," http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.6638 (1994). === p-adaptive cubature === The p-adaptive cubature routine uses a tensor product of Clenshaw-Curtis quadrature rules (using the embedded half-order rules for error estimation). The degree of the rules is doubled along each dimension (one dimension at a time) until convergence is achieved. This is often superior to h-adaptive cubature for smooth (especially analytic) integrands in low dimensions (particularly 1 or 2 dimensions), especially when high accuracy is needed. It is not usually a good idea in more than 3 dimensions. The quadrature weights and points are precomputed with a program clencurt_gen.c and stored in a file clencurt.h. The default clencurt.h file includes Clenshaw-Curtis rules up to order 2^20 (about 10^6) along each dimension, which should normally suffice for any smooth function that is not extremely oscillatory. (If not, pcubature will return a nonzero result, indicating failure.) If you need higher-order rules, you can either switch to the h-adaptive routine or recompile clencurt_gen (which requires FFTW, from www.fftw.org) and run it with a larger argument m > 19 (corresponding to degree 2^(m+1) rules). ----------------------------------------------------------------------- == Usage == You should compile either hcubature.c or pcubature.c (or both), depending on whether you are using h-adaptive or p-adaptive cubature (or both). Then link it with your program and #include the header file cubature.h. The central subroutine you will be calling is probably: int hcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, unsigned maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); for h-adaptive cubature, or pcubature (same arguments) for p-adaptive cubature. (See also the vectorized interface below.) This integrates a function F(x) returning a vector of FDIM integrands, where x is a DIM-dimensional vector ranging from XMIN to XMAX (i.e. in a hypercube XMIN[i] <= x[i] <= XMAX[i]). MAXEVAL specifies a maximum number of function evaluations (0 for no limit). Otherwise, the integration stops when the estimated |error| is less than REQABSERROR (the absolute error requested), or when the estimated |error| is less than REQRELERROR * |integral value| (the relative error requested). For vector-valued integrands (FDIM > 1), NORM specifies the norm that is used to measure the error and determine convergence properties. (The NORM argument is irrelevant for FDIM<=1 and is ignored.) Given vectors v and e of estimated integrals and errors therein, respectively, the NORM argument takes on one of the following enumerated constant values: * ERROR_L1, ERROR_L2, ERROR_LINF: the absolute error is measured as |e| and the relative error as |e|/|v|, where |...| is the L1, L2, or L-infinity norm, respectively. (|x| in the L1 norm is the sum of the absolute values of the components, in the L2 norm is the root mean square of the components, and in the L-infinity norm is the maximum absolute value of the components) * ERROR_INDIVIDUAL: Convergence is achieved only when each integrand (each component of v and e) individually satisfies the requested error tolerances. * ERROR_PAIRED: Like ERROR_INDIVIDUAL, except that the integrands are grouped into consecutive pairs, with the error tolerance applied in an L2 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 you only care about the error in the complex plane rather than the error in the real and imaginary parts separately. VAL and ERR are arrays of length FDIM, which upon return are the computed integral values and estimated errors, respectively. (The estimated errors are based on an embedded cubature rule of lower order; for smooth functions, this estimate is usually conservative.) The return value of hcubature is 0 on success and nonzero if there was an error (currently, only out-of-memory situations or when the integrand signals an error). For a nonzero return value, the contents of the VAL and ERR arrays are undefined. The integrand function F should be a function of the form: int F(unsigned ndim, const double *x, void *fdata, unsigned fdim, double *fval); Here, the input is an array X of length NDIM (the point to be evaluated), the output is an array FVAL of length FDIM (the vector of function values at the point X). The return value should be 0 on success or a nonzero value if an error occurred and the integration is to be terminated immediately (hcubature will then return a nonzero error code). The FDATA argument of F is equal to the FDATA argument passed to cubature -- this can be used by the caller to pass any additional information through to F as needed (rather than using global variables, which are not re-entrant). If F does not need any additional data, you can just pass FDATA = NULL and ignore the FDATA argument to F. ----------------------------------------------------------------------- == "Vectorized" interface == For parallelization and other purposes, it is useful to call a single integrand function with an array of points to evaluate, rather than a single point at a time. (e.g. you may wish to evaluate the different points in parallel.) This is accomplished by calling: int hcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, unsigned maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err); (and similarly for pcubature_v) where all the arguments are the same as above except that the integrand function F should now be a function of the form: int F(unsigned ndim, size_t npts, const double *x, void *fdata, unsigned fdim, double *fval); Now, x is not a single point, but an array of npts points, and upon return the values of all fdim integrands at all npts points should be stored in fval. In particular, x[i*ndim + j] is the j-th coordinate of the i-th point (i < npts and j < ndim), and the k-th function evaluation (k < fdim) for the i-th point is returned in fval[i*fdim + k]. Again, the return value should be 0 on success or nonzero to terminate the integration immediately (e.g. if an error occurred). The size of npts will vary with the dimensionality of the problem; higher-dimensional problems will have (exponentially) larger npts, allowing for the possibility of more parallelism. Also npts will vary between calls to your integrand---subsequent calls (if the first batch of points did not achieve the desired accuracy) will have larger and larger values of npts. ----------------------------------------------------------------------- Test cases: To compile a test case, just compile cubature.c while #defining TEST_INTEGRATOR, e.g. (on Unix or GNU/Linux) via: cc -DHCUBATURE -o htest hcubature.c test.c -lm cc -DPCUBATURE -o ptest pcubature.c test.c -lm The usage is then: ./htest ./ptest where = # dimensions, = relative tolerance, is 0-7 for one of eight possible test integrands (see below) and is the maximum # function evaluations (0 for none, the default). The different test integrands are: 0: a product of cosine functions 1: a Gaussian integral of exp(-x^2), remapped to [0,infinity) limits 2: volume of a hypersphere (integrating a discontinuous function!) 3: a simple polynomial (product of coordinates) 4: a Gaussian centered in the middle of the integration volume 5: a sum of two Gaussians 6: an example function by Tsuda, a product of terms with near poles 7: a test integrand by Morokoff and Caflisch, a simple product of dim-th roots of the coordinates (weakly singular at the boundary) For example: ./htest 3 1e-5 4 integrates the Gaussian function (4) to a desired relative error tolerance of 1e-5 in 3 dimensions. The output is: 3-dim integral, tolerance = 1e-05 integrand 4: integral = 1, est err = 9.99952e-06, true err = 2.54397e-08 #evals = 82203 Notice that it finds the integral after 82203 function evaluations with an estimated error of about 1e-5, but the true error (compared to the exact result) is much smaller (2.5e-8): the error estimation is typically conservative when applied to smooth functions like this. cubature/src/cubature-1.0.2/vwrapper.h0000644000176200001440000000100113133476200017127 0ustar liggesusers/* vectorized wrapper around non-vectorized integrands */ typedef struct fv_data_s { integrand f; void *fdata; } fv_data; static int fv(unsigned ndim, size_t npt, const double *x, void *d_, unsigned fdim, double *fval) { fv_data *d = (fv_data *) d_; integrand f = d->f; void *fdata = d->fdata; unsigned i; /* printf("npt = %u\n", npt); */ for (i = 0; i < npt; ++i) if (f(ndim, x + i*ndim, fdata, fdim, fval + i*fdim)) return FAILURE; return SUCCESS; } cubature/src/cubature-1.0.2/clencurt.h0000644000176200001440002337747213133476200017141 0ustar liggesusers/* AUTOMATICALLY GENERATED -- DO NOT EDIT */ static const int clencurt_M = 16; static const double clencurt_x[65536] = { /* length 2^M */ 1, 0.707106781186547524, 0.923879532511286756, 0.382683432365089772, 0.980785280403230449, 0.831469612302545237, 0.555570233019602225, 0.195090322016128268, 0.995184726672196886, 0.956940335732208865, 0.88192126434835503, 0.773010453362736961, 0.634393284163645498, 0.471396736825997649, 0.290284677254462368, 0.0980171403295606019, 0.998795456205172393, 0.989176509964780973, 0.970031253194543993, 0.941544065183020778, 0.903989293123443332, 0.85772861000027207, 0.80320753148064491, 0.740951125354959091, 0.671558954847018401, 0.595699304492433343, 0.514102744193221727, 0.427555093430282094, 0.336889853392220051, 0.24298017990326389, 0.146730474455361752, 0.0490676743274180143, 0.99969881869620422, 0.997290456678690216, 0.992479534598709998, 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0.0209229298156301767, 0.0208750033857799259, 0.0208270769079600372, 0.0207791503822806435, 0.0207312238088518769, 0.0206832971877838705, 0.0206353705191867567, 0.0205874438031706689, 0.0205395170398457397, 0.0204915902293221025, 0.0204436633717098903, 0.0203957364671192367, 0.0203478095156602749, 0.0202998825174431387, 0.0202519554725779616, 0.0202040283811748776, 0.0201561012433440202, 0.0201081740591955237, 0.0200602468288395222, 0.0200123195523861498, 0.0199643922299455408, 0.0199164648616278296, 0.0198685374475431509, 0.0198206099878016391, 0.0197726824825134291, 0.0197247549317886556, 0.0196768273357374536, 0.0196288996944699581, 0.0195809720080963044, 0.0195330442767266276, 0.0194851165004710631, 0.0194371886794397463, 0.0193892608137428129, 0.0193413329034903985, 0.0192934049487926388, 0.0192454769497596697, 0.0191975489065016272, 0.0191496208191286474, 0.0191016926877508664, 0.0190537645124784205, 0.0190058362934214461, 0.0189579080306900796, 0.0189099797243944578, 0.018862051374644717, 0.0188141229815509943, 0.0187661945452234264, 0.0187182660657721504, 0.0186703375433073033, 0.0186224089779390222, 0.0185744803697774446, 0.0185265517189327076, 0.018478623025514949, 0.0184306942896343059, 0.0183827655114009164, 0.018334836690924918, 0.0182869078283164487, 0.0182389789236856463, 0.0181910499771426491, 0.0181431209887975949, 0.0180951919587606224, 0.0180472628871418694, 0.0179993337740514748, 0.0179514046195995768, 0.0179034754238963144, 0.0178555461870518258, 0.0178076169091762503, 0.0177596875903797265, 0.0177117582307723937, 0.0176638288304643906, 0.0176158993895658567, 0.0175679699081869312, 0.0175200403864377535, 0.0174721108244284629, 0.0174241812222691992, 0.0173762515800701019, 0.0173283218979413107, 0.0172803921759929657, 0.0172324624143352065, 0.0171845326130781735, 0.0171366027723320064, 0.0170886728922068457, 0.0170407429728128315, 0.0169928130142601044, 0.0169448830166588046, 0.016896952980119073, 0.0168490229047510498, 0.0168010927906648762, 0.0167531626379706928, 0.0167052324467786405, 0.0166573022171988604, 0.0166093719493414936, 0.0165614416433166811, 0.0165135112992345644, 0.0164655809172052847, 0.0164176504973389836, 0.0163697200397458025, 0.0163217895445358832, 0.0162738590118193671, 0.0162259284417063962, 0.0161779978343071124, 0.0161300671897316576, 0.0160821365080901738, 0.0160342057894928033, 0.0159862750340496881, 0.0159383442418709706, 0.0158904134130667933, 0.0158424825477472985, 0.0157945516460226289, 0.015746620708002927, 0.0156986897337983358, 0.0156507587235189977, 0.0156028276772750559, 0.0155548965951766532, 0.0155069654773339328, 0.0154590343238570377, 0.0154111031348561112, 0.0153631719104412965, 0.0153152406507227372, 0.0152673093558105766, 0.0152193780258149582, 0.0151714466608460257, 0.0151235152610139229, 0.0150755838264287934, 0.0150276523572007812, 0.0149797208534400301, 0.0149317893152566843, 0.0148838577427608878, 0.0148359261360627848, 0.0147879944952725195, 0.0147400628205002365, 0.0146921311118560798, 0.0146441993694501942, 0.0145962675933927242, 0.0145483357837938143, 0.0145004039407636096, 0.0144524720644122545, 0.0144045401548498941, 0.0143566082121866732, 0.0143086762365327371, 0.0142607442279982306, 0.0142128121866932992, 0.0141648801127280878, 0.014116948006212742, 0.0140690158672574071, 0.0140210836959722287, 0.0139731514924673522, 0.0139252192568529233, 0.0138772869892390877, 0.0138293546897359912, 0.0137814223584537796, 0.013733489995502599, 0.0136855576009925951, 0.0136376251750339143, 0.0135896927177367024, 0.0135417602292111059, 0.0134938277095672709, 0.0134458951589153439, 0.0133979625773654712, 0.0133500299650277995, 0.0133020973220124751, 0.0132541646484296449, 0.0132062319443894554, 0.0131582992100020534, 0.013110366445377586, 0.0130624336506261998, 0.0130145008258580421, 0.0129665679711832596, 0.0129186350867119998, 0.0128707021725544096, 0.0128227692288206365, 0.0127748362556208276, 0.0127269032530651305, 0.0126789702212636925, 0.0126310371603266614, 0.0125831040703641844, 0.0125351709514864096, 0.0124872378038034844, 0.0124393046274255568, 0.0123913714224627745, 0.0123434381890252857, 0.0122955049272232381, 0.0122475716371667799, 0.0121996383189660592, 0.0121517049727312243, 0.0121037715985724233, 0.0120558381965998047, 0.0120079047669235167, 0.0119599713096537079, 0.0119120378249005266, 0.0118641043127741217, 0.0118161707733846415, 0.0117682372068422349, 0.0117203036132570507, 0.0116723699927392375, 0.0116244363453989445, 0.0115765026713463204, 0.0115285689706915144, 0.0114806352435446754, 0.0114327014900159526, 0.0113847677102154952, 0.0113368339042534525, 0.0112889000722399737, 0.0112409662142852083, 0.0111930323304993056, 0.0111450984209924153, 0.0110971644858746867, 0.0110492305252562696, 0.0110012965392473135, 0.0109533625279579683, 0.0109054284914983837, 0.0108574944299787096, 0.0108095603435090958, 0.0107616262321996924, 0.0107136920961606492, 0.0106657579355021166, 0.0106178237503342444, 0.010569889540767183, 0.0105219553069110824, 0.0104740210488760933, 0.0104260867667723656, 0.01037815246071005, 0.0103302181307992969, 0.0102822837771502568, 0.0102343493998730803, 0.0101864149990779179, 0.0101384805748749206, 0.0100905461273742387, 0.0100426116566860234, 0.00999467716292042515, 0.0099467426461875952, 0.00989880810659768424, 0.00985087354426084348, 0.00980293895928722375, 0.00975500435178697638, 0.00970706972187025231, 0.00965913506964720296, 0.00961120039522797938, 0.00956326569872273308, 0.00951533098024161522, 0.00946739623989477742, 0.00941946147779237093, 0.00937152669404454747, 0.00932359188876145839, 0.00927565706205325552, 0.0092277222140300903, 0.00917978734480211465, 0.00913185245447948012, 0.00908391754317233873, 0.00903598261099084211, 0.00898804765804514228, 0.0089401126844453913, 0.00889217769030174094, 0.00884424267572434347, 0.00879630764082335076, 0.00874837258570891516, 0.00870043751049118865, 0.00865250241528032368, 0.00860456730018647229, 0.00855663216531978704, 0.00850869701079042007, 0.00846076183670852401, 0.00841282664318425111, 0.00836489143032775408, 0.00831695619824918526, 0.00826902094705869744, 0.00822108567686644306, 0.00817315038778257501, 0.00812521507991724578, 0.00807727975338060837, 0.00802934440828281537, 0.00798140904473401984, 0.00793347366284437446, 0.00788553826272403238, 0.00783760284448314637, 0.00778966740823186965, 0.00774173195408035508, 0.00769379648213875597, 0.00764586099251722525, 0.0075979254853259162, 0.00754998996067498217, 0.00750205441867457619, 0.0074541188594348518, 0.0074061832830659621, 0.00735824768967806071, 0.00731031207938130083, 0.00726237645228583614, 0.00721444080850181991, 0.0071665051481394059, 0.00711856947130874748, 0.00707063377811999847, 0.0070226980686833123, 0.00697476234310884288, 0.00692682660150674372, 0.0068788908439871688, 0.0068309550706602717, 0.00678301928163620647, 0.00673508347702512677, 0.00668714765693718671, 0.00663921182148254003, 0.00659127597077134093, 0.00654334010491374319, 0.00649540422401990108, 0.00644746832819996846, 0.00639953241756409968, 0.00635159649222244866, 0.00630366055228516981, 0.00625572459786241712, 0.00620778862906434495, 0.00615985264600110771, 0.00611191664878285948, 0.00606398063751975484, 0.00601604461232194795, 0.00596810857329959344, 0.00592017252056284555, 0.00587223645422185895, 0.00582430037438678795, 0.0057763642811677873, 0.00572842817467501136, 0.00568049205501861494, 0.00563255592230875245, 0.00558461977665557878, 0.0055366836181692484, 0.00548874744695991624, 0.00544081126313773685, 0.0053928750668128652, 0.0053449388580954559, 0.00529700263709566399, 0.00524906640392364413, 0.00520113015868955142, 0.00515319390150354056, 0.00510525763247576672, 0.00505732135171638466, 0.00500938505933554959, 0.00496144875544341632, 0.00491351244015014013, 0.00486557611356587588, 0.00481763977580077878, 0.00476970342696500408, 0.00472176706716870671, 0.00467383069652204208, 0.00462589431513516517, 0.00457795792311823144, 0.00453002152058139593, 0.00448208510763481412, 0.00443414868438864112, 0.00438621225095303247, 0.00433827580743814329, 0.00429033935395412919, 0.00424240289061114535, 0.0041944664175193474, 0.00414652993478889056, 0.00409859344252993053, 0.00405065694085262257, 0.0040027204298671224, 0.00395478390968358535, 0.00390684738041216717, 0.00385891084216302323, 0.00381097429504630933, 0.00376303773917218087, 0.0037151011746507937, 0.00366716460159230326, 0.00361922802010686543, 0.0035712914303046357, 0.00352335483229576998, 0.0034754182261904238, 0.00342748161209875301, 0.00337954499013091348, 0.00333160836039706079, 0.00328367172300735095, 0.00323573507807193957, 0.0031877984257009827, 0.00313986176600463599, 0.00309192509909305551, 0.00304398842507639695, 0.00299605174406481641, 0.00294811505616846962, 0.00290017836149751272, 0.00285224166016210144, 0.00280430495227239197, 0.00275636823793854008, 0.00270843151727070198, 0.00266049479037903347, 0.00261255805737369079, 0.00256462131836482975, 0.00251668457346260665, 0.00246874782277717731, 0.00242081106641869806, 0.00237287430449732475, 0.00232493753712321371, 0.00227700076440652085, 0.00222906398645740251, 0.00218112720338601463, 0.00213319041530251357, 0.00208525362231705528, 0.00203731682453979606, 0.00198938002208089219, 0.00194144321505049966, 0.0018935064035587749, 0.00184556958771587392, 0.00179763276763195315, 0.00174969594341716863, 0.00170175911518167682, 0.00165382228303563376, 0.00160588544708919594, 0.00155794860745251942, 0.0015100117642357607, 0.00146207491754907585, 0.00141413806750262141, 0.00136620121420655345, 0.00131826435777102851, 0.0012703274983062027, 0.00122239063592223257, 0.00117445377072927423, 0.00112651690283748425, 0.00107858003235701877, 0.00103064315939803435, 0.000982706284070687145, 0.000934769406485133732, 0.000886832526751530269, 0.000838895644980033352, 0.000790958761280799146, 0.000743021875763984253, 0.000695084988539744848, 0.000647148099718237435, 0.000599211209409618519, 0.000551274317724044287, 0.000503337424771671359, 0.000455400530662655928, 0.00040746363550715462, 0.000359526739415323632, 0.000311589842497319596, 0.000263652944863298712, 0.000215716046623417615, 0.000167779147887832508, 0.000119842248766700031, 7.19053493701763875e-05, 2.39684498084182178e-05 }; static const double clencurt_w[131088] = { /* length M+2^(M+1) */ /* m = 0: */ 1.33333333333333333, 0.333333333333333333, /* m = 1: */ 0.8, 0.0666666666666666667, 0.533333333333333333, /* m = 2: */ 0.393650793650793651, 0.015873015873015873, 0.279365079365079365, 0.146218649216018155, 0.361717858720489781, /* m = 3: */ 0.196410125821890528, 0.00392156862745098039, 0.138956468368233074, 0.0754823315431518344, 0.181473784236493357, 0.0373687028372056103, 0.10890555258189093, 0.163172664281703303, 0.192513864612925647, /* m = 4: */ 0.0981785777817682968, 0.000977517106549364616, 0.0694275756304354509, 0.037594341914047206, 0.0907061128677209987, 0.0192342451326811492, 0.0545550163039803104, 0.0816348176549385102, 0.0962923259454881792, 0.0093931979629550147, 0.0284579166772336901, 0.0462627628377517495, 0.0622721095452940046, 0.0758838004413884705, 0.0865775384418274354, 0.0939432444387687357, 0.0976981882080555818, /* m = 5: */ 0.0490876235149424558, 0.0002442002442002442, 0.0347104981809251143, 0.0187865297417957835, 0.0453511095516606722, 0.00958233879528379039, 0.0272722571414683869, 0.0408150134003578338, 0.0481444325725122034, 0.00483146544879091264, 0.0142520604323519968, 0.0231406949343581985, 0.0311412971040676245, 0.0379454599212848171, 0.0432915149616908293, 0.0469739590466141487, 0.0488512566430660937, 0.00235149067531170332, 0.00719269316173611402, 0.0119233947142127716, 0.0165349876572895897, 0.0209862744297374338, 0.0252350649817547659, 0.0292406531974683377, 0.03296454656997633, 0.0363709202866391831, 0.0394269887129560998, 0.042103331111418102, 0.0443741792392573158, 0.0462176675109255768, 0.047616044585250193, 0.0485558448571410527, 0.0490280184310255529, /* m = 6: */ 0.0245437075055141826, 6.10388817676860156e-05, 0.0173550412541139496, 0.00939256583934814871, 0.0226754327045667172, 0.00478862143341336762, 0.0136357932129377205, 0.0204073561200386786, 0.0240721079232785, 0.00240715327877140915, 0.00712483332325489785, 0.0115698834829084997, 0.0155703907389942596, 0.0189725558706794843, 0.0216456235671288244, 0.0234868657119316351, 0.0244255230615670869, 0.00120930061875273991, 0.00360197835812614147, 0.00596388034730799521, 0.00826865154203087108, 0.0104938620257689212, 0.0126180359797774327, 0.0146207025463435021, 0.0164825695622037791, 0.0181857037792633948, 0.0197137018370015572, 0.0210518475900201113, 0.0221872535589719509, 0.0231089849162740717, 0.0238081647302425897, 0.0242780594207574592, 0.0245141435888156829, 0.000588072153828697545, 0.0018030812669536236, 0.00300345869904497128, 0.00419553798718534675, 0.00537724746840184621, 0.00654590843862298929, 0.00769875778896082811, 0.00883303867470133581, 0.00994602784923457905, 0.0110350487742725418, 0.0120974805280716411, 0.0131307651669397463, 0.0141324143785309413, 0.0151000157247926678, 0.0160312385874505792, 0.0169238398584649937, 0.0177756693887528, 0.0185846751956690866, 0.0193489084239245184, 0.020066528051983576, 0.0207358053349014782, 0.0213551279742597073, 0.0219230040059875689, 0.0224380653972263018, 0.0228990713439060588, 0.0233049112613114327, 0.0236546074605776652, 0.023947317504769015, 0.0241823362389314757, 0.0243590974892764318, 0.0244771754274344428, 0.0245362855965149547, /* m = 7: */ 0.0122718472343792895, 1.52590218966964215e-05, 0.00867750760083810363, 0.00469623861722940691, 0.0113377087165893661, 0.00239414280403931247, 0.00681787552618365369, 0.0102036686348348107, 0.01203604718557155, 0.00120294646714375788, 0.00356233997347505809, 0.00578491249038589019, 0.00778517919237115442, 0.00948626703268709977, 0.0108228034048098232, 0.0117434257383023345, 0.0122127549492255464, 0.000602512672975876653, 0.00180069673385486407, 0.00298183115090367422, 0.00413426869582077584, 0.00524689548303089543, 0.0063089933825440137, 0.00731033293112933667, 0.00824127034187625554, 0.00909284002451894917, 0.00985684081921393574, 0.0105259149374630678, 0.0110936188044194495, 0.0115544851060064148, 0.0119040754381081509, 0.0121390230486920665, 0.0122570652603110472, 0.000302414838661812207, 0.000902939741087453963, 0.0015022664164212447, 0.002098049694820088, 0.00268879607402742587, 0.0032730710274964018, 0.00384946355518490976, 0.00441658375488667577, 0.00497306478685625395, 0.00551756573726307471, 0.00604877468922575561, 0.0065654118160297822, 0.00706623243296879161, 0.00755002998009127418, 0.00801563892042517457, 0.00846193754306058625, 0.00888785066261381639, 0.00929235220770270495, 0.00967446769173887088, 0.0100332765598313116, 0.010367914406002221, 0.0106775750552874484, 0.010961512505653061, 0.0112190427250168183, 0.0114495452990242433, 0.011652464925595819, 0.0118273127526355869, 0.0119736675556722427, 0.0120911767525914032, 0.0121795572530115018, 0.0122385961402550421, 0.0122681511842708724, 0.000147030506832164583, 0.000451077372229244745, 0.000752281785192075262, 0.00105276786505253903, 0.00135255443179264086, 0.00165150284956579013, 0.00194944607190546715, 0.00224620972036610005, 0.00254161735307662264, 0.00283549220412886478, 0.00312765790293549498, 0.00341793884114908965, 0.00370616040049264394, 0.00399214911961078728, 0.004275732831708586, 0.00455674078702335156, 0.00483500376677199114, 0.00511035419189020943, 0.00538262622829102142, 0.00565165588957047701, 0.00591728113766767264, 0.00617934198175585931, 0.00643768057551113637, 0.00669214131282928684, 0.00694257092201595646, 0.00718881855844773661, 0.00743073589568460597, 0.00766817721500350575, 0.00790099949331615045, 0.00812906248943005855, 0.00835222882860933065, 0.00857036408539034803, 0.00878333686460695575, 0.00899101888057959938, 0.00919328503442315097, 0.00939001348942868645, 0.00958108574447519333, 0.00976638670542804712, 0.00994580475448206433, 0.0101192318174079948, 0.0102865634286624412, 0.0104476987943223746, 0.0106025408528066438, 0.0107509963333481484, 0.0108929758121816487, 0.0110283937664135255, 0.0111571686255411694, 0.0112792228205910675, 0.0113944828308460764, 0.0115028792281338011, 0.0116043467186494621, 0.0116988241822881031, 0.0117862547094624871, 0.0118665856353845362, 0.0119397685717896904, 0.012005759436085101, 0.0120645184779041161, 0.0121160103030510822, 0.0121602038948220479, 0.0121970726326885401, 0.0122265943083331659, 0.0122487511390273882, 0.0122635297783434225, 0.0122709213241938074, /* m = 8: */ 0.00613592320974978177, 3.81471181759573962e-06, 0.00433875298570438529, 0.00234811652971850094, 0.00566885388096578707, 0.00119706075108121382, 0.00340893644365496405, 0.00510183372812415509, 0.00601802316922848958, 0.000601431670945399972, 0.00178116516212281629, 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0.00306796157940925433, 9.53675225901819082e-07, 0.00216937644192350024, 0.00117405809102033294, 0.00283442691064830389, 0.000598529707307292322, 0.0017044681393325414, 0.00255091682722781769, 0.00300901155814108463, 0.000300713198517770007, 0.000890582279019380145, 0.00144622709189688665, 0.00194629422878071689, 0.00237156637467456469, 0.00270570055655371402, 0.00293585618430481857, 0.003053188505898008, 0.000150539243380972161, 0.000450163626509508183, 0.000745453917225398022, 0.00103356515752577883, 0.00131172261806940045, 0.00157724747894740859, 0.00182758258714816814, 0.00206031707740227148, 0.00227320958873988216, 0.00246420984959170787, 0.00263147842286544927, 0.00277340442066312602, 0.00288862101798071648, 0.00297601861596435403, 0.0030347555279455064, 0.00306426608533916945, 7.52974768597021835e-05, 0.000225693925292063497, 0.000375551490577499891, 0.000524504630059486425, 0.000672194261770552425, 0.000818264542834653271, 0.000962363563533659274, 0.00110414417148513244, 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65517 65519 65521 65523 65525 65527 65529 65531 65533 65535 */ cubature/src/cubature-1.0.2/test.c0000644000176200001440000002060713133476200016250 0ustar liggesusers/* Test program for hcubature/pcubature. * * Copyright (c) 2005-2013 Steven G. Johnson * * Portions (see comments) based on HIntLib (also distributed under * the GNU GPL, v2 or later), copyright (c) 2002-2005 Rudolf Schuerer. * (http://www.cosy.sbg.ac.at/~rschuer/hintlib/) * * Portions (see comments) based on GNU GSL (also distributed under * the GNU GPL, v2 or later), copyright (c) 1996-2000 Brian Gough. * (http://www.gnu.org/software/gsl/) * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * */ /* Usage: ./test where = # dimensions, = relative tolerance, is either 0/1/2 for the three test integrands (see below), and is the maximum # function evaluations (0 for none). Compile with -DSCUBATURE to test scubature instead of cubature. */ #include #include #include #include #define VERBOSE 0 #if defined(PCUBATURE) # define cubature pcubature #else # define cubature hcubature #endif int count = 0; unsigned integrand_fdim = 0; int *which_integrand = NULL; const double radius = 0.50124145262344534123412; /* random */ /* Simple constant function */ double fconst (double x[], size_t dim, void *params) { return 1; } /*** f0, f1, f2, and f3 are test functions from the Monte-Carlo integration routines in GSL 1.6 (monte/test.c). Copyright (c) 1996-2000 Michael Booth, GNU GPL. ****/ /* Simple product function */ double f0 (unsigned dim, const double *x, void *params) { double prod = 1.0; unsigned int i; for (i = 0; i < dim; ++i) prod *= 2.0 * x[i]; return prod; } #define K_2_SQRTPI 1.12837916709551257390 /* Gaussian centered at 1/2. */ double f1 (unsigned dim, const double *x, void *params) { double a = *(double *)params; double sum = 0.; unsigned int i; for (i = 0; i < dim; i++) { double dx = x[i] - 0.5; sum += dx * dx; } return (pow (K_2_SQRTPI / (2. * a), (double) dim) * exp (-sum / (a * a))); } /* double gaussian */ double f2 (unsigned dim, const double *x, void *params) { double a = *(double *)params; double sum1 = 0.; double sum2 = 0.; unsigned int i; for (i = 0; i < dim; i++) { double dx1 = x[i] - 1. / 3.; double dx2 = x[i] - 2. / 3.; sum1 += dx1 * dx1; sum2 += dx2 * dx2; } return 0.5 * pow (K_2_SQRTPI / (2. * a), dim) * (exp (-sum1 / (a * a)) + exp (-sum2 / (a * a))); } /* Tsuda's example */ double f3 (unsigned dim, const double *x, void *params) { double c = *(double *)params; double prod = 1.; unsigned int i; for (i = 0; i < dim; i++) prod *= c / (c + 1) * pow((c + 1) / (c + x[i]), 2.0); return prod; } /* 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. */ static double morokoff(unsigned dim, const double *x, void *params) { double p = 1.0 / dim; double prod = pow(1 + p, dim); unsigned int i; for (i = 0; i < dim; i++) prod *= pow(x[i], p); return prod; } /*** end of GSL test functions ***/ int f_test(unsigned dim, const double *x, void *data_, unsigned fdim, double *retval) { double val; unsigned i, j; ++count; (void) data_; /* not used */ for (j = 0; j < fdim; ++j) { double fdata = which_integrand[j] == 6 ? (1.0+sqrt (10.0))/9.0 : 0.1; switch (which_integrand[j]) { case 0: /* simple smooth (separable) objective: prod. cos(x[i]). */ val = 1; for (i = 0; i < dim; ++i) val *= cos(x[i]); break; case 1: { /* integral of exp(-x^2), rescaled to (0,infinity) limits */ double scale = 1.0; val = 0; for (i = 0; i < dim; ++i) { if (x[i] > 0) { double z = (1 - x[i]) / x[i]; val += z * z; scale *= K_2_SQRTPI / (x[i] * x[i]); } else { scale = 0; break; } } val = exp(-val) * scale; break; } case 2: /* discontinuous objective: volume of hypersphere */ val = 0; for (i = 0; i < dim; ++i) val += x[i] * x[i]; val = val < radius * radius; break; case 3: val = f0(dim, x, &fdata); break; case 4: val = f1(dim, x, &fdata); break; case 5: val = f2(dim, x, &fdata); break; case 6: val = f3(dim, x, &fdata); break; case 7: val = morokoff(dim, x, &fdata); break; default: fprintf(stderr, "unknown integrand %d\n", which_integrand[j]); exit(EXIT_FAILURE); } #if VERBOSE if (count < 100) { printf("%d: f(%g", count, x[0]); for (i = 1; i < dim; ++i) printf(", %g", x[i]); printf(") = %g\n", val); } #endif retval[j] = val; } return 0; } #define K_PI 3.14159265358979323846 /* surface area of n-dimensional unit hypersphere */ static double S(unsigned n) { double val; int fact = 1; if (n % 2 == 0) { /* n even */ val = 2 * pow(K_PI, n * 0.5); n = n / 2; while (n > 1) fact *= (n -= 1); val /= fact; } else { /* n odd */ val = (1 << (n/2 + 1)) * pow(K_PI, n/2); while (n > 2) fact *= (n -= 2); val /= fact; } return val; } static double exact_integral(int which, unsigned dim, const double *xmax) { unsigned i; double val; switch(which) { case 0: val = 1; for (i = 0; i < dim; ++i) val *= sin(xmax[i]); break; case 2: val = dim == 0 ? 1 : S(dim) * pow(radius * 0.5, dim) / dim; break; default: val = 1.0; } return val; } #include int main(int argc, char **argv) { double *xmin, *xmax; double tol, *val, *err; unsigned i, dim, maxEval; if (argc <= 1) { fprintf(stderr, "Usage: %s [dim] [reltol] [integrand] [maxeval]\n", argv[0]); return EXIT_FAILURE; } dim = argc > 1 ? atoi(argv[1]) : 2; tol = argc > 2 ? atof(argv[2]) : 1e-2; maxEval = argc > 4 ? atoi(argv[4]) : 0; /* parse: e.g. "x/y/z" is treated as fdim = 3, which_integrand={x,y,z} */ if (argc <= 3) { integrand_fdim = 1; which_integrand = (int *) malloc(sizeof(int) * integrand_fdim); which_integrand[0] = 0; /* default */ } else { unsigned j = 0; integrand_fdim = 1; for (i = 0; argv[3][i]; ++i) if (argv[3][i] == '/') ++integrand_fdim; if (!integrand_fdim) { fprintf(stderr, "invalid which_integrand \"%s\"", argv[3]); return EXIT_FAILURE; } which_integrand = (int *) malloc(sizeof(int) * integrand_fdim); which_integrand[0] = 0; for (i = 0; argv[3][i]; ++i) { if (argv[3][i] == '/') which_integrand[++j] = 0; else if (isdigit(argv[3][i])) which_integrand[j] = which_integrand[j]*10 + argv[3][i] - '0'; else { fprintf(stderr, "invalid which_integrand \"%s\"", argv[3]); return EXIT_FAILURE; } } } val = (double *) malloc(sizeof(double) * integrand_fdim); err = (double *) malloc(sizeof(double) * integrand_fdim); xmin = (double *) malloc(dim * sizeof(double)); xmax = (double *) malloc(dim * sizeof(double)); for (i = 0; i < dim; ++i) { xmin[i] = 0; xmax[i] = 1; } printf("%u-dim integral, tolerance = %g\n", dim, tol); cubature(integrand_fdim, f_test, NULL, dim, xmin, xmax, maxEval, 0, tol, ERROR_INDIVIDUAL, val, err); for (i = 0; i < integrand_fdim; ++i) { printf("integrand %d: integral = %0.11g, est err = %g, true err = %g\n", which_integrand[i], val[i], err[i], fabs(val[i] - exact_integral(which_integrand[i], dim, xmax))); } printf("#evals = %d\n", count); free(xmax); free(xmin); free(err); free(val); free(which_integrand); return EXIT_SUCCESS; } cubature/src/cubature-1.0.2/ChangeLog0000644000176200001440000002133013133172427016675 0ustar liggesusersFri Nov 7 09:10:17 EST 2014 stevenj@alum.mit.edu * silence compiler warning M ./hcubature.c +1 Fri Nov 7 09:08:21 EST 2014 stevenj@alum.mit.edu * fix memory leak in BSD realloc(p, 0); fix stevengj/Cubature.jl#7" M ./NEWS +5 M ./hcubature.c -1 +7 Thu May 23 15:07:44 EDT 2013 stevenj@alum.mit.edu * 1.0.1 version bump M ./NEWS +4 Thu May 23 15:05:51 EDT 2013 stevenj@alum.mit.edu * cubature.h should include stdlib.h to make sure it has the definition of size_t M ./cubature.h +2 Fri Mar 1 22:08:46 EST 2013 stevenj@alum.mit.edu * add version number 1.0 and NEWS M ./Makefile -8 +6 A ./NEWS Thu Feb 28 23:13:23 EST 2013 stevenj@alum.mit.edu * dylib rules M ./Makefile +14 Thu Feb 28 23:05:04 EST 2013 stevenj@alum.mit.edu * tarball directory no longer includes date M ./Makefile -1 +1 Thu Feb 28 22:49:41 EST 2013 stevenj@alum.mit.edu * dll tweak M ./Makefile -2 +2 Thu Feb 28 21:21:25 EST 2013 stevenj@alum.mit.edu * makefile updates M ./Makefile -1 +17 Thu Feb 28 16:07:47 EST 2013 stevenj@alum.mit.edu * whoops, fix convergence test for negative integrals M ./converged.h -2 +2 Wed Feb 27 17:33:07 EST 2013 stevenj@alum.mit.edu * comment fix M ./cubature.h -1 +1 Wed Feb 27 17:14:17 EST 2013 stevenj@alum.mit.edu * check for integrand error in 0-dimensional case M ./hcubature.c -1 +1 M ./pcubature.c -1 +1 Wed Feb 27 16:51:32 EST 2013 stevenj@alum.mit.edu * since I'm changing the API, take advantage and change a few parameters to size_t for 64-bit systems (although it still seems unlikely that > 2^32 points will ever be used for integration) M ./README -1 +1 M ./cubature.h -8 +8 M ./hcubature.c -17 +19 M ./pcubature.c -18 +22 M ./vwrapper.h -1 +1 Wed Feb 27 14:29:34 EST 2013 stevenj@alum.mit.edu * increase max clencurt rule (scientific applications can afford a few MB of lookup table, especially since we access it in a cache-friendly way) M ./Makefile -1 +1 M ./README -8 +9 Tue Feb 26 14:31:19 EST 2013 stevenj@alum.mit.edu * whoops, fix bug in 1d hcubature introduced by recent re-indexing M ./hcubature.c -16 +17 Tue Feb 26 13:49:53 EST 2013 stevenj@alum.mit.edu * rename adapt_integrate -> cubature M ./README -14 +14 M ./cubature.h -34 +32 M ./hcubature.c -24 +24 M ./pcubature.c -20 +20 M ./test.c -5 +5 Tue Feb 26 13:48:04 EST 2013 stevenj@alum.mit.edu * more updates for release M ./Makefile -3 +3 M ./README -4 +16 M ./clencurt_gen.c +26 M ./cubature.h -1 +1 M ./hcubature.c -1 +1 M ./pcubature.c +20 M ./test.c -2 +2 M ./vwrapper.h -1 +1 Tue Feb 26 13:17:23 EST 2013 stevenj@alum.mit.edu * converged.c -> converged.h ./converged.c -> ./converged.h M ./hcubature.c -1 +1 M ./pcubature.c -1 +1 Tue Feb 26 13:14:46 EST 2013 stevenj@alum.mit.edu * make hcubature and pcubature use the same vectorized format (transpose the former) M ./Makefile -2 +2 M ./README -3 +3 M ./hcubature.c -47 +25 M ./pcubature.c -16 +1 A ./vwrapper.h Tue Feb 26 12:35:47 EST 2013 stevenj@alum.mit.edu * updated docs M ./Makefile -4 +4 M ./README -45 +105 M ./hcubature.c +1 M ./pcubature.c +1 Tue Feb 26 10:40:21 EST 2013 stevenj@alum.mit.edu * rename adapt_integrate -> hadapt_integrate, support different error norms for vector integrands, support user termination of integration from the integrand function ./cubature.c -> ./hcubature.c M ./Makefile -8 +5 A ./converged.c M ./cubature.h -28 +46 M ./hcubature.c -32 +45 M ./pcubature.c -47 +56 M ./test.c -7 +8 Wed Feb 22 17:43:43 EST 2012 stevenj@alum.mit.edu * whoops, handle > MAXDIM integrands M ./pcubature.c -3 +6 Wed Feb 22 16:07:59 EST 2012 stevenj@alum.mit.edu * link -lm in pcubature test M ./Makefile -2 +4 Wed Feb 22 16:04:24 EST 2012 stevenj@alum.mit.edu * allow pcubature_v_buf to take m[] as argument M ./cubature.h +1 M ./pcubature.c -11 +15 Wed Feb 22 15:59:01 EST 2012 stevenj@alum.mit.edu * bug fix in pcubature M ./pcubature.c -1 +1 Mon Feb 13 13:12:01 EST 2012 stevenj@alum.mit.edu * bug fixes in pcubature, now seems to work in >1d M ./pcubature.c -6 +6 Sun Feb 12 21:57:31 EST 2012 stevenj@alum.mit.edu * initial stab at C-C pcubature (not working in > 1d yet) M ./Makefile -2 +16 A ./clencurt_gen.c M ./cubature.h +18 A ./pcubature.c M ./test.c -1 +3 Sun Oct 31 19:35:41 EDT 2010 stevenj@alum.mit.edu * one byte per j (4 bits is only enough for j < 16, not 32, duh, and that's too small); another attempt at an error estimate (still not right) M ./scubature.c -156 +154 Sun Oct 31 14:36:20 EDT 2010 stevenj@alum.mit.edu * turn off VERBOSE, don't evaluate endpoints in singular case M ./test.c -4 +10 Sun Oct 31 14:35:29 EDT 2010 stevenj@alum.mit.edu * bug fixes in Jp_set and evaluation count M ./scubature.c -10 +9 Sun Oct 31 13:54:48 EDT 2010 stevenj@alum.mit.edu * eval missing J -- tentatively looking good in > 1d! M ./scubature.c -1 +4 Sun Oct 31 13:50:32 EDT 2010 stevenj@alum.mit.edu * some bugfixes, > 1d sort of works, but is not yet doing all the J's it should M ./scubature.c -5 +5 Sun Oct 31 13:44:57 EDT 2010 stevenj@alum.mit.edu * various bugfixes in scubature; 1d case now seems to work, > 1d still untested M ./scubature.c -12 +30 M ./test.c -1 +9 Sun Oct 31 12:27:03 EDT 2010 stevenj@alum.mit.edu * separate test program from cubature.c, add scubature API (untested) M ./Makefile -3 +11 M ./cubature.c -268 M ./cubature.h +7 M ./scubature.c -36 +108 A ./test.c Sun Oct 31 11:40:53 EDT 2010 stevenj@alum.mit.edu * always set N = |M|_\infty M ./scubature.c -7 +7 Sun Oct 31 11:38:29 EDT 2010 stevenj@alum.mit.edu * make redblack.c a header file so users won't try to compile it alone ./redblack.c -> ./redblack.h M ./scubature.c -1 +1 Sun Oct 31 03:27:38 EDT 2010 stevenj@alum.mit.edu * first stab at sparse-grid cubature (not yet finishedd, lonly 1d C-C part is tested) A ./redblack.c A ./scubature.c Mon Oct 18 00:35:54 EDT 2010 stevenj@alum.mit.edu * updated TODO etc. M ./cubature.c -6 +6 Sun Oct 17 22:10:01 EDT 2010 stevenj@alum.mit.edu * make sure it compiles in pedantic ANSI mode M ./Makefile -1 +1 M ./cubature.c -6 +10 Sun Oct 17 20:58:49 EDT 2010 stevenj@alum.mit.edu * more parallelism in adapt_integrate_v, using an algorithm by Gladstone M ./README -1 +19 M ./cubature.c -88 +152 Sun Oct 17 01:42:50 EDT 2010 stevenj@alum.mit.edu * spelling M ./cubature.c -1 +1 Sat Oct 16 22:24:45 EDT 2010 stevenj@alum.mit.edu * added ChangeLog to dist M ./Makefile -2 +7 Sat Oct 16 22:15:22 EDT 2010 stevenj@alum.mit.edu * use SUCCESS/FAILURE return codes consistently M ./cubature.c -42 +42 Sat Oct 16 21:59:16 EDT 2010 stevenj@alum.mit.edu * check for more malloc failures M ./cubature.c -19 +22 Sat Oct 16 21:50:10 EDT 2010 stevenj@alum.mit.edu * expose a bit more parallelism by evaluating two regions at once (thanks to Alejandro Rodriguez for the suggestion); also, more extensive checking for out-of-memory situations M ./cubature.c -177 +261 Tue Oct 12 01:05:33 EDT 2010 stevenj@alum.mit.edu * correct error return val for 0-dimensional integral with multiple integrands M ./cubature.c -3 +3 Mon Oct 11 22:03:12 EDT 2010 stevenj@alum.mit.edu * documentation of new feature, acknowledgements M ./README +37 M ./cubature.c +8 Mon Oct 11 21:50:53 EDT 2010 stevenj@alum.mit.edu * whoops, correct error estimation in 1d case; now matches old cod M ./cubature.c -5 +15 Mon Oct 11 21:30:07 EDT 2010 stevenj@alum.mit.edu * bug fix in new vectorized code; now gives identical results to before M ./cubature.c -6 +6 Mon Oct 11 19:02:22 EDT 2010 stevenj@alum.mit.edu * clean up where pts/vals are allocated M ./cubature.c -49 +57 Mon Oct 11 18:45:36 EDT 2010 stevenj@alum.mit.edu * first stab at vectorized version (thanks to Dmitry Turbiner for hacking an initial prototype) M ./Makefile -1 +3 M ./cubature.c -185 +225 M ./cubature.h +13 Fri Oct 2 19:35:53 EDT 2009 stevenj@alum.mit.edu * use .tgz, not .tar.gz M ./Makefile -1 +1 Fri Oct 2 19:25:53 EDT 2009 stevenj@alum.mit.edu * tweak M ./Makefile -1 +1 Fri Oct 2 19:25:26 EDT 2009 stevenj@alum.mit.edu * simple Makefile A ./Makefile Fri Oct 2 19:19:35 EDT 2009 stevenj@alum.mit.edu * document test program in README M ./README +43 Fri Oct 2 19:07:34 EDT 2009 stevenj@alum.mit.edu * added web link M ./README -4 +9 Fri Oct 2 19:05:01 EDT 2009 stevenj@alum.mit.edu * initial darcs import A ./COPYING A ./README A ./cubature.c A ./cubature.h cubature/src/cubature-1.0.2/pcubature.c0000644000176200001440000003240313133476200017260 0ustar liggesusers/* Adaptive multidimensional integration of a vector of integrands. * * Copyright (c) 2005-2013 Steven G. Johnson * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA * */ /* p-adaptive cubature (adaptive by increasing the degree of the cubature rule rather than subdividing the domain), using products of Clenshaw-Curtis rules. This algorithm may be superior to Genz-Malik for smooth integrands lacking strongly-localized features, in moderate dimensions. */ #include #include #include #include /* error return codes */ #define SUCCESS 0 #define FAILURE 1 /* pre-generated Clenshaw-Curtis rules and weights */ #include "clencurt.h" /* no point in supporting very high dimensional integrals here */ #define MAXDIM (20U) /***************************************************************************/ /* For adaptive cubature, thanks to the nesting of the C-C rules, we can re-use the values from coarser grids for finer grids, and the coarser grids are also used for error estimation. A grid is determined by an m[dim] array, where m[i] denotes 2^(m[i]+1)+1 points in the i-th dimension. */ /* cache of the values for the m[dim] grid. If mi < dim, then we only store the values corresponding to the difference between the m grid and the grid with m[mi] -> m[mi]-1. (m[mi]-1 == -1 corresponds to the trivial grid of one point in the center.) */ typedef struct cacheval_s { unsigned m[MAXDIM]; unsigned mi; double *val; } cacheval; /* array of ncache cachevals c[i] */ typedef struct valcache_s { size_t ncache; cacheval *c; } valcache; static void free_cachevals(valcache *v) { if (!v) return; if (v->c) { size_t i; for (i = 0; i < v->ncache; ++i) free(v->c[i].val); free(v->c); v->c = NULL; } v->ncache = 0; } /***************************************************************************/ /* recursive loop over all cubature points for the given (m,mi) cache entry: add each point to the buffer buf, evaluating all at once whenever the buffer is full or when we are done */ static int compute_cacheval(const unsigned *m, unsigned mi, double *val, size_t *vali, unsigned fdim, integrand_v f, void *fdata, unsigned dim, unsigned id, double *p, const double *xmin, const double *xmax, double *buf, size_t nbuf, size_t *ibuf) { if (id == dim) { /* add point to buffer of points */ memcpy(buf + (*ibuf)++ * dim, p, sizeof(double) * dim); if (*ibuf == nbuf) { /* flush buffer */ if (f(dim, nbuf, buf, fdata, fdim, val + *vali)) return FAILURE; *vali += *ibuf * fdim; *ibuf = 0; } } else { double c = (xmin[id] + xmax[id]) * 0.5; double r = (xmax[id] - xmin[id]) * 0.5; const double *x = clencurt_x + ((id == mi) ? (m[id] ? (1 << (m[id] - 1)) : 0) : 0); unsigned i, nx = (id == mi ? (m[id] ? (1 << (m[id] - 1)) : 1) : (1 << (m[id]))); if (id != mi) { p[id] = c; if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, dim, id + 1, p, xmin, xmax, buf, nbuf, ibuf)) return FAILURE; } for (i = 0; i < nx; ++i) { p[id] = c + r * x[i]; if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, dim, id + 1, p, xmin, xmax, buf, nbuf, ibuf)) return FAILURE; p[id] = c - r * x[i]; if (compute_cacheval(m, mi, val, vali, fdim, f, fdata, dim, id + 1, p, xmin, xmax, buf, nbuf, ibuf)) return FAILURE; } } return SUCCESS; } static size_t num_cacheval(const unsigned *m, unsigned mi, unsigned dim) { unsigned i; size_t nval = 1; for (i = 0; i < dim; ++i) { if (i == mi) nval *= m[i] == 0 ? 2 : (1 << (m[i])); else nval *= (1 << (m[i] + 1)) + 1; } return nval; } static int add_cacheval(valcache *vc, const unsigned *m, unsigned mi, unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, double *buf, size_t nbuf) { size_t ic = vc->ncache; size_t nval, vali = 0, ibuf = 0; double p[MAXDIM]; vc->c = (cacheval *) realloc(vc->c, sizeof(cacheval) * ++(vc->ncache)); if (!vc->c) return -1; vc->c[ic].mi = mi; memcpy(vc->c[ic].m, m, sizeof(unsigned) * dim); nval = fdim * num_cacheval(m, mi, dim); vc->c[ic].val = (double *) malloc(sizeof(double) * nval); if (!vc->c[ic].val) return FAILURE; if (compute_cacheval(m, mi, vc->c[ic].val, &vali, fdim, f, fdata, dim, 0, p, xmin, xmax, buf, nbuf, &ibuf)) return FAILURE; if (ibuf > 0) /* flush remaining buffer */ return f(dim, ibuf, buf, fdata, fdim, vc->c[ic].val + vali); return SUCCESS; } /***************************************************************************/ /* recursive loop to evaluate the integral contribution from the cache entry c, accumulating in val, for the given m[] except with m[md] -> m[md] - 1 if md < dim, using the cached values (cm,cmi,cval). id is the current loop dimension (from 0 to dim-1). */ static unsigned eval(const unsigned *cm, unsigned cmi, double *cval, const unsigned *m, unsigned md, unsigned fdim, unsigned dim, unsigned id, double weight, double *val) { size_t voff = 0; /* amount caller should offset cval array afterwards */ if (id == dim) { unsigned i; for (i = 0; i < fdim; ++i) val[i] += cval[i] * weight; voff = fdim; } else if (m[id] == 0 && id == md) /* using trivial rule for this dim */ { voff = eval(cm, cmi, cval, m, md, fdim, dim, id+1, weight*2, val); voff += fdim * (1 << cm[id]) * 2 * num_cacheval(cm + id+1, cmi - (id+1), dim - (id+1)); } else { unsigned i; unsigned mid = m[id] - (id == md); /* order of C-C rule */ const double *w = clencurt_w + mid + (1 << mid) - 1 + (id == cmi ? (cm[id] ? 1 + (1 << (cm[id]-1)) : 1) : 0); unsigned cnx = (id == cmi ? (cm[id] ? (1 << (cm[id]-1)) : 1) : (1 << (cm[id]))); unsigned nx = cm[id] <= mid ? cnx : (1 << mid); if (id != cmi) { voff = eval(cm, cmi, cval, m, md, fdim, dim, id + 1, weight * w[0], val); ++w; } for (i = 0; i < nx; ++i) { voff += eval(cm, cmi, cval + voff, m, md, fdim, dim, id + 1, weight * w[i], val); voff += eval(cm, cmi, cval + voff, m, md, fdim, dim, id + 1, weight * w[i], val); } voff += (cnx - nx) * fdim * 2 * num_cacheval(cm + id+1, cmi - (id+1), dim - (id+1)); } return voff; } /* loop over all cache entries that contribute to the integral, (with m[md] decremented by 1) */ static void evals(valcache vc, const unsigned *m, unsigned md, unsigned fdim, unsigned dim, double V, double *val) { size_t i; memset(val, 0, sizeof(double) * fdim); for (i = 0; i < vc.ncache; ++i) { if (vc.c[i].mi >= dim || vc.c[i].m[vc.c[i].mi] + (vc.c[i].mi == md) <= m[vc.c[i].mi]) eval(vc.c[i].m, vc.c[i].mi, vc.c[i].val, m, md, fdim, dim, 0, V, val); } } /* evaluate the integrals for the given m[] using the cached values in vc, storing the integrals in val[], the error estimate in err[], and the dimension to subdivide next (the largest error contribution) in *mi */ static void eval_integral(valcache vc, const unsigned *m, unsigned fdim, unsigned dim, double V, unsigned *mi, double *val, double *err, double *val1) { double maxerr = 0; unsigned i, j; evals(vc, m, dim, fdim, dim, V, val); /* error estimates along each dimension by comparing val with lower-order rule in that dimension; overall (conservative) error estimate from maximum error of lower-order rules. */ memset(err, 0, sizeof(double) * fdim); *mi = 0; for (i = 0; i < dim; ++i) { double emax = 0; evals(vc, m, i, fdim, dim, V, val1); for (j = 0; j < fdim; ++j) { double e = fabs(val[j] - val1[j]); if (e > emax) emax = e; if (e > err[j]) err[j] = e; } if (emax > maxerr) { maxerr = emax; *mi = i; } } /* printf("eval: %g +/- %g (dim %u)\n", val[0], err[0], *mi); */ } /***************************************************************************/ static int converged(unsigned fdim, const double *vals, const double *errs, double reqAbsError, double reqRelError, error_norm norm) #define ERR(j) errs[j] #define VAL(j) vals[j] #include "converged.h" /***************************************************************************/ /* Vectorized version with user-supplied buffer to store points and values. The buffer *buf should be of length *nbuf * dim on entry (these parameters are changed upon return to the final buffer and length that was used). The buffer length will be kept <= max(max_nbuf, 1) * dim. Also allows the caller to specify an array m[dim] of starting degrees for the rule, which upon return will hold the final degrees. The number of points in each dimension i is 2^(m[i]+1) + 1. */ int pcubature_v_buf(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, unsigned *m, double **buf, size_t *nbuf, size_t max_nbuf, double *val, double *err) { int ret = FAILURE; double V = 1; size_t numEval = 0, new_nbuf; unsigned i; valcache vc = {0, NULL}; double *val1 = NULL; if (fdim <= 1) norm = ERROR_INDIVIDUAL; /* norm is irrelevant */ #ifndef R_PACKAGE if (norm < 0 || norm > ERROR_LINF) return FAILURE; /* invalid norm */ #endif if (fdim == 0) return SUCCESS; /* nothing to do */ if (dim > MAXDIM) return FAILURE; /* unsupported */ if (dim == 0) { /* trivial case */ if (f(0, 1, xmin, fdata, fdim, val)) return FAILURE; for (i = 0; i < fdim; ++i) err[i] = 0; return SUCCESS; } for (i = 0; i < fdim; ++i) { val[i] = 0; err[i] = HUGE_VAL; } for (i = 0; i < dim; ++i) V *= (xmax[i] - xmin[i]) * 0.5; /* scale factor for C-C volume */ new_nbuf = num_cacheval(m, dim, dim); if (max_nbuf < 1) max_nbuf = 1; if (new_nbuf > max_nbuf) new_nbuf = max_nbuf; if (*nbuf < new_nbuf) { free(*buf); *buf = (double *) malloc(sizeof(double) * (*nbuf = new_nbuf) * dim); if (!*buf) goto done; } /* start by evaluating the m=0 cubature rule */ if (add_cacheval(&vc, m, dim, fdim, f, fdata, dim, xmin, xmax, *buf, *nbuf) != SUCCESS) goto done; val1 = (double *) malloc(sizeof(double) * fdim); while (1) { unsigned mi; eval_integral(vc, m, fdim, dim, V, &mi, val, err, val1); if (converged(fdim, val, err, reqAbsError, reqRelError, norm) || (numEval > maxEval && maxEval)) { ret = SUCCESS; goto done; } m[mi] += 1; if (m[mi] > clencurt_M) goto done; /* FAILURE */ new_nbuf = num_cacheval(m, mi, dim); if (new_nbuf > *nbuf && *nbuf < max_nbuf) { *nbuf = new_nbuf; if (*nbuf > max_nbuf) *nbuf = max_nbuf; free(*buf); *buf = (double *) malloc(sizeof(double) * *nbuf * dim); if (!*buf) goto done; /* FAILURE */ } if (add_cacheval(&vc, m, mi, fdim, f, fdata, dim, xmin, xmax, *buf, *nbuf) != SUCCESS) goto done; /* FAILURE */ numEval += new_nbuf; } done: free(val1); free_cachevals(&vc); return ret; } /***************************************************************************/ #define DEFAULT_MAX_NBUF (1U << 20) int pcubature_v(unsigned fdim, integrand_v f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { int ret; size_t nbuf = 0; unsigned m[MAXDIM]; double *buf = NULL; memset(m, 0, sizeof(unsigned) * dim); ret = pcubature_v_buf(fdim, f, fdata, dim, xmin, xmax, maxEval, reqAbsError, reqRelError, norm, m, &buf, &nbuf, DEFAULT_MAX_NBUF, val, err); free(buf); return ret; } #include "vwrapper.h" int pcubature(unsigned fdim, integrand f, void *fdata, unsigned dim, const double *xmin, const double *xmax, size_t maxEval, double reqAbsError, double reqRelError, error_norm norm, double *val, double *err) { int ret; size_t nbuf = 0; unsigned m[MAXDIM]; double *buf = NULL; fv_data d; d.f = f; d.fdata = fdata; memset(m, 0, sizeof(unsigned) * dim); ret = pcubature_v_buf( fdim, fv, &d, dim, xmin, xmax, maxEval, reqAbsError, reqRelError, norm, m, &buf, &nbuf, 16 /* max_nbuf > 0 to amortize function overhead */, val, err); free(buf); return ret; } cubature/src/Rcpp-cubature.cpp0000644000176200001440000001035213133476177016110 0ustar liggesusers// -*- mode: C++; c-indent-level: 4; c-basic-offset: 4; indent-tabs-mode: nil; -*- // Copyright (C) 2016 Balasubramanian Narasimhan // // We need both R and C/C++ interfaces! #include // need to include the main Rcpp header file only #include SEXP fun; /* The function itself */ int count; /* Count of function evaluations */ int fWrapper(unsigned ndim, const double *x, void *fdata, unsigned fdim, double *fval) { // Rcpp::Rcout<<"In Wrapper" < do not edit by hand // Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393 #include "../inst/include/cubature.h" #include using namespace Rcpp; // doHCubature Rcpp::List doHCubature(int fDim, SEXP f, Rcpp::NumericVector xLL, Rcpp::NumericVector xUL, int maxEval, double absErr, double tol, int vectorInterface, unsigned norm); RcppExport SEXP _cubature_doHCubature(SEXP fDimSEXP, SEXP fSEXP, SEXP xLLSEXP, SEXP xULSEXP, SEXP maxEvalSEXP, SEXP absErrSEXP, SEXP tolSEXP, SEXP vectorInterfaceSEXP, SEXP normSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< int >::type fDim(fDimSEXP); Rcpp::traits::input_parameter< SEXP >::type f(fSEXP); Rcpp::traits::input_parameter< Rcpp::NumericVector >::type xLL(xLLSEXP); Rcpp::traits::input_parameter< Rcpp::NumericVector >::type xUL(xULSEXP); Rcpp::traits::input_parameter< int >::type maxEval(maxEvalSEXP); Rcpp::traits::input_parameter< double >::type absErr(absErrSEXP); Rcpp::traits::input_parameter< double >::type tol(tolSEXP); Rcpp::traits::input_parameter< int >::type vectorInterface(vectorInterfaceSEXP); Rcpp::traits::input_parameter< unsigned >::type norm(normSEXP); rcpp_result_gen = Rcpp::wrap(doHCubature(fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm)); return rcpp_result_gen; END_RCPP } // doPCubature Rcpp::List doPCubature(int fDim, SEXP f, Rcpp::NumericVector xLL, Rcpp::NumericVector xUL, int maxEval, double absErr, double tol, int vectorInterface, unsigned norm); RcppExport SEXP _cubature_doPCubature(SEXP fDimSEXP, SEXP fSEXP, SEXP xLLSEXP, SEXP xULSEXP, SEXP maxEvalSEXP, SEXP absErrSEXP, SEXP tolSEXP, SEXP vectorInterfaceSEXP, SEXP normSEXP) { BEGIN_RCPP Rcpp::RObject rcpp_result_gen; Rcpp::RNGScope rcpp_rngScope_gen; Rcpp::traits::input_parameter< int >::type fDim(fDimSEXP); Rcpp::traits::input_parameter< SEXP >::type f(fSEXP); Rcpp::traits::input_parameter< Rcpp::NumericVector >::type xLL(xLLSEXP); Rcpp::traits::input_parameter< Rcpp::NumericVector >::type xUL(xULSEXP); Rcpp::traits::input_parameter< int >::type maxEval(maxEvalSEXP); Rcpp::traits::input_parameter< double >::type absErr(absErrSEXP); Rcpp::traits::input_parameter< double >::type tol(tolSEXP); Rcpp::traits::input_parameter< int >::type vectorInterface(vectorInterfaceSEXP); Rcpp::traits::input_parameter< unsigned >::type norm(normSEXP); rcpp_result_gen = Rcpp::wrap(doPCubature(fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm)); return rcpp_result_gen; END_RCPP } cubature/NAMESPACE0000644000176200001440000000022613133172427013306 0ustar liggesusers# Generated by roxygen2: do not edit by hand export(adaptIntegrate) export(hcubature) export(pcubature) importFrom(Rcpp,evalCpp) useDynLib(cubature) cubature/R/0000755000176200001440000000000013133475305012271 5ustar liggesuserscubature/R/hcubature.R0000644000176200001440000003642413133245617014410 0ustar liggesusers#' 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. #' #' 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 \code{adaptIntegrate} function #' is an alias for the underlying \code{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! #' #' @importFrom Rcpp evalCpp #' @aliases adaptIntegrate pcubature #' #' @param f The function (integrand) to be integrated #' @param lowerLimit The lower limit of integration, a vector for hypercubes #' @param upperLimit The upper limit of integration, a vector for hypercubes #' @param ... All other arguments passed to the function f #' @param tol The maximum tolerance, default 1e-5. #' @param fDim The dimension of the integrand, default 1, bears no relation to #' the dimension of the hypercube #' @param 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. #' @param absError The maximum absolute error tolerated #' @param doChecking A flag to be a bit anal about checking inputs to C #' routines. A FALSE value results in approximately 9 percent speed gain in our #' experiments. Your mileage will of course vary. Default value is FALSE. #' @param vectorInterface A flag that indicates whether to use the vector interface #' and is by default FALSE. See details below #' @param norm For vector-valued integrands, \code{norm} specifies the norm that is #' used to measure the error and determine convergence properties. See below. #' @return The returned value is a list of three items: \item{integral}{the #' value of the integral} \item{error}{the estimated relative error} #' \item{functionEvaluations}{the number of times the function was evaluated} #' \item{returnCode}{the actual integer return code of the C routine} #' #' @details #' #' The \code{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, \code{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 \code{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 \code{norm} argument is irrelevant for scalar integrands and is ignored. #' Given vectors \eqn{v} and \eqn{e} of estimated integrals and errors therein, #' respectively, the \code{norm} argument takes on one of the following values: #' \describe{ #' \item{\code{INDIVIDUAL}}{Convergence is achieved only when each integrand #' (each component of \eqn{v} and \eqn{e}) individually satisfies the requested #' error tolerances} #' \item{\code{L1}, \code{L2}, \code{LINF}}{The absolute error is measured as #' \eqn{|e|} and the relative error as \eqn{|e|/|v|}, where \eqn{|...|} is the #' \eqn{L_1}, \eqn{L_2}, or \eqn{\L_{\infty}} norm, respectively} #' \item{\code{PAIRED}}{Like \code{INDIVIDUAL}, except that the integrands are #' grouped into consecutive pairs, with the error tolerance applied in an #' \eqn{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 #' @keywords math #' @examples #' #' \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) #' } #' #' @export hcubature adaptIntegrate pcubature #' hcubature <- adaptIntegrate <- function(f, lowerLimit, upperLimit, ..., tol = 1e-5, fDim = 1, maxEval = 0, absError = 0, doChecking = FALSE, vectorInterface = FALSE, norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")) { NORM_CODES <- c(INDIVIDUAL = 0L, PAIRED = 1L, L2 = 2L, L1 = 3L, LINF = 4L) norm <- as.integer(NORM_CODES[match.arg(norm)]) nL = length(lowerLimit); nU = length(upperLimit) if (fDim <= 0 || nL <= 0 || nU <= 0) { stop("Both f and x must have dimension >= 1") } if (nL != nU) { stop("lowerLimit and upperLimit must have same length") } if (tol <= 0) { stop("tol should be positive!") } f <- match.fun(f) if (doChecking) { fnF <- function(x) { fx <- f(x, ...) if(!is.numeric(fx) || length(fx) != fDim) { cat("hcubature: Error in evaluation function f(x) for x = ", x, "\n") stop("hcubature: Result f(x) is not numeric or has wrong dimension") } fx } } else { fnF <- function(x) { f(x, ...) } } .Call("_cubature_doHCubature", as.integer(fDim), fnF, as.double(lowerLimit), as.double(upperLimit), as.integer(maxEval), as.double(absError), as.double(tol), as.integer(vectorInterface), norm, PACKAGE="cubature") } #' @rdname hcubature pcubature <- function(f, lowerLimit, upperLimit, ..., tol = 1e-5, fDim = 1, maxEval = 0, absError = 0, doChecking = FALSE, vectorInterface = FALSE, norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")) { NORM_CODES <- c(INDIVIDUAL = 0L, PAIRED = 1L, L2 = 2L, L1 = 3L, LINF = 4L) norm <- as.integer(NORM_CODES[match.arg(norm)]) nL = length(lowerLimit); nU = length(upperLimit) if (fDim <= 0 || nL <= 0 || nU <= 0) { stop("Both f and x must have dimension >= 1") } if (nL != nU) { stop("lowerLimit and upperLimit must have same length") } if (nL > 3) { warning("pcubature not recommended for dimensions > 3!") } if (tol <= 0) { stop("tol should be positive!") } f <- match.fun(f) if (doChecking) { fnF <- function(x) { fx <- f(x, ...) if(!is.numeric(fx) || length(fx) != fDim) { cat("pcubature: Error in evaluation function f(x) for x = ", x, "\n") stop("pcubature: Result f(x) is not numeric or has wrong dimension") } fx } } else { fnF <- function(x) { f(x, ...) } } .Call("_cubature_doPCubature", as.integer(fDim), fnF, as.double(lowerLimit), as.double(upperLimit), as.integer(maxEval), as.double(absError), as.double(tol), as.integer(vectorInterface), norm, PACKAGE="cubature") } cubature/R/cubature-package.R0000644000176200001440000000145513133475305015624 0ustar liggesusers #' Cubature is a package for adaptive multidimensional integration over #' hypercubes #' #' Cubature is a package for adaptive multidimensional integration over #' hypercubes. It is a wrapper around the pure C, GPLed implementation by #' Steven G. Johnson available. #' #' \tabular{ll}{ Package: \tab cubature\cr Type: \tab Package\cr Version: \tab #' 1.0\cr Date: \tab 2009-12-17\cr License: \tab GPL V2 or later\cr LazyLoad: #' \tab yes\cr } There is only one function in the package called #' \link[cubature]{adaptIntegrate}. #' #' @name cubature-package #' @useDynLib cubature #' @aliases cubature-package cubature #' @docType package #' @author C code by Steven G. Johnson, R by Balasubramanian Narasimhan #' #' Maintainer: Balasubramanian Narasimhan #' @keywords package NULL cubature/R/RcppExports.R0000644000176200001440000000105313133443711014700 0ustar liggesusers# Generated by using Rcpp::compileAttributes() -> do not edit by hand # Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393 doHCubature <- function(fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm) { .Call('_cubature_doHCubature', PACKAGE = 'cubature', fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm) } doPCubature <- function(fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm) { .Call('_cubature_doPCubature', PACKAGE = 'cubature', fDim, f, xLL, xUL, maxEval, absErr, tol, vectorInterface, norm) } cubature/vignettes/0000755000176200001440000000000013133476177014110 5ustar liggesuserscubature/vignettes/cubature.Rmd0000644000176200001440000003360313133475417016367 0ustar liggesusers--- title: "Cubature Vectorization Results" author: "Balasubramanian Narasimhan" date: '`r Sys.Date()`' output: html_document: fig_caption: yes theme: cerulean toc: yes toc_depth: 2 vignette: > %\VignetteIndexEntry{Cubature Vectorization Efficiency} %\VignetteEngine{knitr::rmarkdown} \usepackage[utf8]{inputenc} --- ```{r echo=FALSE} knitr::opts_chunk$set( message = FALSE, warning = FALSE, error = FALSE, tidy = FALSE, cache = FALSE ) ``` ## Introduction Starting with version 1.2, `cubature` now uses `Rcpp`. Also, version 1.3 uses the newer version (1.0.2) of Steven G. Johnson's `hcubature` and `pcubature` routines, including the vectorized interface. Per the documentation, use of `pcubature` is advisable only for smooth integrands in dimesions up to three at most. In fact, the `pcubature` routines perform significantly worse than the vectorized `hcubature` in inappropriate cases. So when in doubt, you are better off using `hcubature`. The main point of this note is to examine the difference vectorization makes. My recommendations are below in the summary section. ## A Timing Harness Our harness will provide timing results for `hcubature`, `pcubature` (where appropriate) and `R2Cuba` calls. We begin by creating a harness for these calls. ```{r} loadedSuggested <- c(benchr = FALSE, R2Cuba = FALSE) if (requireNamespace("benchr", quietly = TRUE)) { loadedSuggested["benchr"] <- TRUE } if (requireNamespace("R2Cuba", quietly = TRUE)) { loadedSuggested["R2Cuba"] <- TRUE } library(cubature) harness <- function(which = NULL, f, fv, lowerLimit, upperLimit, tol = 1e-3, times = 20, ...) { fns <- c(hc = "Non-vectorized Hcubature", hc.v = "Vectorized Hcubature", pc = "Non-vectorized Pcubature", pc.v = "Vectorized Pcubature") if (loadedSuggested["R2Cuba"]) { fns <- c(fns, cc = "R2Cuba::cuhre") cc <- function() R2Cuba::cuhre(ndim = ndim, ncomp = 1, integrand = f, lower = lowerLimit, upper = upperLimit, flags = list(verbose = 0, final = 1), rel.tol = tol, max.eval = 10^6, ...) } hc <- function() cubature::hcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) hc.v <- function() cubature::hcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) pc <- function() cubature::pcubature(f = f, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, ...) pc.v <- function() cubature::pcubature(f = fv, lowerLimit = lowerLimit, upperLimit = upperLimit, tol = tol, vectorInterface = TRUE, ...) ndim = length(lowerLimit) if (is.null(which)) { fnIndices <- seq_along(fns) } else { fnIndices <- na.omit(match(which, names(fns))) } fnList <- lapply(names(fns)[fnIndices], function(x) call(x)) if (loadedSuggested["benchr"]) { argList <- c(fnList, times = times, progress = FALSE) result <- do.call(benchr::benchmark, args = argList) d <- summary(result)[seq_along(fnIndices), ] d$expr <- fns[fnIndices] d } else { d <- data.frame(expr = names(fns)[fnIndices], timing = NA) } } ``` We reel off the timing runs. ## Example 1. ```{r} func <- function(x) sin(x[1]) * cos(x[2]) * exp(x[3]) func.v <- function(x) { matrix(apply(x, 2, function(z) sin(z[1]) * cos(z[2]) * exp(z[3])), ncol = ncol(x)) } d <- harness(f = func, fv = func.v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), tol = 1e-5, times = 100) knitr::kable(d, digits = 3, row.names = FALSE) ``` ## Multivariate Normal Using `cubature`, we evaluate $$ \int_R\phi(x)dx $$ where $\phi(x)$ is the three-dimensional multivariate normal density with mean 0, and variance $$ \Sigma = \left(\begin{array}{rrr} 1 &\frac{3}{5} &\frac{1}{3}\\ \frac{3}{5} &1 &\frac{11}{15}\\ \frac{1}{3} &\frac{11}{15} & 1 \end{array} \right) $$ and $R$ is $[-\frac{1}{2}, 1] \times [-\frac{1}{2}, 4] \times [-\frac{1}{2}, 2].$ We construct a scalar function (`my_dmvnorm`) and a vector analog (`my_dmvnorm_v`). First the functions. ```{r} 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 logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values)) my_dmvnorm <- function (x, mean, sigma, logdet) { x <- matrix(x, ncol = length(x)) distval <- stats::mahalanobis(x, center = mean, cov = sigma) exp(-(3 * log(2 * pi) + logdet + distval)/2) } my_dmvnorm_v <- function (x, mean, sigma, logdet) { distval <- stats::mahalanobis(t(x), center = mean, cov = sigma) exp(matrix(-(3 * log(2 * pi) + logdet + distval)/2, ncol = ncol(x))) } ``` Now the timing. ```{r} d <- harness(f = my_dmvnorm, fv = my_dmvnorm_v, lowerLimit = rep(-0.5, 3), upperLimit = c(1, 4, 2), tol = 1e-5, times = 10, mean = rep(0, m), sigma = sigma, logdet = logdet) knitr::kable(d, digits = 3) ``` The effect of vectorization is huge. So it makes sense for users to vectorize the integrands as much as possible for efficiency. Furthermore, for this particular example, we expect `mvtnorm::pmvnorm` to do pretty well since it is specialized for the multivariate normal. The good news is that the vectorized versions of `hcubature` and `pcubature` are quite competitive if you compare the table above to the one below. ```{r} library(mvtnorm) g1 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = Miwa(), abseps = 1e-5, releps = 1e-5) g2 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = GenzBretz(), abseps = 1e-5, releps = 1e-5) g3 <- function() mvtnorm::pmvnorm(lower = rep(-0.5, m), upper = c(1, 4, 2), mean = rep(0, m), corr = sigma, alg = TVPACK(), abseps = 1e-5, releps = 1e-5) knitr::kable(summary(benchr::benchmark(g1(), g2(), g3(), times = 20, progress = FALSE)), digits = 3, row.names = FALSE) ``` ## Product of cosines ```{r} testFn0 <- function(x) prod(cos(x)) testFn0_v <- function(x) matrix(apply(x, 2, function(z) prod(cos(z))), ncol = ncol(x)) d <- harness(f = testFn0, fv = testFn0_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 1000) knitr::kable(d, digits = 3) ``` ## Gaussian function ```{r} testFn1 <- function(x) { val <- sum(((1 - x) / x)^2) scale <- prod((2 / sqrt(pi)) / x^2) exp(-val) * scale } testFn1_v <- function(x) { val <- matrix(apply(x, 2, function(z) sum(((1 - z) / z)^2)), ncol(x)) scale <- matrix(apply(x, 2, function(z) prod((2 / sqrt(pi)) / z^2)), ncol(x)) exp(-val) * scale } d <- harness(f = testFn1, fv = testFn1_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 10) knitr::kable(d, digits = 3) ``` ## Discontinuous function ```{r} testFn2 <- function(x) { radius <- 0.50124145262344534123412 ifelse(sum(x * x) < radius * radius, 1, 0) } testFn2_v <- function(x) { radius <- 0.50124145262344534123412 matrix(apply(x, 2, function(z) ifelse(sum(z * z) < radius * radius, 1, 0)), ncol = ncol(x)) } d <- harness(which = c("hc", "hc.v", "cc"), f = testFn2, fv = testFn2_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 10) knitr::kable(d, digits = 3) ``` ## A Simple Polynomial (product of coordinates) ```{r} testFn3 <- function(x) prod(2 * x) testFn3_v <- function(x) matrix(apply(x, 2, function(z) prod(2 * z)), ncol = ncol(x)) d <- harness(f = testFn3, fv = testFn3_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 50) knitr::kable(d, digits = 3) ``` ## Gaussian centered at \frac{1}{2} ```{r} testFn4 <- function(x) { a <- 0.1 s <- sum((x - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(x) * exp (-s / (a * a)) } testFn4_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s <- sum((z - 0.5)^2) ((2 / sqrt(pi)) / (2. * a))^length(z) * exp (-s / (a * a)) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn4, fv = testFn4_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ``` ## Double Gaussian ```{r} testFn5 <- function(x) { a <- 0.1 s1 <- sum((x - 1 / 3)^2) s2 <- sum((x - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) } testFn5_v <- function(x) { a <- 0.1 r <- apply(x, 2, function(z) { s1 <- sum((z - 1 / 3)^2) s2 <- sum((z - 2 / 3)^2) 0.5 * ((2 / sqrt(pi)) / (2. * a))^length(z) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a))) }) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn5, fv = testFn5_v, lowerLimit = rep(0, 2), upperLimit = rep(1, 2), times = 20) knitr::kable(d, digits = 3) ``` ## Tsuda's Example ```{r} testFn6 <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 prod( a / (a + 1) * ((a + 1) / (a + x))^2) } testFn6_v <- function(x) { a <- (1 + sqrt(10.0)) / 9.0 r <- apply(x, 2, function(z) prod( a / (a + 1) * ((a + 1) / (a + z))^2)) matrix(r, ncol = ncol(x)) } d <- harness(f = testFn6, fv = testFn6_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ``` ## Morokoff & Calflish Example ```{r} testFn7 <- function(x) { n <- length(x) p <- 1/n (1 + p)^n * prod(x^p) } testFn7_v <- function(x) { matrix(apply(x, 2, function(z) { n <- length(z) p <- 1/n (1 + p)^n * prod(z^p) }), ncol = ncol(x)) } d <- harness(f = testFn7, fv = testFn7_v, lowerLimit = rep(0, 3), upperLimit = rep(1, 3), times = 20) knitr::kable(d, digits = 3) ``` ## Wang-Landau Sampling 1d, 2d Examples ```{r} 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)) } d <- harness(f = I.1d, fv = I.1d_v, lowerLimit = -2, upperLimit = 2, times = 100) knitr::kable(d, digits = 3) ``` ```{r} 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) } I.2d_v <- function(x) { matrix(apply(x, 2, function(z) { x1 <- z[1]; x2 <- z[2] sin(4 * x1 + 1) * cos(4 * x2) * x1 * (x1 * (x1 * x1)^2 - x2 * (x2 * x2 - x1) +2) }), ncol = ncol(x)) } d <- harness(f = I.2d, fv = I.2d_v, lowerLimit = rep(-1, 2), upperLimit = rep(1, 2), times = 100) knitr::kable(d, digits = 3) ``` ## An implementation note About the only real modification we have made to the underlying `cubature-1.0.2` library is that we use `M = 16` rather than the default `M = 19` suggested by the original author for `pcubature`. This allows us to comply with CRAN package size limits and seems to work reasonably well for the above tests. Future versions will allow for such customization on demand. ### Apropos the `Cuba` library The package [`R2Cuba`](https://cran.r-project.org/package=R2Cuba) provides a suite of cubature and other useful Monte Carlo integration routines linked against version 1.6 of the C library. The authors of `R2Cuba` have obviously put a lot of work has into it since it uses C-style R API. This approach also means that it is harder to keep the R package in sync with new versions of the underlying C library. In fact, the Cuba C library has marched on now to version 4.2. In a matter of a couple of hours, I was able to link the latest version (4.2) of the [Cuba libraries](http://www.feynarts.de/cuba/) with R using Rcpp; you can see it on the `Cuba` branch of my [Github repo](https://github.com/bnaras/cubature). This branch package builds and installs in R on my Mac and Ubuntu machines and gives correct answers at least for `cuhre`. The 4.2 version of the Cuba library also has vectorized versions of the routines that can be gainfully exploited (not implemented in the branch). As of this writing, I have also not yet carefully considered the issue of parallel execution (via `fork()`) which might be problematic in the Windows version. In addition, my timing benchmarks showed very disappointing results. For the above reasons, I decided not to bother with Cuba for now, but if there is enough interest, I might consider rolling `Cuba-4.2+` into this `cubature` package in the future. ## Summary The following is therefore my recommendation. 1. _Vectorize_ your function. The time spent in so doing pays back enormously. This is easy to do and the examples above show how. 2. Vectorized `hcubature` seems to be a good starting point. 3. For smooth integrands in low dimensions ($\leq 3$), `pcubature` might be worth trying out. Experiment before using in a production package. cubature/README.md0000644000176200001440000000100713133211455013337 0ustar liggesuserscubature ======== Updated this package in several ways: - Now using `Rcpp` (as of version 1.2) - Synced up to newer version of [cubature](http://ab-initio.mit.edu/wiki/index.php/Cubature) library (version 1.0.2) by Steven Johnson so that both `hcubature` `pcubature` are now available. - Added vectorized interface to `hcubature` and `pcubature` - Added a vignette to show the great gains from vectorization - Added tests to avoid screw-ups - Fixed up clencurt.h so that CRAN size limitations are not violated cubature/MD50000644000176200001440000000407313133704177012406 0ustar liggesusersd225d5c4743455f34dcbf24ad0f0d3a2 *ChangeLog d1648823a9219061d74b79b814acbc62 *DESCRIPTION 8df3f49918a800e7b888bad4cc989d17 *NAMESPACE 2aefbfddaefcea13068f51831e586cf8 *R/RcppExports.R 64a8aae99b4968e149b8494f66f58d02 *R/cubature-package.R 63a1be59257b342d3ba289f682925f8b *R/hcubature.R 2bfd14df6fa13626d8f3d3e264730f3d *README.md abd1df6eedd6e5939c527a2dedaedb0f *build/vignette.rds 4e4ec707b1aef4578d056033f879c6bf *inst/doc/cubature.R 9a409afdd8059f593970380258d92b4a *inst/doc/cubature.Rmd 49b104359cc43cd2fdfc91f2d9e92f0f *inst/doc/cubature.html 8f9a5a70456a0b11ab7244b4beb521f0 *inst/include/cubature.h 7d2e2edf6d3dedd86dab67ba85725b2b *inst/include/exp_cubature.h c25f03d1f6f706441867fc3b6caab050 *inst/include/exp_cubature_typedefs.h adb8e7b6c3ef32f0b86f7e6113266007 *man/cubature-package.Rd 67fa63c02b5bb1f6d69192adbd369fa2 *man/hcubature.Rd 35b485b4c72e2589d48f2c3ee792f003 *src/Makevars 6b56291adbef285ccf45d7400adce9d5 *src/Makevars.win 094ed9ab6791fc498868529307c37190 *src/Rcpp-cubature.cpp 9ab34335565b2acc577f0e32d3cbfaa7 *src/RcppExports.cpp d32239bcb673463ab874e80d47fae504 *src/cubature-1.0.2/COPYING 6a66c0d9f6bd91f4d0155b8d688a9467 *src/cubature-1.0.2/ChangeLog 55cbec04f280a64eb409f582ba56ba5d *src/cubature-1.0.2/Makefile 17ade8e017b1c8e355b26b3a604083f2 *src/cubature-1.0.2/NEWS 506b496da441e7ee8e69425a09c23a7c *src/cubature-1.0.2/README b4087fd2bb56324894b9c6a7353cabdc *src/cubature-1.0.2/README.md a90b0b738722cca97e72b67f68830bbc *src/cubature-1.0.2/clencurt.h 60ebd9f387ca8472b4084fc2fa5d853f *src/cubature-1.0.2/clencurt_gen.c 987ae5b854301ef719b013bb50740220 *src/cubature-1.0.2/converged.h 67a2a806e286b271602375ac2d6fd358 *src/cubature-1.0.2/hcubature.c 7687624e2e1807af545188b058d39b34 *src/cubature-1.0.2/pcubature.c d3a89f700d2fc68445eef66ca4682dda *src/cubature-1.0.2/test.c b76332176f1b8acd24eab4ecb6bb30f0 *src/cubature-1.0.2/vwrapper.h a5cc0cd1bfcd34a92c5bb3d67396269a *src/cubature_init.c aa87bab397dc5c414d91c14693ab51f5 *tests/testthat.R 206c39f849829f3f8d838251551229ae *tests/testthat/test_integration.R 9a409afdd8059f593970380258d92b4a *vignettes/cubature.Rmd cubature/build/0000755000176200001440000000000013133476177013177 5ustar liggesuserscubature/build/vignette.rds0000644000176200001440000000033513133476177015537 0ustar liggesusers‹mQK‚0-Qˆ$&\/À%D6n 1Æm-%6jJ Á•'”ê$3ö½7Ÿôä „td˜Ò H ‚¾׉l8—¤:cY Äy¢`ëMùGJ$ì†%ã…¿MSF-H£ܱØEæ™:ŸN0–÷BÞ 0SÿbÎ œÓR‘ÏCz¥EÒ=?þëµV‘,v´©¹4ŽÕs̈etè{`r¼û0êS-Vgø©?™ß¼†ÜîîZ0uQ’áR]ÔI°ÄA*@ßÍý³ôÚ…Æcubature/DESCRIPTION0000644000176200001440000000214113133704177013576 0ustar liggesusersPackage: cubature Type: Package Title: Adaptive Multivariate Integration over Hypercubes Version: 1.3-11 VignetteBuilder: knitr URL: https://github.com/bnaras/cubature Authors@R: c(person("Balasubramanian", "Narasimhan", role=c("aut", "cre"), email = "naras@stat.stanford.edu"), person("Manuel", "Koller", role=c("ctb"), email = "koller.manuel@gmail.com"), person(c("Steven", "G."), "Johnson", role=c("aut"))) Description: R wrapper around the cubature C library of Steven G. Johnson for adaptive multivariate integration over hypercubes. This version provides both hcubature and pcubature routines in addition to a vector interface that results in substantial speed gains. License: GPL-3 LinkingTo: Rcpp Imports: Rcpp NeedsCompilation: yes RoxygenNote: 6.0.1 Suggests: testthat, knitr, mvtnorm, R2Cuba, benchr Packaged: 2017-07-18 21:24:15 UTC; naras Author: Balasubramanian Narasimhan [aut, cre], Manuel Koller [ctb], Steven G. Johnson [aut] Maintainer: Balasubramanian Narasimhan Repository: CRAN Date/Publication: 2017-07-19 16:27:43 UTC cubature/ChangeLog0000644000176200001440000000463213133476114013646 0ustar liggesusersChanges for package cubature Version 1.0 (2009-12-18) - Original version of package based on Steven G. Johnson's cubature routines at http://ab-initio.mit.edu/wiki/index.php/Cubature Version 1.1 (2011-11-10) Changes from previous versions - Synced up to SGJ cubature routines dated 2010-10-18 on his website http://ab-initio.mit.edu/wiki/index.php/Cubature - Bugfix: potential memory leak fixed up in heap routine (my oversight!) - Routine adaptIntegrate gains ... argument (request of Baptiste Auguie) - Corrected radius constant in testFn2 to match cubature output exactly Version 1.1-1 (2011-12-01) - Added doChecking argument (default FALSE) to save some computation time in evaluating integrand (9% speedup). Version 1.1-2 (2011-12-08) - Fixed typo in doc for function adaptIntegrate; default value for doChecking was incorrectly stated as TRUE Version 1.1-2 (2013-02-24) - Registered native cubature functions adapt_integrate and adapt_integrate_v so that they are directly callable from C (courtesy of Simen Gaure) Version 1.1-3 (2015-07-15) - Roxygenized in preparation for upgrade to newer version of cubature library on abinitio website Version 1.2-0 (2016-11-20) - Moved to Rcpp framework - Added tests in preparation for SGJ cubature-1.0.2 version Version 1.3-0 (2016-11-30) - Moved to SGJ cubature-1.0.2 version. - adaptIntegrate and hcubature are aliases - hcubature function gains a norm argument that is set to a sensible default so as not to affect depending packages Version 1.3-1 (2016-12-05) - Added pcubature - Added vector versions Version 1.3-2 (2016-12-05) - Added vignette showing huge gains due to vectorization Version 1.3-3 (2016-12-09) - Moved the cubature-1.0.2 C library to github to avoid hitting the CRAN size limit. Version 1.3-4 up (2016-12-11) - Generated smaller pcubature header (clencurt.h) using M = 16 and put back C cubature source in tree. Version 1.3-7 (2017-05-09) - Registered .Call stuff and removed microbenchmark suggestion in favor of benchr. Version 1.3-8 through 1.3-11 (2017-07-17) - Merged Manuel Koller's registration for C code and vignette fix for NA. - Moved cubature header and exp_cubature headers to inst/include for linking to other packages - Renamed Readme.md to README.md - Removed references to ab-initio website that caused some hassle due to misconfigured site. cubature/man/0000755000176200001440000000000013133172427012642 5ustar liggesuserscubature/man/cubature-package.Rd0000644000176200001440000000154413133475470016344 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/cubature-package.R \docType{package} \name{cubature-package} \alias{cubature-package} \alias{cubature} \title{Cubature is a package for adaptive multidimensional integration over hypercubes} \description{ Cubature is a package for adaptive multidimensional integration over hypercubes. It is a wrapper around the pure C, GPLed implementation by Steven G. Johnson available. } \details{ \tabular{ll}{ Package: \tab cubature\cr Type: \tab Package\cr Version: \tab 1.0\cr Date: \tab 2009-12-17\cr License: \tab GPL V2 or later\cr LazyLoad: \tab yes\cr } There is only one function in the package called \link[cubature]{adaptIntegrate}. } \author{ C code by Steven G. Johnson, R by Balasubramanian Narasimhan Maintainer: Balasubramanian Narasimhan } \keyword{package} cubature/man/hcubature.Rd0000644000176200001440000002727413133234463015126 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/hcubature.R \name{hcubature} \alias{hcubature} \alias{adaptIntegrate} \alias{pcubature} \alias{pcubature} \title{Adaptive multivariate integration over hypercubes (hcubature and pcubature)} \usage{ hcubature(f, lowerLimit, upperLimit, ..., tol = 1e-05, fDim = 1, maxEval = 0, absError = 0, doChecking = FALSE, vectorInterface = FALSE, norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")) pcubature(f, lowerLimit, upperLimit, ..., tol = 1e-05, fDim = 1, maxEval = 0, absError = 0, doChecking = FALSE, vectorInterface = FALSE, norm = c("INDIVIDUAL", "PAIRED", "L2", "L1", "LINF")) } \arguments{ \item{f}{The function (integrand) to be integrated} \item{lowerLimit}{The lower limit of integration, a vector for hypercubes} \item{upperLimit}{The upper limit of integration, a vector for hypercubes} \item{...}{All other arguments passed to the function f} \item{tol}{The maximum tolerance, default 1e-5.} \item{fDim}{The dimension of the integrand, default 1, bears no relation to the dimension of the hypercube} \item{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.} \item{absError}{The maximum absolute error tolerated} \item{doChecking}{A flag to be a bit anal about checking inputs to C routines. A FALSE value results in approximately 9 percent speed gain in our experiments. Your mileage will of course vary. Default value is FALSE.} \item{vectorInterface}{A flag that indicates whether to use the vector interface and is by default FALSE. See details below} \item{norm}{For vector-valued integrands, \code{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 three items: \item{integral}{the value of the integral} \item{error}{the estimated relative error} \item{functionEvaluations}{the number of times the function was evaluated} \item{returnCode}{the actual integer return code of the C routine} } \description{ 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. } \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 \code{adaptIntegrate} function is an alias for the underlying \code{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 \code{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, \code{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 \code{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 \code{norm} argument is irrelevant for scalar integrands and is ignored. Given vectors \eqn{v} and \eqn{e} of estimated integrals and errors therein, respectively, the \code{norm} argument takes on one of the following values: \describe{ \item{\code{INDIVIDUAL}}{Convergence is achieved only when each integrand (each component of \eqn{v} and \eqn{e}) individually satisfies the requested error tolerances} \item{\code{L1}, \code{L2}, \code{LINF}}{The absolute error is measured as \eqn{|e|} and the relative error as \eqn{|e|/|v|}, where \eqn{|...|} is the \eqn{L_1}, \eqn{L_2}, or \eqn{\L_{\infty}} norm, respectively} \item{\code{PAIRED}}{Like \code{INDIVIDUAL}, except that the integrands are grouped into consecutive pairs, with the error tolerance applied in an \eqn{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} } } \examples{ \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) } } \author{ Balasubramanian Narasimhan } \keyword{math}