# Cubature Vectorization Results

#### Balasubramanian Narasimhan

#### 2025-11-21

## Introduction

This R `cubature` package exposes both the `hcubature` and `pcubature` routines of the underlying C `cubature` library, including the vectorized interfaces.

Per the documentation, use of `pcubature` is advisable only for smooth integrands in dimensions 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`.

Version 2.0 of this package integrates the `Cuba` library as well, once again providing vectorized interfaces.

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 Cuba `cuhre` calls. We begin by creating a harness for these calls.

    library(bench)
    library(cubature)

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

      fns <- c(hc = "Non-vectorized Hcubature",
               hc.v = "Vectorized Hcubature",
               pc = "Non-vectorized Pcubature",
               pc.v = "Vectorized Pcubature",
               cc = "Non-vectorized cubature::cuhre",
               cc_v = "Vectorized cubature::cuhre")
      cc <- function() cubature::cuhre(f = f,
                                       lowerLimit = lowerLimit, upperLimit = upperLimit,
                                       relTol = tol,
                                       ...)
      cc_v <- function() cubature::cuhre(f = fv,
                                         lowerLimit = lowerLimit, upperLimit = upperLimit,
                                         relTol = tol,
                                         nVec = 1024L,
                                         ...)
      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))

      argList <- c(fnList, iterations = iterations, check = FALSE)
      result <- do.call(bench::mark, args = argList)
      d <- result[seq_along(fnIndices), 1:9]
      d$expression <- fns[fnIndices]
      d
    }

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,
                 iterations = 100)[, 1:9]

    knitr::kable(d, digits = 3, row.names = FALSE)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 120µs | 130µs | 7677.496 | 61.8KB | 77.550 | 99 | 1 | 12.9ms
Vectorized Hcubature | 132µs | 138µs | 7154.789 | 54.5KB | 72.271 | 99 | 1 | 13.8ms
Non-vectorized Pcubature | 350µs | 361µs | 2697.865 | 31.2KB | 55.058 | 98 | 2 | 36.3ms
Vectorized Pcubature | 360µs | 372µs | 2665.535 | 58.4KB | 54.399 | 98 | 2 | 36.8ms
Non-vectorized cubature::cuhre | 227µs | 232µs | 4284.183 | 40.5KB | 43.275 | 99 | 1 | 23.1ms
Vectorized cubature::cuhre | 179µs | 185µs | 5323.173 | 39.8KB | 53.769 | 99 | 1 | 18.6ms

## 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,
                 iterations = 10,
                 mean = rep(0, m), sigma = sigma, logdet = logdet)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 222.9ms | 227.9ms | 4.360 | 139.58KB | 55.370 | 10 | 127 | 2.29s
Vectorized Hcubature | 719.9µs | 734.2µs | 1354.788 | 1.89MB | 0.000 | 10 | 0 | 7.38ms
Non-vectorized Pcubature | 95.3ms | 97.1ms | 10.281 | 0B | 55.515 | 10 | 54 | 972.71ms
Vectorized Pcubature | 477µs | 495.3µs | 1990.374 | 810.26KB | 0.000 | 10 | 0 | 5.02ms
Non-vectorized cubature::cuhre | 91.4ms | 93.5ms | 10.517 | 0B | 54.689 | 10 | 52 | 950.84ms
Vectorized cubature::cuhre | 929.5µs | 941.8µs | 1047.349 | 898.41KB | 0.000 | 10 | 0 | 9.55ms

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 vectorized versions of `hcubature` and `pcubature` seem competitive and in some cases better, if you compare the table above to the one below.

    library(mvtnorm)
    g1 <- function() 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() 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() 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(bench::mark(g1(), g2(), g3(), iterations = 20, check = FALSE)[, 1:9],
                 digits = 3, row.names = FALSE)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
g1() | 323µs | 581µs | 1845.746 | 355.03KB | 0 | 20 | 0 | 10.8ms
g2() | 301µs | 553µs | 1820.018 | 2.49KB | 0 | 20 | 0 | 11ms
g3() | 286µs | 526µs | 1994.843 | 2.49KB | 0 | 20 | 0 | 10ms

## 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), iterations = 1000)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 15.1µs | 17.1µs | 58334.985 | 7.66KB | 58.393 | 999 | 1 | 17.1ms
Vectorized Hcubature | 19.8µs | 22.1µs | 45567.457 | 1.16KB | 45.613 | 999 | 1 | 21.9ms
Non-vectorized Pcubature | 17.6µs | 18.5µs | 53472.985 | 0B | 53.527 | 999 | 1 | 18.7ms
Vectorized Pcubature | 35.6µs | 36.7µs | 26822.919 | 18.68KB | 53.753 | 998 | 2 | 37.2ms
Non-vectorized cubature::cuhre | 121µs | 126.6µs | 7776.688 | 0B | 62.715 | 992 | 8 | 127.6ms
Vectorized cubature::cuhre | 105.3µs | 110.2µs | 8816.691 | 16.38KB | 71.102 | 992 | 8 | 112.5ms

## 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), iterations = 10)

    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 1.01ms | 1.02ms | 975.748 | 67.3KB | 108.416 | 9 | 1 | 9.22ms
Vectorized Hcubature | 1.45ms | 1.48ms | 673.010 | 290.5KB | 74.779 | 9 | 1 | 13.37ms
Non-vectorized Pcubature | 26.12µs | 27.12µs | 36251.868 | 0B | 0.000 | 10 | 0 | 275.85µs
Vectorized Pcubature | 46.25µs | 47.17µs | 21038.756 | 4.1KB | 0.000 | 10 | 0 | 475.31µs
Non-vectorized cubature::cuhre | 4.82ms | 4.95ms | 203.222 | 0B | 87.095 | 7 | 3 | 34.45ms
Vectorized cubature::cuhre | 6.03ms | 6.04ms | 165.472 | 971.5KB | 110.315 | 6 | 4 | 36.26ms

## 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", "cc_v"),
                 f = testFn2, fv = testFn2_v,
                 lowerLimit = rep(0, 2), upperLimit = rep(1, 2), iterations = 10)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 14.3ms | 14.8ms | 64.635 | 17.7KB | 96.953 | 10 | 15 | 154.72ms
Vectorized Hcubature | 14.1ms | 14.4ms | 69.021 | 1011.8KB | 96.630 | 10 | 14 | 144.88ms
Non-vectorized cubature::cuhre | 271.6ms | 276.6ms | 3.553 | 0B | 74.979 | 10 | 211 | 2.81s
Vectorized cubature::cuhre | 255.6ms | 262.6ms | 3.798 | 21.2MB | 69.887 | 10 | 184 | 2.63s

## 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), iterations = 50)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 24.3µs | 26.1µs | 37488.813 | 6.75KB | 0.00 | 50 | 0 | 1.33ms
Vectorized Hcubature | 30.9µs | 31.6µs | 31024.512 | 2.91KB | 0.00 | 50 | 0 | 1.61ms
Non-vectorized Pcubature | 21.6µs | 22.3µs | 43750.825 | 0B | 0.00 | 50 | 0 | 1.14ms
Vectorized Pcubature | 28.2µs | 29.2µs | 33357.477 | 19.12KB | 0.00 | 50 | 0 | 1.5ms
Non-vectorized cubature::cuhre | 264.5µs | 270.2µs | 3644.598 | 0B | 74.38 | 49 | 1 | 13.45ms
Vectorized cubature::cuhre | 190.5µs | 199.8µs | 4948.027 | 39.84KB | 100.98 | 49 | 1 | 9.9ms

## Gaussian centered at \(\frac{1}{2}\)

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

    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), iterations = 20)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 480.23µs | 495.07µs | 2005.473 | 85.2KB | 105.551 | 19 | 1 | 9.47ms
Vectorized Hcubature | 546.53µs | 576.62µs | 1740.550 | 147.2KB | 91.608 | 19 | 1 | 10.92ms
Non-vectorized Pcubature | 734.64µs | 763.17µs | 1306.317 | 0B | 68.754 | 19 | 1 | 14.54ms
Vectorized Pcubature | 815.65µs | 837.1µs | 1189.970 | 68.9KB | 62.630 | 19 | 1 | 15.97ms
Non-vectorized cubature::cuhre | 1.22ms | 1.24ms | 805.426 | 0B | 42.391 | 19 | 1 | 23.59ms
Vectorized cubature::cuhre | 1.11ms | 1.14ms | 859.780 | 125.6KB | 95.531 | 18 | 2 | 20.94ms

## 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), iterations = 20)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 1.32ms | 1.34ms | 740.809 | 133KB | 82.312 | 18 | 2 | 24.3ms
Vectorized Hcubature | 1.46ms | 1.54ms | 659.112 | 249KB | 73.235 | 18 | 2 | 27.3ms
Non-vectorized Pcubature | 885.97µs | 911.39µs | 1099.329 | 0B | 57.859 | 19 | 1 | 17.3ms
Vectorized Pcubature | 1.04ms | 1.07ms | 933.854 | 69KB | 103.762 | 18 | 2 | 19.3ms
Non-vectorized cubature::cuhre | 2.55ms | 2.6ms | 385.834 | 0B | 68.088 | 17 | 3 | 44.1ms
Vectorized cubature::cuhre | 2.67ms | 2.77ms | 362.021 | 224KB | 90.505 | 16 | 4 | 44.2ms

## 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), iterations = 20)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 566.87µs | 591.3µs | 1697.030 | 64.6KB | 89.317 | 19 | 1 | 11.2ms
Vectorized Hcubature | 655.67µs | 680.97µs | 1470.313 | 156KB | 77.385 | 19 | 1 | 12.9ms
Non-vectorized Pcubature | 3.05ms | 3.1ms | 322.840 | 0B | 80.710 | 16 | 4 | 49.6ms
Vectorized Pcubature | 3.09ms | 3.22ms | 312.259 | 386.2KB | 78.065 | 16 | 4 | 51.2ms
Non-vectorized cubature::cuhre | 1.58ms | 1.61ms | 618.790 | 0B | 68.754 | 18 | 2 | 29.1ms
Vectorized cubature::cuhre | 1.49ms | 1.53ms | 642.970 | 225.8KB | 71.441 | 18 | 2 | 28ms

## 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), iterations = 20)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 1.21ms | 1.24ms | 811.394 | 32.89KB | 90.155 | 18 | 2 | 22.2ms
Vectorized Hcubature | 1.21ms | 1.22ms | 815.052 | 205.61KB | 90.561 | 18 | 2 | 22.1ms
Non-vectorized Pcubature | 2.99ms | 3.08ms | 325.866 | 0B | 108.622 | 15 | 5 | 46ms
Vectorized Pcubature | 2.72ms | 2.94ms | 342.490 | 386.24KB | 85.622 | 16 | 4 | 46.7ms
Non-vectorized cubature::cuhre | 14.61ms | 14.68ms | 68.110 | 0B | 612.994 | 2 | 18 | 29.4ms
Vectorized cubature::cuhre | 12.26ms | 12.36ms | 80.381 | 2.04MB | 455.490 | 3 | 17 | 37.3ms

## 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, iterations = 100)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 56.2µs | 57.9µs | 16640.94 | 53.7KB | 168.090 | 99 | 1 | 5.95ms
Vectorized Hcubature | 72.5µs | 74µs | 13381.10 | 68.5KB | 0.000 | 100 | 0 | 7.47ms
Non-vectorized Pcubature | 20.8µs | 21.6µs | 46197.96 | 0B | 0.000 | 100 | 0 | 2.17ms
Vectorized Pcubature | 50.3µs | 52.3µs | 19181.88 | 0B | 193.756 | 99 | 1 | 5.16ms
Non-vectorized cubature::cuhre | 95.2µs | 98µs | 10188.71 | 0B | 0.000 | 100 | 0 | 9.81ms
Vectorized cubature::cuhre | 159.6µs | 165.8µs | 5925.25 | 0B | 59.851 | 99 | 1 | 16.71ms

    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), iterations = 100)
    knitr::kable(d, digits = 3)

expression | min | median | itr/sec | mem_alloc | gc/sec | n_itr | n_gc | total_time
--- | --- | --- | --- | --- | --- | --- | --- | ---
Non-vectorized Hcubature | 1.81ms | 1.87ms | 533.812 | 78.4KB | 65.977 | 89 | 11 | 166.7ms
Vectorized Hcubature | 1.64ms | 1.72ms | 580.982 | 304.7KB | 71.807 | 89 | 11 | 153.2ms
Non-vectorized Pcubature | 161.29µs | 167.28µs | 5846.189 | 0B | 59.052 | 99 | 1 | 16.9ms
Vectorized Pcubature | 191.92µs | 204.88µs | 4873.923 | 18.3KB | 49.232 | 99 | 1 | 20.3ms
Non-vectorized cubature::cuhre | 480.27µs | 502.62µs | 1970.367 | 0B | 60.939 | 97 | 3 | 49.2ms
Vectorized cubature::cuhre | 419.68µs | 444.28µs | 2246.679 | 60.1KB | 45.851 | 98 | 2 | 43.6ms

## Implementation Notes

About the only real modification we have made to the underlying `cubature` 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.

The changes made to the `Cuba` library are managed in a Github repo branch: each time a new release is made, we update the main branch, and keep all changes for Unix platforms in a branch named `R_pkg` against the current main branch. Customization for windows is done in the package itself using the `Makevars.win` script.

## Summary

The recommendations are:

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.

## Session Info

    sessionInfo()

    ## R version 4.5.2 (2025-10-31)
    ## Platform: aarch64-apple-darwin20
    ## Running under: macOS Tahoe 26.1
    ##
    ## Matrix products: default
    ## BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
    ## LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1
    ##
    ## locale:
    ## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    ##
    ## time zone: America/Los_Angeles
    ## tzcode source: internal
    ##
    ## attached base packages:
    ## [1] stats     graphics  grDevices utils     datasets  methods   base
    ##
    ## other attached packages:
    ## [1] mvtnorm_1.3-3    cubature_2.1.4-1 bench_1.1.4
    ##
    ## loaded via a namespace (and not attached):
    ##  [1] digest_0.6.38     R6_2.6.1          fastmap_1.2.0     xfun_0.54
    ##  [5] magrittr_2.0.4    glue_1.8.0        cachem_1.1.0      tibble_3.3.0
    ##  [9] knitr_1.50        pkgconfig_2.0.3   htmltools_0.5.8.1 rmarkdown_2.30
    ## [13] lifecycle_1.0.4   profmem_0.7.0     cli_3.6.5         vctrs_0.6.5
    ## [17] sass_0.4.10       jquerylib_0.1.4   compiler_4.5.2    tools_4.5.2
    ## [21] pillar_1.11.1     evaluate_1.0.5    bslib_0.9.0       Rcpp_1.1.0
    ## [25] yaml_2.3.10       rlang_1.1.6       jsonlite_2.0.0