# Cuba

## Introduction

Cuba.jl is a Julia library for multidimensional numerical integration of real-valued functions of real arguments, using different algorithms.

This is just a Julia wrapper around the C Cuba library, version 4.2, by Thomas Hahn. All the credits goes to him for the underlying functions, blame me for any problem with the Julia interface.

All algorithms provided by Cuba library are supported in Cuba.jl:

For more details on the algorithms see the manual included in Cuba library and available in deps/usr/share/cuba.pdf after successful installation of Cuba.jl.

Integration is always performed on the $n$-dimensional unit hypercube $[0, 1]^{n}$.

Integrate over different domains

If you want to compute an integral over a different set, you have to scale the integrand function in order to have an equivalent integral on $[0, 1]^{n}$ using substitution rules. For example, recall that in one dimension

$$$\int_{a}^{b} f(x)\,\mathrm{d}x = \int_{0}^{1} f(a + (b - a) y) (b - a)\,\mathrm{d}y$$$

where the final $(b - a)$ is the one-dimensional version of the Jacobian.

Integration over a semi-infinite or an inifite domain is a bit trickier, but you can follow this advice from Steven G. Johnson: to compute an integral over a semi-infinite interval, you can perform the change of variables $x=a+y/(1-y)$:

$$$\int_{a}^{\infty} f(x)\,\mathrm{d}x = \int_{0}^{1} f\left(a + \frac{y}{1 - y}\right)\frac{1}{(1 - y)^2}\,\mathrm{d}y$$$

For an infinite interval, you can perform the change of variables $x=(2y - 1)/((1 - y)y)$:

$$$\int_{-\infty}^{\infty} f(x)\,\mathrm{d}x = \int_{0}^{1} f\left(\frac{2y - 1}{(1 - y)y}\right)\frac{2y^2 - 2y + 1}{(1 - y)^2y^2}\,\mathrm{d}y$$$

In addition, recall that for an even function $\int_{-\infty}^{\infty} f(x)\,\mathrm{d}x = 2\int_{0}^{\infty}f(x)\,\mathrm{d}x$, while the integral of an odd function over the infinite interval $(-\infty, \infty)$ is zero.

All this generalizes straightforwardly to more than one dimension. In Examples section you can find the computation of a 3-dimensional integral with non-constant boundaries using Cuba.jl and two integrals over infinite domains.

Cuba.jl is available on all platforms supported by Julia.

## Installation

The latest version of Cuba.jl is available for Julia 1.3 and later versions, and can be installed with Julia built-in package manager. In a Julia session run the commands

pkg> update
pkg> add Cuba

Older versions are also available for Julia 0.4-1.2.

## Usage

After installing the package, run

using Cuba

or put this command into your Julia script.

Cuba.jl provides the following functions to integrate:

Cuba.vegasFunction
vegas(integrand, ndim=1, ncomp=1[, keywords]) -> integral, error, probability, neval, fail, nregions

Calculate integral of integrand over the unit hypercube in ndim dimensions using Vegas algorithm. integrand is a vectorial function with ncomp components. ndim and ncomp default to 1.

Accepted keywords:

• userdata
• nvec
• rtol
• atol
• flags
• seed
• minevals
• maxevals
• nstart
• nincrease
• nbatch
• gridno
• statefile
• spin
Cuba.suaveFunction
suave(integrand, ndim=1, ncomp=1[, keywords]) -> integral, error, probability, neval, fail, nregions

Calculate integral of integrand over the unit hypercube in ndim dimensions using Suave algorithm. integrand is a vectorial function with ncomp components. ndim and ncomp default to 1.

Accepted keywords:

• userdata
• nvec
• rtol
• atol
• flags
• seed
• minevals
• maxevals
• nnew
• nmin
• flatness
• statefile
• spin
Cuba.divonneFunction
divonne(integrand, ndim=2, ncomp=1[, keywords]) -> integral, error, probability, neval, fail, nregions

Calculate integral of integrand over the unit hypercube in ndim dimensions using Divonne algorithm. integrand is a vectorial function with ncomp components. ncomp defaults to 1, ndim defaults to 2 and must be ≥ 2.

Accepted keywords:

• userdata
• nvec
• rtol
• atol
• flags
• seed
• minevals
• maxevals
• key1
• key2
• key3
• maxpass
• border
• maxchisq
• mindeviation
• ngiven
• ldxgiven
• xgiven
• nextra
• peakfinder
• statefile
• spin
Cuba.cuhreFunction
cuhre(integrand, ndim=2, ncomp=1[, keywords]) -> integral, error, probability, neval, fail, nregions

Calculate integral of integrand over the unit hypercube in ndim dimensions using Cuhre algorithm. integrand is a vectorial function with ncomp components. ncomp defaults to 1, ndim defaults to 2 and must be ≥ 2.

Accepted keywords:

• userdata
• nvec
• rtol
• atol
• flags
• minevals
• maxevals
• key
• statefile
• spin

Large parts of the following sections are borrowed from Cuba manual. Refer to it for more information on the details.

Cuba.jl wraps the 64-bit integers functions of Cuba library, in order to push the range of certain counters to its full extent. In detail, the following arguments:

• for Vegas: nvec, minevals, maxevals, nstart, nincrease, nbatch, neval,
• for Suave: nvec, minevals, maxevals, nnew, nmin, neval,
• for Divonne: nvec, minevals, maxevals, ngiven, nextra, neval,
• for Cuhre: nvec, minevals, maxevals, neval,

are passed to the Cuba library as 64-bit integers, so they are limited to be at most

julia> typemax(Int64)
9223372036854775807

There is no way to overcome this limit. See the following sections for the meaning of each argument.

### Arguments

The only mandatory argument of integrator functions is:

• integrand (type: Function): the function to be integrated

Optional positional arguments are:

• ndim (type: Integer): the number of dimensions of the integratation domain. If omitted, defaults to 1 in vegas and suave, to 2 in divonne and cuhre. Note: ndim must be at least 2 with the latest two methods.
• ncomp (type: Integer): the number of components of the integrand. Default to 1 if omitted

integrand should be a function integrand(x, f) taking two arguments:

• the input vector x of length ndim
• the output vector f of length ncomp, used to set the value of each component of the integrand at point x

x and f are matrices with dimensions (ndim, nvec) and (ncomp, nvec), respectively, when nvec > 1. See the Vectorization section below for more information.

Also anonymous functions are allowed as integrand. For those familiar with Cubature.jl package, this is the same syntax used for integrating vector-valued functions.

For example, the integral

$$$\int_{0}^{1} \cos (x) \,\mathrm{d}x = \sin(1) = 0.8414709848078965$$$

can be computed with one of the following commands

julia> vegas((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.841491000525961 ± 5.2708169786483034e-5 (prob.: 0.028607201258847748)
Integrand evaluations: 13500
Number of subregions:  0
Note: The desired accuracy was reached

julia> suave((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.8413748866950329 ± 7.772872640815592e-5 (prob.: 1.0)
Integrand evaluations: 23000
Number of subregions:  23
Note: The desired accuracy was reached

julia> divonne((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.841468071955942 ± 5.3955070531551656e-5 (prob.: 1.1102230246251565e-16)
Integrand evaluations: 1686
Number of subregions:  14
Note: The desired accuracy was reached

julia> cuhre((x, f) -> f[1] = cos(x[1]))
Component:
1: 0.8414709848078967 ± 2.304857594221477e-15 (prob.: 4.869900880782919e-5)
Integrand evaluations: 195
Number of subregions:  2
Note: The desired accuracy was reached

In section Examples you can find more complete examples. Note that x and f are both arrays with type Float64, so Cuba.jl can be used to integrate real-valued functions of real arguments. See how to work with a complex integrand.

Limit on number of components

The Cuba C library has a hard limit on the number of components that you can use for your integrand function, which is set at compile time. The build of the library currently used by Cuba.jl has this limit set to 1024, the default. If you use an integrand with a larger number of components, the integration will fail, which you can detect by checking the fail field of the Integral output object, which will be non-zero, see the "Output" section below.

If you need to integrate functions with components larger than 1024, consider using other packages which don't have this limitation, like HCubature.jl or Cubature.jl.

Compatibility

If you used Cuba.jl until version 0.0.4, be aware that the user interface has been reworked in version 0.0.5 in a backward incompatible way.

### Optional Keywords

All other arguments required by Cuba integrator routines can be passed as optional keywords. Cuba.jl uses some reasonable default values in order to enable users to invoke integrator functions with a minimal set of arguments. Anyway, if you want to make sure future changes to some default values of keywords will not affect your current script, explicitely specify the value of the keywords.

#### Common Keywords

These are optional keywords common to all functions:

• nvec (type: Integer, default: 1): the maximum number of points to be given to the integrand routine in each invocation. Usually this is 1 but if the integrand can profit from e.g. Single Instruction Multiple Data (SIMD) vectorization, a larger value can be chosen. See Vectorization section.

• rtol (type: Real, default: 1e-4), and atol (type: Real, default: 1e-12): the requested relative ($\varepsilon_{\text{rel}}$) and absolute ($\varepsilon_{\text{abs}}$) accuracies. The integrator tries to find an estimate $\hat{I}$ for the integral $I$ which for every component $c$ fulfills $|\hat{I}_c - I_c|\leq \max(\varepsilon_{\text{abs}}, \varepsilon_{\text{rel}} |I_c|)$.

• flags (type: Integer, default: 0): flags governing the integration:

• Bits 0 and 1 are taken as the verbosity level, i.e. 0 to 3, unless the CUBAVERBOSE environment variable contains an even higher value (used for debugging).

Level 0 does not print any output, level 1 prints "reasonable" information on the progress of the integration, level 2 also echoes the input parameters, and level 3 further prints the subregion results (if applicable).

• Bit 2 = 0: all sets of samples collected on a subregion during the various iterations or phases contribute to the final result.

Bit 2 = 1, only the last (largest) set of samples is used in the final result.

• (Vegas and Suave only)

Bit 3 = 0, apply additional smoothing to the importance function, this moderately improves convergence for many integrands.

Bit 3 = 1, use the importance function without smoothing, this should be chosen if the integrand has sharp edges.

• Bit 4 = 0, delete the state file (if one is chosen) when the integration terminates successfully.

Bit 4 = 1, retain the state file.

• (Vegas only)

Bit 5 = 0, take the integrator's state from the state file, if one is present.

Bit 5 = 1, reset the integrator's state even if a state file is present, i.e. keep only the grid. Together with Bit 4 this allows a grid adapted by one integration to be used for another integrand.

• Bits 8–31 =: level determines the random-number generator.

To select e.g. last samples only and verbosity level 2, pass 6 = 4 + 2 for the flags.

• seed (type: Integer, default: 0): the seed for the pseudo-random-number generator. This keyword is not available for cuhre. The random-number generator is chosen as follows:

seedlevel (bits 8–31 of flags)Generator
zeroN/ASobol (quasi-random)
non-zerozeroMersenne Twister (pseudo-random)
non-zeronon-zeroRanlux (pseudo-random)

Ranlux implements Marsaglia and Zaman's 24-bit RCARRY algorithm with generation period $p$, i.e. for every 24 generated numbers used, another $p - 24$ are skipped. The luxury level is encoded in level as follows:

• Level 1 ($p = 48$): very long period, passes the gap test but fails spectral test.
• Level 2 ($p = 97$): passes all known tests, but theoretically still defective.
• Level 3 ($p = 223$): any theoretically possible correlations have very small chance of being observed.
• Level 4 ($p = 389$): highest possible luxury, all 24 bits chaotic.

Levels 5–23 default to 3, values above 24 directly specify the period $p$. Note that Ranlux's original level 0, (mis)used for selecting Mersenne Twister in Cuba, is equivalent to level = 24.

• minevals (type: Real, default: 0): the minimum number of integrand evaluations required

• maxevals (type: Real, default: 1000000): the (approximate) maximum number of integrand evaluations allowed

• statefile (type: AbstractString, default: ""): a filename for storing the internal state. To not store the internal state, put "" (empty string, this is the default) or C_NULL (C null pointer).

Cuba can store its entire internal state (i.e. all the information to resume an interrupted integration) in an external file. The state file is updated after every iteration. If, on a subsequent invocation, a Cuba routine finds a file of the specified name, it loads the internal state and continues from the point it left off. Needless to say, using an existing state file with a different integrand generally leads to wrong results.

This feature is useful mainly to define "check-points" in long-running integrations from which the calculation can be restarted.

Once the integration reaches the prescribed accuracy, the state file is removed, unless bit 4 of flags (see above) explicitly requests that it be kept.

• spin (type: Ptr{Void}, default: C_NULL): this is the placeholder for the "spinning cores" pointer. Cuba.jl does not support parallelization, so this keyword should not have a value different from C_NULL.

• userdata (arbitrary julia type, default: missing): user data passed to the integrand. See Passing data to the integrand function for a usage example.

Compatibility

The keyword userdata is only supported in Cuba.jl after the version 2.3.0.

#### Vegas-Specific Keywords

These optional keywords can be passed only to vegas:

• nstart (type: Integer, default: 1000): the number of integrand evaluations per iteration to start with

• nincrease (type: Integer, default: 500): the increase in the number of integrand evaluations per iteration

• nbatch (type: Integer, default: 1000): the batch size for sampling

Vegas samples points not all at once, but in batches of size nbatch, to avoid excessive memory consumption. 1000 is a reasonable value, though it should not affect performance too much

• gridno (type: Integer, default: 0): the slot in the internal grid table.

It may accelerate convergence to keep the grid accumulated during one integration for the next one, if the integrands are reasonably similar to each other. Vegas maintains an internal table with space for ten grids for this purpose. The slot in this grid is specified by gridno.

If a grid number between 1 and 10 is selected, the grid is not discarded at the end of the integration, but stored in the respective slot of the table for a future invocation. The grid is only re-used if the dimension of the subsequent integration is the same as the one it originates from.

In repeated invocations it may become necessary to flush a slot in memory, in which case the negative of the grid number should be set.

#### Suave-Specific Keywords

These optional keywords can be passed only to suave:

• nnew (type: Integer, default: 1000): the number of new integrand evaluations in each subdivision
• nmin (type: Integer, default: 2): the minimum number of samples a former pass must contribute to a subregion to be considered in that region's compound integral value. Increasing nmin may reduce jumps in the $\chi^2$ value
• flatness (type: Real, default: .25): the type of norm used to compute the fluctuation of a sample. This determines how prominently "outliers", i.e. individual samples with a large fluctuation, figure in the total fluctuation, which in turn determines how a region is split up. As suggested by its name, flatness should be chosen large for "flat" integrands and small for "volatile" integrands with high peaks. Note that since flatness appears in the exponent, one should not use too large values (say, no more than a few hundred) lest terms be truncated internally to prevent overflow.

#### Divonne-Specific Keywords

These optional keywords can be passed only to divonne:

• key1 (type: Integer, default: 47): determines sampling in the partitioning phase: key1 $= 7, 9, 11, 13$ selects the cubature rule of degree key1. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions.

For other values of key1, a quasi-random sample of $n_1 = |\verb|key1||$ points is used, where the sign of key1 determines the type of sample,

• key1 $> 0$, use a Korobov quasi-random sample,

• key1 $< 0$, use a "standard" sample (a Sobol quasi-random sample if seed $= 0$, otherwise a pseudo-random sample).

• key2 (type: Integer, default: 1): determines sampling in the final integration phase:

key2 $= 7, 9, 11, 13$ selects the cubature rule of degree key2. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions.

For other values of key2, a quasi-random sample is used, where the sign of key2 determines the type of sample,

• key2 $> 0$, use a Korobov quasi-random sample,
• key2 $< 0$, use a "standard" sample (see description of key1 above),

and $n_2 = |\verb|key2||$ determines the number of points,

• $n_2\geq 40$, sample $n_2$ points,
• $n_2 < 40$, sample $n_2\,n_{\text{need}}$ points, where $n_{\text{need}}$ is the number of points needed to reach the prescribed accuracy, as estimated by Divonne from the results of the partitioning phase
• key3 (type: Integer, default: 1): sets the strategy for the refinement phase:

key3 $= 0$, do not treat the subregion any further.

key3 $= 1$, split the subregion up once more.

Otherwise, the subregion is sampled a third time with key3 specifying the sampling parameters exactly as key2 above.

• maxpass (type: Integer, default: 5): controls the thoroughness of the partitioning phase: The partitioning phase terminates when the estimated total number of integrand evaluations (partitioning plus final integration) does not decrease for maxpass successive iterations.

A decrease in points generally indicates that Divonne discovered new structures of the integrand and was able to find a more effective partitioning. maxpass can be understood as the number of "safety" iterations that are performed before the partition is accepted as final and counting consequently restarts at zero whenever new structures are found.

• border (type: Real, default: 0.): the width of the border of the integration region. Points falling into this border region will not be sampled directly, but will be extrapolated from two samples from the interior. Use a non-zero border if the integrand function cannot produce values directly on the integration boundary

• maxchisq (type: Real, default: 10.): the $\chi^2$ value a single subregion is allowed to have in the final integration phase. Regions which fail this $\chi^2$ test and whose sample averages differ by more than mindeviation move on to the refinement phase.

• mindeviation (type: Real, default: 0.25): a bound, given as the fraction of the requested error of the entire integral, which determines whether it is worthwhile further examining a region that failed the $\chi^2$ test. Only if the two sampling averages obtained for the region differ by more than this bound is the region further treated.

• ngiven (type: Integer, default: 0): the number of points in the xgiven array

• ldxgiven (type: Integer, default: 0): the leading dimension of xgiven, i.e. the offset between one point and the next in memory

• xgiven (type: AbstractArray{Real}, default: zeros(Cdouble, ldxgiven, ngiven)): a list of points where the integrand might have peaks. Divonne will consider these points when partitioning the integration region. The idea here is to help the integrator find the extrema of the integrand in the presence of very narrow peaks. Even if only the approximate location of such peaks is known, this can considerably speed up convergence.

• nextra (type: Integer, default: 0): the maximum number of extra points the peak-finder subroutine will return. If nextra is zero, peakfinder is not called and an arbitrary object may be passed in its place, e.g. just 0

• peakfinder (type: Ptr{Void}, default: C_NULL): the peak-finder subroutine

#### Cuhre-Specific Keyword

This optional keyword can be passed only to cuhre:

• key (type: Integer, default: 0): chooses the basic integration rule:

key $= 7, 9, 11, 13$ selects the cubature rule of degree key. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions.

For other values, the default rule is taken, which is the degree-13 rule in 2 dimensions, the degree-11 rule in 3 dimensions, and the degree-9 rule otherwise.

### Output

The integrating functions vegas, suave, divonne, and cuhre return an Integral object whose fields are

integral :: Vector{Float64}
error    :: Vector{Float64}
probl    :: Vector{Float64}
neval    :: Int64
fail     :: Int32
nregions :: Int32

The first three fields are arrays with length ncomp, the last three ones are scalars. The Integral object can also be iterated over like a tuple. In particular, if you assign the output of integrator functions to the variable named result, you can access the value of the i-th component of the integral with result[1][i] or result.integral[i] and the associated error with result[2][i] or result.error[i].

• integral (type: Vector{Float64}, with ncomp components): the integral of integrand over the unit hypercube
• error (type: Vector{Float64}, with ncomp components): the presumed absolute error for each component of integral
• probability (type: Vector{Float64}, with ncomp components): the $\chi^2$ -probability (not the $\chi^2$ -value itself!) that error is not a reliable estimate of the true integration error. To judge the reliability of the result expressed through prob, remember that it is the null hypothesis that is tested by the $\chi^2$ test, which is that error is a reliable estimate. In statistics, the null hypothesis may be rejected only if prob is fairly close to unity, say prob $>.95$
• neval (type: Int64): the actual number of integrand evaluations needed
• fail (type: Int32): an error flag:
• fail = 0, the desired accuracy was reached
• fail = -1, dimension out of range
• fail > 0, the accuracy goal was not met within the allowed maximum number of integrand evaluations. While Vegas, Suave, and Cuhre simply return 1, Divonne can estimate the number of points by which maxevals needs to be increased to reach the desired accuracy and returns this value.
• nregions (type: Int32): the actual number of subregions needed (always 0 in vegas)

## Vectorization

Vectorization means evaluating the integrand function for several points at once. This is also known as Single Instruction Multiple Data (SIMD) paradigm and is different from ordinary parallelization where independent threads are executed concurrently. It is usually possible to employ vectorization on top of parallelization.

Cuba.jl cannot automatically vectorize the integrand function, of course, but it does pass (up to) nvec points per integrand call (Common Keywords). This value need not correspond to the hardware vector length –computing several points in one call can also make sense e.g. if the computations have significant intermediate results in common.

When nvec > 1, the input x is a matrix of dimensions (ndim, nvec), while the output f is a matrix with dimensions (ncomp, nvec). Vectorization can be used to evaluate more quickly the integrand function, for example by exploiting parallelism, thus speeding up computation of the integral. See the section Vectorized Function below for an example of a vectorized funcion.

Disambiguation

The nbatch argument of vegas is related in purpose but not identical to nvec. It internally partitions the sampling done by Vegas but has no bearing on the number of points given to the integrand. On the other hand, it it pointless to choose nvec > nbatch for Vegas.

## Examples

### One dimensional integral

The integrand of

$$$\int_{0}^{1} \frac{\log(x)}{\sqrt{x}} \,\mathrm{d}x$$$

has an algebraic-logarithmic divergence for $x = 0$, but the integral is convergent and its value is $-4$. Cuba.jl integrator routines can handle this class of functions and you can easily compute the numerical approximation of this integral using one of the following commands:

julia> vegas( (x,f) -> f[1] = log(x[1])/sqrt(x[1]))
Component:
1: -3.9981623937128448 ± 0.00044066437168409865 (prob.: 0.28430529712907515)
Integrand evaluations: 1007500
Number of subregions:  0
Note: The accuracy was not met within the maximum number of evaluations

julia> suave( (x,f) -> f[1] = log(x[1])/sqrt(x[1]))
Component:
1: -4.000246664970977 ± 0.00039262438882794375 (prob.: 1.0)
Integrand evaluations: 50000
Number of subregions:  50
Note: The desired accuracy was reached

julia> divonne( (x,f) -> f[1] = log(x[1])/sqrt(x[1]), atol = 1e-8, rtol = 1e-8)
Component:
1: -3.999999899620808 ± 2.1865962888458758e-7 (prob.: 0.0)
Integrand evaluations: 1002059
Number of subregions:  1582
Note: The accuracy was not met within the maximum number of evaluations
Hint: Try increasing maxevals to 4884287

julia> cuhre( (x,f) -> f[1] = log(x[1])/sqrt(x[1]))
Component:
1: -4.000000355067185 ± 0.00033954840286260385 (prob.: 0.0)
Integrand evaluations: 5915
Number of subregions:  46
Note: The desired accuracy was reached

### Vector-valued integrand

Consider the integral

$$$\int\limits_{\Omega} \boldsymbol{f}(x,y,z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$$

where $\Omega = [0, 1]^{3}$ and

$$$\boldsymbol{f}(x,y,z) = \left(\sin(x)\cos(y)\exp(z), \,\exp(-(x^2 + y^2 + z^2)), \,\frac{1}{1 - xyz}\right)$$$

In this case it is more convenient to write a simple Julia script to compute the above integral

julia> using Cuba, SpecialFunctions

julia> function integrand(x, f)
f[1] = sin(x[1])*cos(x[2])*exp(x[3])
f[2] = exp(-(x[1]^2 + x[2]^2 + x[3]^2))
f[3] = 1/(1 - prod(x))
end
integrand (generic function with 1 method)

julia> result, err = cuhre(integrand, 3, 3, atol=1e-12, rtol=1e-10);

julia> answer = ((ℯ-1)*(1-cos(1))*sin(1), (sqrt(pi)*erf(1)/2)^3, zeta(3));

julia> for i = 1:3
println("Component ", i)
println(" Result of Cuba: ", result[i], " ± ", err[i])
println(" Actual error:   ", abs(result[i] - answer[i]))
end
Component 1
Result of Cuba: 0.6646696797813743 ± 1.0083313461375621e-13
Exact result:   0.6646696797813771
Actual error:   2.886579864025407e-15
Component 2
Result of Cuba: 0.4165383858806458 ± 2.9328672381493003e-11
Exact result:   0.41653838588663805
Actual error:   5.992262241960589e-12
Component 3
Result of Cuba: 1.202056903164971 ± 1.195855757269273e-10
Exact result:   1.2020569031595951
Actual error:   5.375921929839933e-12

### Integral with non-constant boundaries

The integral

$$$\int_{-y}^{y}\int_{0}^{z}\int_{0}^{\pi} \cos(x)\sin(y)\exp(z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$$

has non-constant boundaries. By applying the substitution rule repeatedly, you can scale the integrand function and get this equivalent integral over the fixed domain $\Omega = [0, 1]^{3}$

$$$\int\limits_{\Omega} 2\pi^{3}yz^2 \cos(\pi yz(2x - 1)) \sin(\pi yz) \exp(\pi z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$$

that can be computed with Cuba.jl using the following Julia script

julia> using Cuba

julia> function integrand(x, f)
f[1] = 2pi^3*x[2]*x[3]^2*cos(pi*x[2]*x[3]*(2*x[1] - 1.0))*
sin(pi*x[2]*x[3])*exp(pi*x[3])
end
integrand (generic function with 1 method)

julia> result, err = cuhre(integrand, 3, 1, atol=1e-12, rtol=1e-10);

julia> answer = pi*ℯ^pi - (4ℯ^pi - 4)/5;

julia> begin
println("Result of Cuba: ", result[1], " ± ", err[1])
println("Actual error:   ", abs(result[1] - answer))
end
Result of Cuba: 54.98607586826152 ± 5.460606620717379e-9
Exact result:   54.98607586789537
Actual error:   3.6614977716453723e-10

### Integrals over Infinite Domains

Cuba.jl assumes always as integration domain the hypercube $[0, 1]^n$, but we have seen that using integration by substitution we can calculate integrals over different domains as well. In the Introduction we also proposed two useful substitutions that can be employed to change an infinite or semi-infinite domain into a finite one.

As a first example, consider the following integral with a semi-infinite domain:

$$$\int_{0}^{\infty}\frac{\log(1 + x^2)}{1 + x^2}\,\mathrm{d}x$$$

whose exact result is $\pi\log 2$. This can be computed as follows:

julia> using Cuba

julia> # The function we want to integrate over [0, ∞).

julia> func(x) = log(1 + x^2)/(1 + x^2)
func (generic function with 1 method)

julia> # Scale the function in order to integrate over [0, 1].

julia> function integrand(x, f)
f[1] = func(x[1]/(1 - x[1]))/(1 - x[1])^2
end
integrand (generic function with 1 method)

julia> result, err = cuhre(integrand, atol = 1e-12, rtol = 1e-10);

julia> begin
println("Result of Cuba: ", result[1], " ± ", err[1])
println("Actual error:   ", abs(result[1] - answer))
end
Result of Cuba: 2.1775860903056903 ± 2.153947023352171e-10
Exact result:   2.177586090303602
Actual error:   2.0881074647149944e-12

Now we want to calculate this integral, over an infinite domain

$$$\int_{-\infty}^{\infty} \frac{1 - \cos x}{x^2}\,\mathrm{d}x$$$

which gives $\pi$. You can calculate the result with the code below. Note that integrand function has value $1/2$ for $x=0$, but you have to inform Julia about this.

julia> using Cuba

julia> # The function we want to integrate over (-∞, ∞).

julia> func(x) = x==0 ? 0.5*one(x) : (1 - cos(x))/x^2
func (generic function with 1 method)

julia> # Scale the function in order to integrate over [0, 1].

julia> function integrand(x, f)
f[1] = func((2*x[1] - 1)/x[1]/(1 - x[1])) *
(2*x[1]^2 - 2*x[1] + 1)/x[1]^2/(1 - x[1])^2
end
integrand (generic function with 1 method)

julia> result, err = cuhre(integrand, atol = 1e-7, rtol = 1e-7);

julia> begin
println("Result of Cuba: ", result[1], " ± ", err[1])
println("Actual error:   ", abs(result[1] - answer))
end
Result of Cuba: 3.1415928900554886 ± 2.050669142055499e-6
Exact result:   3.141592653589793
Actual error:   2.3646569546897922e-7

### Complex integrand

As already explained, Cuba.jl operates on real quantities, so if you want to integrate a complex-valued function of complex arguments you have to treat complex quantities as 2-component arrays of real numbers. For example, if you do not remember Euler's formula, you can compute this simple integral

$$$\int_{0}^{\pi/2} \exp(\mathrm{i} x)\,\mathrm{d}x$$$

with the following code

julia> using Cuba

julia> function integrand(x, f)
# Complex integrand, scaled to integrate in [0, 1].
tmp = cis(x[1]*pi/2)*pi/2
# Assign to two components of "f" the real
# and imaginary part of the integrand.
f[1], f[2] = reim(tmp)
end
integrand (generic function with 1 method)

julia> result = cuhre(integrand, 2, 2);

julia> begin
println("Result of Cuba: ", complex(result[1]...))
println("Exact result:   ", complex(1.0, 1.0))
end
Result of Cuba: 1.0000000000000002 + 1.0000000000000002im
Exact result:   1.0 + 1.0im

### Passing data to the integrand function

Cuba Library allows program written in C and Fortran to pass extra data to the integrand function with userdata argument. This is useful, for example, when the integrand function depends on changing parameters.

For example, the cumulative distribution function $F(x;k)$ of chi-squared distribution is defined by

$$$F(x; k) = \int_{0}^{x} \frac{t^{k/2 - 1}\exp(-t/2)}{2^{k/2}\Gamma(k/2)} \,\mathrm{d}t = \frac{x}{2^{k/2}\Gamma(k/2)} \int_{0}^{1} (xt)^{k/2 - 1}\exp(-xt/2) \,\mathrm{d}t$$$

The integrand depends on user-defined parameters $x$ and $k$. One option is passing a tuple (x, k) to the integrand using the userdata keyword argument in vegas, suave, divonne or cuhre. The following julia script uses this trick to compute $F(x = \pi; k)$ for different $k$ and compares the result with more precise values, based on the analytic expression of the cumulative distribution function, provided by GSL.jl package.

julia> using Cuba, GSL, Printf, SpecialFunctions

julia> function integrand(t, f, userdata)
# Chi-squared probability density function, without constant denominator.
# The result of integration will be divided by that factor.
# userdata is a tuple (x, k/2), see below
x, k = userdata
f[1] = (t[1]*x)^(k/2 - 1.0)*exp(-(t[1]*x)/2)
end

julia> chi2cdf(x::Real, k::Real) = x*cuhre(integrand; userdata = (x, k))[1][1]/(2^(k/2)*gamma(k/2))
chi2cdf (generic function with 1 method)

julia> x = float(pi);

julia> begin
@printf("Result of Cuba: %.6f %.6f %.6f %.6f %.6f\n",
map((k) -> chi2cdf(x, k), collect(1:5))...)
@printf("Exact result:   %.6f %.6f %.6f %.6f %.6f\n",
map((k) -> cdf_chisq_P(x, k), collect(1:5))...)
end
Result of Cuba: 0.923681 0.792120 0.629694 0.465584 0.321833
Exact result:   0.923681 0.792120 0.629695 0.465584 0.321833

An alternative solution to pass the user-defined data is implementing the integrand as a nested inner function. The inner function can access any variable visible in its scope.

julia> using Cuba, GSL, Printf, SpecialFunctions

julia> function chi2cdf(x::Real, k::Real)
k2 = k/2
# Chi-squared probability density function, without constant denominator.
# The result of integration will be divided by that factor.
function chi2pdf(t::Float64)
# "k2" is taken from the outside.
return t^(k2 - 1.0)*exp(-t/2)
end
# Neither "x" is passed directly to the integrand function,
# but is visible to it.  "x" is used to scale the function
# in order to actually integrate in [0, 1].
x*cuhre((t,f) -> f[1] = chi2pdf(t[1]*x))[1][1]/(2^k2*gamma(k2))
end
chi2cdf (generic function with 1 method)

julia> x = float(pi);

julia> begin
@printf("Result of Cuba: %.6f %.6f %.6f %.6f %.6f\n",
map((k) -> chi2cdf(x, k), collect(1:5))...)
@printf("Exact result:   %.6f %.6f %.6f %.6f %.6f\n",
map((k) -> cdf_chisq_P(x, k), collect(1:5))...)
end
Result of Cuba: 0.923681 0.792120 0.629694 0.465584 0.321833
Exact result:   0.923681 0.792120 0.629695 0.465584 0.321833

### Vectorized Function

Consider the integral

$$$\int\limits_{\Omega} \prod_{i=1}^{10} \cos(x_{i}) \,\mathrm{d}\boldsymbol{x} = \sin(1)^{10} = 0.1779883\dots$$$

where $\Omega = [0, 1]^{10}$ and $\boldsymbol{x} = (x_{1}, \dots, x_{10})$ is a 10-dimensional vector. A simple way to compute this integral is the following:

julia> using Cuba, BenchmarkTools

julia> cuhre((x, f) -> f[] = prod(cos.(x)), 10)
Component:
1: 0.17798706658707045 ± 1.070799596273229e-6 (prob.: 0.2438374079277991)
Integrand evaluations: 7815
Number of subregions:  2
Note: The desired accuracy was reached

julia> @benchmark cuhre((x, f) -> f[] = prod(cos.(x)), 10)
BenchmarkTools.Trial: 2448 samples with 1 evaluation.
Range (min … max):  1.714 ms …   7.401 ms  ┊ GC (min … max): 0.00% … 75.31%
Time  (median):     1.820 ms               ┊ GC (median):    0.00%
Time  (mean ± σ):   2.035 ms ± 858.367 μs  ┊ GC (mean ± σ):  9.08% ± 14.32%

██▅▅▄▁                                                   ▁  ▁
███████▇▆▆▄▁▄▁▁▁▄▃▃▅▁▁▃▃▁▃▃▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▅▆▇▇██▇ █
1.71 ms      Histogram: log(frequency) by time      5.97 ms <

Memory estimate: 2.03 MiB, allocs estimate: 39078.

We can use vectorization in order to speed up evaluation of the integrand function.

julia> function fun_vec(x,f)
f[1,:] .= 1.0
for j in 1:size(x,2)
for i in 1:size(x, 1)
f[1, j] *= cos(x[i, j])
end
end
end
fun_vec (generic function with 1 method)

julia> cuhre(fun_vec, 10, nvec = 1000)
Component:
1: 0.17798706658707045 ± 1.070799596273229e-6 (prob.: 0.2438374079277991)
Integrand evaluations: 7815
Number of subregions:  2
Note: The desired accuracy was reached

julia> @benchmark cuhre(fun_vec, 10, nvec = 1000)
BenchmarkTools.Trial: 4971 samples with 1 evaluation.
Range (min … max):  951.837 μs …  1.974 ms  ┊ GC (min … max): 0.00% … 0.00%
Time  (median):     981.597 μs              ┊ GC (median):    0.00%
Time  (mean ± σ):     1.001 ms ± 69.101 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%

█▂  ▁▆▇▁▂▂▂▆▅▂▁▁▁▁                                           ▁
██▇▆█████████████████████▇██▇████▇▆▇█▆▆▆▇▆▅▆▆█▇▅▅▅▅▄▅▆▄▅▄▂▃▂ █
952 μs        Histogram: log(frequency) by time      1.24 ms <

Memory estimate: 1.59 KiB, allocs estimate: 39.

A further speed up can be gained by running the for loop in parallel with Threads.@threads. For example, running Julia with 4 threads:

julia> function fun_par(x,f)
f[1,:] .= 1.0
for i in 1:size(x, 1)
f[1, j] *= cos(x[i, j])
end
end
end
fun_par (generic function with 1 method)

julia> cuhre(fun_par, 10, nvec = 1000)
Component:
1: 0.17798706658707045 ± 1.070799596273229e-6 (prob.: 0.2438374079277991)
Integrand evaluations: 7815
Number of subregions:  2
Note: The desired accuracy was reached

julia> @benchmark cuhre(fun_par, 10, nvec = 1000)
BenchmarkTools.Trial: 6198 samples with 1 evaluation.
Range (min … max):  587.456 μs …   7.076 ms  ┊ GC (min … max): 0.00% … 0.00%
Time  (median):     798.933 μs               ┊ GC (median):    0.00%
Time  (mean ± σ):   801.498 μs ± 149.908 μs  ┊ GC (mean ± σ):  0.18% ± 1.50%

▇█▇▃
▂▄▆▃▂▃▃▂▂▂▂▂▂▁▂▁▁▁▁▂▄▅▃▂▅██████▆▆▆▆▆█▇▇▆▄▄▃▂▃▂▂▂▂▂▂▁▂▁▁▁▁▁▁▁▁ ▃
587 μs           Histogram: frequency by time         1.03 ms <

Memory estimate: 19.31 KiB, allocs estimate: 228.

## Performance

Cuba.jl cannot (yet?) take advantage of parallelization capabilities of Cuba Library. Nonetheless, it has performances comparable with equivalent native C or Fortran codes based on Cuba library when CUBACORES environment variable is set to 0 (i.e., multithreading is disabled). The following is the result of running the benchmark present in test directory on a 64-bit GNU/Linux system running Julia 0.7.0-beta2.3 (commit 83ce9c7524) equipped with an Intel(R) Core(TM) i7-4700MQ CPU. The C and FORTRAN 77 benchmark codes have been compiled with GCC 7.3.0.

$CUBACORES=0 julia -e 'using Pkg; import Cuba; include(joinpath(dirname(dirname(pathof(Cuba))), "benchmarks", "benchmark.jl"))' [ Info: Performance of Cuba.jl: 0.257360 seconds (Vegas) 0.682703 seconds (Suave) 0.329552 seconds (Divonne) 0.233190 seconds (Cuhre) [ Info: Performance of Cuba Library in C: 0.268249 seconds (Vegas) 0.682682 seconds (Suave) 0.319553 seconds (Divonne) 0.234099 seconds (Cuhre) [ Info: Performance of Cuba Library in Fortran: 0.233532 seconds (Vegas) 0.669809 seconds (Suave) 0.284515 seconds (Divonne) 0.195740 seconds (Cuhre) Of course, native C and Fortran codes making use of Cuba Library outperform Cuba.jl when higher values of CUBACORES are used, for example: $ CUBACORES=1 julia -e 'using Pkg; cd(Pkg.dir("Cuba")); include("test/benchmark.jl")'
[ Info: Performance of Cuba.jl:
0.260080 seconds (Vegas)
0.677036 seconds (Suave)
0.342396 seconds (Divonne)
0.233280 seconds (Cuhre)
[ Info: Performance of Cuba Library in C:
0.096388 seconds (Vegas)
0.574647 seconds (Suave)
0.150003 seconds (Divonne)
0.102817 seconds (Cuhre)
[ Info: Performance of Cuba Library in Fortran:
0.094413 seconds (Vegas)
0.556084 seconds (Suave)
0.139606 seconds (Divonne)
0.107335 seconds (Cuhre)

Cuba.jl internally fixes CUBACORES to 0 in order to prevent from forking julia processes that would only slow down calculations eating up the memory, without actually taking advantage of concurrency. Furthemore, without this measure, adding more Julia processes with addprocs() would only make the program segfault.

There are other Julia packages for multidimenensional numerical integration:

## Development

Cuba.jl is developed on GitHub: https://github.com/giordano/Cuba.jl. Feel free to report bugs and make suggestions at https://github.com/giordano/Cuba.jl/issues.

### History

The ChangeLog of the package is available in NEWS.md file in top directory. There have been some breaking changes from time to time, beware of them when upgrading the package.