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https://github.com/SciML/Integrals.jl

A common interface for quadrature and numerical integration for the SciML scientific machine learning organization
https://github.com/SciML/Integrals.jl

algorithmic-differentiation automatic-differentiation differentiable-programming integration julia julia-language julialang numerical-integration quadrature scientific-machine-learning sciml

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A common interface for quadrature and numerical integration for the SciML scientific machine learning organization

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README 

        
# Integrals.jl



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Integrals.jl is an instantiation of the SciML common `IntegralProblem`

interface for the common numerical integration packages of Julia, including

both those based upon quadrature as well as Monte-Carlo approaches. By using

Integrals.jl, you get a single predictable interface where many of the

arguments are standardized throughout the various integrator libraries. This

can be useful for benchmarking or for library implementations, since libraries

which internally use a quadrature can easily accept a integration method as an

argument.



## Tutorials and Documentation



For information on using the package,

[see the stable documentation](https://docs.sciml.ai/Integrals/stable/). Use the

[in-development documentation](https://docs.sciml.ai/Integrals/dev/) for the version of

the documentation, which contains the unreleased features.



## Examples



To perform one-dimensional quadrature, we can simply construct an `IntegralProblem`. The code below evaluates $\int_{-2}^5 \sin(xp)~\mathrm{d}x$ with $p = 1.7$. This argument $p$ is passed

into the problem as the third argument of `IntegralProblem`.



```julia

using Integrals

f(x, p) = sin(x * p)

p = 1.7

domain = (-2, 5) # (lb, ub)

prob = IntegralProblem(f, domain, p)

sol = solve(prob, QuadGKJL())

```



For basic multidimensional quadrature we can construct and solve a `IntegralProblem`. Since we are using no arguments `p` in this example, we omit the third argument of `IntegralProblem`

from above. The lower and upper bounds are now passed as vectors, with the `i`th elements of

the bounds giving the interval of integration for `x[i]`.



```julia

using Integrals

f(x, p) = sum(sin.(x))

domain = (ones(2), 3ones(2)) # (lb, ub)

prob = IntegralProblem(f, domain)

sol = solve(prob, HCubatureJL(), reltol = 1e-3, abstol = 1e-3)

```



If we would like to parallelize the computation, we can use the batch interface

to compute multiple points at once. For example, here we do allocation-free

multithreading with Cubature.jl:



```julia

using Integrals, Cubature, Base.Threads

function f(dx, x, p)

    Threads.@threads for i in 1:size(x, 2)

        dx[i] = sum(sin, @view(x[:, i]))

    end

end

domain = (ones(2), 3ones(2)) # (lb, ub)

prob = IntegralProblem(BatchIntegralFunction(f, zeros(0)), domain)

sol = solve(prob, CubatureJLh(), reltol = 1e-3, abstol = 1e-3)

```



If we would like to compare the results against Cuba.jl's `Cuhre` method, then

the change is a one-argument change:



```julia

using Cuba

sol = solve(prob, CubaCuhre(), reltol = 1e-3, abstol = 1e-3)

```