# How to call multi-dimensional vector value Vectorized Cubature numerical integration?

**URL:** <https://discourse.julialang.org/t/how-to-call-multi-dimensional-vector-value-vectorized-cubature-numerical-integration/90200>\
**Category:** General Usage\
**Tags:** package, integral\
**Created:** [November 13, 2022, 6:26pm UTC](https://discourse.julialang.org/t/how-to-call-multi-dimensional-vector-value-vectorized-cubature-numerical-integration/90200 "2022-11-13T18:26:03Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![swish47](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/swish47/32/43921_2.png) [@swish47](https://discourse.julialang.org/u/swish47)\
**Post date:** [November 13, 2022, 6:26pm UTC](https://discourse.julialang.org/t/how-to-call-multi-dimensional-vector-value-vectorized-cubature-numerical-integration/90200/1 "2022-11-13T18:26:03Z")

</div>

As the topic, I want to use parallelized numerical integration. However, since the special receives of `hcubature` with vector-valued integrands, and I don’t know how to set each components of v respectively, there is a problem for calling higher dimensional case. Here are my attempts. First I use f(x,y)=x^2+y^2 as an 1D example:

```julia
using Cubature,BenchmarkTools
@btime hcubature(1,(x,v) -> v[1]= x[1]^2+x[2]^2, (0,0),(1,1))
  2.222 μs (73 allocations: 2.42 KiB)
([0.6666666666666665], [2.220446049250313e-16])
@btime hcubature(1,(x,v) -> v[1]= x[1]^2+x[2]^2, (0,0),(1,1))
  2.222 μs (73 allocations: 2.42 KiB)
([0.6666666666666665], [2.220446049250313e-16])
@btime hcubature_v((x,v) -> v[:]= x[1,:].^2+x[2,:].^2, (0,0),(1,1))
  1.020 μs (14 allocations: 1.38 KiB)
(0.6666666666666665, 2.220446049250313e-16)
@btime hcubature_v(1,(x,v) -> v[:]= x[1,:].^2+x[2,:].^2, (0,0),(1,1))
1.020 μs (14 allocations: 1.39 KiB)
([0.6666666666666665], [2.220446049250313e-16])

```

Both normal version and vectorized version works well.  
Now let’s turn to a 2D example, f(x,y)=(x^2+y^2,x^3+y^3):

```julia
@btime hcubature(2,(x,v) -> v[:]=[x[1]^2+x[2]^2,x[1]^3+x[2]^3], (0,0),(1,1))
  2.756 μs (90 allocations: 3.78 KiB)
([0.6666666666666665, 0.4999999999999999], [2.220446049250313e-16, 1.1102230246251565e-16])
@btime hcubature_v(2,(x,v) -> v[1:2,:]=[x[1,:].^2+x[2,:].^2;x[1,:].^3+x[2,:].^3], (0,0),(1,1))
@btime hcubature_v(2,(x,v) -> v[:,:]=[x[1,:].^2+x[2,:].^2;x[1,:].^3+x[2,:].^3], (0,0),(1,1))

```

Unparallel integration is collect but vectorized versions return errors. I also tried `let` and to define a `function` as an integrand outside then use. But all of them failed. I think I should know how to set integrand `(x,v) -> v[1,:],v[2,:]...` respectively but not define as a list. Then I can find a way to use.

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<div class="post-metadata">

**Author:** ![swish47](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/swish47/32/43921_2.png) [@swish47](https://discourse.julialang.org/u/swish47)\
**Post date:** [November 13, 2022, 7:00pm UTC](https://discourse.julialang.org/t/how-to-call-multi-dimensional-vector-value-vectorized-cubature-numerical-integration/90200/2 "2022-11-13T19:00:34Z")

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I find one method, by using `begin` and `end`.

```julia
using Cubature,BenchmarkTools
@btime hcubature(2,((x,v) -> begin v[1]=x[1]^2+x[2]^2;v[2]=x[1]^3+x[2]^3;v end), (0,0),(1,1))
2.244 μs (73 allocations: 2.45 KiB)
([0.6666666666666665, 0.4999999999999999], [2.220446049250313e-16, 1.1102230246251565e-16])
@btime hcubature_v(2,(x,v) -> begin v[1,:]=x[1,:].^2+x[2,:].^2; v[2,:]=x[1,:].^3+x[2,:].^3;v end, (0,0),(1,1))
  1.230 μs (19 allocations: 2.36 KiB)
([0.6666666666666665, 0.4999999999999999], [2.220446049250313e-16, 1.1102230246251565e-16])

```

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<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [November 13, 2022, 9:16pm UTC](https://discourse.julialang.org/t/how-to-call-multi-dimensional-vector-value-vectorized-cubature-numerical-integration/90200/3 "2022-11-13T21:16:56Z")

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> [@swish47](#):
>
> `[x[1,:].^2+x[2,:].^2;x[1,:].^3+x[2,:].^3]`

This won’t work — it just appends everything into a long 1d vector, because a slice like `x[1,:]` is a 1d array (“column vector”), not a 1-row matrix as in Matlab.

(Note also that for efficiency I would tend to try to work in-place in your integrands rather than generating lots of temporary arrays.)

Might be easier to write things out as explicit loops. See, for example, [this test integrand](https://github.com/JuliaMath/Cubature.jl/blob/7a3d8e030c7b0f37d1fc8d74dc0833ca1c7e4586/test/runtests.jl#L53-L57) which is used with `hcubature_v`.