# \[NLsolve\] Providing the Jacobian of a system when variables are provided in an Array?

**URL:** <https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823>\
**Category:** New to Julia\
**Tags:** nlsolve\
**Created:** [May 9, 2021, 1:49pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823 "2021-05-09T13:49:12Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![dadada\_dadada](https://avatars.discourse-cdn.com/v4/letter/d/ce7236/32.png) [@dadada\_dadada](https://discourse.julialang.org/u/dadada_dadada)\
**Post date:** [May 9, 2021, 1:49pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/1 "2021-05-09T13:49:12Z")

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When solving systems of equation, sometimes, it’s better to have the variables in Array format, instead of a vector.

That said, can we provide multidimensional Jacobians to the solvers of the `NLsolve` package?

For example, I want to solve the system:

\begin{array}{r} 3 x\_{11}^{3}+2 x\_{11} x\_{21}=5 \\ x\_{21}^{3}+2 x\_{22}=3 \\ 2 x\_{12}+3 x\_{22}^{3}=5 \\ 2 x\_{12}+3 x\_{21}=5 \end{array}

The variables are represented in array format (instead of a vector).

x=\left[\begin{array}{ll} x\_{11} & x\_{12} \\ x\_{21} & x\_{22} \end{array}\right]

The solution is:

x^{\*}=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]

Solving the system in vector format looks like this.

```julia
using NLsolve

# solution [1,1,1,1]

function f!(F, x)
    F[1]= 3*x[1]^3 + 2*x[1]*x[2] - 5.
    F[2]= x[2]^3 + 2*x[4] - 3.
    F[3]= 2*x[3] + 3*x[4]^3 - 5.
    F[4]= 2*x[3] + 3*x[2] - 5.
end

function j!(J, x)
    J[1, 1] = 9*x[1]^2 + 2*x[2]
    J[1, 2] = 2*x[1]
    J[1, 3] = 0.
    J[1, 4] = 0.

    J[2, 1] = 0.
    J[2, 2] = 3*x[2]^2
    J[2, 3] = 0.
    J[2, 4] = 2.

    J[3, 1] = 0.
    J[3, 2] = 0.
    J[3, 3] = 2.
    J[3, 4] = 9*x[4]^2

    J[4, 1] = 0.
    J[4, 2] = 3.
    J[4, 3] = 2.
    J[4, 4] = 0.
end

nlsolve(f!,j!,[0., 0., 0., 0.])

```

Solving it while representing the variables in an array looks like this.

```julia
## system in "matrix form"

function f!(F, x)
    F[1,1] = 3*x[1,1]^3 + 2*x[1,1]*x[2,1] - 5.
    F[2,1] = x[2,1]^3 + 2*x[2,2] - 3.
    F[1,2] = 2*x[1,2] + 3*x[2,2]^3 - 5.
    F[2,2] = 2*x[1,2] + 3*x[2,1] - 5.
end

nlsolve(f!, [0. 0.; 0. 0.])

```

Is there any way to provide the Jacobian to `NLsolve` in the second case?

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**Author:** ![jishnub](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jishnub/32/33620_2.png) [@jishnub](https://discourse.julialang.org/u/jishnub)\
**Post date:** [May 9, 2021, 7:34pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/2 "2021-05-09T19:34:50Z")

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Why don’t you pass a vector of parameters and reshape it to a matrix within `f!`? This way the Jacobian is still a matrix. You may reshape the vector to a matrix once you have your solution

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**Author:** ![dadada\_dadada](https://avatars.discourse-cdn.com/v4/letter/d/ce7236/32.png) [@dadada\_dadada](https://discourse.julialang.org/u/dadada_dadada)\
**Post date:** [May 9, 2021, 11:09pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/3 "2021-05-09T23:09:12Z")

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For convenience. The actual inputs of my non-linear system are multidimensional arrays that represent the value functions of different states of the world. To reshape that is very cumbersome.

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**Author:** ![amrods](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/amrods/32/2543_2.png) [@amrods](https://discourse.julialang.org/u/amrods)\
**Post date:** [May 9, 2021, 11:27pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/4 "2021-05-09T23:27:40Z")

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If you “flatten” the vectors with `vec`, and then reshape with `reshape`, it should yield the same results. Although, `NLsolve` takes any `AbstractArray`.

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**Author:** ![dadada\_dadada](https://avatars.discourse-cdn.com/v4/letter/d/ce7236/32.png) [@dadada\_dadada](https://discourse.julialang.org/u/dadada_dadada)\
**Post date:** [May 9, 2021, 11:37pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/5 "2021-05-09T23:37:13Z")

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@amrods, yes… `NLsolve` takes any `AbstractArray` and works fine with arrays as inputs if you don’t provide the Jacobian. I didn’t see anything in the Docs about multidimensional Jacobians tho.

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**Author:** ![amrods](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/amrods/32/2543_2.png) [@amrods](https://discourse.julialang.org/u/amrods)\
**Post date:** [May 9, 2021, 11:40pm UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/6 "2021-05-09T23:40:48Z")

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What if you supply the Jacobian with autodiff? Even if you really want to supply analytical expressions, it can serve you to compare what is the structure of a multidimensional Jacobian.

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**Author:** ![pkofod](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/pkofod/32/2179_2.png) [@pkofod](https://discourse.julialang.org/u/pkofod)\
**Post date:** [May 11, 2021, 5:53am UTC](https://discourse.julialang.org/t/nlsolve-providing-the-jacobian-of-a-system-when-variables-are-provided-in-an-array/60823/7 "2021-05-11T05:53:41Z")

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The structure with AD is a matrix for the Jaobian and `vec`ed input. I’m not sure there’s a another/better way to do it, because the linear algebra has to work out. Currently, all input is `vec`ed where it’s used. If it’s an `AbstractVector` it’s a noop. There no similar `mat` that makes it NxM as far as I’m aware, and it’d be ambiguous either way because there are potentially many possible matrices that fit the number of elements.

The “input a containter of a different shape as your variables” is a convenience layer. It doesn’t work well for Jacobians. We have the same in Optim where you can input non-`Vector` variables and gradients, but if you want to use a Hessian-based optimizer, you have to use a Matrix.