# Solving a nonlinear system of equations in Julia

**URL:** <https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261>\
**Category:** General Usage\
**Created:** [August 7, 2017, 3:56pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261 "2017-08-07T15:56:20Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Petre\_C](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petre_c/32/36167_2.png) [@Petre\_C](https://discourse.julialang.org/u/Petre_C)\
**Post date:** [August 7, 2017, 3:56pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261/1 "2017-08-07T15:56:20Z")

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I can write in function in Matlab in this way:

function res=resid(theta,alpha,beta);  
RHS=;  
LHS=;  
RHS= theta-alpha;  
LHS= theta\*beta;  
res = (LHS-RHS);

We set the parameters, call the function:

alpha=0.3;beta=0.95;  
a01=[1.0;1.0];  
th=fsolve(‘resid’,a01,,alpha,beta)  
This will return [6.0;6.0]. Does the option"" signal to fsolve that the input is a vector?

Anyway, how can I implement this in Julia using a NLsolve, Optim or JuMP? The original problem has more than 10 variables, so I would prefer a vector approach.

I can implement the function in Julia:

h! =function (theta)  
RHS=;  
LHS=;  
RHS= theta-alpha;  
LHS= theta\*beta;  
res= (LHS-RHS);  
return res;  
end  
But simply using NLsolve:

a01 = [1.0;1.0];  
res = nlsolve(h!,a01)  
Returns:

MethodError: no method matching (::##17#18)(::Array{Float64,1}, ::Array{Float64,1})  
Closest candidates are:  
#17(::Any) at In[23]:3  
If I alternatively use Optim, I get:

using Optim  
optimize(h!, a01)  
which returns:

MethodError: Cannot `convert` an object of type Array{Float64,1} to an object of type Float64  
This may have arisen from a call to the constructor Float64(…),  
since type constructors fall back to convert methods.

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**Author:** ![ohsonice](https://avatars.discourse-cdn.com/v4/letter/o/d6d6ee/32.png) [@ohsonice](https://discourse.julialang.org/u/ohsonice)\
**Post date:** [August 7, 2017, 5:04pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261/2 "2017-08-07T17:04:11Z")

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A couple things to point out

1. Semi colons are not necessary within functions
2. In julia, when a function ends in an explamation point `!` such as `h!`, that _should_ signify that it mutates (changes) one of its inputs ( I say should since this is not enforced in the language but is a naming convention)
3. When showing code on this forum, you should put your code in “backticks” (the key under tilda) for readability: `this is enclosed in backticks `

Now for the code, the error message you get from nlsolve has to do with point #2. NLsolve expects your function to take the residual vector and simply mutate it as opposed to returning a new vector. Here’s an example that works.

```julia
julia> using NLsolve

julia> function h!(theta, resvec, alpha, beta)
       resvec[1] = theta[1]*beta - (theta[1] - alpha)
       resvec[2] = theta[2]*beta - (theta[2] - alpha)
       end

julia> function f!(theta, resvec)
           h!(theta,resvec, 0.3, 0.95)
       end

julia> a01 = [1.0,1.0];

julia> nlsolve(f!,a01)
Results of Nonlinear Solver Algorithm
 * Algorithm: Trust-region with dogleg and autoscaling
 * Starting Point: [1.0,1.0]
 * Zero: [6.0,6.0]
 * Inf-norm of residuals: 0.000000
 * Iterations: 3
 * Convergence: true
   * |x - x'| < 0.0e+00: false
   * |f(x)| < 1.0e-08: true
 * Function Calls (f): 4
 * Jacobian Calls (df/dx): 4

```

I’m sure there are other ways to handle the alpha and beta parameters besides making another function.

Good reading materials:

> **[GitHub - JuliaNLSolvers/NLsolve.jl: Julia solvers for systems of nonlinear...](https://github.com/JuliaNLSolvers/NLsolve.jl)**
>
> Julia solvers for systems of nonlinear equations and mixed complementarity problems - GitHub - JuliaNLSolvers/NLsolve.jl: Julia solvers for systems of nonlinear equations and mixed complementarity ...

[https://docs.julialang.org/en/release-0.4/manual/noteworthy-differences/](https://docs.julialang.org/en/release-0.4/manual/noteworthy-differences/)

ETA: And to clarify for your other example, `Optim` is used to optimize a function, this is different than solving it, the error you get is because you are returning a vector, while optimization is performed on a scalar (cost) quantity). Hope this helps!

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**Author:** ![smldis](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/smldis/32/1735_2.png) [@smldis](https://discourse.julialang.org/u/smldis)\
**Post date:** [August 7, 2017, 5:04pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261/3 "2017-08-07T17:04:42Z")

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With Optim, from the error I see, I think you need to return just a number instead of an array. You didn’t specified exactly what you want to minimize, but I assume it’s some norm of resulting array. Try to return the norm [https://docs.julialang.org/en/latest/stdlib/linalg/#Base.LinAlg.norm](https://docs.julialang.org/en/latest/stdlib/linalg/#Base.LinAlg.norm).

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**Author:** ![mkborregaard](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mkborregaard/32/556_2.png) [@mkborregaard](https://discourse.julialang.org/u/mkborregaard)\
**Post date:** [August 7, 2017, 5:11pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261/4 "2017-08-07T17:11:10Z")

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You should cross-refer when double-posting: [Solving a nonlinear system of equations in Julia - Stack Overflow](https://stackoverflow.com/questions/45545275/solving-a-nonlinear-system-of-equations-in-julia)

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<div class="post-metadata">

**Author:** ![Petre\_C](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petre_c/32/36167_2.png) [@Petre\_C](https://discourse.julialang.org/u/Petre_C)\
**Post date:** [August 7, 2017, 7:48pm UTC](https://discourse.julialang.org/t/solving-a-nonlinear-system-of-equations-in-julia/5261/5 "2017-08-07T19:48:46Z")

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Hello!

Thanks everybody for helping!

Sorry for not cross-refering. Anyway, I posted this before getting to solution. Thanks ohsonice for your solution too.