## Generate Code for `fsolve`

This example shows how to generate C code for solving systems of nonlinear equations with `fsolve`.

### Equation to Solve

The system of nonlinear equations to solve is

`$\begin{array}{c}{e}^{-{e}^{-\left({x}_{1}+{x}_{2}\right)}}={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}.\end{array}$`

Convert the equations to the form F(x) = 0.

`$\begin{array}{c}{e}^{-{e}^{-\left({x}_{1}+{x}_{2}\right)}}-{x}_{2}\left(1+{x}_{1}^{2}\right)=0\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)-\frac{1}{2}=0.\end{array}$`

### Code Generation Steps

1. Write a function that computes the left side of the two equations. For code generation, your program must allocate all arrays when they are created, and must not change their sizes after creation.

```function F = root2d(x) F = zeros(2,1); % Allocate return array F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5; end```
2. Write a function that sets up the problem and calls `fsolve`. The function must refer to `root2d` as a function handle, not as a name.

```function [x,fval] = solveroot options = optimoptions('fsolve','Algorithm','levenberg-marquardt','Display','off'); fun = @root2d; rng default x0 = rand(2,1); [x,fval] = fsolve(fun,x0,options); end```
3. Create a configuration for code generation. In this case, use `'mex'`.

`cfg = coder.config('mex');`
4. Generate code for the `solveroot` function.

`codegen -config cfg solveroot`
5. Test the generated code by running the generated file, which is named `solveroot_mex.mexw64` or similar.

`[x,fval] = solveroot_mex`
```x = 0.3532 0.6061 fval = 1.0e-14 * -0.1998 -0.1887```