# fsolve

Find a solution of a system of real nonlinear equations.

## Syntax

x = fsolve(@func,x0)

x = fsolve(@func,x0,options)

[x,fval,info,output] = fsolve(...)

## Inputs

`func`- The system to solve. See the optimset option Jacobian for details.
`x0`- An estimate of the solution.
`options`- A struct containing option settings.

## Outputs

- x
- A solution of the system.
- fval
- The value of
`func`evaulated at`x`. - info
- The convergence status flag.
- info = 4
- Relative step size converged to within tolX.
- info = 3
- Relative function value converged to within tolFun.
- info = 2
- Step size converged to within tolX.
- info = 1
- Function value converged to within tolFun.
- info = 0
- Reached maximum number of iterations or function calls, or the algorithm aborted because it was not converging.
- info = -3
- Trust region became too small to continue.

- output
- A struct containing iteration details. The members are as follows:
- iterations
- The number of iterations.
- nfev
- The number of function evaluations.
- xiter
- The candidate solution at each iteration.
- fvaliter
- The objective function value at each iteration.

## Examples

Solve the system of equations
SysFunc.

```
function res = SysFunc(x)
% intersection of two paraboloids and a plane
v1 = (x(1))^2 + (x(2))^2 + 6;
v2 = 2*(x(1))^2 + 2*(x(2))^2 + 4;
v3 = 5*x(1) - 5*x(2);
res = zeros(2,1);
res(1,1) = v1 - v3;
res(2,1) = v2 - v3;
end
x0 = [1; 2];
[x,fval] = fsolve(@SysFunc,x0)
```

```
x = [Matrix] 2 x 1
1.40000
-0.20000
fval = [Matrix] 2 x 1
3.67339e-07
7.34677e-07
```

Modify the previous example to pass an extra parameter to the function using a function
handle.

```
function res = SysFunc(x, offset)
% intersection of two paraboloids and a plane
v1 = (x(1))^2 + (x(2))^2 + offset(1);
v2 = 2*(x(1))^2 + 2*(x(2))^2 + offset(2);
v3 = 5*x(1) - 5*x(2) + offset(3);
res = zeros(2,1);
res(1,1) = v1 - v3;
res(2,1) = v2 - v3;
end
handle = @(x) SysFunc(x, [8,6,4]);
[x,fval] = fsolve(handle,x0)
```

```
x = [Matrix] 2 x 1
1.40000
0.20000
fval = [Matrix] 2 x 1
0
0
```

## Comments

fsolve uses a modified Gauss-Netwon algorithm with a trust region method.

Options for convergence tolerance controls and analytical derivatives are specified with optimset.

If fsolve converges to a solution that is not a zero of the system, it will produce a warning indicating that a best fit value is being returned.

To pass additional parameters to a function argument, use an anonymous function.

The optimset options and defaults are as follows.

- MaxIter: 400
- MaxFunEvals: 1,000,000
- TolFun: 1.0e-7
- TolX: 1.0e-7
- Jacobian: 'off'
- Display: 'off'