# 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.
See optimset for details.

## 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 user 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

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'