$($b$).(20$ marks$)$ Implement the Gauss$-$Newton method to solve the minimization problem

without line search in Matlab. The function head for the Gauss$-$Newton method is

function $[x,it,r]=$ Gauss$\_$Newton$($fun$,x0,$ itmax, tol$)$

$\%$

$\%$ Gauss$-$Newton method for solving nonlinear least squares problem without line search

$\%$

$\%$ Input

$\%$ fun $-$ function $F(x),$ the objective function is $\frac{1}{2}**F^{\text{'}}**F,$ which returns

$\%F(x)$ and corresponding Jacobian matrix $J.$

$\%0-$ initial value of $x$

$\%$ itmax $-$ max number of iteration

$\%$ tol $-$ stopping tolerance

$\%$

$\%$ Output

$\%x-$ final result

$\%$ it $-$ number of iterations

$\%r-$ objective function value $\frac{1}{2}**F^{\text{'}}**F$