Modify the gradient$-$descent code to implement Newton's method:

$x^{i+1}=x^i-[{}^{2}f(x^i){]}^{-1}(gradf(x^i))$

where $$ is the step$-$size $($you may choose $=1$ for this assignment$).$ Here, ${}^{2}f$ is the Hessian $($i$.$e$.,$

second derivative$)$ and gradf is the gradient $($i$.$e$.,$ first derivative$)$ of $f.$

Make sure the parameters are set to the following in your code:

$\%$ define starting point

x$0=6\mathrm{.}5$;

$\%$ termination tolerance

tol $=1$e$-6$;

$\%$ maximum number of allowed iterations

maxiter $=1000$;

$\%$ minimum allowed perturbation

dxmin $=1$e$-6$;

$\%$ step size

alpha $=\mathrm{.}02$;

$($a$)$ Solve the optimization problem $x^2+\frac{3}{2}si{n}^{2}x$ using your developed Newton method. Report the

solution $x^{**}.$ Comment whether the solution changes when you vary the initial guess $x_0.$

$($b$)$ Compare the number of iterations taken by the Newton's method and the standard gradient descent

method.