Program the gradient descent and Newton algorithms using the backtracking

line search $($Procedure below, set $=0\mathrm{.}9,c=0\mathrm{.}6).$ Use them to minimize the

Rosenbrock function

$f(x)=100(x_2-x_1^2{)}^2+(1-x_1{)}^2.$

First try the initial point $x_0=(1\mathrm{.}2$;$1\mathrm{.}2)$ and then the more difficult point

$x_0=(-1\mathrm{.}2$;$1).$ Plot the contour of the function and then plot the path of

iterates obtained by each method on the contour using matlab or python. $($You can use 'help contour'

and 'help plot' to find instructions of using 'contour' and 'plot'.$)$

Procedure $($Backtracking Line Search$).$

Choose $,$cin$(0,1)$; Set $larr1$;

repeat until $f(x_k+p_k)f(x_k)+cgradf(x_k{)}^T{p}_k$

$,larr$;

end $(repeat)$

Terminate with ${}_k=.$