Please provide Matlab code program AND the $2$ contour plots for the two different initial starting points. Make sure your code can be run too. Also, figure $1$ is the sample plot for a different f$($x$).$

Program the gradient descent and Newton algorithms using the backtracking line search $($Procedure below, set $=0\mathrm{.}9,=0\mathrm{.}6).$ 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. $($You can use 'help contour'

and 'help plot' to find instructions of using 'contour' and 'plot'. A possible

way to plot with one initial point is given below.$)$

Procedure $($Backtracking Line Search$).$

Choose $,in(0,1)$; Set $=1$;

repeat until $f(x_k+t{p}_k)f(x_k)+tgradf(x_k{)}^T{p}_k$

$t=t$;

end $($repeat$)$

Terminate with $t_k=t.$

Figure $1$: Q$3$: Plot with $f(x,y)=x^2+y^4$