Write a numerical code to minimize the Rosenbrock function using a slew of gradient based optimization below. The function is a non$-$convex function and has a global minimum at

$(1,1).$ The minimum is at the bottom of a narrow parabolic valley that is curved on the x$-$y plane.

The function is often used to test the performance of optimization algorithms. Employ a constant step

length throughout the iterations. Experiment to obtain a reasonable value for the step length $\backslash$alpha $.$

f$($x$,$ y$)=(1$ x$)^2+100($y $$ x$^2)^2$

Write three codes using the nonlinear conjugate gradient, quasi$-$Newton, and Newton method to

solve for the minimum of the function. Use a simple backtracking line search method to compute

the step length. You may choose one particular algorithm for the conjugate gradient $($either the

Hestenes$-$Stiefel, Polak$-$Ribiere, or Fletcher$-$Reeves$)$ and quasi$-$Newton $($DFP or BFGS$)$ method

Provide the following

$($a$)$ Convergence of the gradient $($y$-$axis: log of the gradient, x$-$axis: iteration$).$

$($b$)$ Contour plot of the path of the optimization algorithm