$(3)$ Write a Matlab routine for implementing the steepest descent method. $($The routine should

be written from scratch

you to write a generalized function that works for any function to optimize. Its inputs could be

"initial vector" $x^{(0)}$ and the function to be optimized, and outputs can be the optimal values and

error through iterations.$)$

Your routine should be able to find the minimum of a function with multiple variables using the

following step size methods:

i$)$ constant step size

ii$)$ variable step size

iii$)$ quadratic fitting

$($A$)$ Execute your routine for minimizing Rosenbrock function :

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

Terminate the algorithm when If the number of iterations exceeds $15000,$

terminate it regardless of the norm of the gradient. Initialize the input vector of the first

iteration as $x^{(0)}=[35].$

$($B$)$ Plot the error as $||gradf||$ with respect to the iteration number for each step size methods. $,$

$($C$)$ Compare the convergence rates, computation times, final error and final iteration number

for each method in a neat Table. Give your brief comments.