Minimizer of a multivariable function and iteration through Newton's method

Question

I got stuck on the following question.

Find the minimizer for $$f (x_1,x_2) = \frac 12 (x_1^2 - x_2)^2 + \frac 12 (1-x_1)^2$$ and compute one iteration for minimizing $f$ from point $(2,2)$. Also, is this a good step?

Answer 1

$f(x_1,x_2)$ is a well known test function. It is the Rosenbrock function. The Newton method

$$ X_{k+1}=X_k - H^{-1}_k \nabla f_k $$

gives

$$ \left[ \begin{array}{ccc} x_1 & x_2 & f(x_1,x_2)\\ 2 & 2 & 2.5 \\ 1.8 & 2. & 0.3208 \\ 1.05926 & 3.2 & 0.15229 \\ 1.03101 & 0.573333 & 0.000480989 \\ 1.00005 & 1.06217 & \text{4.603702725}*{10}^{-7} \\ 1. & 0.999141 & \text{4.47192928}*{10}^{-15} \\ \end{array} \right] $$