Q:- Minimize $f(x_1,x_2)=2x_1-2x_2+2x_1^2+2x_1x_2+x_2^2 $ starting from the point

$\begin{equation}X_1=\begin{bmatrix}0\\0\end{bmatrix}\end{equation}$

Solution:- $X_{1+i} = X_i-[J_i]^{-1}\nabla f_i$

and

$\begin{equation} \nabla f=\begin{bmatrix} \frac{\partial f}{\partial x_1}\\ \frac{\partial f}{\partial x_2}\end{bmatrix}=\begin{bmatrix} 2+4x_1+2x_2\\-2+2x_1+2x_2\end{bmatrix}\end{equation}$

Hence $\begin{equation} [J_1] =\begin{bmatrix} \frac {\partial^2 f}{\partial x_1^2}&\frac {\partial^2 f}{\partial x_1\partial x_2}\\\frac {\partial^2 f}{\partial x_1\partial x_2}&\frac{\partial^2 f}{\partial x_2^2} \end{bmatrix} _{X_1}=\begin{bmatrix} 4&2\\2&2 \end{bmatrix} \end{equation}$

$\begin{aligned} &=[J_1]^{-1}=1/4\begin{bmatrix} …