$($a$)$ Compute the Newton step, $s_k^N.$ Show that even though $B_k$ is positive def$-$

inite $s_k^N$ does not satisfy the trust$-$region constraint. $($Therefore$,$ the sub$-$

problem can't be "short$-$circuited".$)$

$($b$)$ Derive a $1$ dimensional Newton iteration for solving the secular$-$equation

$()=0$ for the optimal shift $0$ where

$()=\frac{1}{||s()|{|}_2}-\frac{1}{{}_k}$

$($c$)$ Solve the trust$-$region subproblem in eq$.(1).$ You can use a convergence

tolerance of $lon={10}^{-5}$ for $|({}_i)|lon.$ What is the computed shift ${}_i,$ and

what is the solution vector of the subproblem.