The algorithm below describes the Inverse Power Iteration method, which converges to the eigenvector associated to the eigenvalue hat$({)}_j$ with the smallest distance to the shift $s.$

$(($see iamge$))$

$($a$)$ Modify the algorithm so it uses the Rayleigh quotient ${}_{j-1}=$vec$(u{)}_{j-1}^T$Avec$(u{)}_{j-1}$ as the updated shift at the $j-$th iteration $($leading to the so$-$called Rayleigh Quotient Iteration$).(($see attached image$)$

$($b$)$ Explain why the algorithm in $($a$)$ becomes singular after convergence unless a stopping criterion is included. Insert that criterion to prevent it from crashes.