 Solution

The update in Newton's method in optimization is ${\mathbf{z}}_{k+1}={\mathbf{z}}_k-H_f({\mathbf{z}}_k{)}^{-1}\nabla f({\mathbf{z}}_k)$. Substituting the previous formulas givesThus the distance from the unique minimum obeys $r_{k+1}=r_k^3$, where $r_k=\Vert {\mathbf{z}}_k\Vert$. In particular,For $0<r_0<1$, the error ratio $r_{k+1}/r_k^3$ is exactly $1$, proving exact cubic convergence. If $r_0=0$, the initial point already attains the minimum. All updates are defined because the Hessian is invertible everywhere. This is Newton minimization of the square-root norm; its special update also shows why the given initial condition matters: $r_0=1$ produces an alternating two-cycle, and $r_0>1$ diverges.