= 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 gives
$$
\mathbf z_{k+1}=\mathbf z_k-(I+\mathbf z_k\mathbf z_k^T)\mathbf z_k=-\|\mathbf z_k\|^2\mathbf z_k.
$$
Thus the distance from the unique <minimum> obeys \b[$r_{k+1}=r_k^3$], where $r_k=\|\mathbf z_k\|$. In particular,
$$
r_k=r_0^{3^k}\longrightarrow0\qquad(0\leq r_0<1).
$$
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.
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