Newton minimization of the square-root norm ($f(z)=\sqrt{1+\Vert z{\Vert }^2},z_{k+1}=-\Vert z_k{\Vert }^2{z}_k$)

For the Euclidean norm on ${\mathbb{R}}^d$, the regularized radial function $f(z)=\sqrt{1+\Vert z{\Vert }^2}$ has gradient $z/f$ and Hessian $I/f-z{z}^T/f^3$. The Cauchy-Schwarz inequality makes its Hessian a positive-definite matrix, so $f$ is strictly convex with unique minimum at zero. Its inverse matrix is $f(I+z{z}^T)$: multiplication verifies this using $(z{z}^T{)}^2=\Vert z{\Vert }^{2}z{z}^T$ and $f^2=1+\Vert z{\Vert }^2$. Substitution in Newton's method in optimization therefore gives $z_{k+1}=-\Vert z_k{\Vert }^2{z}_k$. Consequently $\Vert z_{k+1}\Vert =\Vert z_k{\Vert }^3$: nonzero starts in the open unit ball have exact cubic convergence, starts on the unit sphere alternate between opposite points, and starts outside it diverge. This example separates strict convexity from global convergence of an undamped iteration.