I'd like to prove that the following non-linear system
$ F(x) = \begin{pmatrix} x_1^3 + x_2^3 - 4 \\ x_1^3 - x_2^3 \\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \end{pmatrix} $ will converge when using the Newton's Method for the start vectors $x^{(0)} = (x_1^{(0)}, x_2^{(0)})^T \in [1,2] \times [1, 2] \subset \mathbb R^2$.
I'm trying to use this lemma I learned on class
Let $f: D \subset \mathbb R^n \rightarrow \mathbb R^n$ be continuously differentiable and let $D$ be convex as well open, $f'$ is the jacobian matrix. If $q = \sup_{x\in D}||f'(x)|| < 1$ then $f$ is a contraction mapping using the Norm $|| \cdot ||$ with a contraction rate $q$.
I have calculated the Jacobian matrix, and it is as follows $ \begin{pmatrix} 3x_1^2 & 3x_2^2 \\ 3x_1^2 & - 3x_2^2 \\ \end{pmatrix} $ Using the maximum norm (sum of lines) for matrixes, I get: $||F' (x_1, x_2)||_\infty = \max \{3x_1^2 + 3x_2^2, 3x_1^2 - 3x_2^2\} = 3x_1^2 + 3x_2^2$ which is for sure not $< 1$ in the desired interval. Where have I gone wrong? Thanks a lot guys!