Rate of convergence of fixed-point iteration in higher dimensions Consider the fixed-point iteration process in $\mathbb{R}^n$.
Given a sufficiently smooth function $f:\mathbb{R}^n\to\mathbb{R}^n$ and an initial value $x_0\in\mathbb{R}^n$, define the iteration sequence $x_{k+1}=f(x_k)$. Suppose that
$$\lim_{k\to\infty}x_k=x^*,$$
then apparently $x^*$ is a fixed point of $f(x)$. I'm familiar with the case in $\mathbb{R}^1$ where the sequence is (generally) linearly convergent with rate
$$\lim_{k\to\infty}\frac{|x_{k+1}-x^*|}{|x_k-x^*|}=|f'(x^*)|<1.$$
And I thought the analogy of this constant $|f'(x^*)|$ in $\mathbb{R}^n$ case would be $\|J_f(x^*)\|$, where $J_f(x^*)$ denotes the Jacobian matrix $(\partial f_i(x^*)/\partial x_j)_{n\times n}$ and $\|\cdot\|$ the operator norm induced by vector norm.
But the results of my numerical experiments proved me wrong, and I found the following claim on some website:
$$\lim_{k\to\infty}\frac{\|x_{k+1}-x^*\|}{\|x_k-x^*\|}=\rho(J_f(x^*))<1,$$
where $\rho$ the spectral radius of a matrix. Indeed the claim fits well my experiment results.
It does surprise me that the rate of convergence is independent of the vector norm, but I could not find a proper proof either by myself or by online materials.
Any help or link on its proof would be appreciated.

It's well known that  \ru(J) <=||J|| for every norm. so the the fixed point iteration will be convergence if the second relation be valid. The proof can be found in basic linear algebra books.
