The basic iterative Method to solve linear system $Ax=b$ is: $$Mx_{k+1} = Nx_k+b$$ We define that $G = M^{-1}N$
Show that the spectral radius approximately satisfies $$\rho(G) \approx \frac{||x_{k+1}-x_{k}||}{||x_{k}-x_{k-1}||}$$
I have no idea about the relationship between spectral radius and the ratio, could anyone help me?
We have $x_{k+1}=Gx_k+M^{-1}b$ and $x_k=Gx_{k-1}+M^{-1}b$, so $$ x_{k+1}-x_k=G(x_k-x_{k-1}). $$ If we define $y_k:=x_{k+1}-x_k$, this can be written as $y_k=Gy_{k-1}$ so $y_k=G^ky_0$. You can see that the vectors $y_k$ correspond to eigenvector estimates of $G$ computed by the power method with the initial vector $y_0$ corresponding to the dominating eigenvalue $\lambda$ for which $|\lambda|=\rho(G)$. If $y$ is the eigenvector corresponding to this eigenvalue and we have eventually $y_{k-1}\approx y$, which means that $$ y_k=Gy_{k-1}\approx Gy=\lambda y\approx \lambda y_{k-1} $$ so $$ \rho(G)=|\lambda|\approx \frac{\|y_k\|}{\|y_{k-1}\|}. $$