With basic iterative method $Mx_{k+1} = Nx_k + b$, show that $||x_k-x||_2 \leq \dfrac{\rho(G)}{1-\rho(G)}||x_k-x_{k-1}||_2$

Question

With $Mx_{k+1} = Nx_k + b$, define $G=M^{-1}N$

show that $$||x_k-x||_2 \leq \dfrac{\rho(G)}{1-\rho(G)}||x_k-x_{k-1}||_2$$

where $\rho(G)$ is the spectral radius of G

Could anyone help me?

Answer 1

HINT:

You already know that $$ x_{k+1}−x_k=G(x_k−x_{k−1}). $$ Since $x$ is the fixed point, it satisfies $(I-G)x=M^{-1}b$ so we have $$ x_{k+1}=Gx_k+M^{-1}b=Gx_k+(I-G)x. $$ Put these two together to get $$ x-x_k=(I-G)^{-1}G(x_k−x_{k−1}). $$