Iterative method for solving a linear system - convergence

I have to solve the following linear system using an iterative method : $$\begin{cases} &4x_1+x_2 = b_1 \\ &x_{i-1}+4x_i + x_{i+1}=b_i,\ i=2 \cdots n-1\\ &x_{n-1} + 4x_n=b_n \end{cases}$$

The iterative method is

$$\begin{cases} x_1^{k+1} = \alpha x_1^{(k)} + \frac{\alpha-1}{4}(x_2^{(k)}-b_1)\\ x_i^{k+1} = \alpha x_i^{(k)} + \frac{\alpha-1}{4}(x_{i-1}^{(k)}+x_{i+1}^{(k)}-b_i),\ i=2 \cdots n-1\\ x_n^{k+1} = \alpha x_n^{(k)} + \frac{\alpha-1}{4}(x_{n-1}^{(k)}-b_n)\\ \end{cases}$$

where $$\alpha$$ is a real and $$x^{(0)}=0$$.

I have to prove first that $$\parallel x^{(k+1)}-x \parallel_\infty \leq \left(|\alpha| + \left|\frac{\alpha-1}{2}\right|\right) \parallel x^{(k)}-x \parallel_\infty$$

I tried lot of things but without success. I am sure that I have to use a proof by induction, but I cannot prove that the initialization is true for $$k=0$$. Can you help me?

• Begin your work by expressing your iteration using matrices and vectors. Exploit that your matrix is strictly diagonally dominant. Commented Feb 20, 2022 at 1:36

The $$\|\cdot\|_{\infty}$$ norm of you iteration matrix is precisely $$|\alpha| +\frac{|\alpha-1|}{2}$$, so the result comes directly from the fixed point theorem.