Studying exercises and past exam questions for my Numerical Linear Algebra semester exams, I came upon an exercise which I have very little knowledge/experience on how to handle and that we haven't discussed a lot, despite being given as a revision exercise.
Exercise :
Let $A$ be the matrix : $$A=\begin{bmatrix} 3 & 2 & 0 \\ 10^{-4} & 6 & 1 \\ 0 & 1 & 6 \end{bmatrix}$$ For the iterative method Gauss-Seidel regarding the solution of the system $Ax=b$ with $b=[5,7.001, 7]^T$, calculate an approximation of the norm of the recursive matrix $B_{GS}$ after two iterations, with the help of the differentials $\delta_j=\|x^{(j)}-x^{(j-1)}\|$. Calculate an a-posteriori estimation for the error based on the different values of $\delta_j$ after two iterations and find the number of iterations needed so that the error is less than $10^{-5}$.
Discussion-Question :
To start off, I have very little experience with such exercises, especially regarding the differentials $\delta_j = \|x^{(j)}-x^{(j-1)}\|$, since we haven't elaborated as much and we have never solved such an exercise during lectures.
Regarding Gauss-Seidel, the matrices of the iteration method are given as :
$$T=-L^{-1}U \space \space \text{and} \space \space C= L^{-1}b$$
where the recursive equation $x^{(k+1)}=Tx^{(k)}+C$ is used for the iterations of the method. The matrices $L$ and $U$ are defined as :
$$L = \begin{bmatrix} 3 & 0 & 0 \\ 10^{-4} & 6 & 0 \\ 0 & 1 & 6 \end{bmatrix} \quad \text{and} \quad U = \begin{bmatrix} 0 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}$$
Now, I don't even understand how to proceed with the question. I guess the recursive matrix $B_{GS}$ is the matrix $T$ mentioned above, but to even proceed I should calculate it and that's a very long/costly calculation due to the inverse and the multiplications (same as $C$).
If that's the case, then I understand we're asking an approximation of the norm of $T$, but what does that even have to do with the differentials mentioned ? How would I derive an approximation of the norm from that ?
Finally, what does the a-posteriori estimation for the error means ? For the iterations required, I know that finding the expression for the error which involves the steps number, I would be able to set the error at $10^{-5}$ and solve the inequality for the steps so to calculate the number of them required.
I would really appreciate an elaboration regarding the questions of the exercise, since I really need to understand such a problem and grasp how it works. Any thorough explanation and solution will be very, very appreciated.