Consider the linear system $A \mathbf{u}=\mathbf{b}$, where $A=\left(\begin{array}{ll}2 & 1 \\ 1 & 3\end{array}\right), \mathbf{b}=\left(\begin{array}{l}3 \\ 2\end{array}\right)$.
(a) What is the solution? (b) Discuss the convergence of the Jacobi iteration method. (c) Discuss the convergence of the Gauss-Seidel iteration method. (d) Write down the explicit formulas for the SOR Method. (e) What is the optimal value of the relaxation parameter $\omega$ for this system? How much faster is the convergence as compared to the Jacobi and Gauss-Seidel Methods? $(f)$ Suppose your initial guess is $\mathbf{u}^{(0)}=0$. Give an estimate as to how many steps each iterative method (Jacobi, Gauss-Seidel, SOR) would require in order to approximate the solution to the system to within 5 decimal places.
$(g)$ Verify your answer by direct computation.