(Solved): 6. Iterative Methods: Given the matrix and right hand side vector, A=(4114),b=(55) (a) Per ...

6. Iterative Methods: Given the matrix and right hand side vector, $A=\left(\begin{array}{ll}4& 1\\ 1& 4\end{array}\right),b=\left(\begin{array}{l}5\\ 5\end{array}\right)$ (a) Perform 3 iterations by hand using Jacobi and Gauss-Seidel methods. Start from initial guess $x_0=(0,0{)}^T$. (b) Write down the iteration matrices for $M_J$ and $M_{GS}$ and compute the $l_{?}$ norm and spectral radius. Which of the two iterateions converges from arbitrary initial guess? Why? (c) Write a program to compute the spectral radius of $M_{?}$ for $?$ values ranging from 0 to 2 , in increments of 0.001 . Plot $?\left(M_{?}\right)$ against $?$. By inspecting the graph, find the optimal value of the relaxation parameter, ${?}^{?}$. Compare with the theoretical value discussed in class. Attach your code and plots. (d) Write a program to solve the linear system numerically, using Jacobi, Gauss-Seidel, and optimal SOR, using $x_0=(0,0{)}^T$. Terminate the iteration after step $k$ if ${?x^{(k)}?x^{(k?1)}?}_{?}<10^{?5}$. Attach your code. (e) Make a semilogy plot that compares the evolution of the error ${?x?x^{(k)}?}_{?}$ for all three methods. Discuss your results.