(Solved): 5. [18 points] This question is applying iterative (the Jacobi and Gauss-Seidel) methods to find t ...

5. [18 points] This question is applying iterative (the Jacobi and Gauss-Seidel) methods to find the approximated solution of a linear system A linear system is given as follows: \[ \begin{array}{r} x_{1}+2 x_{2}-2 x_{3}=7 \\ x_{1}+x_{2}+x_{3}=2 \\ 2 x_{1}+2 x_{2}+x_{3}=5 \end{array} \] (1). [5 points] We use Jacobi's iterative method to find the approximated solution of the system. Determine the matrix form of the iteration formula. 2 (2). [7 points] We use the Gauss-Seidel iterative method to solve the system. Determine the matrix form of the iteration formula. (3). [3 points] Given the initial iterate \[ \mathbf{x}^{(\mathbf{0})}=\left[\begin{array}{c} 0.5 \\ 1 \\ -0.5 \end{array}\right] \] use the Gauss-Seidel iteration formula obtained in Part (2) to determine the 1st iterate \( \mathbf{x}^{(1)} \) (3). [3 points] The exact solution vector of the system is known \[ \mathbf{x}^{*}=\left[\begin{array}{c} 1 \\ 2 \\ -1 \end{array}\right] \] Compute \( \left\|\mathbf{x}^{(1)}-\mathbf{x}^{*}\right\|_{1},\left\|\mathbf{x}^{(1)}-\mathbf{x}^{*}\right\|_{\infty} \) and \( \left\|\mathbf{x}^{(1)}-\mathrm{x}^{*}\right\|_{2} \), respecyively.