91Ó°ÊÓ

Find \(l_{\infty}\) and \(l_{2}\) norms of the vectors. a. \(\quad \mathbf{x}=\left(3,-4,0, \frac{3}{2}\right)^{t}\) b. \(\quad \mathbf{x}=(2,1,-3,4)^{t}\) c. \(\mathbf{x}=\left(\sin k, \cos k, 2^{k}\right)^{t}\) for a fixed positive integer \(k\) d. \(\mathbf{x}=\left(4 /(k+1), 2 / k^{2}, k^{2} e^{-k}\right)^{t}\) for a fixed positive integer \(k\)

Find the first two iterations of the Jacobi method for the following linear systems, using \(\mathbf{x}^{(0)}=\mathbf{0}\) : a. \(\begin{aligned} 4 x_{1}+x_{2}-x_{3} &=5 \\\\-x_{1}+3 x_{2}+x_{3} &=-4 \\\ 2 x_{1}+2 x_{2}+5 x_{3} &=1 \end{aligned}\) b. \(\begin{aligned}-2 x_{1}+x_{2}+\frac{1}{2} x_{3} &=4 \\ x_{1}-2 x_{2}-\frac{1}{2} x_{3} &=-4 \\ x_{2}+2 x_{3} &=0 \end{aligned}\) c. \(\begin{aligned} 4 x_{1}+x_{2}-x_{3}+x_{4} &=-2 \\ x_{1}+4 x_{2}-x_{3}-x_{4} &=-1 \\\\-x_{1}-x_{2}+5 x_{3}+x_{4} &=0 \\\ x_{1}-x_{2}+x_{3}+3 x_{4} &=1 \end{aligned}\) d. \(\quad 4 x_{1}-x_{2} \quad-x_{4} \quad=0\), \(-x_{1}+4 x_{2}-x_{3} \quad-x_{5} \quad=5\) \(-x_{2}+4 x_{3} \quad-x_{6}=0\), \(-x_{1} \quad+4 x_{4}-x_{5} \quad=6\) \(-x_{2} \quad-x_{4}+4 x_{5}-x_{6}=-2\), \(-x_{3} \quad-x_{5}+4 x_{6}=6 .\)

Use the SOR method to solve the linear system \(A \mathbf{x}=\mathbf{b}\) to within \(10^{-5}\) in the \(l_{\infty}\) norm, where the entries of \(A\) are $$ a_{i, j}= \begin{cases}2 i, \quad \text { when } j=i \text { and } i=1,2, \ldots, 80, \\ 0.5 i, & \text { when }\left\\{\begin{array}{l} j=i+2 \text { and } i=1,2, \ldots, 78 \\ j=i-2 \text { and } i=3,4, \ldots, 80 \end{array}\right. \\ 0.25 i, & \text { when }\left\\{\begin{array}{l} j=i+4 \text { and } i=1,2, \ldots, 76 \\ j=i-4 \text { and } i=5,6, \ldots, 80 \end{array}\right. \\ 0, & \text { otherwise }\end{cases} $$ and those of \(\mathbf{b}\) are \(b_{i}=\pi\), for each \(i=1,2, \ldots, 80\).

The linear system $$ \begin{aligned} x_{1}-x_{3} &=0.2 \\ -\frac{1}{2} x_{1}+x_{2}-\frac{1}{4} x_{3} &=-1.425 \\ x_{1}-\frac{1}{2} x_{2}+x_{3} &=2 \end{aligned} $$ has the solution \((0.9,-0.8,0.7)^{t}\). a. Is the coefficient matrix $$ A=\left[\begin{array}{rrr} 1 & 0 & -1 \\ -\frac{1}{2} & 1 & -\frac{1}{4} \\ 1 & -\frac{1}{2} & 1 \end{array}\right] $$ strictly diagonally dominant? b. Compute the spectral radius of the Gauss-Seidel matrix \(T_{g}\). c. Use the Gauss-Seidel iterative method to approximate the solution to the linear system with a tolerance of \(10^{-2}\) and a maximum of 300 iterations. d. What happens in part (c) when the system is changed to $$ \begin{aligned} x_{1}-2 x_{3} &=0.2 \\ -\frac{1}{2} x_{1}+x_{2}-\frac{1}{4} x_{3} &=-1.425 \\ x_{1}-\frac{1}{2} x_{2}+x_{3} &=2 \end{aligned} $$

In Exercise 17 of Section \(7.3\) a technique was outlined to prove that the Gauss-Seidel method converges when \(A\) is a positive definite matrix. Extend this method of proof to show that in this case there is also convergence for the SOR method with \(0<\omega<2\).

Show that the characteristic polynomial \(p(\lambda)=\operatorname{det}(A-\lambda I)\) for the \(n \times n\) matrix \(A\) is an \(n\) th-degree polynomial. [Hint: Expand \(\operatorname{det}(A-\lambda I)\) along the first row, and use mathematical induction on \(n\).]