Linear Algebra with Applications, 10th Edition

Section 7.8 Exercises

1. Let

   $$\begin{array}{cc}A\begin{array}{cc}=\left[\begin{array}{cc}10& 1\\ 2& 10\end{array}\right],& \mathbf{b}=\left[\begin{array}{c}11\\ 12\end{array}\right],\end{array}& \begin{array}{cc}\text{and}& {\mathbf{x}}^{\left(0\right)}=\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}\end{array}$$

   Use Jacobi iteration to compute ${\mathbf{x}}^{\left(1\right)}$ and ${\mathbf{x}}^{\left(2\right)}$. [The exact solution is $\mathbf{x}=(1,1{)}^T\mathrm{.}$]

2. Let

   $$\begin{array}{cc}A\begin{array}{cc}=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],& b=\left[\begin{array}{c}\begin{array}{c}3\\ 2\end{array}\\ 1\end{array}\right],\end{array}& \begin{array}{cc}\text{and}& {\mathbf{x}}^{\left(0\right)}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\end{array}\end{array}$$

   Use Jacobi iteration to compute ${\mathbf{x}}^{\left(1\right)}$, ${\mathbf{x}}^{\left(2\right)}$, ${\mathbf{x}}^{\left(3\right)}$, and ${\mathbf{x}}^{\left(4\right)}$.

3. Repeat Exercise 1 using Gauss–Seidel iteration.

4. Let

   $$\begin{array}{cc}A\begin{array}{cc}=\left[\begin{array}{ccc}10& 1& 1\\ 1& 10& 1\\ 1& 1& 10\end{array}\right],& b=\left[\begin{array}{c}\begin{array}{c}12\\ 12\end{array}\\ 12\end{array}\right],\end{array}& \begin{array}{cc}\text{and}& {\mathbf{x}}^{\left(0\right)}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\end{array}\end{array}$$

   1. Calculate ${\mathbf{x}}^{\left(1\right)}$ using Jacobi iteration.

   2. Calculate ${\mathbf{x}}^{\left(1\right)}$ using Gauss–Seidel iteration.

   3. Compare your answers to (a) and (b) with the correct solution $\mathbf{x}=(1,1,1{)}^T$. Which is closer?

5. For which of the following matrices, will the iteration scheme

   $${\mathbf{x}}^{(k+1)}={\mathbf{B}\mathbf{x}}^{\left(k\right)}+\mathbf{c}$$

   converge to a solution of $\mathbf{x}=B\mathbf{x}+\mathbf{c}?$ Explain.

   1. $B=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$

   2. $B=\left[\begin{array}{ccc}0.9& 1& 1\\ 0& 0.9& 1\\ 0& 0& 0.9\end{array}\right]$

   3. $B=\left[\begin{array}{ccc}\frac{1}{2}& 10& 100\\ 0& \frac{1}{2}& 10\\ 0& 0& \frac{1}{2}\end{array}\right]$

   4. $B=\left[\begin{array}{ccc}\frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{8}\\ \frac{1}{2}& \frac{1}{4}& \frac{1}{8}\end{array}\right]$

   5. $B=\left[\begin{array}{ccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{6}\\ 0& \frac{1}{6}& \frac{1}{3}\end{array}\right]$

6. Let ...