Let

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

Let

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

Repeat Exercise 1 using Gauss–Seidel iteration.

Let

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?

For which of the following matrices, will the iteration scheme

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

Let x be the solution of $\mathbf{x}=B_{\mathbf{x}}+\mathbf{c}\mathrm{.}$ Let ${\mathbf{x}}^{\left(0\right)}$ be an arbitrary vector in $R^n$ and define

for $k=0,1,\dots$. prove that if B^m is the zero matrix, then ${\mathbf{x}}^{\left(m\right)}=\mathbf{x}\mathrm{.}$

Let A be a nonsingular upper triangular matrix. Show that if the Jacobi iteration is carried out using exact arithmetic, it will produce the exact solution to $A\mathbf{x}=\mathbf{b}$ after n iterations.

For an iterative method based on the splitting $A=C-M$, C nonsingular, show that

where ${\mathbf{r}}^{\left(k\right)}$ denotes the residual $\mathbf{b}-A{\mathbf{x}}^{\left(k\right)}\mathrm{.}$

Let $A=D-L-U$, where D, L, and U are defined as in Gauss–Seidel iteration and let $\omega$ be a nonzero scalar. The system $\omega A\mathbf{x}=\omega \mathbf{b}$ can be solved iteratively by splitting ωA into $C-M$, where $C=D-\omega L$. Determine the B and c corresponding to this splitting. (The constant $\omega$ is called a relaxation parameter. The case $\omega =1$ corresponds to Gauss–Seidel iteration.)

Let x be the solution to $\mathbf{x}=B\mathbf{x}+\mathbf{c}\mathrm{.}$ Let ${\mathbf{x}}^{\left(0\right)}$ be an arbitrary vector in $R^n$ and define

for $i=0,1,\mathrm{\dots .}\text{If }\left|\right|B\left|\right|=\alpha <1$, show that