Compare the rate of convergence of Jacobi and Gauss-Seidel method for the matrices
$$ A=\left(\begin{array}{rrr} 1 & 2 & -2 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & 1 & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 1 \end{array}\right) . $$
My solutions is as follows:
Matrix A
Jacobi
$$ \begin{gathered} x^{k+1}=G x^{k}+b \\ G=-D^{-1}(A-D) \\ G=\left(\begin{array}{ccc} 0 & -2 & 2 \\ -1 & 0 & -1 \\ -2 & -2 & 0 \end{array}\right) \end{gathered} $$
$$ |G-\lambda E|=\left|\begin{array}{ccc} -\lambda & -2 & 2 \\ -1 & -\lambda & -1 \\ -2 & -2 & -\lambda \end{array}\right|=-\lambda^{3} \rightarrow r(G)=0 $$
Method is convergenced.
Seidel-Gauss
$$ \begin{gathered} x^{k+1}=G x^{k}+b \\ G=-(D+L)^{-1} U \\ \left(\begin{array}{ccc} 0 & -2 & 2 \\ 0 & 2 & -3 \\ 0 & 0 & 2 \end{array}\right) \rightarrow r(G)=2 \rightarrow \text { so method isn'tconvergenced. } \end{gathered} $$
Matrix B
Jacobi
Method is convergenced.
Seidel-Gauss
$$ x^{k+1}=G x^{k}+b $$
$$ \begin{aligned} & G=-(D+L)^{-1} U \end{aligned} $$
$G=\left(\begin{array}{ccc}0 & -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \\ 0 & \frac{1}{3} & \frac{1}{3}-\frac{1}{\sqrt{3}} \\ 0 & \frac{1}{9}(3-\sqrt{3}) & \frac{1}{9}(6-\sqrt{3})\end{array}\right) \rightarrow r(G)=0.45<1 \rightarrow$ so method istconvergenced.
$$ \begin{aligned} & x^{k+1}=G x^{k}+b \\ & G=-D^{-1}(A-D) \\ & G=\left(\begin{array}{ccc}0 & -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \\-\frac{1}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} \\-\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & 0\end{array}\right) \\ & \lambda E|=| \begin{array}{ccc}-\lambda & -\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \\-\frac{1}{\sqrt{3}} & -\lambda & -\frac{1}{\sqrt{3}} \\-\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & -\lambda\end{array} \mid \end{aligned} $$
Just want to check if this is correct.
