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Help me:

$2x-y=7\\ -x+2y-z=1\\ -y-2z=1$

Show Gauss-Seidel iteration scheme converges and find the rate of convergence.

My Attempt: The iteration matrix for this system of equation is

$$ H = \left( \begin{array}{ccc} 0 & 1/2 & 0 \\ 0 & 1/4 & 1/2 \\ 0 & -1/8 & -1/4 \end{array} \right)$$

All the eignvalues for this iterative matrix is 0. So the spectral radius $\rho(H)<1$. Thus Gauss-Seidel iteration scheme converges. But what can we say about rate of convergence.

Thank you all.

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up vote 1 down vote accepted

You are almost there. All eigenvalues being zero means that the matrix is nilpotent: one of its powers is the zero matrix. In this case $H^3=0$. And since the error at $N$th step is controlled by $\sum_{n\ge N}\|H^n\|$, the error becomes zero at the third step.

Let's check this explicitly, writing $v=L_*^{-1}b$ for the vector added at each iteration:

$$\begin{align}x^{1}&=Hx^{0}+v \\ x^{2}&=Hx^{1}+C = H^2 x^0+Hv+v\\ x^{3}&=Hx^{2}+C = H^3 x^0+H^2v+Hv+v = \color{red}{H^2v+Hv+v} \\ x^{4}&=Hx^{3}+C = H^3v+H^2v+Hv+v = \color{red}{H^2v+Hv+v} \\ \dots \end{align}$$

You may want check that the vector in red is indeed the solution, even thought the general theorem tells you so.

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