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Given the linear equation system; $\begin{bmatrix} A_1 & B\\ B^T & A_2 \end{bmatrix}$$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$=$\begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ where $A_1 \in \mathbb{R}^{n_1\times n_1} $,$A_2 \in \mathbb{R}^{n_2\times n_2}$ and $B\in \mathbb{R}^{n_1\times n_2}$, I've got to find the condition for convergence of followig iteration method:

$(1)$ $A_1x_1^{(k+1)}+Bx_2^{(k)}=b_1$

$(2)$ $B^Tx_1^{(k)}+A_2x_2^{(k+1)}=b_2$

I've brought the two equations together:

(1) $x_1^{(k+1)}=A_1^{-1}(b_1-Bx_2^{(k)})$

(2) $x_2^{(k+1)}=A_2^{-1}(b_2-B^Tx_1^{(k)})$ $\rightarrow$ $x_2^{(k)}=A_2^{-1}(b_2-B^Tx_1^{(k-1)})$

i.e.

$\rightarrow$ (1) $x_1^{(k+1)}=A_1^{-1}(b_1-BA_2^{-1}(b_2-B^Tx_1^{(k-1)}))$

Is it all in order? My question is how to apply the theorem of convergence for iteration methods, where the spectral radius of the iteration matrix has to be less one?

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The matrix of the linear part of the last equation is $$A_1^{-1}BA_2^{-1}B^T,$$ this has to be contractive for the method to work.

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  • $\begingroup$ so to be sure you mean the spectral radius has to fulfill $\rho(A_1^{-1}BA_2^{-1}B^T)<1$ , right? $\endgroup$ – Thesinus May 22 '17 at 21:59

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