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


$\rightarrow$ (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?


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.

  • $\begingroup$ 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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