Suppose, I have :
$$
\begin{aligned}
3x+y-z&=7\\
3x+4y-6z&=8\\
x-4y+2z&=-4
\end{aligned}
$$
We know it's not diagonally matrix, since
$$|4|<|3|+|-6|$$
For this, the matrix is said to be diagonally dominant if:
$$|a_{ii}|>\sum_{j\ne i} |a_{ij}|\tag 1$$
I have a problem with the second row. Because, all possibility didn't satisfy equation $(1)$
My goal is solve the system with jacobi iteration. And that method is work (converge) when the matrix is diagonally dominant.
Somehow, when the matrix is "Not" doagonally dominant, sometimes it's converge, is there another condition that make it's converge (another condition of equation $(1)$).
Then, how do i arrange these rows of system, which can be solved with jacobi iteration?
Please help.
I have a test in 2 hours.
Best Answer
You can rearrange your system of equations as \begin{equation*} \begin{aligned} 3x+y-z&=7 \\x-4y+2z&=-4 \\3x+4y-6z&=8 \end{aligned} \end{equation*} Now the first and second rows are diagonally dominant. The issue is the third row. Subtract the first equation from the third and you get, \begin{equation*} \begin{aligned} 3x+y-z&=7 \\x-4y+2z&=-4 \\3y-5z&=1 \end{aligned} \end{equation*} which is diagonally dominant. Hope your test went well!