For iterative methods for solving systems of equations, we may not always get convergence and it can depend simply on the way in which we write the equations.
I understand there are tests which will check if a given system will converge, for example the Frobenius norm being less than one. However, I can only use this to check if a system we already have will converge.
Is there a straightforward way to figure out the correct way of writing the system of equations or must we simply write each one separately and trial each one until we get one that converges?
For clarification, this is the question I am trying to solve, which showed up in a past exam paper I am doing.
Basically, you want the coefficient matrix to be close to $I$. In the three equations, the terms with the biggest coefficients are $4x_3$ in the first equation, $4 x_2$ in the second, $4 x_1$ in the third. You rearrange the equations so the order is $4 x_1$, $4 x_2$, $4 x_3$ and then divide everything by $4$.