I have a $4\times 4$ matrix and I want to use Jacobi iteration on it. Right now the spectral radius is higher than $1$. I know that the method is guaranteed to converge if the matrix is diagonally dominant. I can't do any reordering that would make the matrix diagonally dominant. Is it sensible to try to get as close as possible to diagonally dominant matrix and if the spectral radius is still greater than $1$, I can state that I can not use Jacobi iteration? Or is it possible that there can still exist some reordering of the columns and rows, that would have spectral radius lesser than $1$?
Edit: In other words, is there some correlation between some of the matrix parameters and the size of the spectral radius?