Strategy for finding optimal omega in SOR method

I had written an algorithm that searches for the optimal weight parameter to be implemented in the successive-over relaxation (SOR) method which worked cleanly by vectorizing the interval and for each ω the spectral radius of the iteration matrix is computed.

However, I was advised not to use this approach for large sparse matrices as it is expensive to compute (the same way computing condition number of a large matrix is unfeasible) and rather use it as a demonstration tool. Therefore, I was wondering what strategy is the best to approximate the optimal weight parameter for large sparse systems () that would allow the best convergence of the SOR.

Furthermore, as a result of my question I was wondering if classical iterative stationary methods such as Jacobi, Gauss-Seidel, and the SOR are worthy to be used nowadays in dealing with large sparse systems or is the default preference Krylov methods?

Answer 1

[nour]

A = [5 2 -1; 2 6 3; 1 4 -8];

b = [1; 2; 1];

x0 = [0; 0; 0];

e = 0.001;