Hi,

I have written a little program which solves an equation system (3x3) with a Jacobi- or Gauss-Seidel-procedure.

There is no doubt that the Jacobi-solver needs much more iterations to solve the problem than Gauss-Seidel.

I wonder that a successive over relaxation with Gauss-Seidel (SOR) does not lead to a further reduction of iterations. What could be the reason?

Thanks for help.

As you said you tried it with 3x3 matrices. Try it with some 100x100 matrices and see what happens. There can be many causes for the problem you described.

Dear

Probably you don't use optimum over-relaxation parameter, this parameter usually is found with numerical experiment but 1.7 is usually good choice.

good luck

omega=1.7 ???

I thought it should be around between 0 and 1. I tested it but it does not improve my convergency.

I cannot see any reason why 10x10 should lead to a different behaviour than 3x3. Could you try to give me an explanation?

Thanks.

Having omega between 0 and 1 is under-relaxation, so omega should definately be >1 (and as said 1.7 usually does ok).

What is your matrix?? Iterative methods generally get used for sparse, diagonally dominant problems (like discrete PDE operators).

Also as said, the difference between SOR and GS is going to become more noticable as the size of the matrix increases (like 100's by 100's).

The convergence of iterative methods has alot to do with the spectral properties of the matrix (eigenvalues)

when omega = 1 . it is simply as GS. value greater than 1 means overrelaxation and less mean uder relaxation.

For better understanding the effect of omega (0<omega<2 is convergent) on system behavior, i suggest you to vary omega with small step size, between 0 and 2, and investigate variation of number of iterations with variation of omega, so select optimom value.

Also the number of iterations is completely function of system behavior e.g., when we deal with ill-posed systems (have large condition number) usually we don't acheive performance when we switch from GS to SOR. For such systems preconditionnig and application of non-stationary iterative methods (e.g. CG, BICGSTAB or GMRES) is good cure.