I'm not sure what values you were using for c and r, but I suspect you were working with extremely ill-conditioned matrices. This side of things isn't really my speciality, but I don't think this is a super-fair test for many realistic applications. Legend('Gauss-Seidel','SOR','location','southwest') Title(sprintf('Gauss-Seidel approximate solution, iteration %d', iter)) %initial guess (must match boundary conditions) % Right hand side for the Poisson equation (computed from U to produce theį(Ix,Iy) = (-4*U(Ix,Iy)+U(Ix-1,Iy)+U(Ix+1,Iy)+U(Ix,Iy-1)+U(Ix,Iy+1)) = ndgrid(linspace(-pi,pi,N),linspace(-pi,pi,N)) If you want to play around with this example, I've included the MATLAB code I used below. Doing so for a simple case with a known solution gives the following error versus iteration number for these two methods:Īs you can see, SOR reaches machine precision in about 100 iterations at which point Gauss-Seidel is about 25 orders of magnitude worse.
Where $\Delta x$ is the grid spacing and $L$ is the domain size. I was experimenting with SOR for matrix of this king:
I would like something much simpler - just few examples of matrices (problems) for which SOR converge faster. I don't see simple heuristics how to estimate spectral radius just looking on the matrix ( or problem which it represents ). I mean just by looking on the matrix, or knowledge of particular problem the matrix represents?Īre there any heuristics for optimizing the successive over-relaxation (SOR) method?īut it's a bit too sophisticated. (48) #\[T_\) using the Gauss-Seidel method with a relaxation parameter \(\omega\).Is there any simple rule of thumb to say if it is worth to do SOR instead of Gauss-Seidel? ( and possible way how to estimate realxation parameter $\omega$) The Successive Over Relaxation (SOR) method Stability function of a Runge-Kutta method
Solving initial value problems using implicit Runge-Kutta methods Solving initial value problems using explicit Runge-Kutta methods Higher-order Ordinary Differential Equationsĭerivation of Explicit Runge-Kutta Methodsĭerivation of a fourth-order explicit Runge-Kutta method
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