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Gauss-Seidel vs. SOR

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Old   April 15, 2006, 12:26
Default Gauss-Seidel vs. SOR
  #1
pXYZ
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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.
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Old   April 16, 2006, 09:15
Default Re: Gauss-Seidel vs. SOR
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johannes
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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.
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Old   April 16, 2006, 10:54
Default Re: Gauss-Seidel vs. SOR
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rt
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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
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Old   April 17, 2006, 17:01
Default Re: Gauss-Seidel vs. SOR
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pXYZ
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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.
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Old   April 17, 2006, 21:40
Default Re: Gauss-Seidel vs. SOR
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Darren
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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)
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Old   April 19, 2006, 11:01
Default Re: Gauss-Seidel vs. SOR
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sam
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when omega = 1 . it is simply as GS. value greater than 1 means overrelaxation and less mean uder relaxation.
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Old   April 19, 2006, 13:47
Default Re: Gauss-Seidel vs. SOR
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rt
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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.
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