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February 17, 2000, 23:18 
GaussSeidel

#1 
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Hello Can anybody tell 1>whether GaussSedel method is the fast iterative method(convergence)? 2>are there any other iterative method which is faster than GS? 3>In general, how many number of iteration it need, say the number of unknowns are 100x100? Thank You very much


February 18, 2000, 08:08 
Re: GaussSeidel

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Gaus Seidel is a rather slow iterative Method. Actually I am not aware of any other Method that is slower. (Jordan Method is a little bit slower).
2. There is an other Method that is faster than GausSeidel: the TDMA (Thomas Algorithm) Itīs quite simple to programm, you may find many references to this Algorithm (netlib etc). Some faster Methods use Decompostion (ILU, LU) or Multigrid. Start with the Thomas Algorithm. Itīs as easy to program as the GausSeidel. 3. It depends on your Problem and the special occasion you are working on. Actually you can calculate the difference between two runs and so decide when your solution is the nummerical approximation of your problem. Some times you donīt have to solve your equations up to convergence. You save much time if you let your solver do some iterations and then go on with the next equation. This is the case if you have material properties that show a strong temperature dependance while you solve a set of equations. Because of the iterative Nature of the problem you are allowed to do so. best Regards Yiannis Dotsikas 

February 18, 2000, 11:26 
Re: GaussSeidel

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There are definitely faster methods to solve a linear system than GS. If you have banded matrices you can use the Thomas algorithm. For multidimensions you can do a linebyline approach which is an iterative application of the Thomas algorithm. These only work for structured grid applications, for unstructured you generally have to use a pointbypoint method.
For a math class project I took a CFD code that I had already written and used a number of different techniques to solve the linear system, including the "new" Krylov subspace methods. I found that the driver in flow problems was the nonlinear (outer) iteration convergence, and not solving the linear (inner) system. With GS, you don't have to completely converge the inner system before you relinearize at the outer step, but you still get benefit by "pushing" the result in the right direction. As such, in terms of overall computation time, nothing outperformed good old GaussSeidel. One caveat: You have to make sure that your coefficient matrix is diagonally dominant. As for iterations to convergence, that will depend on just how diagonally dominant your coefficient matrix is. Keith 

February 18, 2000, 12:27 
Re: GaussSeidel

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It greatly depends upon the structure of your matrix.
In general, GaussSeidel method is slow. There are much better iterative methods. In order to get an answer to your question, you need to provide us with at least the following information: (*) is your matrix symmetric; (*) is it positive definite; (*) is it banded (blockbanded). 

February 18, 2000, 22:17 
Re: GaussSeidel

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Hi, COBOK For my case, the matrix have all the three properties you mentioned,1>symmetric, 2>positive definite, 3>band matrix


February 19, 2000, 12:15 
Re: GaussSeidel

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The Cholessky decomposition method may okay for your problem, but since the matrix is banded... BTW, what is the bandwidth of your matrix? As for starter, I'd recommend to go through the books/reports/proceedinds (some of them are good): 1)Rheinboldt, W.C., Methods for solving systems of nonlinear equations,2nd ed., Philadelphia : Society for Industrial and Applied Mathematics,1998. 2)Greenbaum, A.,Iterative methods for solving linear systems, Philadelphia, PA : Society for Industrial and Applied Mathematics, 1997. 3)Kelley, C. T., Iterative methods for linear and nonlinear equations, Philadelphia, Society for Industrial and Applied Mathematics,1995. (4)Tidriri, M. D., Krylov methods for compressible flows, Hampton, VA : Institute for Computer Applications in Science and Engineering, NASA Langley Research Center,<Springfield, Va. : National Technical Information Service, distributor, 1995>, NASACR198176 (5)Iterative methods for large linear systems, (eds)D.R. Kincaid, L.J.Hayes., Boston : Academic Press, c1990. (6)An iterative implicit diagonallydominant factorization algorithm for solving the NavierStokes equations, ShuCheng Chen, NanSuey Liu, and Hyun Dae Kim., NASATM105259 (7) Iterative methods of large linear systems and their applications, (eds) M.Natori and T. Nodera.<Yokohama, Japan> : Keio University, 1989. (8) Iterative solution of large linear systems, D.M.Young., New York, Academic Press, 1971. (9)Varga, R.S., Matrix iterative analysis, Englewood Cliffs, N.J., PrenticeHall, 1962
I'd advise you to read some chapters in these references, so that you get an idea where to start. After that you can eihter implement some algorithm that you think is the best you found, or try to find a free code on the net. Unfortunately, I do not have any, but maybe someone else could direct you. 

February 23, 2000, 15:34 
Re: GaussSeidel

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The Jacobi iterative method is slower than GaussSeidel. The S.O.R (Successive Over Relaxation method) is an accelerated, in terms of convergence, GaussSeidel method.


February 25, 2000, 11:52 
Re: GaussSeidel

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Nobody is using GaussSeidel method lonely if he wants to solve some practical problems. However, combined with multigrid algorithm, GaussSeidel is still a good one, since it is easy to implement compared with other algorithm, such as ILU, CGStab, GMRes, SSOR, etc.
Good luck. 

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