TAILIEUCHUNG - Parallel Programming: for Multicore and Cluster Systems- P43

Parallel Programming: for Multicore and Cluster Systems- P43: Innovations in hardware architecture, like hyper-threading or multicore processors, mean that parallel computing resources are available for inexpensive desktop computers. In only a few years, many standard software products will be based on concepts of parallel programming implemented on such hardware, and the range of applications will be much broader than that of scientific computing, up to now the main application area for parallel computing | Iterative Methods for Linear Systems 403 r k 1 _ m xi an n bi - aijx kJ 1 - m x k i 1 . n . More popular is the modification with a relaxation parameter for the Gauss-Seidel method the SOR method. SOR Method The SOR method or successive over-relaxation is a modification of the Gauss-Seidel iteration that speeds up the convergence of the Gauss-Seidel method by introducing a relaxation parameter v e R. This parameter is used to modify the way in which the combination of the previous approximation x k and the components of the V k V current approximation a . x 1 are combined in the computation or x . The k 1 th approximation computed according to the Gauss-Seidel iteration is now considered as intermediate result x k 1 and the next approximation x k 1 of the SOR method is computed from both vectors x k 1 and x k 1 in the following way i 1 n x k 1 I bi aijX k 1 aijX k I i 1 . n i 1 j i 1 x k 1 x k x - x k i 1 . n . Substituting Eq. into Eq. results in the iteration Ak 1 m xi an i -1 n bi - aijX k 1 - aijx k I 1 - m x k j 1 j i 1 for i 1 . n. The corresponding splitting of the matrix A is A D L R 1 2 D and an iteration step in matrix form is D - wL x k 1 1 - m Dx k mRx k mb . The convergence of the SOR method depends on the properties of A and the value chosen for the relaxation parameter a . For example the following property holds If A is symmetric and positive definite and rn e 0 2 then the SOR method converges for every start vector x 0 . For more numerical properties see books on numerical linear algebra . 23 61 71 166 . Implementation Using Matrix Operations The iteration computing x k 1 for a given vector x k consists of 404 7 Algorithms for Systems of Linear Equations a matrix-vector multiplication of the iteration matrix C with x k and a vector-vector addition of the result of the multiplication with vector d. The specific structure of the iteration matrix . CJa for the Jacobi iteration and CGa .

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