TAILIEUCHUNG - Parallel Programming: for Multicore and Cluster Systems- P44

Parallel Programming: for Multicore and Cluster Systems- P44: 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 413 Dr F Xr bl E Db xb b2 where xR denotes the subvector of size nR of the first red unknowns and xB denotes the subvector of size nB of the last black unknowns. The right-hand side b of the original equation system is reordered accordingly and has subvector b1 for the first nR equations and subvector b2 for the last nB equations. The matrix A consists of four blocks DR e RnRxnR DB e RnB xnB E e RnB xnR and F e RnRxnB. The submatrices DR and DB are diagonal matrices and the submatrices E and F are sparse banded matrices. The structure of the original matrix of the discretized Poisson equation in Fig. in Sect. is thus transformed into a matrix A with the structure shown in Fig. c . The diagonal form of the matrices DR and DB shows that a red unknown x i e 1 . nR does not depend on the other red unknowns and a black unknown xj j e nR 1 . nR nB does not depend on the other black unknowns. The matrices E and F specify the dependences between red and black unknowns. The row i of matrix F specifies the dependences of the red unknowns xi i nR on the black unknowns x j j nR 1 . nR nB. Analogously a row of matrix E specifies the dependences of the corresponding black unknowns on the red unknowns. The transformation of the original linear equation system Ax b into the equivalent system Ax b can be expressed by a permutation n 1 . n 1 . n . The permutation maps a node i e 1 . n of the rowwise numbering onto the number n i of the red-black numbering in the following way xi xn i T T 1 r 1 T r T bi bn i i 1 . n or x Px and b Pb with a permutation matrix P Pij i j 1 . n Pij j 0 othefwise j i . For the matrices A and A the equation A PT AP holds. Since for a permutation matrix the inverse is equal to the transposed matrix . PT P-1 this leads to Ax PT APPtx PTb b. The easiest way to exploit the red-black ordering is to use an iterative solution method as discussed earlier in this section. Gauss-Seidel Iteration

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