TAILIEUCHUNG - Parallel Programming: for Multicore and Cluster Systems- P38

Parallel Programming: for Multicore and Cluster Systems- P38: 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 | Gaussian Elimination 363 can be used to solve several linear systems with the same matrix A and different right-hand side vectors b without repeating the elimination process. Pivoting Forward elimination and LU decomposition require the division by afk and so these methods can only be applied when a 0. That is even if det A 0 and the system Ax y is solvable there does not need to exist a decomposition A LU when akk is a zero element. However for a solvable linear system there exists a matrix resulting from permutations of rows of A for which an LU decomposition is possible . BA LU with a permutation matrix B describing the permutation of rows of A. The permutation of rows of A if necessary is included in the elimination process. In each elimination step a pivot element is determined to substitute a kk. A pivot element is needed when akk 0 and when akk is very small which would induce an elimination factor which is very large leading to imprecise computations. Pivoting strategies are used to find an appropriate pivot element. Typical strategies are column pivoting row pivoting and total pivoting. __ __ k a7 _P__1 Column pivoting considers the elements akk a k of column k and determines the element a k r n with the maximum absolute value. If r k the rows r and k of matrix A k and the values b and b of the vector b k are exchanged. Row 11 tl 1 iT l l rrrti ll U 1 TA1X7 vt x 1 x 1Y1 11 t 7 1 it 11 71 til 1 1 1 t 11 vl Yi Yl Y1Y 11 1 7 7 pivoting determines a pivot element akr k r n within the elements akk akn of row k of matrix A k with the maximum absolute value. If r k the columns k and r of A k are exchanged. This corresponds to an exchange of the enumeration of the unknowns xk and xr of vector x. Total pivoting determines the element with the maximum absolute value in the matrix A k a k i j n and exchanges columns and rows of A k depending on i k and j k. In practice row or column pivoting is used instead of total pivoting since they have smaller .

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