Parallel Algorithms for Matrix Computations by K. A. Gallivan

By K. A. Gallivan

Describes a variety of vital parallel algorithms for matrix computations. reports the present prestige and offers an total viewpoint of parallel algorithms for fixing difficulties bobbing up within the significant parts of numerical linear algebra, together with (1) direct answer of dense, based, or sparse linear platforms, (2) dense or dependent least squares computations, (3) dense or based eigenvaluen and singular price computations, and (4) swift elliptic solvers. The publication emphasizes computational primitives whose effective execution on parallel and vector desktops is vital to acquire excessive functionality algorithms.

Consists of 2 finished survey papers on very important parallel algorithms for fixing difficulties coming up within the significant components of numerical linear algebra--direct answer of linear platforms, least squares computations, eigenvalue and singular worth computations, and speedy elliptic solvers, plus an in depth up to date bibliography (2,000 goods) on similar examine.

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Version 3 of the algorithm can be viewed as a hybrid of the first two versions. Like Version 2, it is assumed that the first (i — l)u columns of L and rows of U are known at the start of step i. It also assumes, like Version 1, that the transformations that produced these known columns and rows must be applied elements of A which are to be transformed into the next u> columns and rows of L and U. As a result, Version 3 does not update the remainder of the matrix at every step. Consider the factorization: where AH is a square matrix of order (i — l)u and the rest are partitioned conformally.

As a result, Version 3 does not update the remainder of the matrix at every step. Consider the factorization: where AH is a square matrix of order (i — l)u and the rest are partitioned conformally. By our assumptions, LH, I/2i, t/n, and U\2 are known and the first u columns of L22 and the first u) rows of U^i are to be computed. Since Version 3 assumes that none of the update A 2 2 <— ^22 — £21^12 has occurred in the first i — 1 steps of the algorithm, the first part of step i is to perform the update to the portion upon which the desired columns of Z/22 and rows of t/22 depend.

For a discussion of the numerical stability of this algorithm see [187]. Note that thus far we have assumed only one right-hand side vector. The BLAS3 primitive triangular solver assumes that multiple right-hand side vectors and solutions are required. This, of course, provides the necessary data locality for high performance on a hierarchical memory system. The generalization of the algorithms above are straightforward and the blocksizes (the number and order of right-hand sides solved in a stage of the algorithm) can be analyzed in a fashion similar to the matrix multiplication primitives.

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