By David E. Keyes (auth.), David E. Keyes, Ahmed Sameh, V. Venkatakrishnan (eds.)
In this quantity, designed for computational scientists and engineers engaged on purposes requiring the stories and processing charges of large-scale parallelism, best algorithmicists survey their very own field-defining contributions, including adequate historic and bibliographical viewpoint to allow operating one's technique to the frontiers.
This e-book is uncommon from prior surveys in parallel numerical algorithms by means of its extension of assurance past center linear algebraic tools into instruments extra at once linked to partial differential and critical equations - although nonetheless with an attractive generality - and through its specialise in functional medium-granularity parallelism, approachable via conventional programming languages.
a number of of the authors used their invitation to take part as an opportunity to face again and create a unified review, which nonspecialists will appreciate.
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1993a,b) for more details on software implementations of these algorithms. 5. Gaussian Elimination We now show how to put together the tools and techniques we have described and design high performance Gaussian elimination software both for shared memory and distributed memory machines. We begin by establishing basic "Matlab-style" notation. A( i : j, k : 1) will refer to the submatrix of A lying in rows i through j and columns k through 1. A( i : j, :) will refer to the entirety of rows i through j.
1990. "Exploiting fast matrix multiplication within the Level 3 BLAS," ACM Trans. Math. Soft. 16, pp. 352-368. 54 Hong, X. and Kung, H. , 1981. "I/O complexity: the red blue pebble game," in Pmc. 18th Symp. on the Theory of Computing, pp. 326-334, ACM. , 1979. "Basic Linear Algebra Subprograms for Fortran usage. ACM Trans. Math. Soft. 5, pp. 308-323. , 1994. "A serial implementation of Cuppen's divide and conquer algorithm for the symmetric eigenvalue problem," Mathematics Dept. ps, University of California.
The idea is to process the matrix b columns at a time rather than one at a time (see Figure 7). So for any particular value of the index i, the rectangular matrix lying in rows i through n and columns i through i + b - 1 are LU factorized using unbloc,ked Gaussian elimination to get their LU factors (step (1) in Figure 7; the submatrix of A modified here is also labeled (1) in the figure). Then, in step (2), the upper b by b triangular L' of L is used to update the subblock of U lying directly to its right.