Two enhancements in a partitioned sparse code
Coarse grain parallel codes for solving systems of linear algebraic equations whose coefficient matrices are sparse can be developed in several different ways. The following procedure is suitable for some parallel computers. A preliminary reordering device is first applied to move as many zero elements as possible to the lower left corner of the matrix. After that the matrix is partitioned into large blocks. The blocks in the lower left corner contains only zero elements. An attempt to obtain a good load-balance is carried out by allowing the diagonal blocks to be rectangular.
While the algorithm based on the above ideas has good parallel properties, some stability problems may arise during the factorization (because the pivotal search is restricted to the diagonal blocks). A simple a priori procedure has been used in a previous version of the partitioned algorithm in an attempt to stabilize the algorithm. It will be shown in this paper that two enhanced stability devices can successfully be incorporated in the algorithm so that it is further stabilized and, moreover, the parallel properties of the original algorithm are preserved. The first device is based on a dynamic check of the stability. In the second device a slightly modified reordering is used in an attempt to get more non-zero elements in the diagonal blocks (the number of candidates for pivots tends to increase in this situation and, therefore, there is a better chance to select more stable pivots).
Some numerical results obtained by using the two devices will be presented in the last section. The well-known sparse matrices from the Harwell-Boeing set will be used in the experiments.
AMS Subject Classifications65F05 65Y05
Keywordssparse matrix general sparsity Gaussian elimination drop-tolerance reordering block algorithm coarse-grain parallelism speed-up
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