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For multiprocessor systems with two level memory hierarchy; the communication requirements of parallel Cholesky factorization of dense and sparse symmetric, positive definite matrices are analyzed. The data traffic associated with computing the Chloesky factor of an nn ×n dense matrix using n&agr; processors, &agr; ≤ 2, is shown to be &OHgr;(n2+&agr;/2), assuming that the computational load is uniformly distributed. For an nn ×n sparse matrix, representing a √n × √n regular grid graph, the corresponding data traffic is shown to be &OHgr;(n1+&agr;/2), &agr; ≤ 1.Partitioning schemes that are variations of block assignment scheme are described. The data traffic generated by these schemes are asymptotically optimal and these schemes allow efficient use of up to &Ogr;(n2) and &Ogr;(n) processors in the dense and the sparse case, respectively. The block based partitioning schemes are shown to provide a better utilization of the data accessed from the shared memory and reduce the total data traffic as compared to the schemes based on the column-wise wrap around assignment.

Data traffic reduction schemes for Cholesky factorization on asynchronous multiprocessor systems