Sparse Orthogonal Decomposition on a Hypercube Multiprocessor

Abstract

In this article the orthogonal decomposition of large sparse matrices on a hypercube multiprocessor is considered. The proposed algorithm offers a parallel implementation of the general row merging scheme for sparse Givens transformations recently developed by Joseph Liu. The proposed parallel algorithm is novel in several aspects. First, a new mapping strategy whose goal is to reduce the communication cost and balance the work load during the entire computing process is proposed. Second, a new sequential algorithm for merging two upper trapezoidal matrices (possibly of different dimensions) is described, wherein the order of computation is different from the standard Givens scheme, and is more suitable for parallel implementation. Third, it is shown that the hypercube network can be employed as a multi-loop multiprocessor. The performance of the parallel algorithm applied to a model problem is analyzed and computation/communication complexity results are presented. Finally it is shown that the parallel submatrix merging algorithm can be viewed as a special case of a more general scheme and it is indicated how the generalized scheme may further reduce the communication cost.