A Large-Grain Parallel Sparse System Solver

The eeciency of solving sparse linear systems on parallel processors and more complex multicluster architectures such as Cedar is greatly enhanced if relatively large grain computational tasks can be assigned to each cluster or processor. The ordering of a system into a bordered block upper triangular form facilitates a reasonable large-grain partitioning. A new algorithm which produces this form for unsymmetric sparse linear systems is considered and the associated factorization algorithm is presented. Computational results are presented for the Cedar multiprocessor. Several techniques have been proposed to solve large sparse systems of linear equations on parallel processors. A key task which determines the eeectiveness of these techniques is the identiication and exploitation of the computational granularity appropriate for the target multiprocessor architecture. Many algorithms assume special properties such as symmetric positive deeniteness or exploit knowledge of the application from which the system arises e.g., nite element problems. In this paper, we give an overview of the use of a new ordering technique, the hybrid ordering (H*), and an associated factorization algorithm for unsymmetric unstructured sparse linear systems. More detail on the reordering can be found in 7] and on the merger of the reordering and the factorization algorithm for multicluster architectures in 4]. 1. The Hybrid Ordering. The hybrid ordering H* is composed of two diierent types of orderings: unsymmetric and symmetric. The unsymmetric ordering changes the associated graph of the matrix, mostly by row or column interchanges. The symmetric orderings only relabel the nodes of the associated graphs and maintain certain properties of the system, e.g., symmetry, diagonal dominance. The symmetric orderings are used to obtain a bordered block triangular matrix. The unsymmetric ordering is used to enhance the numerical properties of the matrix.