Fast and Efficient Parallel Solution of Sparse Linear Systems

Fast and Efficient Parallel Solution of Sparse Linear Systems

This paper presents a parallel algorithm for the solution of a linear system Ax = b with a sparse n n symmetric positive deenite matrix A, associated with the graph G(A) that has n vertices and has an edge for each nonzero entry of A. If G(A) has an s(n)-separator family and a known s(n)-separator tree, then the algorithm requires only O(log 3 n) time and (jEj+M(s(n)))= log n processors for the...

The Efficient Parallel Iterative Solution of Large Sparse Linear Systems

The development of eecient, general-purpose software for the iterative solution of sparse linear systems on a parallel MIMD computer requires an interesting combination of expertise. Parallel graph heuristics, convergence analysis, and basic linear algebra implementation issues must all be considered. In this paper, we discuss how we have incorporated recent results in these areas into a genera...

Fast and Efficient Parallel Solution of Dense Linear Systems

The most efficient previously known parallel algorithms for the inversion ofa nonsingular n x n matrix A or solving a linear system Ax = b over the rational numbers require O(log2n) time and M(n).~ processors [provided that M(n) processors suffice in order to multiply two n × n rational matrices in time O (log n)]. Furthermore, the known polylog arithmetic time algorithms for those problems are...

Parallel Solution of Sparse Linear

Structure-adaptive parallel solution of sparse triangular linear systems

Solving sparse triangular systems of linear equations is a performance bottleneck in many methods for solving more general sparse systems. Both for direct methods and for many iterative preconditioners, it is used to solve the system or improve an approximate solution, often across many iterations. Solving triangular systems is notoriously resistant to parallelism, however, and existing paralle...