A parallel Gauss-Seidel algorithm for sparse power system matrices

We describe the implementation and performance of an efficient parallel Gauss-Seidel algorithm that has been developed for irregular, sparse matrices from electrical power systems applications. Although, Gauss-Seidel algorithms are inherently sequential, by performing specialized orderings on sparse matrices, it is possible to eliminate much of the data dependencies caused by precedence in the calculations. A two-part matrix ordering technique has been developed-first to partition the matrix into block-diagonal-bordered form using diakoptic techniques and then to multi-color the data in the last diagonal block using graph coloring techniques. The ordered matrices often have extensive parallelism, while maintaining the strict precedence relationships in the Gauss-Seidel algorithm. We present timing results for a parallel Gauss-Seidel solver implemented on the Thinking Machines CM-5 distributed memory multi-processor. The algorithm presented here requires active message remote procedure calls in order to minimize communications overhead and obtain good relative speedup.