A new gaussian elimination-based algorithm for parallel solution of linear equations

AbstractIn this paper, a variant of Gaussian Elimination (GE) called Successive Gaussian Elimination (SGE) algorithm for parallel solution of linear equations is presented. Unlike the conventional GE algorithm, the SGE algorithm does not have a separate back substitution phase, which requires O(N) steps using O(N) processors or O(log22 N) steps using O(N3) processors, for solving a system of linear algebraic equations. It replaces the back substitution phase by only one step division and possesses numerical stability through partial pivoting. Further, in this paper, the SGE algorithm is shown to produce the diagonal form in the same amount of parallel time required for producing triangular form using the conventional parallel GE algorithm. Finally, the effectiveness of the SGE algorithm is demonstrated by studying its performance on a hypercube multiprocessor system

A Low Communication Condensation-based Linear System Solver Utilizing Cramer\u27s Rule

Systems of linear equations are central to many science and engineering application domains. Given the abundance of low-cost parallel processing fabrics, the study of fast and accurate parallel algorithms for solving such systems is receiving attention. Fast linear solvers generally use a form of LU factorization. These methods face challenges with workload distribution and communication overhead that hinder their application in a true broadcast communication environment. Presented is an efficient framework for solving large-scale linear systems by means of a novel utilization of Cramer\u27s rule. While the latter is often perceived to be impractical when considered for large systems, it is shown that the algorithm proposed has an order N^3 complexity with pragmatic forward and backward stability. To the best of our knowledge, this is the first time that Cramer\u27s rule has been demonstrated to be an order N^3 process. Empirical results are provided to substantiate the stated accuracy and computational complexity, clearly demonstrating the efficacy of the approach taken. The unique utilization of Cramer\u27s rule and matrix condensation techniques yield an elegant process that can be applied to parallel computing architectures that support a broadcast communication infrastructure. The regularity of the communication patterns, and send-ahead ability, yields a viable framework for solving linear equations using conventional computing platforms. In addition, this dissertation demonstrates the algorithm\u27s potential for solving large-scale sparse linear systems

A new algorithm for parallel solution of linear equations

Abstract is not available