Solving dense linear systems by Gauss-Huard's method on a distributed memory system

Introduction In this paper we present a modification of Gauss-Huard's method for solving dense linear systems that allows an efficient implementation on machines with a hierarchical memory structure. GaussHuard's method resembles Gauss-Jordan's method in the fact that it reduces the given system by elementary transformations to a diagonal system and it resembles regular Gaussian elimination in the fact that it only uses 2 3 3 + O(n2) floating-point operations to calculate the solution, where GaussJordan's method uses n3 + O(n2) floating-point operations to do so. The method of Gauss-Huard was introduced in 1979 in a version that not included a stabilizing pivoting strategy [6]. After the stability of Gauss-Jordan's method had been properly established in 1989 [1], a stabilizing pivoting strategy for Gauss-Huard's method could be introduced. In 1992 it was proven that Gauss-Huard's method in combination with this pivoting strategy is numerically stable [3]. Our new formulation of Gauss-Huard's method allows an implementation with high data locality; that is to say, this variant is very efficient on machines with a hierarchical memory structure where transport of data to processors may take non-negligible time. The possibility of a better data utilization permits a better performance in comparison with Gaussian elimination on this type of machines. On distributed memory systems with a fast local memory, the performance of our algorithm approaches the performance of Gaussian elimination for large matrices.