An efficient variant of gauss-jordan type algorithms for direct and parallel solution of dense linear systems

Abstract

Two variants of the Gauss-Jordan type methods, the Purcell method and the Gauss -Huard method, solve a linear system with a similar number of flop counts to the Gauss elimination method (i.e., less than the basic Gauss-Jordan method). In this paper, we first show that the unpivoted versions of the two variants are actually equivalent and their pivoted versions are in general different. Then we demonstrate how one can reproduce a Gauss-Huard method with partial pivoting by modifying the Purcell method and conversely recover the Purcell method by modifying the Gauss-Huard method. It turns out that the latter relationship gives rise to an efficient variant of the Purcell algorithm in terms of memory and an improved variant of the Gauss-Huard method in terms of a smaller growth factor (stability). Some parallel algorithms and results are presented for the new variant