The improved CGS method for large and sparse linear systems on bulk synchronous parallel architectures

Abstract

We propose an improved version of the conjugate gradient squared (CGS) method for the solutions of large and sparse linear systems of equations with unsymmetric coefficient matrices. The proposed method combines elements of numerical stability and parallel algorithm design without increasing computational costs. The algorithm is derived such that all matrix-vector multiplication, inner products and vector updates of a single iteration step are independent and the communication time required for inner product can be overlapped efficiently with the computation time of vector updates. Therefore, the cost of global communication which represents the bottleneck of the performance can be significantly reduced. In this paper, the bulk synchronous parallel model is used to design a fully efficient, scalable and portable parallel proposed algorithm and to provide accurate performance prediction of the algorithm for a wide range of architectures including the Cray T3D, the Parsytec, and a cluster of workstations connected by an Ethernet. This performance model uses only a few system dependent parameters based on a simple and accurate cost modelling to provide useful insight in the time complexity of the method. The theoretical performance prediction are compared with some preliminary measured timing results of a numerical application from the ocean flow simulation.