Developing variable s-step CGNE and CGNR algorithms for non-symmetric linear systems

Abstract

As is well known, CGNE and CGNR methods as powerful techniques are now commonly used for solving sparse non-symmetric linear systems. In this article, we propose the variable s-step versions of CGNE and CGNR methods to solve non-symmetric linear systems. By reducing communication through data locality, the proposed methods reduce the run time. It is known that sudden increases in s during s-step algorithms can cause the algorithms to lose convergence. This issue is exacerbated for CGNE and CGNR methods by normalizing the system. Two s-step algorithms are also introduced for the methods and a step-by-step and gradual increase of s is suggested to maintain convergence. Some features of CGNE and CGNR extend to their s-step algorithms. Numerical results demonstrate the effectiveness of the methods in maintaining convergence with increasing s.