Sgmres Algorithm Convergence Behavior Research

SGMRES algorithm is a simpler GMRES algorithm on the basis of thegeneralized minimum residual method, which is widely used in solving largenon-symmetric sparse system of linear equations. This paper conducts a numberof experimental studies for SGMRES algorithm based on the deepunderstanding of the algorithm. SGMRES algorithm are analyzed and correctedwith the help of specific models, followed by some analysis on the convergencebehavior of the improved SGMRES algorithm. This thesis has completed thefollowing work:1. The SGMRES algorithm is closely related with the GMRES algorithm.This thesis presents a summary introduction of the GMRES method for itshistorical development and practical applications, with an emphasis on its recentstatus. We start with a summary on the origin of the method, followed by somenotable variants, together with some recent developments. Then, we introducesome recent applications of the GMRES method in various research fields,pointing out its connection to and impact on these fields. We provide an outlookon the further development and applications of the GMRES method.2. From the basic idea of SGMRES algorithm, SGMRES algorithm isillustrated through detailed theoretical derivation. We propose another SGMRESalgorithm which can be implemented directly. Then, we summarize theprinciples of the two kinds of algorithm, followed by the specificimplementation steps of the Simpler Gram-Schmidt GMRES(m) algorithm andDirect SGMRES algorithm.3. We have implemented the SGMRES algorithm with the Matlab language.In particular, we found the complex residual norm in numerical experiment. Itprompted us to improve the SGMRES algorithm, proposing four differentcorrections, and make a further discussion to the stability of SGMRESalgorithms. After that, we put forward the fifth correction of SGMRESalgorithm and analyze convergence behavior of the improved SGMRESalgorithm. Experimental show that the SGMRES algorithm is simpler, withsmaller computational cost, and the improved SGMRES algorithm has a better convergence. The improved version does not stagnate during the iteration, andits stability has been improved significantly.