A modified CMRH method for solving large nonsymmetric linear systems

Abstract

This paper presents a modified CMRH method for solving large nonsymmetric linear systems. Based on the Hessenberg process, the proposed method requires less computation and storage compared to GMRES and QMR methods. The improvement involves utilizing the $(m+1)$-th vector generated during the Hessenberg process to optimize the approximate solution through a linear combination, thereby reducing the number of matrix–vector multiplications and inner products. Theoretical analysis demonstrates that the improved CMRH method is more cost-effective than the original $(m+1)$-step method. Numerical experiments validate the efficiency of the modified CMRH method across various test problems, showing fewer iterations and less computation time, particularly in large-scale problems.