Research On Several Kinds Of GMRES (M) Algorithm Based On Galerkin Principle

In various fields such as science and technology,engineering calculations,many problems are ultimately solving large sparse linear equations.The most common iterative algorithm for solving such equations is the GMRES(m)algorithm based on the Galerkin principle.However,in the solution of practical problems,the number of iterations will increase with the increase of the number of conditions of the coefficient matrix,which will lead to the increase of the calculation amount of the algorithm and the storage space required by the computer,thus slowing down the convergence speed of the algorithm or causing stagnation of convergence.Based on this,the main work of the paper is as follows:First,the paper discusses the basic idea of Galerkin principle,the domestic and international research status of GMRES(m)algorithm and improved algorithm,research significance and related theoretical basic knowledge,which lays the foundation for the in-depth study of GMRES(m)algorithm based on Galerkin principle.Secondly,based on the variable parameter H-GMRES(m)algorithm,using the truncation technique and incomplete orthogonal Householder transformation,a truncated H-IGMRES(m)algorithm is proposed.The feasibility and convergence of the H-IGMRES(m)algorithm are verified by theoretical analysis and numerical examples.The H-GMRES(m)algorithm and the H-IGMRES(m)algorithm are compared in terms of calculation accuracy and efficiency.The efficiency and superiority of H-IGMRES(m)algorithm are proved.Theoretical analysis and research results show that the proposed truncated H-IGMRES(m)algorithm improves the calculation efficiency and reduces the number of iterations while ensuring the calculation accuracy.Finally,a symmetric successive super-relaxation iteration method(SSOR method)is used to propose a preprocessor,and the VRPGMRES(m)algorithm is preprocessed to form a new algorithm,namely SORVGMRES(m)algorithm.The convergence of the algorithm is analyzed theoretically,and the factors that affect the accuracy and efficiency of SORVGMRES(m)algorithm are analyzed.At the same time,the SORVGMRES(m)algorithm is compared with the existing VRPGMRES(m)algorithm.Theoretical analysis and numerical examples show that the SORVGMRES(m)algorithm is not only feasible,but also reduces the calculation time and the number of iterations on the premise of ensuring the calculation accuracy,which has a certain effect on the practical solution of large linear equations.