The Study Of Iterative Methods For Large-scale Sparse Linear Systems

Large-scale sparse linear systems play an important role in many fields of science and engineering.How to quickly and effectively solve large-scale sparse linear systems has become one of the most important research topics.The solution method of linear system includes direct method and iterative method.The direct method is more convenient for solving small-scale linear systems,and the iterative method is more suitable for solving large-scale sparse linear equations.In 2003,Bai proposed the Hermitian and skew-Hermitian(HSS)iterative method,which has the advantages of simple form,wide application range,and unconditional convergence.As soon as the HSS method was proposed,it attracted wide attention.At present,HSS-like methods have become one of the mainstream methods for solving large-scale sparse linear systems.This thesis proposes three iterative methods for solving large sparse linear systems.First,this thesis proposes an extrapolated positive definite and skew-Hermitian(EPSS)iterative method to solve large-scale sparse positive definite linear equations,and analyzes the convergence of the EPSS iterative method.When dealing with certain problems,the EPSS iterative method is more effective than the PSS iterative method.At the same time,this thesis also provides an extrapolated generalized Hermitian and skew-Hermitian(EGHSS)iterative method for the solution of large sparse non-Hermitian positive definite linear equations.Then the theory analyzed the convergence of the EGHSS method.Numerical experiments show that in dealing with some practical problems,the EHSS iterative method is more effective than the GHSS iterative method and EHSS iterative method.In addition,this thesis proposes a new generalized over-relaxation(PSS-GSOR)iterative method for solving saddle point problems.Theoretical analysis shows that the PSS-GSOR method converges to the only solution of linear equations under certain conditions.Numerical calculations prove the effectiveness of the PSS-GSOR method.Finally,the new methods are summarized,and the future research direction is given.