| The solution of large linear systems has become increasingly important in scientific and engineering computing.In this dissertation,by using Krylov subspace,matrix splitting and preconditioning techniques,we consider some iterative methods and their preconditioned methods for solving non-Hermitian linear systems.The main contributions are as follows.For complex nonsymmetric linear systems,firstly,the coupled two-term biconjugate A-orthonormalization process is proposed,and a new quasi-minimal residual(QMOR)method is derived based on this process.Then,some convergence properties of the new method and the relation between the new method and the GMRES mathod are analyzed.To accelerate the convergence rate of the QMOR method,its two-sided preconditioned version is established.Secondly,to try and remedy the irregular convergence behavior of the conjugated A-orthogonal residual squared(CORS)method,a transpose-free quasi-minimal residual variant of the CORS method(TFQMORS)is developed by using quasi-smoothing techniques.Furthermore,the relation between TFQMORS method and GMRES method and the finite termination of the TFQMORS iteration are given.Some convergence properties of the TFQMORS method are presented.To accelerate the convergence rate of the TFQMORS method and improve its stability and robustness,the two-sided preconditioned TFQMORS method is designed.Finally,another way to improve the convergence and smoothness of the CORS method is considered.The generalized CORS(GCORS)method is proposed based on products of two nearby biconjugate A-orthogonal residual(BiCOR)polynomials instead of squaring the BiCOR polynomial.The new method can effectively improve the convergence behavior of the CORS method.The GCORS2 method and its preconditioned version are presented.For complex symmetric linear systems,firstly,the QMOR method is generalized to solve complex symmetric linear systems,and then a complex symmetric QMOR(CSQMOR)method is derived.By using indefinite inner product,a symmetric QMOR(SQMOR)method and its two-sided preconditioned version are established based on the connection of the quasi-minimal residual(QMR)and the BiCOR method.Secondly,for complex symmetric indefinite linear systems,preconditioned simplified Hermitian normal splitting(PSHNS)iteration method and PSHNS preconditioner are constructed,and the convergence of the PSHNS iteration method is analyzed.The optimal parameter,the upper bounds of radius of the iterative matrix and the spectral distribution of the preconditioned matrix are given.Finally,the solution of block two-by-two real linear systems arising from the complex symmetric linear systems is considered,and a new block preconditioner is established based on the special splitting of coefficient matrix and relaxing techniques.Some spectral properties of the preconditioned matrix are analyzed,an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained and the detailed implementation of the new preconditioner is derived.For complex linear systems with multiple right-hand siedes,firstly,complex global Bi CG method and complex global BiCGSTAB method are provided.Secondly,a new global generalized product-type Bi CG(Gl-GPBiCG)method based on the complex global Bi CG method and its preconditioned version are constructed.Finally,the variant of the Gl-GPBiCG method and its preconditioned version are developed by studying the reverse-ordered recurrence and the instability of the auxiliary polynomial during the iterative process of the Gl-GPBi CG method.Numerical results show that the proposed methods for solving non-Hermitian linear systems are efficient. |
