Study On HSS-bascd Iteration Methods And Accelerated Techniques For Solving Some Linear And Nonlinear Systems And A Class Of Continuous Sylvester Equations

In this thesis, we discuss some iterative methods and accelerated techniques based on Hermitian and skew-Hermitian splitting (HSS-based) for solving linear and nonlinear systems and a class of continuous Sylvester equations. We consider complex symmetric linear systems, non-Hermitian positive semidefinite linear saddle-point systems, systems of nonlinear equations with positive Jacobian matrices and a class of continuous Sylvester equations. These problems arise in many application areas, it is meaningful to construct robust and effective numerical schemes for these problems.The thesis consists of five chapters:In Chapter One, we describe the research background and progress of iterative meth-ods for solving linear and nonlinear systems, and point out the problems we will consider and the main work of the thesis.In Chapter Two, based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems.The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter a, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter a*which minimizes the above upper bound is also obtained.Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.In Chapter Three, by utilizing the preconditioned Hermitian and skew-Hermitian split-ting (PHSS) iteration technique, we establish a parameterized PHSS (PPHSS) iteration method for non-Hermitian positive semidefinite linear saddle-point systems. The PPHSS method is essentially a two-parameter iteration which covers standard PHSS iteration and can extend the possibility to optimize the iterative process. The iterative sequence produced by the PPHSS method is proved to be convergent to the unique solution of the saddle-point problem when the iteration parameters satisfy a proper condition. In addition, for a spe-cial case of the PPHSS iteration method, we derive the optimal iteration parameter and the corresponding optimal convergence factor. Numerical experiments demonstrate the effec-tiveness and robustness of the PPHSS method both used as a solver and as a preconditioner for Krylov subspace methods.In Chapter Four, by utilizing generalized Hermitian and skew-Hermitian splitting (GHSS) iteration and generalized positive-definite and skew-Hermitian splitting (GPSS) iteration as the inner solver of inexact Newton method, inexact Newton-GHSS and Newton-GPSS methods for solving systems of nonlinear equations with positive Jacobian matrices are proposed, respectively. We discuss the local and semilocal convergence properties un-der some proper assumptions. Moreover, an accelerated Newton-GPSS method is estab-lished and its convergence behavior is analyzed. Numerical results demonstrate that our three methods considerably outperforms the Newton-HSS method in the sense of number of iterations and CPU time.In Chapter Five, by utilizing the Hermitian and skew-Hermitian splitting (HSS) iter-ation technique, we establish a four-parameter HSS (FPHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi-definite matrices. The FPHSS method is essentially a four-parameter iteration which covers standard HSS iteration and can extend the possibility to optimize the iterative process.An exact parameter region of convergence for the method is strictly proved and a minimum value for the upper bound of the iterative spectrum is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the FPHSS (IFPHSS) iteration method and study its convergence property. Numerical experiments demonstrate the effectiveness and robustness of the FPHSS iteration method and its inexact variant.