The Study Of Splitting Iteration Methods For The Non-Hermitian Positive Definite Linear Systems Of Equations And A Kind Of Matrix Equations With Complex Coefficient Matrices

Linear system of equations and matrix equation often appear in many fields of scientific computation and engineering application, such as light scattering imaging, structural dynamics, signal processing, control theory, quantum chemistry and eddy current problems, the neural network, and the numerical solution of partial differential equation. Therefore, the research of numerical method for this kind of problem has high practical significance and application value.This dissertation focuses on numerical method for the linear system of equations with coefficient matrix being non-Hermitian positive definite and the numerical method for the linear matrix equation AXB =C with complex coefficient matrices. Firstly, based on the Hermitian and skew-Hermitian(HSS) iteration method and the positive definite and skew-Hermitian(PSS) iteration method, a modified PSS(MPSS) method for solving the large sparse non-Hermitian positive definite linear systems of equations Ax =b has been proposed, and the unconditional convergence of the new method has been proved. The results of numerical experiments also confirm that the MPSS is better than the GPSS iteration method. Secondly, a modified HSS(MHSS) iteration method has been presented for complex linear matrix equation AXB =C, which is effective to avoid the solution of two matrix equations with complex coefficient matrices in each iteration step. Moreover, the convergence of the new method is analysized. As the numerical experiments show, the new method is effective. This dissertation includes four chapters, which is organized as follows:In Chapter 1, the research background and preliminary knowledge for solving linear system of equations and linear matrix equation have been introduced. The main content of this dissertation has also been introduced.In Chapter 2, based on the HSS iteration method and PSS iteration method, a modified PSS iteration method for solving the non-Hermitian positive definite linear system of equations Ax =b has been presented, and the unconditional convergence of the method has been proved. In addition, several numerical experiments are given to illustrate the efficiency of the new method. According to the results of numerical experiments, we can find that the MPSS iteration method is better than the GPSS and the PSS iteration method.In the third chapter, numerical method of complex linear matrix equation AXB =C has been studied. Based on HSS iteration method, a modified HSS(MHSS) iteration method has been proposed. Moreover, the convergence of the new method has been proved. Finally, some numerical experiments are given to show the efficiency of the method.In the end, the research work of this dissertation is summarized and the future further research work is discussed.