The Study Of Numerical Methods For A Kind Of Linear Systems Of Equations And Matrix Equations

Linear systems of equations and matrix equations often arise in areas ofscientific computing and engineering application, and iterative methods for solvingthese problem have been gaining comprehensive applications in many areas, such aselectrics, mechanics, theory of vibration, automatic control theory and PDEs.Therefore, it is of great practical value to study this problem.In this dissertation, fast iterative algorithms for solving the solutions of twoclasses of matrix equations have been studied, i.e., the large sparse non-Hermitianpositive definite linear systems Ax band the Sylvester equations AX XB C，respectively. Firstly, by modifying the PSS iterative method and the GPSS iterativemethod available now, a MGPSS iterative method for solving non-Hermitian positivedefinite linear systems has been proposed. Secondly, we presented a PSS iterativemethod for the above-mentioned Sylvester equations, based on the PSS iterativemethod for the large sparse non-Hermitian positive definite linear systems. Numericalexamples are given to confirm the efficiency of the proposed methods. Thisdissertation includes four chapters, which is organized as follows:In Chapter1, the background of the research and the preliminary knowledge andthe main work of this paper have been illustrated.In Chapter2, we give a modified version of the generalization of Positive-definedand skew-Hermitian splitting iteration (MGPSS) to solve the non-Hermitian positivedefinite linear systems. The theoretical analysis shows that the MGPSS iterationmethod will converge unconditionally. Numerical examples are reported to confirmthe efficiency of the proposed methods.A positive-definite and skew-Hermitian splitting (PSS) iteration method forcontinuous Sylvester equationsAX XB Chas been proposed in Chapter3. Theanalysis shows that the PSS iteration method will converge under certain assumptions.Numerical results show that this new method is more efficient and robust than theexisting ones.In the end, the research work of this dissertation is summarized and the possible research lines are discussed.