Research On Numerical Methods For Solving Sylvester Equations And Preconditioning Technology

The efficient solution of the matrix equation is an extremely important problem in computational mathematics.The matrix equations has a widely application in the field of scientific computing and engineering applications,such as light scattering imaging,structural dynamics,digital signal processing,digital image processing,computational fluid mechanics and electromagnetic field calculation,oil exploration data processing,seismic data processing,numerical simulation,numerical weather prediction and explosion control theory,quantum chemistry,and eddy current problems neural network,as well as the numerical solution of partial differential equations.Therefore,it is an important and practical problem to design an effective method for solving the matrix equation.In this paper,we consider two different iterative methods to solve the continuous Sylvester matrix equation:(1)a modified general PSS,called the MGPSS iterative method.In order to solve the continuous Sylvester equations more effectively,we propose the MGPSS iterative method based on the PSS iterative algorithm and the generalized PSS iterative algorithm.And we prove that the MGPSS iteration method will unconditionally converge under certain assumptions.Moreover,the numerical experiments show that the MGPSS iterative method is more effective on the aspect of iterative number of IT and the runtime CPU;(2)Preconditioned PSS,called PPSS.In this paper,a preconditioned positive and skew-Hermitian(PPSS)iterative method has been proposed to solve the Sylvester matrix equation,and its convergence is proved theoretically.An inexact variant of the PPSS iteration method(IPPSS)has also been established,and the analysis of its convergence property in detail has been established.Numerical results show that this new method is more effficient and robust than the existing ones.This dissertation is divided into four chapters,which is organized as follows:In Chapter 1,the background of the research and the preliminary knowledge and the main work of this paper have been illustrated.In Chapter 2,we obtain a modified GPSS(MGPSS)iterative method for solving the continuous Sylvester equations by analyzing the analysis of the generalized PSS.The theoretical analysis shows that the MGPSS iteration method will converge under certain assumptions.Numerical results show that MGPSS iterative method is more efficient than GPSS iteration.In Chapter 3,we propose a preconditioned positive and skew-Hermitian(PPSS)iteration method for solving Sylvester equations.The analysis shows that the PPSS iteration method will converge under certain assumptions.Numerical results show that this new method is more efficient and robust than the existing ones.In the end,the research of this dissertation is summarized and the possible research lines are discussed.