Several Studies On The Iterative Methods For Some Generalized Sylvester Matrix Equations

The fast and effective methods for solving the matrix equations are always an important research topic in the field of numerical algebra. In this thesis, we in-vestigate some generalized Sylvester matrix equations on the aspect of the theory and algorithm in detail. Some satisfactory results are obtained in the thesis. Es-pecially, several algorithms about the numerical simulation results are superior to some efficient methods currently. Therefore, our work can be regard as the effective improvement for the part of the present research work.This paper is organized as follows:Firstly, we briefly review the sources and applications for the Lyapunov equa-tion. Riccati equation, Stein equation, Kalman-Yakubovich equation. We specially introduce the practical applications and the recent research developments for the Sylvester matrix equation. In view of the close relation between the matrix equa-tions and the systems of linear equations, we also, in this part, briefly introduce some efficient methods and their accelerated versions for solving the systems of linear equations.In the first chapter, the iterative method of the generalized coupled Sylvester-transpose linear matrix equations over reflexive and anti-reflexive matrix solution is presented. We prove the conver-gence of the algorithm. The expression forms of the initial iterative matrix is given when the matrix equations are consistent, moreover, the unique minimum Frobenius norm solution is derived. Numerical simulation shows that the proposed algorithm is effective.In the second chapter, we firstly construct the iterative algorithm for solving the matrix equations over centrally symmetric and centrally anti-symmetric matrix solution by extending the matrix equations investigated by some research papers currently, and discuss the convergence of the algorithm on the general complex domain. We show that the exact solution can be obtained within finite iterative steps in the absence bf roundoff error. The special structure of initial iterative matrix is also given to obtain the unique minimum Frobenius norm solution. Some numerical examples are presented to illustrate the effectiveness of the proposed algorithm.In the third chapter, an accelerated gradient based iterative method is con-sidered for solving the generalized Sylvester-transpose matrix equation AXB+ CXTD= F. Not only the iterative information with previous half-step but also a relaxed parameter is fully considered. Then the next iterative step must be a better point close to the exact solution. Under the appropriate assumption, the algorithm converges to the exact solution of the matrix equation. Finally, we show that the effectiveness of the algorithm by comparing three existing methods with some numerical examples, which illustrates the convergence of the AGBI inethod very powerful.In the fourth chapter, by the Kroneeker product, the vec operator and the real representation of the complex matrix, three efficient methods (CGS, Bi-CGSTAB and GPBi-CG) for solving linear system are extended to solve the generalized cou-pled Sylvester-conjugate matrix equation A1XB1+C1YD1=E, A2XB2+C22YD2=F. The proposed methods are comparing with each other, numerical tests show that the extending algorithms are efficient.In the fifth chapter, we investigate the iterative solutions for the two class matrix equations AXB+CXD= E and AiXBi= Fi, (i= 1,2,..., N) based on the method of G.G. We convert the problems of the matrix equations solution into the minimization problems, respectively. Meanwhile, we construct the scaling conjugate gradient (SCG) method with a variable parameter. Under the hypothesis of being consistent, we show the SCG is convergence, namely, the finite termination of the algorithm. At last, the proposed method (SCG) compares with the four existing approaches, that is, GI and LSI methods proposed by Ding et al. [58], also CM and SM methods given by Tang et al. [121]. Many numerical experiments demonstrate that the introduced iterative method is more efficient than the four existing methods...