The Study Of Numerical Algorithms For Two Kinds Of Sylvester Matrix Equations

Numerical methods for solving matrix equations become interesting as soon as they play an important role in various fields, such as control, signal processing, neural network, model reduction and image restoration etc. In this dissertation, we studied the numerical methods for solving two kinds of Sylvester matrix equations AX+XB= C and X+AXB=C. Firstly, we presented a modified gradient based iterative algorithm for Sylvester matrix equations, based on the gradient based iterative algorithm and proved the new algorithm will converge for any initial value under certain condition. Secondly, we proved the existence of the optimal convergence factor and presented a formula for gradient-type based iterative algorithm by detailed theoretical analysis, which is an open problem. Finally, a new algorithm was presented by combining the preconditioned technique with gradient based algorithm for Sylvester matrix equations X+AXB= C. This dissertation includes five chapters, which is organized as follows:Firstly, the research background and research status are given, as well as the preliminary knowledge. Furthermore, the main contents of this paper are briefed.In the second chapter, a modified gradient based iterative algorithm is presented, based on the gradient based iterative algorithm. As the information of the first half iterative step is required to update the solution by the modified method, the new algorithm shows better convergence behavior. Moreover, the new algorithm has been proved to be convergent under certain condition. Numerical results show that the proposed algorithm is efficient than the existing numerical ones.An open-problem, that is, how to choose the optimal convergence factor for gradient based iterative algorithm, has been discussed and solved in the third chapter. The numerical experiments verify the theoretical findings.In Chapter 4, a new algorithm by combining the preconditioned technique with the gradient-type algorithm has been proposed for solving Sylvester matrix equations X+AXB=C and the algorithm is tested on computer. The performance of the preconditioned gradient based iterative method is compared with the existing ones on several numerical examples. The faster convergence behavior is illustrated.Finally, the research work of this dissertation is summarized and the possible direction is further proposed.