On The Study Of Iterative Methods For Matrix Equations

Matrix equations have extensive applications in matrix theory,scientific computation and engineering.In this paper we mainly study the iterative solutions to matrix equations AX+XB=C and AXB=C.First,we study the convergence of the exact ADI iteration for solving the Sylvester equation AX+XB =C and derive the upper bound of the contraction factor of the ADI iterations,where the coefficient matrices A and B are assumed to be positive semi-definite matrices(not necessarily Hermitian),and at least one of them to be positive definite.Then we introduce an inexact version of the ADI(IADI)iteration to reduce its computational complexity.The numerical experiments demonstrate that the ADI iterative method have a faster convergence rate than HSS iterative method and the IADI iterative method is effective than the ADI iteration.Finally,we point out an error in the proof of convergence theorem(theorem 3.3)of iterative method for solving the matrix equation AXB=C in the literature[27],and give a more general convergence theorem.This thesis consists of four chapters:In Chapter 1,we mainly introduce the research background,the related solution and the research status of matrix equations,as well as the innovation point in this paper.In Chapter 2,we review some basic definitions and theorems frequently used in the sequel.In Chapter 3,we study the exact and inexact ADI iteration methods for solving the Sylvester equations.In Chapter 4,we mainly point out the errors in the proof of theorem 3.3 in literature[27],and gives a more general convergence theorem.