A Study Of Global Krylov Subspace Methods For Complex Matrix Equations

Matrix equations are widely used in system theory,image processing,model reduction,numerical methods of differential equations and other fields.According to different problems,different matrix equations can be obtained,such as Lyapunov matrix equation,Sylvester matrix equation,Stein matrix equation and so on.Although many of the matrix equations generated in practical problems are real matrices,there are also some problems that can generate complex matrix equations.For example,in computational electromagnetics,structural eigenvalue problems,linear algebra and other fields,large complex matrix equations are often needed to be solved.Therefore,it is of great theoretical significance and practical value to discuss the numerical solution of complex matrix equations.At present,there are many numerical algorithms for real matrix equations,but few for complex matrix equations.The existing extended GI algorithm(EGI)and extended CGNE(ECGNE)show slow convergence rate in many numerical experiments when solving complex matrix equations.Therefore,it is very important to construct high performance numerical methods to solve complex matrix equations.In this paper,for two kinds of complex matrix equations,the corresponding complex global Krylov subspace methods are established.The structure of the paper is as follows:In chapter 1,we present the background,significance and research status of complex matrix equations.In chapter 2,we introduce the symbolic representation,and the definition and related properties of Kronecker product.In chapter 3,for the complex symmetric Sylvester matrix equations,the global complex symmetric M-Lanczos orthogonalization process is established,and then the global complex symmetric M-Lanczos method is obtained.By using the triangular decomposition of matrix and the properties of residual matrix,the global conjugate M-orthogonal conjugate residual method(Gl-COCR)is obtained.In order to overcome the irregular convergence behavior of residual norm in Gl-COCR method,a smooth Gl-COCR method(SGl-COCR)is proposed by using minimal residual smoothing technique.Numerical results verify the effectiveness of the new method.In chapter 4,for general complex matrix equations,a new complex global M-biorthogonalization process is established by using the real inner product of complex matrices.According to the matrix relations in the biorthogonalization process,the complex global quasi-minimal residual method(CGl-QMR)is constructed.Finally,we give the residual estimation of the new method.Numerical results show that the new method is more effective than other existing methods.In chapter 5,we summarize the research work of this paper and put forward the direction of future research.