Conditional Number Estimation Algorithms For Generalized Sylvester Matrix Equations

Generalized Sylvester matrix equations have important applications in many fields of mathematics.There have been many papers devoted to the study of the existence,uniqueness and numerical algorithms of the solution of the generalized Sylvester matrix equations.The error analysis of such matrix equations is an important research topic in recent years.Conditional number theory characterizes the sensitivity of a solution to the perturbation of input data in the worst case scenario.In this thesis,we propose two error estimation methods for generalized Sylvester matrix equations.One is based on the adjoint method;the other is based on the small sample statistical condition number estimation method(SCE).Using the small sample statistical condition number estimation strategy,we propose the estimation methods for the normwise,mixed and componentwise condition numbers of the generalized Sylvester matrix equations.Since the proposed method can utilize the already computed matrix decomposition of the direct method of the matrix equation for example Hessenberg-Schur method,the estimation methods are relatively inexpensive compared with Hessenberg-Schur method.Finally,we also carried out numerical experiments on the proposed algorithms,and the results show that the proposed algorithms are effective.