Numerical Solutions Of Some Problems On Singular Linear Systems And Matrix Equations

Many fields in scientific computing and engineering applications,such as fluid dy-namics,structure dynamics,constrained optimization,electromagnetism,image restora-tion,signal processing,control and systems theory,are closely related to the solving of linear systems.Thus,it is of great theoretical significance and practical application value to design a kind of robust and efficient numerical algorithm based on the structure and characteristic of problems.This thesis mainly focuses on the solving of two kinds of linear systems,one is numerical solutions of large,sparse system of linear equations with singular and non-Hermitian positive semi-definite coefficient matrix,and another is numerical solutions and applications of linear matrix equations.Firstly,we study numerical solutions of the system of linear equations with singu-lar and non-Hermitian positive semi-definite coefficient matrix.By introducing a class of singular preconditioner for the Hermitian and skew-Hermitian splitting(HSS)itera-tion method,we propose a generalized preconditioned HSS(GPHSS)iteration method.Then,it is proved that under some condition,the GPHSS iteration method converges to the minimum norm least squares solution of both the consistent and inconsistent singu-lar system of linear equations,and the convergence does not depend on the initial guess vector.Meantime,we also analyse the convergence property of the GPHSS precon-ditioned GMRES method.Besides,it is also proved that the GPHSS iteration method converges unconditionally to the minimum norm least squares solution of the singular saddle point problem.Numerical experiments are presented to show the feasibility and the effectiveness of the GPHSS iteration method and the corresponding preconditioner.Next,we extend the PMHSS iteration method for solving a class of non-singular block two-by-two system of linear equations to the singular case,named as generalized PMHSS(GPMHSS)iteration method.Theoretical analyses show that the GPMHSS it-eration method converges unconditionally to the minimum norm least squares solution of both the consistent and inconsistent singular block two-by-two system of linear equa-tions.Moreover,the corresponding preconditioned GMRES method also determines the minimum norm least squares solution of the consistent singular block two-by-two system of linear equations at breakdown.Numerical experiments are used to verify the robustness and the effectiveness of the GPMHSS iteration method and the GPMHSS preconditioner.Then,we analyse how to solve the elliptic partial differential equations defined in a polygonal domain numerically.By using the Schwarz-Christoffel(SC)mappings and the SC toolbox in Matlab,we transform the partial differential equations from a poly-gon domain to a rectangle domain.Then,discretizing the transformed equation by the finite difference method leads to a special kind of matrix equations.Since the complex-ity of the transformed partial differential equation,the corresponding matrix equation involves Hadamard product,such that most of the methods used to solve generalized matrix equations are not applicable.Therefore,we discuss how to solve this kind of matrix equations numerically in detail.Besides,we also analyse the selection of the control vertices of the Schwarz-Christoffel mappings.Numerical experiments verify the feasibility and the effectiveness of our method.Finally,in order to solve the computational drawback of the vectorized Sherman-Morrison-Woodbury(SMW)algorithm for solving matrix equations,we explore how to use the matrix-oriented Sherman-Morrison-Woodbury algorithm to solve small and medium size generalized linear matrix equations,and simply analyse the stability of this method.Then,by combining it with the projection methods,we apply the matrix-oriented SMW algorithm to the solving of large-scale generalized matrix equations.Moreover,based on the matrix-oriented SMW algorithm,a kind of method for solving linear tensor equations is discussed.Numerical experiments are given to illustrate the feasibility and effectiveness of our strategies.