Theories And Algorithms For Several Nonlinear Matrix Equations

Solving the nonlinear matrix equation is one of important problems in the areas of science and engineering computation. The investigation for the nonlinear matrix equations has become a hot topic of numerical algebra. On the base of the exsiting results, we study the following five classes of nonlinear matrix equations systematically.Making use of the properties of the Kronecker product and the principle for the monotonic and bounded sequence in the Banach space, we investigate the positive definite solution of the matrix equation where A is an mn x n matrix, Q is an n x n positive definite matrix, C is an mn x mn positive semidefinite matrix,0<|r|<1 or r=-1. When r=-1, this matrix is generated from a class of interpolation problem. We prove the existence and uniqueness of the positive definite solution of the equation in this situation. When 0<|r|<1, we obtain the same result as the case r=-1 by means of the theory on the monotonic operator defined in a normal cone. Furthermore, the perturbation analysis of the equation is given, the accurate perturbation bound is also presented. Numerical examples illustrate the obtained results.When m=1, r=-1, C=0, the matrix equation has been studied by many authors, some good results has been derived. However, the gen-eralized equation in the case C≠0 or r≠-1 is less studied. In this paper, we investigate the more general equation and obtain some sufficient conditions and necessary conditions for the existence of positive definite solutions of the matrix equation. Simultaneously, the condition that the equation has a unique positive definite solution is also given. In addition, we give the range for the positive definite solution of the matrix equation and perturbation analysis of the equation. The perturbation bound for the positive definite solution is also given.Basing on the elegant properties of the Thomson metric, we study the matrix equa-tion where A is an n×n matrix, Q is an n×n positive definite matrix and q>1. That the equation always has a unique positive definite solution is proved in the case that F(X) is a self-adjoint and nonexpansive map. An iterative method is proposed to compute this unique solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique solution is derived.According to Brouwer's fixed point theorem and Banach's fixed point theorem, we study the positive definite solution of the matrix equation where A is an n×n matrix, Q is an n×n positive definite matrix and s, t are positive integers. We prove the existence of the matrix equation and give the range for the pos-itive definite solution. We also derive some sufficient conditions that the equation has a unique positive definite solution. Moreover, the perturbation analysis and the perturba-tion bound for the equation's positive definite solution are presented. The exsiting results are generalized and improved. The correctness and effectiveness are showed by numerical examples.The following matrix equation is investigated by many authors when 0< r < 1. We continue to study this matrix equation and place emphasis on the case that r > 1. The sufficient conditions and necessary conditions that the matrix has positive definite solutions are obtained. Some sufficient conditions that the equation has a unique positive definite solution are also derived. Furthermore, we give the perturbation analysis for the equation and provide the range for the positive definite solutions of the equation. An iterative method to compute this unique solution is presented. Numerical examples show that the method is feasible and effective.