Algorithms For Several Nonlinear Matrix Equations

Nonlinear matrix equations have many applications in:dynamic programming, control theory, ladder networks, stochastic filtering, statistics and so on. In recent years, the investigation for the nonlinear matrix equations have become a hot topic of numerical algebra. Especially, the problem of solving nonlinear matrix equations has been one of the important issues in the field of non-linear sciences and numerical algebra. There has been many results on positive definite solutions of nonlinear matrix equation in the literature. On the base of the existing results, we study the following three types of nonlinear matrix equation systematically.(1)X+A*X-1A=Q (where A is a nonsingular matrix, Q is a positive definite matrix);(2) X+A*X-αA=Q(α>0);(3) Xs+A*X-tA=Q(s,t are positive integers). Here, we mainly study the existence of a positive definite solution of the above matrix equations and methods for solving these matrix equations.The main results are as follows:1. For the nonlinear matrix equation X+A*X-1A=Q, we consider the conditions for the existence of a positive defisite solution of the matrix equation and two iterative algorithms for obtaining the positive definite solutions of this matrix equation. By the way, the convergence of these iterative algorithms is proposed under certain conditions.2. For the nonlinear matrix equationX+A*X-αA=Q, we give some necessary conditions for the existence of the positive definite solutions of the matrix equation. We also obtain some new sufficient and necessary conditions for the existence of the positive definite solutions of the matrix equation. Next, we respectively propose two iterative methods for computing the positive definite solutions in both cases:α∈(0,1] and ore(1,∞). By the way, the convergence and the error estimations of these iterative methods are found. Then for X+A*X-αA=Q,α∈(0,1], we prove that if it has a positive definite solution, it must has a maximal solution, for X+A*X-αA=Q,α∈(1,∞), we prove that if it has a positive definite solution, it must have a minimal solution.3. For the nonlinear matrix equation Xs+A*X-tA=Q, a new sufficient and necessary condition for the equation to have a positive definite solution X and the sufficient and necessary condition for the equation to have a positive definite solution of the form X=θQs(0<θ<1) are derived. In particular, when A is singular and A, Q satisfy λ1(A*A)≤s/s+t(l/s+t)l/s λnl/s(Q), a new estimate of positive definite solutions is obtained. In the end, we propose the iterative method for computing the positive definite solutions and the convergence of the method. Especially, an iterative algorithm for obtaining the positive definite solution of the equation with Q=I is discussed. The error estimations are found.