Perturbation Analysis Of A Nonlinear Matrix Equation

Nonlinear matrix equation is one of the important substance of the numerical algebra and nonlinear analysis study fields. This kind of equation arises in various areas of application, including dynamic programming, control theory, ladder networks, stochastic filtering statistics, and so on. The Hermitian positive definite solutions of nonlinear matrix equation X+A*X-nA=I(1) is studied, where I is an m×m identity matrix, A is an m×m nonsingular complex matrix,A* denotes the conjugate transpose of the matrix A, and n is a positive integer. In this paper, the definition of the maximal solution of the matrix equation and some new sufficient conditions for the existence of a unique maximal solution of the matrix equation are given, and some new properties of the maximal solution are obtained. Let the coefficient matrix A be slightly perturbed to A=A+ΔA,ΔA∈Cm×m. The perturbed matrix equation is X+A*X-nA=I. (2) Under the condition‖A‖2<nn/((n+1)n+1), a new and better first order perturbation bound for the maximal solution is obtained by means of Schauder fixed point theorem and implicit function theorem, and an explicit expression of its condition number is obtained by the Rice theory. The numerical example is given to illustrate the correctness of theoretical results.Theorem 2.1.1. If‖A‖2≤4/(27), then the maximal solution XL of the matrix equation X+A*X-2A=I satisfies that 1-ζ2≤‖XL-1A‖2≤1-η2 Theorem 2.1.2.If X∈Ω1 such that‖X-1AX-1/2‖2<1/2,then x>2/3I,and X is a unique maximal solution of the matrix equation X+A*X-2A=I.Theorem 2.1.3.If X∈Ω1 such that‖X-1A‖<(?)/3,then X>2/3I,,and X is a unique maximal solution of the matrix equation X+A*X-2A=I.Theorem 2.2.1.If‖A‖2<nn/((n+1)n+1),then matrix equation(1)exists a unique maximal solution.Theorem 2.2.2.If X∈Ω2 such that(1-λmin(X))λmaxn(X)<nn/((n+1)n+1,then X is a unique maximal solution of the matrix equation(1).Theorem 2.2.3.If‖A‖2<nn/((n+1)n+1),then the maximal solution XL of the equation (1)satisfy that 1-α2≤‖XL-n/2A‖2≤1-β2Lemma 3.1.If‖A‖2<nn/((n+1)n+1),then the linear operator is invertible,and‖L-1‖≤1/ρ,whereρ=1-n〔(n+1)/n〕(n+1)‖A‖2.Theorem 3.1.If(1)v.=(?)<1/ξ, then the maximal solution XL of the perturbed matrix Equation(2)exists,and we have‖XL-XL‖≤v.,where v.is the minimal positive solution of the equation Theorem 4.1.The conditon number C(XL)=lim sup(?)for the maximal solution XL of the matrix equation(1)has the explicit expression C(XL)=α/(?)‖Uc‖, where the matrix Uc=(?).