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The Hermitian Positive Definite Solutions Of Nonlinear Equation X+A~*X~qA=I(q>0)

Posted on:2009-09-04Degree:MasterType:Thesis
Country:ChinaCandidate:S L YangFull Text:PDF
GTID:2120360272471980Subject:Applied Mathematics
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The problem of solving the nonlinear matrix equation, is mainly to determine the solution of the equation by the information of parameters of the equation.In practice, the equation X + A*XqA = I arises in various areas of applications, including control theory, dynamic programming, statistic, the finite difference approximation to an elliptic partial differential equation, and so on. From the application point of view, the Hermitian positive solution is more important, about which are we concerned. In the sequel, a solution always means a Hermitian positive definite one. The study of this kind of problem haw three basic problems: (1)the theoretic issue on solvability, ie.. the necessary and sufficient conditions for the existence of the solution; (2)the numerical solution, ie.. the effective numerical ways; (3)the analysis of the perturbation.First, in this paper, we discuss the existence of the solution of the equationX + A*XqA = I, q > 0by the following theorems.Theorem 1 Eq.(1) has a solution if and only if A admits the following factorization:A = (W*W)-q/2Zwhere W is a nonsingular square matrix and columns of (?) are unitary orthononnal.Inthis case X = W*W is a solution and all the solutions can be formed in this way.Theorem 2 For any invertible matrix A∈Cn×n,Eq.(1) has a solution if and only if there exist unitary matrices P and Q and diagonal matrices T > 0 andΣ> 0 withΓ+Σ2 = I such thatA = PΓ-q/2QΣP. In this case X=P*ΓP is a solution of Eq.(1). Theorem 3 There do not exist two compareable solutions to Eq.(1).It is impossible that for any two solution X(1) and X(2) of Eq.(1). sueh thatX(1)≤X(2) or X(1)≥X(2).Theorem 4 If Eq.(1) has a positive definite solution X,then(a)ρ(X(q+1)/2A-A*X(q+1)/2)≤1(b)ρ(X(q+1)/2A + A*X(q+1)/2≤1(c) If A is invertible,thcn X-q > AA*.Theorem 5 Ifηis the minimum characteristic value of X,ξis the maximum characteristic value of X,λis the characteristic value of A ,then we have :Second, this paper offers two different iterative methods to approximate the solutions of the Eq.(1) with q > 0 and the convergence behaviors of the basic fixed point iteration solutions are investigated by the following theorems.Theorem 6 Soppose that A is invertible and ||A*A||≤1/(q+1), 0 < q < l,then(a) Eq.(1) has a unique solution X and the solution satisfies q/(q+1)I≤X≤I.(b) The solution can be obtained by the following matrix sequence :(c) The estimateshold where a = (q/(q+1))q .Theorem 7 If A is invertible, when 0 *A||<(q+1)q-1/qq,(b)ξqA*A + ((1 -ξ)A-*A-1)1/q>I.Eq.(1) has a solution X and it can be obtained by the matrix sequence(3.2).Theorem 8 Soppose that A is invertible and ||A-*A-1||≤(q/(q+1))q , q > l,then(a) Eq.(1) has a unique solution X and the solution satisfies 0≤X≤q/(q+1)I.(b) The solution can be obtained by the following matrix sequence :(c) The estimateshold where b = (1/(q+1))1/q.Theorem 9 If A is invertible, when q > 1,0 <η<1,if(a) (1 -η)I < A*A < (1-η)/ηqI,(b)ηqA*A + ((1 -η)A-*A-1)1/q 1 and being integer. The perturbation analysis for the unique solution of the matrix equation are studied according to the following theorems.Theorem 10 When A*A < I and q being integer,ifthen Eq.(1) has a unique solution X and the solution satisfiesαI≤X≤βI.Theorem 11 Suppose that X and (?) are the solutions of the matrix equationsX + A*XqA = I, andrespectively. Let (?) land one of the condition is well(a) ||(?)||≤||A||,(b) ||(?)|| > ||A|and||△A|| < (?).then||△X||≤σ||△A||whereTheorem 12 Suppose that X and (?) are the solutions of the matrix equationsX + A*XqA = I,andrespectively. Let (?) then||△X||≤0||△A||where...
Keywords/Search Tags:Nonlinear matrix equation, Positive definite solution, Iterative method, Perturbation bound
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