Abstract
In this paper, the nonlinear matrix equation X s - A ∗ X - t A = Q is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given ‖ A ‖ · ‖ Q - 1 ‖ s + t s < s t . In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.
Highlights
Denote the set of all n × n positive definite matrices by P(n).In this paper, we consider the matrix equationXs + A∗f (X) A = Q, (1)where A is nonsingular, Q is a Hermitian positive definite matrix, s is a positive real number, f is a continuous map from P(n) into P(n), and f is either monotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≤ f(Y)) or antimonotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≥ f(Y)).Nonlinear matrix equation of the form (1) often arises in dynamic programming, control theory, stochastic filtering, statistics, and so on
Where A is nonsingular, Q is a Hermitian positive definite matrix, s is a positive real number, f is a continuous map from P(n) into P(n), and f is either monotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≤ f(Y)) or antimonotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≥ f(Y))
For the case s = 1, Ran and Reurings [1] derived some sufficient conditions for the existence and uniqueness of a positive definite solution of (1)
Summary
Denote the set of all n × n positive definite matrices by P(n). where A is nonsingular, Q is a Hermitian positive definite matrix, s is a positive real number, f is a continuous map from P(n) into P(n), and f is either monotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≤ f(Y)) or antimonotone (meaning that 0 ≤ X ≤ Y implies that f(X) ≥ f(Y)). If there is unique Hermitian positive definite matrix T, such that f(X) = T2, X ∈ P(n), we denote that T = f1/2(X). (1) has a Hermitian positive definite solution if and only if there is a nonsingular matrix W, such that W∗W = WW∗, and A = (f1/2(W∗W))−1ZQ1/2, where (Q−1/2)∗(Ws)∗ (Ws) Q−1/2 + Z∗Z = I. In this case, (1) has a Hermitian positive definite solution X = W∗W. F(X) = f(W∗W) ∈ HP(n), and there is unique Hermitian positive definite matrix T, such that f(W∗W) = T2.
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