Abstract
In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I and X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C , and D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.
Highlights
Introduction and PreliminariesConsider the system of matrix equations of the formΩ1(X) + A∗P1(X)A + B∗P2(X)B Q, (1)Ω2(X) + C∗Q1(X)C + D∗Q2(X)D Q, (2)where Q is an n × n Hermitian positive definite matrix (HPD, for short), A, B, C, and D are complex matrices of order n × n, Ω1(X), Ω2(X), P1(X), P2(X), Q1(X), and Q2(X) are mappings from the set of positive definite matrices to itself, and A∗ is the conjugate transpose of A.We can see that the above equations incorporate a few linear as well as nonlinear matrix equations (NMEs, in short)
Based on the numerical results, we have concluded that the new iterative approach is extremely powerful and efficient in finding numerical solutions for a wide range of nonlinear matrix equations including complex ones
It produces very accurate results with less iterations and lower computational costs, compared with the basic fixed-point approach
Summary
Where Q is an n × n Hermitian positive definite matrix (HPD, for short), A, B, C, and D are complex matrices of order n × n, Ω1(X), Ω2(X), P1(X), P2(X), Q1(X), and Q2(X) are mappings from the set of positive definite matrices to itself, and A∗ is the conjugate transpose of A. For different types of applications of the Riccati equation, one can check [4, 12, 13]. K(n) and P(n)) denotes the set of all n × n Hermitian Positive semidefinite and positive definite) matrices over C and M(n) stands for the set of all n × n matrices over C. P > Q) indicates that P − Q is positive semidefinite O and I stand for the zero and unit matrix in H(n), respectively
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