On the existence of extremal positive definite solutions of the nonlinear matrix equation X r + ∑ i = 1 m A i ∗ X δ i A i = I

In the present paper, a necessary condition for the existence of positive definite solutions of the nonlinear matrix equation X r + ∑ i = 1 m A i ∗ X δ i A i = I is derived, where − 1 < δ i < 0 , I is an n × n identity matrix, A i are n × n nonsingular complex matrices and r , m are positive i...

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Bibliographic Details
Published in:Mathematical and computer modelling 2010-05, Vol.51 (9), p.1107-1117
Main Authors: Sarhan, A.M., El-Shazly, Naglaa M., Shehata, Enas M.
Format: Article
Language:English
Subjects:
Extremal positive solution
Iteration
Nonlinear matrix equation
Positive definite matrix
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Summary:In the present paper, a necessary condition for the existence of positive definite solutions of the nonlinear matrix equation X r + ∑ i = 1 m A i ∗ X δ i A i = I is derived, where − 1 < δ i < 0 , I is an n × n identity matrix, A i are n × n nonsingular complex matrices and r , m are positive integers. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed. Furthermore, the rate of convergence of some proposed algorithms is proved. Finally, numerical examples are given to illustrate the performance and effectiveness of the proposed algorithms.
ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2009.12.021