On the nonlinear matrix equation Xp=A+∑i=1mMi∗(B+X−1)−1Mi

Abstract

We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation $X^p=A+\sum _{i=1}^m{M}_i^{\ast }(B+X^{-1}{)}^{-1}{M}_i,$where p, m are positive integers, M_i (i = 1, 2, …, m) are n × n nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.

Keywords:

• HPD solution
• nonlinear matrix equation
• fixed point theorem
• partially ordered set

2010 Mathematics Subject Classifications:

• 15A24
• 47H10
• 47H09

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This paper was supported financially by Shanxi Province Science Foundation [201901D111020] and Graduate Science and Technology Innovation Project of Shanxi [2019BY014].