On the Hermitian positive definite solution and Newton's method for a nonlinear matrix equation

ABSTRACT

In this paper, necessary and sufficient conditions for the existence of the Hermitian positive definite solution for a nonlinear matrix equation are derived. Then, a sufficient condition for the existence of the unique Hermitian positive definite solution is presented. Further, we propose to use Newton's method to solve this nonlinear matrix equation with some constraints. In addition, we prove the convergence of Newton's method. Finally, we present some numerical examples to illustrate the effectiveness of the derived results.

KEYWORDS:

• Nonlinear matrix equation
• Hermitian positive definite solution
• Newton's method
• convergence analysis

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A24

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported in part by National Natural Science Foundation of China (11771370, 11771368, 11571292), Natural Science Foundation of Hunan Province (2018JJ2376, 2017JJ3305), and Youth Project of Education Department of Hunan Province (16B257, 18B057).