Newton's iterative method to solve a nonlinear matrix equation

Abstract

In this paper, Newton's iterative method to solve the nonlinear matrix equation $X+A^{\ast }{X}^{-n}A=Q$ is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation $X+A^{\ast }{X}^{-n}A=Q$ in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.

KEYWORDS:

• Nonlinear matrix equation
• iterative method
• Newton's iterative method
• fixed point iterative method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A39
• 65F30

Acknowledgements

The authors thank professor Ren-cang Li and anonymous referee for many valuable suggestions that helped to improve this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

National Major Scientific Research Instruments Development Program (61627807) and National Natural Science Foundation of China (11261014,11561015).