The inversion-free iterative methods for a system of nonlinear matrix equations

Abstract

In this paper, we consider the positive-definite solution to a system of nonlinear matrix equations, $X_1+A^{\ast }{X}_2^{-{\alpha }_1}A=I$ and $X_2+B^{\ast }{X}_1^{-{\alpha }_2}B=I$, where ${\alpha }_1,{\alpha }_2$ are two positive constants and $A,B\in {\mathbb{C}}^{r\times r}$ are given matrices. We proposed two inversion-free iterative algorithms for obtaining the maximal positive-definite solution of the system. And we prove that the sequences generated by the iterative schemes are monotonically increasing and bounded above. Finally, some numerical examples are given to show the behaviour of the considered algorithms.

Keywords:

• system of nonlinear matrix equations
• inversion-free iterative algorithms
• maximal positive-definite solution
• convergence analysis
• numerical experiment

2000 AMS Subject Classifications:

• 65H10
• 15A24

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of this author was supported by Hu Guozan Overseas Study Grant for Graduates of Fujian Normal University, National Natural Science Foundation of China [11071041,11201074] and Fujian Natural Science Foundation [2013J01006,2015J01578].