A New Inversion-Free Iterative Method for Solving a Class of Nonlinear Matrix Equations

In this paper, we propose a new inversion-free iterative method for computation of positive definite solution of the nonlinear matrix equation Xp=A+M(B+X-1)-1M∗,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X^p=A+M\,(B+X^{-1}\,)^{-1}\,M^{*}, \end{aligned}$$\end{document}where p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document} is a positive integer, A and B are Hermitian positive semidefinite matrices, and M is an arbitrary square complex matrix. This matrix equation has been studied recently in Meng and Kim (J Compt Appl Math 322:139–147, 2017), where the authors proposed an inversion-free algorithm for solving this equation with the hypothesis that the matrix B is nonsingular. For our part, we propose a new algorithm that is applicable for all choices of the positive semidefinite matrix B even if it is singular. To prove the convergence of the proposed algorithm, we prove a new matrix inequality. The efficiency of the proposed algorithm is confirmed by some numerical simulations.