OICC Press Mathematical Sciences 2251-7456 17 4 (December 2023) 2022 01 23 The maximal positive definite solution of the nonlinear matrix equation X+A∗X-1A+B∗X-1B=I 10.1007/s40096-022-00454-4 EN Khosro Sayevand Faculty of Mathematics and Statistics, Malayer University, Malayer, IR Raziyeh Erfanifar Faculty of Mathematics and Statistics, Malayer University, Malayer, IR Hamid Esmaeili Department of Mathematics, Bu-Ali Sina University, Hamedan, IR Journal Article 2022 01 23 Abstract In the current study, we present a new inversion free variant of the fixed point iteration method to find a maximal positive definite solution for the matrix equation X+A∗X-1A+B∗X-1B=I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X + A^{*}X^{-1}A+B^{*}X^{-1}B = I $$\end{document} . The existence conditions of the nonlinear matrix equation are derived. Some numerical examples are presented to show the behavior and efficiency of the considered iterative method. The comparison of the results shows that the new algorithm is more accurate and has fewer operations than the other algorithms. Some conclusions and open problems on possible future research direction close the paper.