10.1007/s40096-022-00454-4

The maximal positive definite solution of the nonlinear matrix equation X+A∗X-1A+B∗X-1B=I

  • Khosro Sayevand1,
  • Raziyeh Erfanifar*1,
  • Hamid Esmaeili2,
  1. Faculty of Mathematics and Statistics, Malayer University, Malayer, IR
  2. Department of Mathematics, Bu-Ali Sina University, Hamedan, IR

Published in Issue 2022-01-23

Copyright (c) -1 Copyright © 2024, The Author(s), under exclusive licence to Islamic Azad University

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This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Sayevand, K., Erfanifar, R., & Esmaeili, H. (2022). The maximal positive definite solution of the nonlinear matrix equation X+A∗X-1A+B∗X-1B=I. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00454-4
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Abstract

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.

Keywords

  • Matrix equation,
  • Convergence rate,
  • Inversion free,
  • Fixed point iteration.

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