The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix Equation X + A H X - 1 A + B H X - 1 B = I

Abstract

We present two inversion-free iterative methods for computing the maximal positive definite solution of the equation $X+A^H{X}^{-1}A+B^H{X}^{-1}B=I$. We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. We also present some numerical results to compare our proposed methods with some previously developed inversion-free techniques for solving the same matrix equation.