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Bull. Korean Math. Soc. 2018; 55(2): 431-448

Online first article February 8, 2018      Printed March 30, 2018

https://doi.org/10.4134/BKMS.b170054

Copyright © The Korean Mathematical Society.

On positive definite solutions of a class of nonlinear matrix equation

Liang Fang, San-Yang Liu, Xiao-Yan Yin

Xidian University, Xidian University, Xidian University

Abstract

This paper is concerned with the positive definite solutions of the nonlinear matrix equation $X-A^{*}\bar{X}^{-1}A=Q$, where $A, Q$ are given complex matrices with $Q$ positive definite. We show that such a matrix equation always has a unique positive definite solution and if $A$ is nonsingular, it also has a unique negative definite solution. Moreover, based on Sherman-Morrison-Woodbury formula, we derive elegant relationships between solutions of $X-A^{*}\bar{X}^{-1}A=I$ and the well-studied standard nonlinear matrix equation $Y+B^{*}Y^{-1}B=Q$, where $B, Q$ are uniquely determined by $A$. Then several effective numerical algorithms for the unique positive definite solution of $X-A^{*}\bar{X}^{-1}A=Q$ with linear or quadratic convergence rate such as inverse-free fixed-point iteration, structure-preserving doubling algorithm, Newton algorithm are proposed. Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.

Keywords: nonlinear matrix equations, Sherman-Morrison-Woodbury formula, Positive definite solution, structure-preserving doubling algorithm, fixed-point iteration, Newton iteration

MSC numbers: 15A24, 65F15, 65F35

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