Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON POSITIVE DEFINITE SOLUTIONS OF A CLASS OF NONLINEAR MATRIX EQUATION

  • Fang, Liang (School of Mathematics and Statistics Xidian University) ;
  • Liu, San-Yang (School of Mathematics and Statistics Xidian University) ;
  • Yin, Xiao-Yan (School of Mathematics and Statistics Xidian University)
  • Received : 2017.01.19
  • Accepted : 2017.10.17
  • Published : 2018.03.31
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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

Acknowledgement

Supported by : Central Universities

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