Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation \(X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q\)

Abstract

In this paper, we study the nonlinear matrix equation \(X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q\), where \(A,Q \in {{\mathbb {C}}}^{n\times n}\), Q is a Hermitian positive definite matrix and \(X \in {{\mathbb {C}}}^{n\times n}\) is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11871444).

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Correspondence to Xiao-Xia Guo.

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On behalf of all authors, Xiao-Xia Guo states that there is no conflict of interest.

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Guo, XX., Wu, HX. Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation \(X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q\). Commun. Appl. Math. Comput. 3, 123–135 (2021). https://doi.org/10.1007/s42967-020-00062-w

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Keywords

  • Positive definite solution
  • Structure-preserving-doubling like algorithm
  • Convergence
  • Numerical experiment

Mathematics Subject Classification

  • 15A24
  • 60J10
  • 65F30
  • 65U05