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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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