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


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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  1. 1.

    Engwerda, J.C.: On the existence of a positive definite solution of the matrix equation \(X+A^{\rm{T}}X^{-1}A=I\). Linear Algebra Appl. 194, 91–108 (1993)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ferrante, A., Levy, B.C.: Hermitian solutions of the equation \(X=Q+NX^{-1}N^{*}\). Linear Algebra Appl. 247, 359–373 (1996)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Guo, C.-H.: Newton’s method for discrete algebraic Riccati equations when the closed-loop matrix has eigenvalues on the unit circle. SIAM J. Matrix Anal. Appl. 20, 279–294 (1998)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Guo, C.-H., Lancaster, P.: Iterative solution of two matrix equations. Math. Comp. 68, 1589–1603 (1999)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Guo, C.-H., Kuo, Y.-C., Lin, W.-W.: Complex symmetric stablizing solution of the matrix equaion \(X+A^{*}X^{-1}A=Q\). Linear Algebra Appl. 435, 1187–1192 (2011)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Guo, C.-H., Kuo, Y.-C., Lin, W.-W.: On a nonlinear matrix equation arising in nano research. SIAM J. Matrix Anal. Appl. 33, 235–262 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Huang, N., Ma, C.-F.: Two structure-preserving-doubling like algorithms for obtaining the positive definite solution to a class of nonlinear matrix equation. Comput. Math. Appl. 69, 494–502 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Levy, B.C., Frezza, R., Krener, A.J.: Modeling and estimation of discrete-time Gaussian reciprocal processes. IEEE Trans. Autom. Control Algebr. 35, 1013–1023 (1990)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lin, W.-W., Xu, S.-F.: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl. 28, 26–39 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Meini, B.: Efficient computation of the extreme solutions of \(X+A^{*}X^{-1}A=Q\) and \(X-A^{*}X^{-1}A=Q\). Math. Comp. 71, 1189–1204 (2001)

    Article  Google Scholar 

  11. 11.

    Xu, S.-F.: On the maximal solution of the matrix equation \(X+A^{*}X^{-1}A=I\). Acta Sci. Nat. Univ. Pekinensis 36, 29–38 (2000)

    MATH  Google Scholar 

  12. 12.

    Zhou, B., Cai, G.-B., Lam, J.: Positive definite solutions of the nonlinear matrix equation \( X +A^{\rm{H}}{\overline{X}}^{-1}A =I\). Appl. Math. Comput. 219, 7377–7391 (2013)

    MathSciNet  MATH  Google Scholar 

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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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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).

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

Mathematics Subject Classification

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