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The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation

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Abstract

In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods

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References

  1. D. A. Paolo, I. Alberto, and R. Antonio, “Realization and structure theory of bilinear dynamical systems,” SIAM J. Control Optim., vol. 12, pp. 517–535, 1974.

    Article  MathSciNet  Google Scholar 

  2. A. Samir, A. L. Baiyat, M. A. Bettayeb, and M. A. L. Saggaf, “New model reduction scheme for bilinear systems,” Int. J. Sys. Scie., vol. 25, pp. 1631–1642, 1994.

    Article  MathSciNet  Google Scholar 

  3. D. L. Kleinman, “On the stability of linear stochastic systems,” IEEE Trans. Automat. Control, vol. 14, pp. 429–430, 1969.

    Article  MathSciNet  Google Scholar 

  4. P. Benner and T. Damm, “Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems,” SIAM J. Control Optim., vol. 49, pp. 686–711, 2011.

    Article  MathSciNet  Google Scholar 

  5. W. S. Gray and J. Mesko, “Energy functions and algebraic Gramians for bilinear systems,” Proc. of IFAC Symposium on Nonlinear Control Systems Design Symposium, The Netherlands, vol. 31, pp. 101–106, 1998.

    Google Scholar 

  6. P. Benner and J. Saak, “Linear-quadratic regulator design for optimal colling of steel profiles,” 2005.

  7. H. Dorissen, “Canonical forms for bilinear systems,” Syst. Control Lett., vol. 13, pp. 153–160, 1989.

    Article  MathSciNet  Google Scholar 

  8. T. Damm, “Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equation,” Numer. Linear Algebra. Appl., vol. 15, pp. 853–871, 2008.

    Article  MathSciNet  Google Scholar 

  9. P. Benner and T. Breiten, “Low rank methods for a class of generalized Lyapunov equations and related issus,” Number Math., vol. 124, pp. 441–470, 2013.

    Article  Google Scholar 

  10. H. Y. Fan, P. Weng, and E. Chu, “Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control,” Numer. Algor., vol. 71, pp. 245–272, 2016.

    Article  MathSciNet  Google Scholar 

  11. S. Y. Li, H. L. Shen, and X. H. Shao, “PHSS iterative method for solving generalized Lyapunov equations,” Matnematics, vol. 7, no. 1, 38, 2019.

    Google Scholar 

  12. Z. Z. Bai, “On Hermition and skew-Hermition splitting itertion methods for continuous Sylvester equations,” J. Comput. Math., vol. 29, pp. 185–198, 2011.

    Article  MathSciNet  Google Scholar 

  13. X. Wang, W. W. Li, and L. Z. Mao, “On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX + XB = C,” Comput. Math. Appl., vol. 66, pp. 2352–2361, 2013.

    Article  MathSciNet  Google Scholar 

  14. D. Zhou, G. L. Chen, and Q. Y. Cai, “On modified HSS iteration methods for continuous Sylvester equations,” Appl. Math. Comput., vol. 263, pp. 84–93, 2015.

    MathSciNet  MATH  Google Scholar 

  15. H. L. Shen, S. Y. Li, and X. H. Shao, “The NMHSS iterative method for the standard Lyapunov equation,” IEEE Acess, vol. 7, pp. 13200–13205, 2019.

    Article  Google Scholar 

  16. M. Dehghan and A. Shirilord, “Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems,” Numer. Algor., vol. 83, pp. 281–304, 2020.

    Article  MathSciNet  Google Scholar 

  17. M. Dehghan and A. Shirilord, “Solving complex Sylvester matrix equation by accelerated double-step scale splitting (ADSS) method,” Engineering with Comput., 2019. DOI: https://doi.org/10.1007/s00366-019-00838-6

  18. M. Dehghan and A. Shirilord, “The double-step scale splitting method for solving complex Sylvester matrix equation,” Comput. Appl. Math., vol. 38, Article number 146, 2019.

  19. M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,” Appl. Math. Comput., vol. 202, pp. 571–588, 2008.

    MathSciNet  MATH  Google Scholar 

  20. M. Dehghan and M. Hajarian, “On the reflexive and anti-reflexive solutions of the generalized coupled Sylvester matrix equations,” Int. J. Sys. Sci., vol. 41, pp. 607–625, 2010.

    Article  Google Scholar 

  21. M. Dehghan and M. Hajarian, “Efficient iterative method for solving the secondorder Sylvester matrix equation EVF2 − AVF − CV = BW,” IET Control Theory Appl., vol. 3, pp. 1401–1408, 2009.

    Article  MathSciNet  Google Scholar 

  22. M. D. Madiseh and M. Dehghan, “Generalized solution sets of the interval generalized Sylvester matrix equation \(\sum\limits_{i = 1}^p {{A_i}{X_i} + \sum\limits_{j = 1}^q {{Y_j}{B_j} = C} } \) and some approaches for inner and outer estimations,” Comput. Math. Appl., vol. 68, pp. 1758–1774, 2014.

    Article  MathSciNet  Google Scholar 

  23. M. Dehghan and A. Shirilord, “A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation,” Appl. Math. Comput., vol. 348, pp. 632–651, 2019.

    MathSciNet  MATH  Google Scholar 

  24. Z. Z. Bai, M. Benzi, and F. Chen, “On preconditioned MHSS iteration methods for complex symmetric linear systems,” Numer. Algorithms, vol. 56, pp. 297–317, 2011.

    Article  MathSciNet  Google Scholar 

  25. L. T. Zhang, X. Y. Zuo, T. X. Gu, and T. Z. Huang, “Conjugate residual squared method and its improvement for non-symmetric linear systems,” Int. J. Comput. Math., vol. 87, pp. 1578–1590, 2010.

    Article  MathSciNet  Google Scholar 

  26. M. Hajarian, “A finite iterative method for solving the general coupled discrete-time periodic matrix equations,” Circuits Syst. Signal Process, vol. 34, pp. 105–125, 2015.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105, P. R. China

    Juan Zhang & Huihui Kang

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  1. Juan Zhang
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  2. Huihui Kang
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Correspondence to Juan Zhang.

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Recommended by Associate Editor Le Van Hien under the direction of Editor Jessie (Ju H.) Park.

This work was supported in part by National Natural Science Foundation of China (11771370, 11771368), Natural Science Foundation of Hunan Province (2018JJ2376), and Project of Education Department of Hunan Province (18B057, 19A500).

Juan Zhang received her B.S., M.S. and Ph.D. degrees in the Department of Mathematics and Computational Science from Xiangtan University, China, in 2006, 2009, and 2013, respectively. Since 2015, she has been an associate professor in the Department of Mathematics and Computational Science at Xiangtan University. Her research interests include control theory and its application, matrix analysis and its application.

Huihui Kang received her B.S. degree in the School of Mathematics and Statistics from Xinyang University, China, in 2016. Now, she is studying for a master degree in the School of Mathematics and Computational Science at Xiangtan University. Her research interests include matrix equation and numerical computation.

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Zhang, J., Kang, H. The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation. Int. J. Control Autom. Syst. 19, 339–349 (2021). https://doi.org/10.1007/s12555-020-0053-1

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  • DOI: https://doi.org/10.1007/s12555-020-0053-1

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