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Extending the GPBiCG algorithm for solving the generalized Sylvester-transpose matrix equation

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Abstract

By applying Kronecker product and vectorization operator, we extend the generalized product bi-conjugate gradient (GPBiCG) algorithm for solving the generalized Sylvester-transpose matrix equation \(\sum\nolimits_{i = 1}^r {(A_i XB_i + C_i X^T D_i ) = E} \). By using numerical results, we compare the new method with other popular iterative solvers in use today.

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References

  1. A. Gerheim, “Numerical solution of the Lyapunov equation for narrow-band digital filters,” IEEE Trans. on Circuits and Systems, vol. 31, no. 11, pp. 991–992, 1984.

    Article  Google Scholar 

  2. J. Feng, J. Lam, and Y. Wei, “Spectral properties of sums of certain Kronecker products,” Linear Algebra Appl., vol. 431, no. 9, pp. 1691–1701, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Sorensen and A. Antoulas, “The Sylvester equation and approximate balanced reduction,” Linear Algebra Appl., vol. 351-352, pp. 671–700, August 2002.

    Article  MathSciNet  Google Scholar 

  4. C. C. Tsui, “New approach to robust observer design,” International Journal of Control, vol. 47, no. 3, pp. 745–751, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Zhou, Z. Lin, and G. R. Duan, “Stabilization of linear systems with input delay and saturation-a parametric Lyapunov equation approach,” International Journal of Robust and Nonlinear Control, vol. 20, no. 13, pp. 1502–1519, 2010.

    MATH  MathSciNet  Google Scholar 

  6. B. Zhou, G. R. Duan, and Z. Lin, “A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation,” Automatica, vol. 47, no. 2, pp. 316–325, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Zhou, Z. Lin, and G. R. Duan, “A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems,” Automatica, vol. 45, no. 1, pp. 238–244, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Zhou, Z. Lin, and G. R. Duan, “Properties of the parametric Lyapunov equation-based low-gain design with applications in stabilization of time-delay systems,” IEEE Trans. Automat. Contr., vol. 54, no. 7, pp. 1698–1704, 2009.

    Article  MathSciNet  Google Scholar 

  9. B. Zhou, G. R. Duan, and Z. Lin, “A parametric Lyapunov equation approach to the design of low gain feedback,” IEEE Trans. Automat. Contr., vol. 53, no. 6, pp. 1548–1554, 2008.

    Article  MathSciNet  Google Scholar 

  10. A. G. Wu, G. R. Duan, and Y. Xue, “Kronecker maps and Sylvester-polynomial matrix equations,” IEEE Trans. Automat. Contr., vol. 52, no. 5, pp. 905–910, 2007.

    Article  MathSciNet  Google Scholar 

  11. A. G. Wu, G. Feng, J. Hu, and G. R. Duan, “Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 442–450, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. G. Wu, G. Feng, G. R. Duan, and W. J. Wu, “Iterative solutions to coupled Sylvester-conjugate matrix equations,” Computers and Mathematics with Applications, vol. 60, no. 1, pp. 54–66, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. G. Wu, X. Zeng, G. R. Duan, and W. J. Wu, “Iterative solutions to the extended Sylvesterconjugate matrix equations,” Applied Mathematics and Computations, vol. 217, no. 1, pp. 130–142, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. G. Wu, L. Tong, and G. R. Duan, “Finite iterative algorithm for solving coupled Lyapunov equations appearing in continuous-time Markov jump linear systems,” Int. J. Systems Science, vol. 44, no. 11, pp. 2082–2093, 2013.

    Article  MathSciNet  Google Scholar 

  15. B. Zhou, G. R. Duan, and Z. Y. Li, “Gradient based iterative algorithm for solving coupled matrix equations,” Syst. Contr. Lett., vol. 58, no. 5, pp. 327–333, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Dehghan and M. Hajarian, “On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations,” Linear and Multilinear Algebra, vol. 59, no. 11, pp. 1281–130, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Dehghan and M. Hajarian, “On the generalized reflexive and anti-reflexive solutions to a system of matrix equations,” Linear Algebra and its Applications, vol. 437, no. 11, pp. 2793–2812, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Dehghan and M. Hajarian, “Solving the generalized Sylvester matrix equation \(\sum\nolimits_{i = 1}^p {A_i XB_i + } \sum\nolimits_{j = 1}^q {C_j YD_j = E} \) over reflexive and anti-reflexive matrices,” International Journal of Control, Automation and Systems, vol. 9, no. 1, pp. 118–124, 2011.

    Article  MathSciNet  Google Scholar 

  19. M. Hajarian and M. Dehghan, “The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + CY T D=E,” Mathematical Methods in the Applied Sciences, vol. 34, no. 13, pp. 1562–1579, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Trans. Autom. Contr., vol. 50, no. 3, pp. 397–402, 2005.

    Article  MathSciNet  Google Scholar 

  21. F. Ding and T. Chen, “Iterative least squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2005.

    Article  MathSciNet  Google Scholar 

  23. A. G. Wu, G. R. Duan, and B. Zhou, “Solution to generalized Sylvester matrix equations,” IEEE Trans. Automat. Contr., vol. 53, no. 3, pp. 811–815, 2008.

    Article  MathSciNet  Google Scholar 

  24. B. Zhou, J. Lam, and G. R. Duan, “On smith-type iterative algorithms for the stein matrix equation,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1038–1044, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  25. T. L. Doug, Conjugate Gradient-Type Product Methods for Solving Nonsymmetric Linear Systems, Ph.D. Thesis, University of California, Los Angeles, 1994.

    Google Scholar 

  26. S. L. Zhang, “GPBi-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems,” SIAM J. Sci. Comput., vol. 18, no. 2, pp. 537–551, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Zhou, J. Lamb, and G. R. Duan, “Toward solution of matrix equation X = Af (X)B + C,” Linear Algebra and its Applications, vol. 435, no. 6, pp. 1370–1398, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  28. S. K. Li and T. Z. Huang, “LSQR iterative method for generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3545–3554, 2012.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran, 19839, Iran

    Masoud Hajarian

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  1. Masoud Hajarian
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Correspondence to Masoud Hajarian.

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Recommended by Associate Editor Choon Ki Ahn under the direction of Editor PooGyeon Park.

The author would like to express his heartfelt thanks to two anonymous referees for their valuable comments and careful reading of the manuscript.

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Hajarian, M. Extending the GPBiCG algorithm for solving the generalized Sylvester-transpose matrix equation. Int. J. Control Autom. Syst. 12, 1362–1365 (2014). https://doi.org/10.1007/s12555-013-0516-8

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  • DOI: https://doi.org/10.1007/s12555-013-0516-8

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