Skip to main content
SpringerLink
Log in

An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The iterative method of the generalized coupled Sylvester-conjugate matrix equations \(\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)\) over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X j (1), j = 1,2,⋯ , l, the exact solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Liao, A.P., Bai, Z.Z., Lei, Y.: Best approximate solution of matrix equation AXB + CXD = E. SIAM J. Matrix Anal. Appl. 27, 675–688 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dehghan, M., Hajarian, M.: Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations. Appl. Math. Modell. 35, 3285–3300 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wu, A.G., Duan, G.R., Fu, Y.M., Wu, W.J.: Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation AX + BY = EXF + S. Computing 89, 147–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wu, A.G., Li, B., Zhang, Y., Duan, G.R.: Finite iterative solutions to coupled Sylvester-conjugate matrix equations. Appl. Math. Modell. 35, 1065–1080 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, N., Ma, C.F.: Modified conjugate gradient method for obtaining the minimum-norm solution of the generalized coupled Sylvester-conjugate matrix equations. Appl. Math. Modell. 40, 1260–1275 (2016)

    Article  MathSciNet  Google Scholar 

  6. Xie, Y.J., Huang, N., Ma, C.F.: Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix. Comput. Math. Appl. 67, 2071–2084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dehghan, M., Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl. Math. Comput. 202, 571–588 (2008)

    MathSciNet  MATH  Google Scholar 

  8. Huang, N., Ma, C.F.: The modified conjugate gradient methods for solving a class of the generalized coupled Sylvester-transpose matrix equations. Comput. Math. Appl. 67, 1545–1558 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dehghan, M., Hajarian, M.: The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra Appl. 432, 1531–1552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dehghan, M., Hajarian, M.: The generalized Sylvester matrix equations over the generalized bisymmetric and skew-symmetric matrices. Int. J. Syst. Sci. 43, 1580–1590 (2012)

    Article  MATH  Google Scholar 

  11. Dehghan, M., Hajarian, M.: On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations. Linear Multilin. Algebra 59, 1281–1309 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dehghan, M., Hajarian, M.: Construction of an iterative method for solving generalized coupled Sylvester matrix equations. Trans. Inst. Meas. Control 35, 961–970 (2013)

    Article  Google Scholar 

  13. Dehghan, M., Hajarian, M.: An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl. Math. Modell. 34, 639–654 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Beik, F.P.A., Salkuyeh, D.K.: The coupled Sylvester-transpose matrix equations over generalized centro-symmetric matrices. Int. J. Comput. Math. 90, 1546–1566 (2013)

    Article  MATH  Google Scholar 

  15. Dehghan, M., Hajarian, M.: The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + CY T D = E. Math. Methods Appl. Sci. 34, 1562–1579 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Beik, F.P.A., Salkuyeh, D.K.: On the global Krylov subspace methods for solving general coupled matrix equations. Comput. Math. Appl. 62, 4605–4613 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, H., Gao, Z., Zhao, D.: Least squares solutions of the matrix equation AXB + CYD = E with the least norm for symmetric arrowhead matrices. Appl. Math. Comput. 226, 719–724 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Deng, Y.B., Bai, Z.Z., Gao, Y.H.: Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations. Numer. Linear Algebra Appl. 13, 801–823 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ding, F., Liu, P.X., Ding, J.: Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl. Math. Comput. 197, 41–50 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Zhou, B., Li, Z.Y., Duan, G.R., Wang, Y.: Solutions to a family of matrix equations by using the Kronecker matrix polynomials. Appl. Math. Comput. 212, 327–336 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Zhang, Z.Z., Hu, X.Y., Zhang, L.: On the Hermitian-generalized hamiltonian solutions of linear matrix equations. SIAM J. Matrix Anal. Appl. 27, 294–303 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker, New York (1990)

    Google Scholar 

  23. Borno, I.: Parallel computation of the solutions of coupled algebraic Lyapunov equations. Automatica 31, 1345–1347 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, S.K.: A Finite iterative method for solving the generalized Hamiltonian solutions of coupled Sylvester matrix equations with conjugate transpose. Int. J. Comput. Math. 94, 757–773 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ke, Y.F., Ma, C.F.: Alternating direction method for generalized Sylvester matrix equation AXB + CYD = E. Appl. Math. Comput. 260, 106–125 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Inman, D.J., Kress, A.: Eigenstructure assignment algorithm for second order systems. J. Guid. Control Dyn. 22, 729–731 (1999)

    Article  Google Scholar 

  27. Zhou, B., Duan, G.R.: On the generalized Sylvester mapping and matrix equations. Syst. Control Lett. 57, 200–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hajarian, M.: Matrix form of the CGS method for solving general coupled matrix equations. Appl. Math. Lett. 34, 37–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hajarian, M.: Developing Bi-CG and Bi-CR methods to solve generalized Sylvester-transpose matrix equations. Bi-CR Int. J. Autom. Comput. 11, 25–29 (2014)

    Article  Google Scholar 

  30. Hajarian, M.: Developing BiCOR and CORS methods for coupled Sylvester-transpose and periodic Sylvester matrix equations. Appl. Math. Model 39, 6073–6084 (2015)

    Article  MathSciNet  Google Scholar 

  31. Hajarian, M.: Matrix GPBiCG algorithms for solving the general coupled matrix equations. IET Control Theory Appl. 9, 74–81 (2015)

    Article  MathSciNet  Google Scholar 

  32. Hajarian, M.: The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations. Appl. Math. Lett. 26, 1013–1017 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hajarian, M.: Developing the CGLS algorithm for the least squares solutions of the general coupled matrix equations. Math. Meth. Appl. Sci. 27, 2782–2798 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jamshidi, M.: An overview on the solutions of the algebraic matrix Riccati equation and related problems. Large Scale Syst. 1, 167–192 (1980)

    MathSciNet  MATH  Google Scholar 

  35. Wimmer, H.K.: Decomposition and parametrization of semidefinite solutions of the continuous time algebraic Riccati equation. SIAM J. Control Optim. 32, 995–1007 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sima, V.: Algorithms for linear-quadratic optimization. Dekker, New York (1996)

    MATH  Google Scholar 

  37. Zhou, K., Doyle, J., Glover, K.: Robust and optimal control. Prentice-Hall, Upper saddle river (1995)

    MATH  Google Scholar 

  38. Bittanti, S., Laub, A.J., Willems, J.C.: The Riccati equation. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  39. Bai, Z.J.: The solvability conditions for the inverse eigenvalue problem of Hermitian and generalized skew-Hamiltonian matrices and its approximation. Inverse Probl. 19, 1185–1194 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hajarian, M.: New finite algorithm for solving the generalized nonhomogeneous Yakubovich-transpose matrix equation. Asian J. Control 19, 164–172 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hajarian, M.: Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices. Numer. Algorithms 73, 591–609 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hajarian, M.: Extending the CGLS algorithm for least squares solutions of the generalized Sylvester-transpose matrix equations. J. Franklin Inst. 353, 1168–1185 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hajarian, M.: Least squares solution of the linear operator equation. J. Optim. Theory Appl. 170, 205–219 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Hajarian, M.: Matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. J. Franklin Inst. 350, 3328–3341 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Dehghan, M., Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl. Math. Comput. 202, 571–588 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Wu, A.G., Lv, L.L., Hou, M.Z.: Finite iterative algorithms for extended Sylvester-conjugate matrix equations. Math. Comput. Model 54, 2363–2384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Informatics, Fujian Key Laborotary of Mathematical Analysis and Applications, Fujian Normal University, Fuzhou, 350117, People’s Republic of China

    Baohua Huang & Changfeng Ma

Authors
  1. Baohua Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changfeng Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changfeng Ma.

Additional information

Supported by Fujian Natural Science Foundation (Grant No. 2016J01005) and Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB18010202).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, B., Ma, C. An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations. Numer Algor 78, 1271–1301 (2018). https://doi.org/10.1007/s11075-017-0423-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0423-9

Keywords

Search

Navigation