Computational Mathematics and Mathematical Physics

, Volume 55, Issue 10, pp 1606–1618 | Cite as

Weighted versions of Gl-FOM and Gl-GMRES for solving general coupled linear matrix equations

  • Fatemeh Panjeh Ali Beik
  • Davod Khojasteh Salkuyeh
Article

Abstract

In this paper, two algorithms called weighted Gl-FOM (WGl-FOM) and weighted Gl-GMRES (WGl-GMRES) are proposed for solving the general coupled linear matrix equations. In order to accelerate the speed of convergence, a new inner product is used. Invoking the new inner product and a new matrix product, the weighted global Arnoldi algorithm is introduced which will be utilized for employing the WGl-FOM and WGl-GMRES algorithms to solve the linear coupled linear matrix equations. After introducing the weighted methods, some relations that link Gl-FOM (Gl-GMRES) to its weighted version are established. Numerical experiments are presented to illustrate the effectiveness of the new algorithms in comparison with Gl-FOM and Gl-GMRES algorithms for solving the linear coupled linear matrix equations.

Keywords

linear matrix equation Krylov subspace weighted methods global FOM global GMRES global Arnoldi 

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References

  1. 1.
    A. Bouhamidi and K. Jbilou, “A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications,” Appl. Math. Comput. 206, 687–694 (2008).MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    X. W. Chang and J. S. Wang, “The symmetric solution of the matrix equations AX + YA = C, AXA T + BYB T = C and (A T XA, B T XB) = (C, D),” Linear Algebra Appl. 179, 171–189 (1993).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Jbilou, A. Messaudi, and H. Sadok, “Global FOM and GMRES algorithms for matrix equations,” Appl. Numer. Math. 31, 49–63 (1999).MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Jbilou and A. J. Riquet, “Projection methods for large Lyapunov matrix equations,” Linear Algebra Appl. 415, 344–358 (2006).MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    D. K. Salkuyeh and F. Toutounian, “New approaches for solving large Sylvester equations,” Appl. Math. Comput. 173, 9–18 (2006).Google Scholar
  6. 6.
    Q. W. Wang, J. H. Sun, and S. Z. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,” Linear Algebra Appl. 353, 169–182 (2002).MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    J. J. Zhang, “A note on the iterative solutions of general coupled matrix equation,” Appl. Math. Comput. 217, 9380–8386 (2011).Google Scholar
  8. 8.
    B. Zhou and G. R. Duan, “On the generalized Sylvester mapping and matrix equation,” Syst. Control Lett.57(3), 200–208 (2008).MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    F. P. A. Beik and D. K. Salkuyeh, “On the global Krylov subspace methods for solving general coupled matrix equations,” Comput. Math. Appl. 62, 4605–4613 (2011).MathSciNetCrossRefGoogle Scholar
  10. 10.
    M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra Appl. 432, 1531–1552 (2010).MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Appl. Math. Comput. 197, 41–50 (2008).MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Y. Saad, Iterative Methods for Sparse linear Systems (PWS, New York, 1995).Google Scholar
  13. 13.
    Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual method for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Y. Q. Lin, “Implicitly restarted global FOM and GMRES for nonsymmetric equations and Sylvester equations,” Appl. Math. Comput. 167, 1004–1025 (2005).MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Essai, “Weighted FOM and GMRES for solving nonsymmetric linear systems,” Numer. Algorithms 18, 277–292(1998).MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Y.-F. Jing and T. Z. Huang, “Restarted weighted full orthogonalization method for shifted linear systems,” Compute. Math. Appl. 57, 1583–1591 (2009).MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Heyouni and A. Essai, “Matrix Krylov subspace methods for linear systems with multiple right-hand sides,” Numer. Algorithms 40, 137–156 (2005).MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Bouyouli, K. Jbilou, R. Sadaka, and H. Sadok, “Convergence properties of some block Krylov subspace methods,” J. Comput. Appl. Math. 196, 498–511 (2006).MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Matrix Market, http://math.nist.gov/MatrixMarket (August, 2005).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Fatemeh Panjeh Ali Beik
    • 1
  • Davod Khojasteh Salkuyeh
    • 2
  1. 1.Department of MathematicsVali-e-Asr University of RafsanjanRafsanjanIran
  2. 2.Faculty of Mathematical SciencesUniversity of GuilanRashtIran

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