Skip to main content
SpringerLink
Log in

Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations

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

Abstract

The solution of a large-scale Sylvester matrix equation plays an important role in control and large scientific computations. In this paper, we are interested in the large Sylvester matrix equation with large dimensionA and small dimension B, and a popular approach is to use the global Krylov subspace method. In this paper, we propose three new algorithms for this problem. We first consider the global GMRES algorithm with weighting strategy, which can be viewed as a precondition method. We present three new schemes to update the weighting matrix during iterations. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm effectively. The deflation strategy is efficient for the solution of large linear systems and large eigenvalue problems; to the best of our knowledge, little work is done on applying deflation to the (weighted) global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. In particular, we are interested in the global GMRES algorithm with deflation, which can be viewed as a special case when the weighted matrix is chosen as the identity. Theoretical analysis is given to show rationality of the new algorithms. Numerical experiments illustrate the numerical behavior of the proposed algorithms.

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. Agoujil, S., Bentbib, A.H., Jabilou, K., Sadek, E.M.: A minimal residual norm method for large-scale Sylvester matrix equations. Electron. Tran. Numer. Anal. 43, 45–59 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Bartels, R., Stewart, G.W.: Algorithm 432: The solution of the matrix equation A X - X B = c. Commun. ACM 8, 820–826 (1972)

    Article  MATH  Google Scholar 

  3. Benner, P., Li, R., Truhar, N.: On the ADI method for Sylvester equations. J. Comput. Appl. Math. 223, 1035–1045 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouhamidi, A., Jbilou, K.: A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications. Appl. Math. Comput. 206, 687–694 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Calvetti, D.: Application of ADI iterative methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl. 17, 165–186 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Datta, B.N.: Numerical Methods for Linear Control Systems Design and Analysis. Elsevier Press (2003)

  7. Datta, B.N., Datta, K.: Theoretical and computational aspects of some linear algebra problems in control theory. In: Byrnes, C. I., Lindquist, A. (eds.) Computational and Combinatorial Methods in Systems Theory, vol. 177, pp 201–212. Elsevier, Amsterdam (1986)

  8. Dehghan, M., Hajarian, M.: Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations. Appl. Math. Lett. 24, 444–449 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duan, C., Jia, Z.: A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems. J. Comput. Appl. Math. 234, 845–860 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Embree, M., Morgan, R.B., Nguyen, H.V.: Weighted inner products for GMRES and GMRES-DR. SIAM J. Sci. Comp. 39, S610–S632 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Essai, A.: Weighted, FOM and GMRES for solving nonsymmetric linear systems. Numer. Alg. 18, 277–292 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. The University of Florida Sparse Matrix Collection, https://www.cise.ufl.edu/research/sparse/matrices

  13. El Guennouni, A., Jbilou, K., Riquet, A.J.: Block Krylov subspace methods for solving large Sylvester equations. Numer. Alg. 29, 75–96 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Güttel, S., Pestana, J.: Some observations on weighted GMRES. Numer. Alg. 67, 733–752 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heyouni, M.: Extended Arnoldi methods for large low-rank Sylvester matrix equations. Appl. Numer. Math. 60, 1171–1182 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heyouni, M., Essai, A.: Matrix Krylov subspace methods for linear systems with multiple right-hand sides. Numer. Alg. 40, 137–156 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jaimoukha, I.M., Kasenally, E.M.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227–251 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jbilou, K., Messaoudi, A., Sadok, H., Global, FOM: GMRES algorithms for matrix equations. Appl. Numer Math. 31, 49–63 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jbilou, K., Riquet, A.J.: Projection methods for large Lyapunov matrix equations. Linear Alg. Appl. 415, 344–358 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, W., Wu, G.: A thick-restarted block Arnoldi algorithm with modified Ritz vectors for large eigenproblems. Comput. Math. Appl. 60, 873–889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Khorsand Zak, M., Toutounian, F.: Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning. Adv. Comput. Math. 40, 865–880 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liesen, J., Strakos, Z.: Krylov Subspace Methods, Principles and Analysis. Oxford University Press (2013)

  23. Lin, Y.: Minimal residual methods augmented with eigenvectors for solving Sylvester equations and generalized Sylvester equations. Appl. Math. Comput. 181, 487–499 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Lin, Y.: Implicitly restarted global, FOM and GMRES for nonsymmetric matrix equations and Sylvester equations. Appl. Math. Comput. 167, 1004–1025 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Lin, Y., Bao, L., Wei, Y.: A projection method based on extended krylov subspaces for solving Sylvester equations. Int. J. Math. Comput. Phys. Elec. Comput. Eng. 5(7), 1064–1070 (2011)

    Google Scholar 

  26. Lin, Y., Simoncini, V.: Minimal residual methods for large scale Lyapunov equations. Appl. Numer. Math. 72, 52–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matrix Market, http://math.nist.gov/matrixMarket/

  28. Morgan, R.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24(1), 20–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Morgan, R.: Restarted block, GMRES with deflation of eigenvalues. Appl. Numer. Math. 54, 222–236 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Morgan, R., Zeng, M.: A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity. Linear Alg. Appl. 415, 96–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mohseni Mohgadam, M., Panjeh Ali Beik, F.: A new weighted global full orthogonalization method for solving nonsymmetric linear systems with multiple right-hand sides. Int. Electron. J. Pure Appl. Math. 2, 47–67 (2010)

    MATH  Google Scholar 

  32. Panjeh Ali Beik, F., Mohseni Mohgadam, M.: Global generalized minimum residual method for solving Sylvester equation. Aust. J. Basic Appl. Sci. 5, 1128–1134 (2011)

    Google Scholar 

  33. Panjeh Ali Beik, F., Khojasteh Salkuyeh, D.: Weighted versions of gl-FOM and gl-GMRES for solving general coupled linear matrix equations. Comput. Math. Math. Phys. 55, 1606–1618 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Penzel, T.: LYAPACK: A MATLAB toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear-quadratic optimal control problems, software available at https://www.tu-chemnitz.de/sfb393/lyapack/

  35. Robbé, M., Sadkane, M.: A convergence analysis of GMRES and FOM methods for Sylvester equations. Numer. Alg. 30, 71–89 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Saberi Najafi, H., Zareamoghaddam, H.: A new computational, GMRES Method. Appl. Math. Comput. 199, 527–534 (2008)

    MathSciNet  MATH  Google Scholar 

  37. Simoncini, V.: The extended Krylov subspace for parameter dependent systems. Appl. Numer. Math. 60, 550–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29, 1268–1288 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58, 377–441 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Sorensen, D.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wan, F.: An in-core finite difference method for separable boundary value problems on a rectangle. Stud. Appl. Math. 52, 103–113 (1973)

    Article  MATH  Google Scholar 

  42. Watkins, D.: The Matrix Eigenvalue Problem. GR and Krylov Subspace Methods. SIAM, Philadelphia (2007)

    Book  MATH  Google Scholar 

  43. Wu, G., Wei, Y.: A Power-Arnoldi algorithm for computing PageRank. Numer. Linear Alg. Appl. 14, 521–546 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wu, K., Simon, H.: Thick-restart Lanczos method for sysmmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 602–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhong, H., Wu, G.: Thick restarting the weighted harmonic Arnoldi algorithm for large interior eigenproblems. Int. J. Comput. Math. 88, 994–1012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the reviewers and our editor Prof. Claude Brezinski for invaluable comments and suggestions that improve the representation of this paper significantly. Moreover, we are grateful to Prof. Yiqin Lin for providing us with the MATLAB codes of the EKS-SYL algorithm.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics & Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran

    Najmeh Azizi Zadeh & Azita Tajaddini

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Gang Wu

Authors
  1. Najmeh Azizi Zadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Azita Tajaddini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Wu.

Additional information

This author is supported by the National Science Foundation of China and the Natural Science Foundation of Jiangsu Province under grant BK20171185.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zadeh, N.A., Tajaddini, A. & Wu, G. Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations. Numer Algor 82, 155–181 (2019). https://doi.org/10.1007/s11075-018-0597-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-018-0597-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation