Skip to main content
SpringerLink
Log in

The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we introduce two new methods for solving large sparse nonsymmetric linear systems with several right-hand sides. These methods are the global Hessenberg and global CMRH methods. Using the global Hessenberg process, these methods are less expensive than the global FOM and global GMRES methods [9]. Theoretical results about the new methods are given, and experimental results that show good performances of these new methods are presented.

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. J.I. Aliaga, D.L. Boley, R.W. Freund and V. Hernandez, A Lanczos-type method for multiple starting vector, to appear.

  2. R.F. Boisvert, R. Pozo, K. Remington, R. Barrett and J.J. Dongarra, The matrix market: A web resource for test matrix collections, in: Quality of Numerical Software: Assesment and Enhancement, ed. R.F. Boisvert (Chapman & Hall, London, 1997) pp. 125–137.

    Google Scholar 

  3. T. Chan and W. Wang, Analysis of projection methods for solving linear systems with multiple righthand sides, SIAM J. Sci. Comput. 18 (1997) 1698–1721.

    Google Scholar 

  4. R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical Analysis, ed. G.A. Watson, Lecture Notes in Mathematics, Vol. 506 (Springer-Verlag, Berlin, 1976) pp. 73–89.

    Google Scholar 

  5. R.W. Freund and N.M. Nachtigal, QMR: A quasi minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339.

    Google Scholar 

  6. R. Freund and M. Malhotra, A block QMR algorithm for non Hermitian systems with multiple righthand sides, Linear Algebra Appl. 254 (1997) 119–157.

    Google Scholar 

  7. M. Heyouni,Méthode de Hessenberg généralisée et applications, Ph.D. thesis, Université des Sciences et Technologies de Lille, France (1996).

  8. M. Heyouni and H. Sadok, On a variable smoothing procedure for Krylov subspace methods, Linear Algebra Appl. 268 (1998) 131–149.

    Google Scholar 

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

    Google Scholar 

  10. K. Jbilou and H. Sadok, Lanczos-based methods for linear systems with multiple right-hand sides, submitted.

  11. P. Joly, Résolution de systèmes linéaires avec plusieurs seconds membres par la méthode du gradient conjugué, Technical Report R-91012, Publications du Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris (March 1991).

    Google Scholar 

  12. S. Kharchenko, P. Kolesnikov, A. Nikishin, A. Yerimin, M. Heroux and Q. Seikh, Iterative solution methods on the Cray/C90. Part II: Dense linear systems, Presented at 1993 Simulation Conference: High Performance Computing Symposium, Washington, DC.

  13. N.M. Nachtigal, A look-ahead variant of the Lanczos algorithm and its application to the quasi minimal residual method for nonhermitian linear systems, Ph.D thesis, Massachusetts Institute of Technology (1991).

  14. N.M. Nachtigal, S.C. Reddy and L.N. Trefethen, A hybrid GMRES algorithm for nonsymmetric matrix iterations, SIAM J. Matrix Anal. Appl. 13 (1992) 796–825.

    Google Scholar 

  15. D. O'Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29 (1980) 293–322.

    Google Scholar 

  16. Y. Saad, Practical use of some Krylov subspace methods for solving indefinite and nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 5 (1984) 203–228.

    Google Scholar 

  17. Y. Saad and M.H. Schultz, GMRES: a Generalized Minimal Residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.

    Google Scholar 

  18. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, New York, 1995).

    Google Scholar 

  19. H. Sadok, CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm, Numer. Algorithms 20 (1999) 303–321.

    Google Scholar 

  20. V. Simoncini and E. Gallopoulos, An iterative method for nonsymmetric systems with multiple righthand sides, SIAM J. Sci. Comput. 16 (1995) 917–933.

    Google Scholar 

  21. V. Simoncini and E. Gallopoulos, Convergence properties of block GMRES and Matrix Polynomials, Linear Algebra Appl. (1996) 97–119.

  22. C. Smith, A. Peterson and R. Mittra, A Conjugate Gradient algorithm for treatement of multiple Incident Electromagnetic Fields, IEEE Trans. Antennas Propagation 37 (1989) 1490–1493.

    Google Scholar 

  23. B. Vital, Etude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseurs, Ph.D thesis, Université de Rennes, France (1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université du Littoral Côte d'Opale, zone universitaire de la Mi-voix, batiment H. Poincaré, 50 rue F. Buisson, BP 699, F-62228, Calais cedex, France

    M. Heyouni

Authors
  1. M. Heyouni
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heyouni, M. The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides. Numerical Algorithms 26, 317–332 (2001). https://doi.org/10.1023/A:1016603612931

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016603612931

Search

Navigation