Skip to main content
Springer Nature Link
Log in

A parallel implementation of the CMRH method for dense linear systems

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

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

This paper presents an implementation of the CMRH (Changing Minimal Residual method based on the Hessenberg process) iterative method suitable for parallel architectures. CMRH is an alternative to GMRES and QMR, the well-known Krylov methods for solving linear systems with non-symmetric coefficient matrices. CMRH generates a (non orthogonal) basis of the Krylov subspace through the Hessenberg process. On dense matrices, it requires less storage than GMRES. Parallel numerical experiments on a distributed memory computer with up to 16 processors are shown on some applications related to the solution of dense linear systems of equations. A comparison with the GMRES method is also provided on those test examples.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bai, Z., Hu, D., Reichel, L.: A Newton basis GMRES implementation. IMA J. Numer. Anal. 14, 563–581 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Hu, D., Reichel, L.: Implementation of GMRES method using QR factorisation. In: Proc. Fifth SIAM Conference on Parallel Processing for Scientific Computing, pp. 84–91 (1992)

  3. Di Brozolo, J.J., Robert, Y.: Parallel conjugate gradient-like algorithms for solving sparse nonsymmetric linear systems on a vector multiprocessor. Parallel Comput. 11, 84–91 (1989)

    Google Scholar 

  4. Erhel, J.: A parallel GMRES version for general sparse matrices. Electron. T. Numer. Anal. 3, 160–176 (1995)

    MathSciNet  MATH  Google Scholar 

  5. Helsing, J.: Approximate inverse preconditioners for some large dense random electrostatic interaction matrices. BIT Numer. Math. 46, 307–323 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heyouni, M., Sadok, H.: A new implementation of the CMRH method for solving dense linear systems. J. Comput. Appl. Math. 213, 387–399 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hessenberg, K.: Behandlung der linearen Eigenwert-Aufgaben mit Hilfe der Hamilton-Cayleychen Gleichung. Darmstadt dissertation (1940)

  8. Message Passing Interface Forum: MPI: a message-passing interface standard. Int. J. Supercomputing Applications and High Performance Computing (1994)

  9. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7, 856–869 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Sadok, H., Szyld, D.B.: A new look at CMRH and its relation to GMRES. BIT Numer. Math. 52, 485–501 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. PWS Publishing, Boston (1996). SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  13. Davis, T.A., Hu, Y.: The University of Florida Sparse Matrix Collection. ACM T. Math. Software 38, 1–25 (2011)

    MathSciNet  Google Scholar 

  14. Alia, A., Sadok, H., Souli, M.: CMRH method as iterative solver for boundary element acoustic systems. Eng. Anal. Bound. Elem. 36, 346–350 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, Centre Universitaire de la Mi-Voix, Batiment H. Poincaré, 50 Rue F. Buisson, BP 699, 62228, Calais cedex, France

    Sébastien Duminil

Authors
  1. Sébastien Duminil
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Sébastien Duminil.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duminil, S. A parallel implementation of the CMRH method for dense linear systems. Numer Algor 63, 127–142 (2013). https://doi.org/10.1007/s11075-012-9616-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-012-9616-4

Keywords