BIT Numerical Mathematics

, Volume 52, Issue 2, pp 485–501 | Cite as

A new look at CMRH and its relation to GMRES

Article

Abstract

CMRH is a Krylov subspace method which uses the Hessenberg process to produce a basis of a Krylov method, and minimizes a quasiresidual. This method produces convergence curves which are very close to those of GMRES, but using fewer operations and storage. In this paper we present new analysis which explains why CMRH has this good convergence behavior. Numerical examples illustrate the new bounds.

Keywords

Krylov subspace Arnoldi Hessenberg CMRH method GMRES 

Mathematics Subject Classification (2010)

65F10 

Notes

Acknowledgements

We thank the referees for their comments and questions, which helped improve our presentation.

D.B. Szyld research supported in part by the U.S. Department of Energy under grant DE-FG02-05ER25672.

References

  1. 1.
    Freund, R.W., Nachtigal, N.M.: QMR: A quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Freund, R.W., Golub, G.H., Nachtigal, N.M.: Iterative solution of linear systems. Acta Numer. 1, 57–100 (1992) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hansen, P.C.: Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6, 1–35 (1994). Software is available in Netlib at http://www.netlib.org MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Helsing, J.: Approximate inverse preconditioners for some large dense random electrostatic interaction matrices. BIT Numer. Math. 46, 307–323 (2006) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Heyouni, M., Sadok, H.: A new implementation of the CMRH method for solving dense linear systems. J. Comput. Appl. Math. 213, 387–399 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Higham, N.J.: The matrix computation toolbox. Available online at http://www.ma.man.ac.uk/~higham/mctoolbox
  7. 7.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing, Boston (1996), 2nd edn. SIAM, Philadelphia (2003) MATHGoogle Scholar
  8. 8.
    Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Sadok, H.: CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm. Numer. Algorithms 20, 303–321 (1999) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Simoncini, V., Szyld, D.B.: The effect of non-optimal bases on the convergence of Krylov subspace methods. Numer. Math. 100, 711–733 (2005) MathSciNetMATHGoogle Scholar
  11. 11.
    Simoncini, V., Szyld, D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 14, 1–59 (2007) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965) MATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du Littoral, Centre Universitaire de la Mi-VoixCalais CedexFrance
  2. 2.Department of MathematicsTemple University (038-16)PhiladelphiaUSA

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