Skip to main content
SpringerLink
Log in

Generalizations and modifications of the GMRES iterative method

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

For solving nonsymmetric linear systems, the well-known GMRES method is considered to be a stable method; however, the work per iteration increases as the number of iterations increases. We consider two new iterative methods GGMRES and MGMRES, which are a generalization and a modification of the GMRES method, respectively. Instead of using a minimization condition as in the derivation of GGMRES, we use a Galerkin condition to derive the MGMRES method. We also introduce another new iterative method, LAN/MGMRES, which is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method. A computer program has been written based on the use of the LAN/MGMRES algorithm for solving nonsymmetric linear systems arising from certain elliptic problems. Numerical tests are presented comparing this algorithm with some other commonly used iterative algorithms. These preliminary tests of the LAN/MGMRES algorithm show that it is comparable in terms of both the approximate number of iterations and the overall convergence behavior.

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. R.O. Abbassian, Lanczos algorithms for the acceleration of nonsymmetrizable iterative methods, Report CNA-193, Center for Numerical Analysis, University of Texas at Austin (1984).

  2. W.E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951) 17–29.

    MATH  MathSciNet  Google Scholar 

  3. A.M. Bruaset, A Survey of Preconditioned Iterative Methods (Longman Scientific & Technical/ Wiley, New York, 1995).

    Google Scholar 

  4. J.-Y. Chen, Iterative solution of large sparse nonsymmetric linear systems, Report CNA-285, Center for Numerical Analysis, University of Texas at Austin (1997) (Ph.D. thesis, Dept. of Mathematics).

  5. R. Fletcher, Conjugate gradient methods for indefinite systems, in: Proc. of the Dundee Conf. on Numerical Analysis (1975), Lecture Notes in Mathematics, Vol. 506 (Springer, New York, 1976) pp. 73–89.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. National Bureau Standards 49(6) (1952) 409–436.

    MATH  MathSciNet  Google Scholar 

  8. K.C. Jea, Generalized conjugate gradient acceleration of iterative methods, Report CNA-176, Center for Numerical Analysis, University of Texas at Austin (1982) (Ph.D. thesis, Dept. of Mathematics).

  9. K.C. Jea and D.M. Young, On the simplification of generalized conjugate gradient methods for nonsymmetrizable linear systems, Linear Algebra Appl. 52/53 (1983) 399–417.

    MATH  MathSciNet  Google Scholar 

  10. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publ., Boston, MA, 1996).

    Google Scholar 

  11. Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7(3) (1986) 856–869.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Sonneveld, CGS: A fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10(1) (1989) 36–52.

    Article  MATH  MathSciNet  Google Scholar 

  13. H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13(2) (1992) 631–644.

    Article  MATH  MathSciNet  Google Scholar 

  14. P.K.W. Vinsome, ORTHOMIN: An iterative method for solving sparse sets of simultaneous linear equations, in: 4th Symp. of Numerical Simulation Reservoir Performance, Society of Petroleum Engineers of the AIME, Paper SPE 5739 (1976).

  15. H.F. Walker, Implementation of the GMRES method using Householder transformations, SIAM J. Sci. Comput. 9 (1988) 152–163.

    Article  MATH  Google Scholar 

  16. H.F. Walker and L. Zhou, A simpler GMRES, Report, Department of Mathematics and Statistics, Utah State University, Logan, UT (1992).

    Google Scholar 

  17. D.M. Young, L.J. Hayes and K.C. Jea, Generalized conjugate gradient acceleration of iterative methods, Part I: The symmetrizable case, Report CNA-162, Center for Numerical Analysis, University of Texas at Austin (1981).

  18. D.M. Young and K.C. Jea, Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl. 34 (1980) 159–194.

    Article  MATH  MathSciNet  Google Scholar 

  19. D.M. Young and K.C. Jea, Generalized conjugate gradient acceleration of iterative methods, Part II: The nonsymmetrizable case, Report CNA-163, Center for Numerical Analysis, University of Texas at Austin (1981).

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung, 840, Taiwan

    Jen-Yuan Chen

  2. Center for Numerical Analysis, University of Texas at Austin, Austin, TX, 78713-8510, USA

    David R. Kincaid & David M. Young

Authors
  1. Jen-Yuan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David R. Kincaid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David M. Young
    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

Chen, JY., Kincaid, D.R. & Young, D.M. Generalizations and modifications of the GMRES iterative method. Numerical Algorithms 21, 119–146 (1999). https://doi.org/10.1023/A:1019105328973

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019105328973

Search

Navigation