Numerische Mathematik

, Volume 66, Issue 1, pp 295–319 | Cite as

An analysis of the composite step biconjugate gradient method

  • Randolph E. Bank
  • Tony F. Chan


The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a “best approximation” result for the method. Some numerical illustrations showing the effect of roundoff error are given.

Mathematics Subject Classifications (1991)

65N20 65F10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aziz, A.K., Babuška, I. (1972): Part I, survey lectures on the mathematical foundations of the finite element method. In: Aziz, A.K., ed., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 1–362. Academic Press, New YorkGoogle Scholar
  2. 2.
    Bank, R.E. (1990): PLTMG: a software package for solving elliptic partial differential equations, Users' Guide 6.0. Front. Appl. Math. SIAM, vol. 7Google Scholar
  3. 3.
    Bank, R.E., Bürgler, J.F., Fichtner, W., Smith, R.K. (1990): Some upwinding techniques for finite element approximations of convection-diffusion equations. Numer. Math.58, 185–202Google Scholar
  4. 4.
    Bank, R.E., Chan, T.F. (1992): A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems. University of California, CaliforniaGoogle Scholar
  5. 5.
    Bunch, J.R. (1974): Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal.11, 521–528Google Scholar
  6. 5a.
    Brezinski, C., Sadok, H. (1993): Lanczos-type algorithms for solving systems of linear equations. Appl. Num. Math.11, 443–473Google Scholar
  7. 5b.
    Brezinski, C., Zaglia, M.R., Sadok, H. (1992): A breakdown-free Lanczos type algorithm for solving linear systems. Numer. Math.63, 29–38Google Scholar
  8. 6.
    Concus, P., Golub, G.H., O'Leary, D.P. (1976): A generalized conjugate gradient method for the numerical o solution of elliptic partial differential equations. In: J.R. Bunch, D.J. Rose, eds., Sparse matrix computations, p. 309–332. Academic Press, New YorkGoogle Scholar
  9. 7.
    Fletcher, R. (1976): Conjugate gradient methods for indefinite systems. In: G.A. Watson, ed., Numerical Analysis. Lecture Notes in Mathematics506, 73–89. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. 8.
    Freund, R.W. (1991): A transpose-free quasi-minimum residual algorithm for non-Hermitian linear systems. Tech. Rep. 91.18, RIACS. Nasa Ames Research Center, Moffett FieldGoogle Scholar
  11. 9.
    Freund, R.W., Golub, G.H., Nachtigal, N.M. (1991): Iterative solution of linear systems. Tech. Rep. NA-91-05. Computer Science Department, Stanford UniversityGoogle Scholar
  12. 10.
    Freund, R.W., Gutknecht, M.H., Nachtigal, N.M. (1991): An implementation of the look-ahead Lanczos algorithm for non-hermitian matrices. Tech. Rep. 91.09, RIACS. Nasa Ames Research Ceneter, Moffett FieldGoogle Scholar
  13. 11.
    Freund, R.W., Nachtigal, N.M.: QMR: a quasi residual residual method for non-Hermetian linear systems. Numer. Math. (to appear)Google Scholar
  14. 12.
    George, A., Liu, J. (1981): Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  15. 13.
    Gutknecht, M.H. (1990): A completed theory of the unsymmetric Lanczos process and related algorithms. Part II, Tech. Rep. 90-16, IPS Research Report, ETH ZürichGoogle Scholar
  16. 14.
    Gutknecht, M.H. (1990): The unsymmetric Lanczoz algorithms and their relations to Páde approximation, continued fractions and the QD algorithm. In: Preliminary Proceedings of the Copper Mountain Conference on Iterative MethodsGoogle Scholar
  17. 15.
    Gutknecht, M.H. (1992): A completed theory of the unsymmetric Lanczos process and related algorithms, part I. SIAM J. Mat. Anal. Appl.13, 594–639Google Scholar
  18. 16.
    Joubert, W. (1992): Lanczos methods for the solution of nonsymmetric systems of linear equations. SIAM J. Matrix Anal. Appl.13, 926–943Google Scholar
  19. 17.
    Manteuffel, T.A. (1977): An Iterative Method for Solving Nonsymmetric Linear Systems with Dynamic Estimation of Parameters, PhD Thesis. University if Illinois, UrbanaGoogle Scholar
  20. 18.
    Manteuffel, T.A. (1977): The Tchebychev iteration for nonsymmetric linear systems. Numer. Math.28, 307–327Google Scholar
  21. 19.
    Nachtigal, N.M.: (1991): A Look-Ahead Variant of the Lanczos Algorithm and its Application to the Quasi-Minimum Residual Methods for Non-Hermitian Linear Systems. PhD Thesis, MIT, CambridgeGoogle Scholar
  22. 20.
    Paige, C.C., Saunders, M.A. (1975): Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal.12, 617–629Google Scholar
  23. 21.
    Parlett, B.N. (1992): Reduction to tridiagonal form and minimal realizations. SIAM J. Matrix Anal. Appl.13, 567–593Google Scholar
  24. 22.
    Parlett, Taylor, D.R., Liu, Z.A. (1985): A look-ahead Lanczos, algorithm for unsymmetric matrices. Mat. Apl. Comput.44, 105–124Google Scholar
  25. 23.
    Saad, Y. (1982): The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems. SIAM J. Numer. Anal.19, 485–506Google Scholar
  26. 24.
    Sonneveld, P. (1989): CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput.10, 36–52Google Scholar
  27. 25.
    Tong, C.H. (1992): A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems. Tech. Rep. SAND91-8402 UC-404. Sandia National Laboratories, AlbuquerqueGoogle Scholar
  28. 26.
    Van Der Vorst, H.A.: BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. (to appear)Google Scholar
  29. 27.
    Wilkinson, J.H. (1965): The Algebraic Eigenvalue Problem. Oxford University Press, OxfordGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Randolph E. Bank
    • 1
  • Tony F. Chan
    • 2
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA

Personalised recommendations