Numerical Algorithms

, Volume 7, Issue 1, pp 1–16

A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems

  • Randolph E. Bank
  • Tony F. Chan


The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.


Biconjugate gradients nonsymmetric linear systems 

AMS(MOS) subject classification

65N20 65F10 


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  1. [1]
    R. Bank and T. Chan, An analysis of the composite step biconjugate gradient method, Numer. Math. 66 (1993) 295–319.CrossRefGoogle Scholar
  2. [2]
    C. Brezinski, M.R. Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algor. 1 (1991) 261–284.Google Scholar
  3. [3]
    C. Brezenski, M.R. Zaglia and H. Sadok, Addendum to: Avoiding breakdown and nearbreakdown in Lanczos type algorithms, Numer. Algor. 2 (1992) 133–136.Google Scholar
  4. [4]
    C. Brezinski, M.R. Zaglia and H. Sadok, Breakdowns in the implementation of the Lanczos method for solving linear systems, Int. J. Math. Appl. (1994), to appear.Google Scholar
  5. [5]
    T. Chan, E. Gallopoulos, V. Simoncini, T. Szeto and C. Tong, QMRCGSTAB: A quasi-minimal residual version of the Bi-CGSTAB algorithm for nonsymmetric systems, Tech. Rep. CAM 92-26, UCLA, Dept. of Math., Los Angeles, CA 90024-1555 (1992), to appear in SIAM J. Sci. Comp.Google Scholar
  6. [6]
    T.F. Chan and T. Szeto, A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems, Numer. Algor. (1994), this issue.Google Scholar
  7. [7]
    A. Draux,Polynômes Orthogonaux Formels. Applications, Lect. Notes in Mathematics, vol. 974 (Springer, Berlin, 1983).Google Scholar
  8. [8]
    V. Faber and T. Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal. 21 (1984) 315–339.CrossRefGoogle Scholar
  9. [9]
    R. Fletcher, Conjugate gradient methods for indefinite systems, in:Numerical Analysis Dundee 1975, Lecture Notes in Mathematics 506, ed. G.A. Watson (Springer, Berlin, 1976) pp. 73–89.Google Scholar
  10. [10]
    R.W. Freund, A transpose-free quasi-minimum residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comp. 14 (1993) 470–482.CrossRefGoogle Scholar
  11. [11]
    R.W. Freund, M.H. Gutknecht and N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comp. 14 (1993) 137–158.CrossRefGoogle Scholar
  12. [12]
    R.W. Freund and N.M. Nachtigal, QMR: quasi residual residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339.CrossRefGoogle Scholar
  13. [13]
    M. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, Part I, SIAM J. Matrix Anal. 13 (1992) 594–639.CrossRefGoogle Scholar
  14. [14]
    W. Joubert, Generalized conjugate gradient and Lanczos methods for the solution of non-symmetric systems of linear equations, PhD thesis, Univ. of Texas at Austin (1990).Google Scholar
  15. [15]
    C. Lanczos, Solution of linear equations by minimized iterations, J. Res. Natl. Bar. Stand. 49 (1952) 33–53.Google Scholar
  16. [16]
    D. Luenberger, Hyperbolic pairs in the method of conjugate gradients, SIAM J. Appl. Math. 17 (1969) 1263–1267.CrossRefGoogle Scholar
  17. [17]
    N. Nachtigal, S. Reddy and L. Trefethen, How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. 13 (1992) 778–795.CrossRefGoogle Scholar
  18. [18]
    C.C. Paige and M.A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975) 617–629.CrossRefGoogle Scholar
  19. [19]
    Y. Saad, The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems, SIAM J. Numer. Anal. 19 (1982) 485–506.CrossRefGoogle Scholar
  20. [20]
    M. Schultz and Y. Saad, GMRES: a generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comp. 7 (1986) 856–869.CrossRefGoogle Scholar
  21. [21]
    P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 10 (1989) 36–52.CrossRefGoogle Scholar
  22. [22]
    C.H. Tong, A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems, Tech. Rep. SAND91-8402 UC-404, Sandia National Laboratories, Albuquerque (1992).Google Scholar
  23. [23]
    H.A. Van Der Vorst, BI-CGSTAB: A fast and smoothly converging varient of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Comp. 13 (1992) 631–644.CrossRefGoogle Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

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

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