Numerical Algorithms

, Volume 29, Issue 1–3, pp 97–105

The Breakdowns of BiCGStab

  • P.R. Graves-Morris


The effects of the three principal possible exact breakdowns which may occur using BiCGStab are discussed. BiCGStab is used to solve large sparse linear systems of equations, such as arise from the discretisation of PDEs. These PDEs often involve a parameter, say γ. We investigate here how the numerical error grows as breakdown is approached by letting γ tend to a critical value, say γc, at which the breakdown is numerically exact. We found empirically in our examples that loss of numerical accuracy due stabilisation breakdown and Lanczos breakdown was discontinuous with respect to variation of γ around γc. By contrast, the loss of numerical accuracy near a critical value γc for pivot breakdown is roughly proportional to |γ−γc|−1.

BiCGStab BiCG Lanczos pivot breakdown Lanczos breakdown LTPM 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. Brezinski and M Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, in: Nonlinear Numerical Methods and Rational Approximation II, ed. A. Cuyt (Kluwer, Dordrecht, 1994) pp. 49–59.Google Scholar
  2. [2]
    C. Brezinski and M Redivo-Zaglia, Look-ahead in BiCGStab and other product methods for linear systems, BIT 35 (1995) 169–201.Google Scholar
  3. [3]
    R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numer. Analysis, Dundee, 1975, ed. G.A. Watson, Lecture Notes in Mathematics, Vol. 506 (Springer, Berlin, 1976) pp. 73–89.Google Scholar
  4. [4]
    R.W. Freund, M.H. Gutknecht and N. Nachtigal. An implementation of look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput. 14 (1993) 137–158.Google Scholar
  5. [5]
    G.H. Golub and H.A. Van der Vorst, Closer to the solution: Iterative linear solvers, in: The State of the Art in Numerical Analysis, eds. I.S. Duff and G.A. Watson (Clarendon Press, Oxford, 1997) pp. 63–92.Google Scholar
  6. [6]
    P.R. Graves-Morris, A 'Look-around Lanczos' algorithm for solving a system of linear equations, Numer. Algorithms 15 (1997) 247–274.Google Scholar
  7. [7]
    P.R. Graves-Morris, VPAStab and its breakdowns, submitted to Numer. Algorithms (2002).Google Scholar
  8. [8]
    P.R. Graves-Morris and A. Salam, Avoiding breakdown in van der Vorst's method, Numer. Algorithms 21 (1999) 205–223.Google Scholar
  9. [9]
    A. Greenbaum, Estimating the attainable accuracy of recursively computed residual methods, SIAM J. Matrix Anal. Appl. 18 (1997) 535–551.Google Scholar
  10. [10]
    M.H. Gutknecht, Lanczos-type solvers for non-symmetric linear systems of equations, Acta Numerica 6 (1997) 271–397.Google Scholar
  11. [11]
    M.H. Gutknecht and K.J. Ressel, Look-ahead procedures for Lanczos-type product methods based on three-term Lanczos recurrences, SIAM J. Matrix Anal. Appl. 21 (2000) 1051–1078.Google Scholar
  12. [12]
    M.H. Gutknecht and Z. Strakoš, Accuracy of two three-term and three two-term recurrences for Krylov space solvers, SIAM J. Matrix Anal. Appl. 22 (2000) 213–229.Google Scholar
  13. [13]
    MATLAB 6.0, The MathWorks Inc., Natick, MA, USA.Google Scholar
  14. [14]
    J.K. Reid, The use of conjugate gradients for systems of equations possessing 'Property A', SIAM J. Numer. Anal. (1972) 325-332.Google Scholar
  15. [15]
    G.L.G. Sleijpen and H.A. van der Vorst, Maintaining convergence properties of BiCGStab methods in finite precision arithmetic, Numer. Algorithms 10 (1995) 203–223.Google Scholar
  16. [16]
    G.L.G. Sleijpen and H.A. van der Vorst, Reliable updated residuals in hybrid Bi-CG methods, Computing 56 (1996) 141–163.Google Scholar
  17. [17]
    H.A. Van der Vorst, Bi-CGStab: A fast and smoothly convergent variant of Bi-CG for the solution of non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631–644.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • P.R. Graves-Morris
    • 1
  1. 1.Computing DepartmentUniversity of BradfordBradford, West YorkshireEngland

Personalised recommendations