The Breakdowns of BiCGStab
- P.R. Graves-Morris
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- 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.
- C. Brezinski and M Redivo-Zaglia, Look-ahead in BiCGStab and other product methods for linear systems, BIT 35 (1995) 169–201.
- 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.
- 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.
- 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.
- P.R. Graves-Morris, A 'Look-around Lanczos' algorithm for solving a system of linear equations, Numer. Algorithms 15 (1997) 247–274.
- P.R. Graves-Morris, VPAStab and its breakdowns, submitted to Numer. Algorithms (2002).
- P.R. Graves-Morris and A. Salam, Avoiding breakdown in van der Vorst's method, Numer. Algorithms 21 (1999) 205–223.
- A. Greenbaum, Estimating the attainable accuracy of recursively computed residual methods, SIAM J. Matrix Anal. Appl. 18 (1997) 535–551.
- M.H. Gutknecht, Lanczos-type solvers for non-symmetric linear systems of equations, Acta Numerica 6 (1997) 271–397.
- 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.
- 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.
- MATLAB 6.0, The MathWorks Inc., Natick, MA, USA.
- J.K. Reid, The use of conjugate gradients for systems of equations possessing 'Property A', SIAM J. Numer. Anal. (1972) 325-332.
- 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.
- G.L.G. Sleijpen and H.A. van der Vorst, Reliable updated residuals in hybrid Bi-CG methods, Computing 56 (1996) 141–163.
- 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.
- The Breakdowns of BiCGStab
Volume 29, Issue 1-3 , pp 97-105
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- pivot breakdown
- Lanczos breakdown
- Industry Sectors
- Author Affiliations
- 1. Computing Department, University of Bradford, Bradford, West Yorkshire, BD7 1DP, England