Treatment of near-breakdown in the CGS algorithm. Lanczos’ method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P k 2 (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.

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  1. Brezinski, C.; Redivo Zaglia, M.; Sadok, H.: A review of formal orthogonality in Lanczos-based methods (2002)
  2. van der Vorst, Henk A.: Computational methods for large eigenvalue problems (2002)
  3. Nyman, Gunnar; Yu, Hua-Gen: Iterative diagonalization of a large sparse matrix using spectral transformation and filter diagonalization (2001)
  4. Gutknecht, Martin H.; Ressel, Klaus J.: Look-ahead procedures for Lanczos-type product methods based on three-term Lanczos recurrences (2000)
  5. Salam, A.: On vector Hankel determinants (2000)
  6. Brezinski, C.; Redivo-Zaglia, M.: Transpose-free Lanczos-type algorithms for nonsymmetric linear systems (1998)
  7. Dongarra, Jack J.; Duff, Iain S.; Sorensen, Danny C.; Van der Vorst, Henk A.: Numerical linear algebra for high-performance computers (1998)
  8. Tichý, Petr; Zítko, Jan: Derivation of BiCG from the conditions defining Lanczos’ method for solving a system of linear equations (1998)
  9. Brezinski, C.; Redivo-Zaglia, M.; Sadok, H.: Breakdowns in the implementation of the Lánczos method for solving linear systems (1997)
  10. Gutknecht, Martin H.: Lanczos-type solvers for nonsymmetric linear systems of equations (1997)
  11. van der Vorst, Henk A.; Chan, Tony F.: Linear system solvers: Sparse iterative methods (1997)
  12. van der Vorst, Henk A.; Golub, Gene H.: 150 years old and still alive: Eigenproblems (1997)
  13. Brezinski, C.: The methods of Vorobyev and Lanczos (1996)
  14. Brezinski, C.: Extrapolation algorithms and Padé approximations: A historical survey (1996)
  15. Brezinski, C.; Redivo-Zaglia, M.: A look-ahead strategy for the implementation of some old and new extrapolation methods (1996)
  16. Chesneaux, Jean-Marie; Matos, Ana C.: Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic (1996)
  17. Brezinski, C.; Redivo-Zaglia, M.: Look-ahead in Bi-CGSTAB and other product methods for linear systems (1995)
  18. Sleijpen, Gerard L. G.; Van der Vorst, Henk A.: Hybrid bi-conjugate gradient methods for CFD problems (1995)
  19. Brezinski, C.; Redivo-Zaglia, M.: Treatment of near-breakdown in the CGS algorithm (1994)
  20. Brezinski, C.; Redivo-Zaglia, M.: Hybrid procedures for solving linear systems (1994)

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