Avoiding breakdown and near-breakdown in Lanczos type algorithms. The paper deals with methods which the authors have developed to avoid breakdown and near-breakdown due to division by a scalar product whose value is zero or is different from zero but small in Lanczos type algorithms for solving linear systems.par In particular, the bulk of the paper concentrates on a method called by the authors the method of recursive zoom (MRZ) and its variants: SMRZ, BMRZ, GMRZ and BSMRZ, where S, B, G stand for symmetric, balancing and general, respectively. It is shown that a breakdown can be avoided by considering only the existing orthogonal polynomials in the Lanczos type algorithms. The methods described in the paper are able to detect if such a polynomial does not exist in order to jump over it.par Pseudo-codes for MRZ and BSMRZ are given and some numerical results are presented. (netlib numeralg na1)

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  1. Alqahtani, Hessah; Reichel, Lothar: Generalized block anti-Gauss quadrature rules (2019)
  2. Jing, Yan-Fei; Huang, Ting-Zhu; Carpentieri, Bruno; Duan, Yong: Exploiting the composite step strategy to the biconjugate (A)-orthogonal residual method for non-Hermitian linear systems (2013)
  3. Farooq, Muhammad; Salhi, Abdellah: New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms (2012)
  4. Golub, Gene H.; Meurant, Gérard: Matrices, moments and quadrature with applications (2010)
  5. Heyouni, M.; Sadok, H.: A new implementation of the CMRH method for solving dense linear systems (2008)
  6. Simoncini, Valeria; Szyld, Daniel B.: Recent computational developments in Krylov subspace methods for linear systems. (2007)
  7. Brezinski, C.; Redivo Zaglia, M.; Sadok, H.: A review of formal orthogonality in Lanczos-based methods (2002)
  8. Bai, Zhaojun (ed.); Demmel, James (ed.); Dongarra, Jack (ed.); Ruhe, Axel (ed.); Van der Vorst, Henk (ed.): Templates for the solution of algebraic eigenvalue problems. A practical guide (2000)
  9. Brezinski, C.; Redivo-Zaglia, M.; Sadok, H.: The matrix and polynomial approaches to Lanczos-type algorithms (2000)
  10. Graves-Morris, Peter R.: Reliability of Lanczos-type product methods from perturbation theory (2000)
  11. Saad, Yousef; van der Vorst, Henk A.: Iterative solution of linear systems in the 20th century (2000)
  12. Brezinski, C.; Chehab, J.-P.: Multiparameter iterative schemes for the solution of systems of linear and nonlinear equations (1999)
  13. Brezinski, C.; Redivo Zaglia, M.; Sadok, H.: New look-ahead Lanczos-type algorithms for linear systems (1999)
  14. El Guennouni, Ahmed: A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms (1999)
  15. Sadok, H.: CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm (1999)
  16. Brezinski, C.; Redivo-Zaglia, M.: Transpose-free Lanczos-type algorithms for nonsymmetric linear systems (1998)
  17. Cao, Zhi-Hao: On the QMR approach for iterative methods including coupled three-term recurrences for solving nonsymmetric linear systems (1998)
  18. Chan, Tony F.; de Pillis, Lisette; van der Vorst, Henk: Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems (1998)
  19. Greenbaum, A.: On the role of the left starting vector in the two-sided Lanczos algorithm and nonsymmetric linear system solvers (1998)
  20. Brezinski, C.; Redivo-Zaglia, M.; Sadok, H.: Breakdowns in the implementation of the Lánczos method for solving linear systems (1997)

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