Numerische Mathematik

, Volume 83, Issue 1, pp 53–85

New look-ahead Lanczos-type algorithms for linear systems

  • C. Brezinski
  • M. Redivo Zaglia
  • H. Sadok

DOI: 10.1007/s002110050439

Cite this article as:
Brezinski, C., Redivo Zaglia, M. & Sadok, H. Numer. Math. (1999) 83: 53. doi:10.1007/s002110050439


A breakdown (due to a division by zero) can arise in the algorithms for implementing Lanczos' method because of the non-existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate. Such a breakdown can be avoided by jumping over the polynomials involved. This strategy was already used in some algorithms such as the MRZ and its variants. In this paper, we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors, independent of the length of the jump. These new algorithms are based on Horner's rule and on a different way for computing the coefficients of the recurrence relationships. Moreover, these new algorithms seem to be more stable than the old ones and they provide better numerical results. Numerical examples and comparisons with other algorithms will be given.

Mathematics Subject Classification (1991):65F10, 65F25

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • C. Brezinski
    • 1
  • M. Redivo Zaglia
    • 2
  • H. Sadok
    • 3
  1. 1. Laboratoire d'Analyse Numérique et d'Optimisation, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France; e-mail: FR
  2. 2. Dipartimento di Elettronica e Informatica, Università degli Studi di Padova, via Gradenigo 6/a, I-35131 Padova, Italy; e-mail: IT
  3. 3. Laboratoire de Mathematiques Appliquées, Université du Littoral, Centre Universitaire de la Mi-Voix, 50 rue F. Buisson, F-62228 Calais, France; e-mail: FR