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A look-ahead strategy for the implementation of some old and new extrapolation methods

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Abstract

Sequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of theG-transformation and the Shanks transformation (that is the ε-algorithm of Wynn), there is a connection with formal orthogonal polynomials. In this paper, this connection is exploited in order to propose a look-ahead strategy for the implementation of these two transformations. This strategy, which is quite similar to the strategy used for treating the same type of problems in Lanczos-based methods for solving systems of linear equations, consists in jumping over the polynomials which do not exist, thus avoiding a division by zero (breakdown) in the algorithms, and over those which could be badly computed (near-breakdown) thus leading to a better numerical stability. Numerical examples illustrate the procedure.

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Brezinski, C., Redivo-Zaglia, M. A look-ahead strategy for the implementation of some old and new extrapolation methods. Numer Algor 11, 35–55 (1996). https://doi.org/10.1007/BF02142487

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Keywords

  • Orthogonal polynomials
  • sequence transformations
  • extrapolation methods
  • numerical stability

AMS subject classification

  • 65B05
  • 65B10
  • 65G05
  • 42C05