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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DOI: https://doi.org/10.1007/BF02142487