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

Sequence transformations, used for accelerating the convergence, are related to biorthogonal polynomials. In the particular cases of the G-transformation and the Shanks transformation (that is the $\varepsilon$-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.