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

摘要

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 $arepsilon$-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.