Abstract
We determine the values of the integer m for which the parametric transformation T+m due Gray and Clark is well conditionned. This process of acceleration being quasilinear transformation, we use an adequate definition of the condition numbers that apply to the real sequences of linear convergence. The results obtained for this set are enough meaning
Highlights
Nonlinear sequences transformations are generally used for solve the extrapolation of the limit or to accelerate slowly convergent sequences
We are interested in this work in the conditioning of the transformation T+m of Gray and Clark, which generalizes the famous ∆2Aitken's process
In the case where the sequence of the punctual condition numbers converges, we can define the asymptotic conditioning of the transformation Φ, applied to the sequence x, by
Summary
Nonlinear sequences transformations are generally used for solve the extrapolation of the limit or to accelerate slowly convergent sequences. Definition 2: The punctual condition number i th step of Φ for a sequence x is given by. In the case where the sequence of the punctual condition numbers converges, we can define the asymptotic conditioning of the transformation Φ , applied to the sequence x, by. This is the transformation that defines the Aitken's ∆2 process. We observe that the process of Aitken is well conditioned when applied to elements of LIN− (−1 < ρ < 0) It is ill conditioned if the asymptotic ratio of the sequence (which we have to accelerate) is close to one.
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