Extrapolation methods and derivatives of limits of sequences

13 years 10 months ago

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Let {Sm} be an infinite sequence whose limit or antilimit S can be approximated very efficiently by applying a suitable extrapolation method E0 to {Sm}. Assume that the Sm and hence also S are differentiable functions of some parameter , d d S being the limit or antilimit of { d d Sm}, and that we need to approximate d d S. A direct way of achieving this would be by applying again a suitable extrapolation method E1 to the sequence { d d Sm}, and this approach has often been used efficiently in various problems of practical importance. Unfortunately, as has been observed at least in some important cases, when d d Sm and Sm have essentially different asymptotic behaviors as m , the approximations to d d S produced by this approach, despite the fact that they are good, do not converge as quickly as those obtained for S, and this is puzzling. In this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which E0 is the Richardson extrapolation process...

Avram Sidi

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