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A convergence and stability study of the iterated Lubkin transformation and the $\theta$-algorithm

Author: Avram Sidi
Journal: Math. Comp. 72 (2003), 419-433
MSC (2000): Primary 65B05, 65B10, 40A05, 40A25, 41A60
Published electronically: May 1, 2002
MathSciNet review: 1933829
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Abstract: In this paper we analyze the convergence and stability of the iterated Lubkin transformation and the $\theta$-algorithm as these are being applied to sequences $\{A_n\}$ whose members behave like $A_n\sim A+\zeta^n/(n!)^r \sum^{\infty}_{i=0}\alpha_in^{\gamma-i}$ as $n\to\infty$, where $\zeta$ and $\gamma$ are complex scalars and $r$ is a nonnegative integer. We study the three different cases in which (i) $r=0$, $\zeta=1$, and $\gamma\neq 0,1,\ldots$ (logarithmic sequences), (ii) $r=0$ and $\zeta\neq 1$ (linear sequences), and (iii) $r=1,2,\ldots$ (factorial sequences). We show that both methods accelerate the convergence of all three types of sequences. We show also that both methods are stable on linear and factorial sequences, and they are unstable on logarithmic sequences. On the basis of this analysis we propose ways of improving accuracy and stability in problematic cases. Finally, we provide a comparison of these results with analogous results corresponding to the Levin $u$-transformation.

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Additional Information

Avram Sidi
Affiliation: Computer Science Department, Technion - Israel Institute of Technology, Haifa 32000, Israel

Received by editor(s): March 21, 2001
Published electronically: May 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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