# Technical Report CS0902

 TR#: CS0902 Class: CS Title: FURTHER CONVERGENCE AND STABILITY RESULTS FOR THE GENERALIZED RICHARDSON EXTRAPOLATION PROCESS GREP$^{(1)}$ WITH AN APPLICATION TO THE $D^{(1)}$-TRANSFORMATION FOR INFINITE INTEGRALS Authors: Avram Sidi PDF Not Available Abstract: Let $a(t) \sim A+\varphi(t) \sum^{\infty}_{i=0} \beta_i t^i$ as $t \rightarrow 0+$, where $a(t)$ and $\varphi(t)$ are known for $00$, but $A$ and the $\beta_i$ are not known. The generalized Richardson extrapolation process GREP$^{(1)}$ is used in obtaining good approximations to $A$, the limit or antilimit of $a(t)$ as $t \rightarrow 0+$. The convergence and stability properties of GREP$^{(1)}$ for the case in which $\varphi(t) \sim \alpha t^{\delta}$ as $t \rightarrow 0+$, $\delta \neq 0,-1,-2,...,$ have been studied to a large extent in a recent work by the author. In the present work we continue this study for the case in which $\delta$ is complex when the set of extrapolation points is $\{t_i=t_0 \omega^i,\ i=0,1,..., n\}$ with $\omega \in (0,1)$. We give a complete convergence and stability analysis under very weak assumptions on $\varphi(t)$. We show that this analysis applies to the Levin-Sidi $D^{(1)}$-transformation that is a GREP$^{(1)}$, as this transformation is used for computing both convergent and divergent infinite-range integrals of functions $f(x)$ that essentially satisfy $f(x)\sim \nu x^{-\delta -1}$ as $x \rightarrow \infty$, with $\delta$ as above. In case of divergence we show that the $D^{(1)}$-transformation produces approximations to the associated Hadamard finite parts. We append numerical examples that demonstrate the theory. Copyright The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information

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