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
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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.
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