New convergence results on the generalized Richardson extrapolation process GREP$^{(1)}$ for logarithmic sequences

Abstract:

Let $a(t)\sim A+\varphi (t)\sum ^\infty _{i=0}\beta _it^i$ as $t\to 0+$, where $a(t)$ and $\varphi (t)$ are known for $0<t\leq c$ for some $c>0$, 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 $t\to 0+$. The approximations $A^{(j)}_n$ to $A$ obtained via GREP$^{(1)}$ are defined by the linear systems $a(t_l)=A^{(j)}_n+\varphi (t_l) \sum ^{n-1}_{i=0}\bar {\beta }_it_l^i$, $l=j,j+1,\ldots ,j+n$, where $\{t_l\}^\infty _{l=0}$ is a decreasing positive sequence with limit zero. The study of GREP$^{(1)}$ for slowly varying functions $a(t)$ was begun in two recent papers by the author. For such $a(t)$ we have $\varphi (t)\sim \alpha t^\delta$ as $t\to 0+$ with $\delta$ possibly complex and $\delta \neq 0$, $-1$, $-2$, …. In the present work we continue to study the convergence and stability of GREP$^{(1)}$ as it is applied to such $a(t)$ with different sets of collocation points $t_l$ that have been used in practical situations. In particular, we consider the cases in which (i) $t_l$ are arbitrary, (ii) $\lim _{l\to \infty }t_{l+1}/t_l=1$, (iii) $t_l\sim cl^{-q}$ as $l\to \infty$ for some $c, q>0$, (iv) $t_{l+1}/t_l\leq \omega \in (0,1)$ for all $l$, (v) $\lim _{l\to \infty }t_{l+1}/t_l= \omega \in (0,1)$, and (vi) $t_{l+1}/t_l=\omega \in (0,1)$ for all $l$.