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

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.