Convergence analysis for a generalized Richardson extrapolation process with an application to the $d^ {(1)}$-transformation on convergent and divergent logarithmic sequences

Abstract:

In an earlier work by the author the Generalized Richardson Extrapolation Process (GREP) was introduced and some of its convergence and stability properties were discussed. In a more recent work by the author a special case of GREP, which we now call ${\text {GREP}^{(1)}}$, was considered and its properties were reviewed with emphasis on oscillatory sequences. In the first part of the present work we give a detailed convergence and stability analysis of ${\text {GREP}^{(1)}}$ as it applies to a large class of logarithmic sequences, both convergent and divergent. In particular, we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table. These theorems are very realistic in the sense that they explain the remarkable efficiency of ${\text {GREP}^{(1)}}$ in a very precise manner. In the second part we apply this analysis to the Levin-Sidi ${d^{(1)}}$-transformation, as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically, or to sum the divergent extensions of such series. This is made possible by the observation that, when the proper analogy is drawn, the ${d^{(1)}}$-transformation is, in fact, a ${\text {GREP}^{(1)}}$. We append numerical examples that demonstrate the theory.