Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences

In this work, we give a detailed convergence and stability analysis for the author's generalized Richardson extrapolation process GREP$^{(m)}$ as this is being applied to linearly convergent or divergent infinite sequences $\{A_n\}$, where $A_n\sim A+\sum^m_{k=1}\zeta_k^n\sum^\infty_{i=0}\beta_{ki}n^{\gamma_k-i}$ as $n\to\infty$, $\zeta_k\neq 1$ being distinct. The quantity we would like to compute is $A$, whether it is the limit or antilimit of $\{A_n\}$. Such sequences arise, for example, as partial sums of power series and of Fourier series of functions that have algebraic and/or logarithmic branch singularities. Specifically, we define the GREP$^{(m)}$ approximation $A^{(m,j)}_n$ to $A$, with $n=(n_1,\ldots,n_m)$ and $\alpha>0$, via the linear systems

$A_l=A^{(m,j)}_n +\sum^m_{k=1}\zeta_k^l\sum^{n_k-1}_{i=0}\bar{\beta}_{ki}(\alpha+l)^{\gamma_k-i}, j\leq l\leq j+\sum^m_{k=1}n_k,$

where $\bar{\beta}_{ki}$ are additional unknowns. We study the convergence and stability properties of $A^{(m,j)}_n$ as $j\to\infty$. We show, in particular, that $A^{(m,j)}_n-A= \sum^m_{k=1}O(\zeta_k^j j^{\gamma_k-2n_k})$ as $j\to\infty$. When compared with $A_j-A=\sum^m_{k=1}O(\zeta_k^j j^{\gamma_k})$ as $j\to\infty$, this result shows that GREP$^{(m)}$ is a true convergence acceleration method for the sequences considered. In addition, we show that GREP$^{(m)}$ is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to GREP$^{(m)}$ with $m>1$.