On a Vectorized Version of a Generalized Richardson Extrapolation Process

Abstract

Let $\{\xx_m\}$ be a vector sequence that satisfies $$ \xx_m\sim \sss+\sum^\infty_{i=1}α_i \gg_i(m)\quad\text{as $m\to\infty$},$$ $\sss$ being the limit or antilimit of $\{\xx_m\}$ and $\{{\gg }_i(m){\}}_{i=1}^{\infty }$ being an asymptotic scale as $m\to \infty$, in the sense that $$\underset{m\to \infty }{\lim }\frac{\Vert {\gg }_{i+1}(m)\Vert }{\Vert {\gg }_i(m)\Vert }=0,i=1,2,\dots .$$ The vector sequences $\{{\gg }_i(m){\}}_{m=0}^{\infty }$, $i=1,2,\dots ,$ are known, as well as $\{\xx_m\}$. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations $$ \sum^k_{i=1}\braket{\yy,Δ\gg_{i}(m)}\widetildeα_i=\braket{\yy,Δ\xx_m},\quad n\leq m\leq n+k-1;\quad \sss_{n,k}=\xx_n+\sum^k_{i=1}\widetildeα_i\gg_{i}(n),$$ $\sss_{n,k}$ being the approximation to $\sss$. Here $\yy$ is some nonzero vector, $⟨\cdot ,\cdot ⟩$ is an inner product, such that $\braket{α\aaa,β\bb}=\barαβ\braket{\aaa,\bb}$, and $Δ\xx_m=\xx_{m+1}-~\xx_m$ and $\Delta {\gg }_i(m)={\gg }_i(m+1)-{\gg }_i(m)$. By imposing a minimal number of reasonable additional conditions on the ${\gg }_i(m)$, we show that the error $\sss_{n,k}-\sss$ has a full asymptotic expansion as $n\to \infty$. We also show that actual convergence acceleration takes place and we provide a complete classification of it.