On a Vectorized Version of a Generalized Richardson Extrapolation Process

Let $\{{\text{xx}}_m\}$ be a vector sequence that satisfies $${\text{xx}}_m\sim \text{sss}+\sum _{i=1}^{\mathrm{\infty }}{\alpha }_i{\gg }_i(m)\text{as }m\to \mathrm{\infty },$$ $\text{sss}$ being the limit or antilimit of $\{{\text{xx}}_m\}$ and $\{{\gg }_i(m){\}}_{i=1}^{\mathrm{\infty }}$ being an asymptotic scale as $m\to \mathrm{\infty }$, in the sense that $$\underset{m\to \mathrm{\infty }}{\lim }\frac{\mathrm{\parallel }{\gg }_{i+1}(m)\mathrm{\parallel }}{\mathrm{\parallel }{\gg }_i(m)\mathrm{\parallel }}=0,i=1,2,\dots \mathrm{.}$$ The vector sequences $\{{\gg }_i(m){\}}_{m=0}^{\mathrm{\infty }}$, $i=1,2,\dots ,$ are known, as well as $\{{\text{xx}}_m\}$. 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 $$\sum _{i=1}^k⟨\text{yy},\mathrm{\Delta }{\gg }_i(m)⟩{\tilde{\alpha }}_i=⟨\text{yy},\mathrm{\Delta }{\text{xx}}_m⟩,n\le m\le n+k-1;{\text{sss}}_{n,k}={\text{xx}}_n+\sum _{i=1}^k{\tilde{\alpha }}_i{\gg }_i(n),$$ ${\text{sss}}_{n,k}$ being the approximation to $\text{sss}$. Here $\text{yy}$ is some nonzero vector, $⟨\cdot ,\cdot ⟩$ is an inner product, such that $⟨\alpha \text{aaa},\beta \text{bb}⟩=\bar{\alpha }\beta ⟨\text{aaa},\text{bb}⟩$, and $\mathrm{\Delta }{\text{xx}}_m={\text{xx}}_{m+1}-{\text{xx}}_m$ and $\mathrm{\Delta }{\gg }_i(m)={\gg }_i(m+1)-{\gg }_i(m)$. By imposing a minimal number of reasonable additional conditions on the ${\gg }_i(m)$, we show that the error ${\text{sss}}_{n,k}-\text{sss}$ has a full asymptotic expansion as $n\to \mathrm{\infty }$. We also show that actual convergence acceleration takes place and we provide a complete classification of it.