On a vectorized version of a generalized Richardson extrapolation process

Abstract

Let {xm} be a vector sequence that satisfies xm~s+?i=1??igi(m)asm??,$$\boldsymbol{x}_{m}\sim \boldsymbol{s}+\sum\limits^{\infty}_{i=1}\alpha_{i} \boldsymbol{g}_{i}(m)\quad\text{as \(m\to\infty\)}, $$s being the limit or antilimit of {xm} and {gi(m)}i=1?$\{\boldsymbol {g}_{i}(m)\}^{\infty }_{i=1}$ being an asymptotic scale as m ? ?, in the sense that limm???gi+1(m)??gi(m)?=0,i=1,2,?.$$\lim\limits_{m\to\infty}\frac{\|\boldsymbol{g}_{i+1}(m)\|}{\|\boldsymbol{g}_{i}(m)\|}=0,\quad i=1,2,\ldots. $$ The vector sequences {gi(m)}m=0?$\{\boldsymbol {g}_{i}(m)\}^{\infty }_{m=0}$, i = 1, 2,?, are known, as well as {xm}. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations ?i=1k?y,Δgi(m)??~i=?y,Δxm?,n≤m≤n+k?1;sn,k=xn+?i=1k?~igi(n),$$\sum\limits^{k}_{i=1}\langle\boldsymbol{y},{\Delta}\boldsymbol{g}_{i}(m)\rangle\widetilde{\alpha}_{i}=\langle\boldsymbol{y},{\Delta}\boldsymbol{x}_{m}\rangle,\quad n\leq m\leq n+k-1;\quad \boldsymbol{s}_{n,k}=\boldsymbol{x}_{n}+\sum\limits^{k}_{i=1}\widetilde{\alpha}_{i}\boldsymbol{g}_{i}(n), $$sn, k being the approximation to s. Here, y is some nonzero vector, ?? ,?? is an inner product, such that ??a,sb?=?¯s?a,b?$\langle \alpha \boldsymbol {a},\beta \boldsymbol {b}\rangle =\overline {\alpha }\beta \langle \boldsymbol {a},\boldsymbol {b}\rangle $, and Δxm = xm + 1? xm and Δgi(m) = gi(m + 1)?gi(m). By imposing a minimal number of reasonable additional conditions on the gi(m), we show that the error sn, k ? s has a full asymptotic expansion as n??. We also show that actual convergence acceleration takes place, and we provide a complete classification of it.