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Some results on convergence acceleration for the $$E$$-algorithm. (English) Zbl 0890.65004
Given the complex number sequences $${\mathcal G} (i):= g(i\mid 0)$$, $$g(i\mid 1), \dots$$ with $$i\geq 0$$ and the further sequence $${\mathcal S}: =s(0)$$, $$s(1), \dots$$, numbers $$e(k,m)$$ $$(k,m \geq 0)$$ may, subject to suitable conditions, be obtained from the system of linear equations $e(k,m)+ \bigl \{\sum \gamma (k,m|i) g(i|n) \mid [0<i\leq k] \bigr\} =s(n)$ with $$m \leq n\leq m+k$$. Reduction of the equations to triangular form produces a simple recursive algorithm for constructing the numbers $$e(k,m)$$. Convergence of sequences of the form $${\mathcal E}(k):=e(k,0)$$, $$e(k,1),\dots$$ to the limit $$S$$ of $${\mathcal S}$$ is investigated. A sample of the results obtained is provided by the treatment of the case in which $$g(i|n+1)=B(n) b(i)^n$$ where $$B(n+1) =o \{B(n)\}$$ as $$n$$ increases, $$b(1)=1$$ and the $$b(i)$$ are distinct.
Assuming that $s(n)- S\sim \Gamma (n)\bigl\{\sum a(j) c(j)^n \mid[j>0] \bigr\}$ where $$a(1)\neq 0$$ and $$|c(j) |$$ $$(j>0)$$ are strictly decreasing, the sequences $${\mathcal E} (k)$$ converge with increasing rapidity to $$S$$ in the sense that $$e(k,n)- S=o \{e(k-1,n)-S\}$$ as $$n$$ increases. Sequences arising from numerical quadrature are treated by the way of illustration.
Reviewer: P.Wynn (México)
##### MSC:
 65B05 Extrapolation to the limit, deferred corrections 40A05 Convergence and divergence of series and sequences 65D32 Numerical quadrature and cubature formulas
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