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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 (MĂ©xico)
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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