On totally monotonic families of sequences
type: Item
instance of: scholarly article
title: On totally monotonic families of sequences
zbMATH Open document ID: 0943.65005
author: P. Mortreux
author: M. Prevost
published in: Numerische Mathematik
publication date: 2000-08-21T00:00:00Z
review text: The authors give some determinantal inequalities which are then used to derive convergence results for the \( E \)-algorithm. The determinants considered have, with \( i,j \geq 1 \), elements of the form [a] \( K(x(i), x(j)) \), where \( x(k)\) \((k \geq 1) \) is a strictly increasing real sequence and \( K \) is real valued, [b] \( g(i|n+j-1) \) where, for each \( i\geq 1\), \(g(i|n)\) \((n\geq 0) \) is a sequence whose members satisfy certain inequalities and, as a special case of this form, [c] \( g(i+n+j-2)\) \((n \geq 0) \) where \( g(n)\) \((n \geq 0) \) is a totally monotone sequence. The theory is illustrated by application to the transformation of the sum \( \{\sum (-1)^{k+1}/\ln(k+2) |k\geq 0\} \), it being necessary to obtain successive derivatives of the function \( f(t) = 1/\ln(1+t) \) which feature in the theory as \( g(1|n) = (-1)^{n}: f(n+1) \) and \( g(i|n) = (-1)^{n+1}: f^{(2i-3)}(n+1)\) \((i \geq 0) \). There is a not entirely successful attempt to contrast the method favorably with the epsilon algorithm. Firstly, application of the latter algorithm to the example considered is stable while the formation of cross products in use of the \( E \)-algorithm is not; secondly, the evaluation of derivatives is unnecessary when using the epsilon algorithm; thirdly, repeated application of the epsilon algorithm (which the authors do not consider) produces estimates which are in the present context considerably better than those displayed in the paper under review.
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 1398748
zbMATH Keywords: \(E\)-algorithm
zbMATH Keywords: total monotonicity
zbMATH Keywords: sequence transformation
zbMATH Keywords: determinantal inequalities
zbMATH Keywords: convergence
zbMATH Keywords: epsilon algorithm
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: s002110050464
OpenAlex ID: W2031702997
DOI: 10.1007/S002110050464
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description: scientific article; zbMATH DE number 1398748