Acceleration methods based on convergence tests
type: Item
instance of: scholarly article
title: Acceleration methods based on convergence tests
zbMATH Open document ID: 0714.65002
DOI: 10.1007/BF01385628
author: Ana C. Matos
published in: Numerische Mathematik
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133504
review text: The following simple theorem offers both a convergence test for a real valued sequence and a derived sequence which converges faster to the same limit. Let \(\{\) S(n)\(\}\) and \(\{\) x(n)\(\}\) be monotone, the latter converging to x. Set \(A(k,n)=\{x(n+1)-x(n)\}/\{S(n+k+1)-S(n+k)\}.\) If, with k fixed, lim A(k,n)\(\neq 0\), then \(\{\) S(n)\(\}\) converges and, setting \(T(n)=S(n)+\{x-x(n-k)\}/A(k,n-k),\lim \{S-T(n)\}/\{S-S(n)\}=0,\) where \(S=\lim S(n).\) The convergence tests of d'Alembert, Cauchy, Kummer, Raabe, Gauss and others are considered in the light of the above result.
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A25
zbMATH DE Number: 4176311
zbMATH Keywords: real sequences
zbMATH Keywords: acceleration
zbMATH Keywords: d'Alembert convergence test
zbMATH Keywords: Cauchy convergence test
zbMATH Keywords: Kummer convergence test
zbMATH Keywords: Raabe convergence test
zbMATH Keywords: Gauss convergence test
reviewed by: Peter Wynn
MaRDI profile type: Publication
cites work: Automatic Selection of Sequence Transformations
cites work: Development of non-linear transformations for improving convergence of sequences
cites work: Q4130834
cites work: Sequence transformations and their applications
description: scientific article; zbMATH DE number 4176311