Convergence acceleration of logarithmic fixed point sequences
type: Item
instance of: scholarly article
title: Convergence acceleration of logarithmic fixed point sequences
zbMATH Open document ID: 0622.65002
DOI: 10.1016/0377-0427(87)90171-3
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: A real sequence \((S_ n)\) which converges to its limit S, verifying: \(\lim (S_{n+1}-S)/(S_ n-S)=1,\) is a sequence converging logarithmically. Such a sequence needs algorithms for accelerating its convergence. So, a sequence \((x_ n)\) generated by: \(x_{n+1}=f(x_ n)\), where \(f(x)=x+\sum_{i\geq 1}\alpha_{p+i}x^{p+i};\) \(p\geq 1\), \(\alpha_{p+1}<0\), for \(x_ 0>0\) small, converges to zero logarithmically i.e. lim \(x_{n+1}/x_ n=1\). The author, using asymptotic expansions of the \(\Delta^ 2\)- and \(\theta\)-algorithms, obtains modified iterated versions of the first one and combinations of it with the iterated \(\theta_ 2\)-algorithm. He explains some superconvergence phenomena and shows that the algorithms are good accelerators for large classes of slowly convergent integrals and series.
reviewed by: A.Donescu
Mathematics Subject Classification ID: 65B05
zbMATH DE Number: 4009628
zbMATH Keywords: logarithmic fixed point sequences
zbMATH Keywords: convergence acceleration
zbMATH Keywords: logarithmical convergence
zbMATH Keywords: delta square algorithm
zbMATH Keywords: theta two algorithm
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: superconvergence
author: P. Sablonnière
MaRDI profile type: Publication
full work available at URL: 0377-0427(87)90171-3
OpenAlex ID: W2008134816
cites work: Accélération de la convergence en analyse numérique
cites work: Q4113343
cites work: Q3219268
cites work: The Set of Logarithmically Convergent Sequences Cannot be Accelerated
cites work: Survey of Extrapolation Processes in Numerical Analysis
cites work: Numerical Quadrature by the ε-Algorithm
cites work: Q3965026
cites work: Development of non-linear transformations for improving convergence of sequences
cites work: Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series
cites work: Numerical Quadrature and Asymptotic Expansions
cites work: Q3228715
cites work: Acceleration of Linear and Logarithmic Convergence
cites work: Numerical Comparisons of Nonlinear Convergence Accelerators
cites work: Sequence transformations and their applications
cites work: Q3236067
description: scientific article; zbMATH DE number 4009628