Comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences
type: Item
instance of: scholarly article
title: Comparison of four nonlinear transforms on some classes of logarithmic fixed point sequences
zbMATH Open document ID: 0840.65002
DOI: 10.1016/0377-0427(96)80013-6
published in: Journal of Computational and Applied Mathematics
publication date: 1996-02-01T00:00:00Z
review text: Let \(D(n) = S(n) - L\) \((n \geq 0)\) be the discrepancies between the members \(S(n)\) of a sequence and its limit \(L\), and let \(F(x) = x + \{\sum \alpha (k) x^{1 + kr}\mid (k \geq 1)\}\) for sufficiently small \(x\), where \(r \geq 1\) is an integer. If the coefficients \(\alpha(k)\) satisfy certain conditions, discrepancies for which \(D(n + 1) = F\{D(n)\}\) have, as \(n\) increases, an asymptotic representation of the form \(D(n) \sim n^{-1/r}\{\sum \beta (k|n) n^{-k+1}\mid (k \geq 1)\}\) where, in particular, \(\beta(k|n)\) is for \(k \geq 1\) a polynomial of degree at most \(k - 1\) in \(\log (n)\). Applications of various nonlinear recursive convergence acceleration algorithms (a modified \(\rho\)-algorithm, the \(\theta\)-algorithm, Lubkin's and Steffenson's methods) to sequences of the above type are studied. Asymptotic error estimates in transformed sequences are given and numerical examples are provided. The methods considered have one affliction in common: gallopping instability -- and no examination of error propagation is conducted. For this reason it might have been of interest to have included in the survey a condensation method due to \textit{A. van Wijngaarden} [Course scientific computing B; process analysis, Math. Centre CR 18, Amsterdam (1965); see also: \textit{J. W. Daniel}, Math. Comp. 23, 91-96 (1969; Zbl 0183.44101)] which is relatively unaffected by instability. (Implementation of the method as a nice exercise in the use of a recursive procedure is given by \textit{P. W. Hemker} (ed.), NUMAL: numerical procedures in Algol 60, MC Syllabus 47. 1-47.7, Math. Centre, Amsterdam (1980); for a C-implementation, see: \textit{H. T. Lau} [A numerical library in C for scientists and engineers, CRC Press (1995; Zbl 0815.65001)]; a necessarily clumsy FORTRAN implementation is given by: \textit{P. Wynn} [Numal in FORTRAN, IIMAS, Universidad Nacional Autonoma de México, Comunicaciones técnicas Nos. 48.0-48.11 (1981)]).
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 825305
zbMATH Keywords: logarithmic fixed point sequences
zbMATH Keywords: nonlinear sequence transformations
zbMATH Keywords: convergence acceleration
zbMATH Keywords: asymptotic error estimates
zbMATH Keywords: Lubkin's method
zbMATH Keywords: nonlinear recursive convergence acceleration algorithms
zbMATH Keywords: Steffenson's methods
zbMATH Keywords: numerical examples
zbMATH Keywords: instability
zbMATH Keywords: error propagation
author: P. Sablonnière
reviewed by: Peter Wynn
describes a project that uses: NUMAL
MaRDI profile type: Publication
full work available at URL: 0377-0427(96)80013-6
OpenAlex ID: W2016735054
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description: scientific article; zbMATH DE number 825305