An acceleration theorem for the \(\rho\)-algorithm
type: Item
instance of: scholarly article
title: An acceleration theorem for the \(\rho\)-algorithm
zbMATH Open document ID: 0858.65001
author: Naoki Osada
published in: Numerische Mathematik
publication date: 1997-03-25T00:00:00Z
review text: The \(\rho\)-algorithm is a sequence transformation introduced by the reviewer [Proc. Camb. Phil. Soc., 52, 663-671 (1956; Zbl 0072.33802)] which involves the construction of numbers \(\rho (m,r)\) \((r>0\), \(m\geq 0)\) form the initial values \(\rho (m,-1)=0\) \((m >0)\), \(\rho(m,0) = S(m)\) \((m\geq 0)\) by use of the recursion \(\rho (m,r+1) = \rho (m+1,r-1)+ (r+ 1)/ \{\rho (m+1,r)-\rho (m,r)\}\) \((r,m\geq 0)\); it is often found that a number of the form \(\rho (m, 2r)\) is a better approximation to \(S= \lim S(m)\) \((m\to\infty)\) than any of the sequence members, \(S(\omega)\) \((m\leq\omega \leq m+2r)\), from which the number is derived, and that this is uniformly true for \(r>0\), \(m\geq 0\). The author shows that if \(S(m)\sim S+m^\theta \{\sum c(\omega) m^{-\omega} \mid\omega \geq 0\}\) for large \(m\) and \(\theta=-k\) is a negative integer, then \(\lim\{\rho (m,2k)-S\}/ \{S(m+2k)-S\} = \mu\) \((m\to\infty)\) where \(\mu=0\) and in this sense the \(\rho\)-algorithm accelerates convergence; if \(\text{Re} (\theta)<0\) and \(\theta\) is not a negative integer, \(\mu\neq 0\) for \(k=1,2,\dots\) and in this sense it does not. A similar negative conclusion is arrived at with respect to sequences for which \(S(m+1)-S = \{\lambda +e(m)\}\) \(\{S(m)-S\}\) where \(-1\leq \lambda<1\) \((\lambda\neq 0)\) and \(\lim e(m)=0\) \((m\to\infty)\). The numerical performance of the \(\rho\)-algorithm is contrasted with that of other algorithms in selected cases and it is in this connection that attention is unwiltingly drawn to a characteristic of the algorithms concerned: all of them (the \(\rho\)-algorithm included) are, used as convergence acceleration devices, afflicted by pronounced growth of rounding error. Thus it would appear from Table IV on p. 530, for example, that as \(r\) increases, with \(m\) fixed, the discrepancies \(|S-\rho(m,2r)|\) decrease initially and then remain constant or even increase slightly. This is not so. During computation rounding errors…
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 932028
zbMATH Keywords: rho-algorithm
zbMATH Keywords: convergence acceleration
zbMATH Keywords: asymptotic estimate
zbMATH Keywords: sequence transformation
zbMATH Keywords: Euler-Maclaurin sum formula
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: s002110050204
OpenAlex ID: W2044607934
DOI: 10.1007/S002110050204
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description: scientific article; zbMATH DE number 932028