Partial Padé prediction
type: Item
instance of: scholarly article
title: Partial Padé prediction
zbMATH Open document ID: 0938.65002
DOI: 10.1023/A:1019187330365
published in: Numerical Algorithms
publication date: 2000-06-06T00:00:00Z
review text: The operation of a sequence transformation may often be referred to numbers placed in columns, each of which is associated with its index \( \tau(r) \), the \( \tau(r)\) \((r \geq 0) \) forming a strictly increasing sequence of nonnegative integers. The first column with index \( \tau(0) = 0 \) contains the successive members \( s(m)\) \((m \geq 0) \) of the sequence \( \mathcal S \) under transformation. The column having index \( \tau(r) \) contains transformed values \( t(r,m|s(m),s(m+1),\ldots,s(m+\tau(r)) \) for \( m \geq 0 \) obtained, in particular, from \( \tau(r) + 1 \) successive members of \( \mathcal S \). The further columns contain auxiliary numbers used in the construction of the transformed numbers. When \( \tau(r) = r\) \((r \geq 0) \) there are no further columns; when \( \tau(r) = 2r\) \((r \geq 0) \) the further columns are interlaced with their counterparts. It often occurs that when \( \mathcal S \) has a special structure, the columns of numbers exhibit a corresponding pattern; for example, for some fixed \( n \geq 0 \) all entries in the column index \( \tau(n) \) are the same, as is so, with \( \tau(n) = 2n \), for the \( \varepsilon \)- and \( \rho \)-algorithms; certain linear processes and the \(q\)-\(d\) algorithm produce other patterns. This property serves as the basis of a sequence extrapolation method: in the case of the \( \varepsilon \)-algorithm, the number \( t(r,0|s(0),\ldots,s(2r)) \) is determined from \( s(0),\ldots,s(2r) \), the column with index \( 2r \) is filled with copies of it, the transformation is put into reverse to construct \( s'(m)\) \((m > 2r) \). The extrapolated sequence \( \mathcal S' \) of terms \( s'(m)\) \((m \geq 0) \) is that sequence of special structure whose first \( 2r + 1 \) members \( s'(m)\) \((0 \leq m \leq 2r) \) agree with those of \( \mathcal S \). The progressive use of the \( \varepsilon \)-algorithm in this way is considered. It is remarked that if the algorithm is applied to the sequence \…
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 1310132
zbMATH Keywords: sequence extrapolation method
zbMATH Keywords: epsilon algorithm
zbMATH Keywords: partial Padé prediction
zbMATH Keywords: convergence acceleration
zbMATH Keywords: quotient-difference algorithm
zbMATH Keywords: rho algorithm
zbMATH Keywords: sequence transformation
zbMATH Keywords: \(q\)-\(d\) algorithm
zbMATH Keywords: stability
zbMATH Keywords: instability
author: M. Prevost
author: Denis Vekemans
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: a:1019187330365
OpenAlex ID: W2293757285
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description: scientific article; zbMATH DE number 1310132