Quasilinear sequence transformations
type: Item
instance of: scholarly article
title: Quasilinear sequence transformations
zbMATH Open document ID: 0906.65002
DOI: 10.1023/A:1019154106316
author: A. Benazzouz
published in: Numerical Algorithms
publication date: 1999-02-14T00:00:00Z
review text: \(p>0\) and \(k\geq 0\) are integers. The elements of \(E\equiv\mathbb{R}^p\) are represented as column vectors. \({\mathfrak G}{\mathfrak L}(E)\) is the linear isomorphism group over \(E\) and \(I\) is its unit member. \(E(p,k)\) is \(E\times\cdots\times E\) (\(k+1\) times). A transformation over \(E\) is extended to \(E(p,k)\) by setting \(A(x,y,\dots, z)= (Ax,Ay,\dots, Az)\). The scalar part \({\mathcal S}\{{\mathfrak D}\}\) of any domain \({\mathfrak D}\) in \(E(p,k)\) is that part consisting of elements of the form \((x,\dots, x)\) alone; with \(x\in E\), \(X= (x,\dots, x)\) is written as \(\text{scal}(x)\); red is the reduction operator from \({\mathcal S}\{E(p,k)\}\) to \(E\) defined by setting \(\text{red}(X)= x\). \(f\) is a transformation from \(E(p, k)\) to \(E\); \(\text{dom}(f)\) and \({\mathcal N}(f)\) are its domain and kernel respectively. \(f\) for which \(X+\text{scal}(b)\) is in \(\text{dom}(f)\) whenever \(X\) is in \(\text{dom}(f)\) and \(b\in E\) and for which also \(f(X+ \text{scal}(b))= f(X)+ b\) is said to be translative. \({\mathfrak M}\) being a set of multipliers in \({\mathfrak G}{\mathfrak L}(E)\), a transformation \(f\) for which (a) \(MX\) is in \(\text{dom}(f)\) whenever \(X\) is in \(\text{dom}(f)\) and (b) \(f(MX)= Mf(X)\), both for all \(M\in{\mathfrak M}\), is said to be \({\mathfrak M}\)-homogeneous. An \({\mathfrak M}\)-homogeneous transformation which is also translative is said to be \({\mathfrak M}\)-quasilinear. The transformation \(\varepsilon(n, 2k)\) over \(E(p,2k)\) determined recursively by a constrained variant of the vector \(\varepsilon\)-algorithm in which \(y(\neq 0)\in E\) is fixed, \(\varepsilon(m,- 1)= 0\in E\) \((n<m\leq n+ 2k)\), \(\varepsilon(m, 0)= x(m)\in E\) \((n\leq m\leq n+ 2k)\), \(\varepsilon(m, 2r+1)= \varepsilon(m+1, 2r-1)+ y/\langle y,\Delta\varepsilon(m, 2r)\rangle\) \((0\leq r<k; n\leq m<n+ 2k- 2r)\), \(\varepsilon(m, 2r+2)= \varepsilon(m+1, 2r)+ \Delta\varepsilon(m, 2r)/\langle\Delta \varepsilon…
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46B15
zbMATH DE Number: 1129530
zbMATH Keywords: quasilinear sequence transformations
zbMATH Keywords: vector sequence transformation
zbMATH Keywords: projection composition
zbMATH Keywords: vector epsilon-algorithm
reviewed by: Peter Wynn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1129530