The epsilon algorithm in a noncommutative algebra
type: Item
instance of: scholarly article
title: The epsilon algorithm in a noncommutative algebra
zbMATH Open document ID: 0631.65053
author: André Draux
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \({\mathcal A}\) be a noncommutative Banach algebra with unit element I over a field \({\mathfrak K}\) and f be the formal power series \(f(t)=\sum^{\infty}_{i=0}c_ it^ i\) where \(c_ i\in {\mathcal A}\) (i\(\geq 0)\) and \(t\in {\mathfrak K}\). The approximating fraction \(R_ k^{(n)}\) of order k, \(n+k-1\) derived from f is given by \(R_ k^{(n)}(t)=N_ k^{(n)}(t)\{D_ n^{(k)}(t)\}^{-1}\) where \(D_ k^{(n)}\), \(N_ k^{(n)}\) are polynomials in t of degrees at most k, \(n+k-1\) respectively, \(D_ k^{(n)}(0)=I\) and \(f(t)-R_ k^{(n)}(t)=O(t^{n+2k}).\) The existence of the fractions \(R_ k^{(n)}\) and relationships obtaining among them are treated with reference to Hankel matrices with elements \(c_{n+i+j}\) (0\(\leq i,j\leq k)\). The theory is also presented in terms of application of the \(\epsilon\)-algorithm to the sequence \(S_ m\in {\mathcal A}\) (m\(\geq 0)\).
Mathematics Subject Classification ID: 65J05
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 65B10
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 4026689
zbMATH Keywords: convergence acceleration
zbMATH Keywords: Padé approximants
zbMATH Keywords: approximating fractions
zbMATH Keywords: Banach algebra
zbMATH Keywords: epsilon-algorithm
zbMATH Keywords: convergence approximation
reviewed by: Peter Wynn
MaRDI profile type: Publication
cites work: Accélération de la convergence en analyse numérique
cites work: Q3326194
cites work: Convergence Theorems for Matrix Continued Fractions
cites work: Q5183947
cites work: Continued fractions whose coefficients obey a non-commutative law of multiplication
cites work: Upon systems of recursions which obtain among the quotients of the Padé table
cites work: Q5650807
description: scientific article; zbMATH DE number 4026689