Continued fractions. Vol. 1: Convergence theory
type: Item
instance of: scholarly article
title: Continued fractions. Vol. 1: Convergence theory
zbMATH Open document ID: 1180.40001
author: Lisa Lorentzen
author: Haakon Waadeland
published in: Atlantis Studies in Mathematics for Engineering and Science
publication date: 2008-06-05T00:00:00Z
review text: The volume under review deals mainly with continued fractions having complex number partial numerators and denominators and offers interpretations of some special results of interest to the authors. In this general account there are heavy casualties. It is difficult, in a short review, to deal adequately with perceived shortcomings, but some specific matters may be mentioned. It is suggested (on p.~11 and on p.~260 et seq.) that to avoid instability in the use of forward recursion in the computation of convergents, systematic use should be made of backward recursion. It is true that instability can occur, but its effect can be monitored. A preliminary error analysis is given in Section~2.6 on p.~277 of a paper by the reviewer [\textit{P. Wynn}, Numer. Math. 1, 272--307, 308--320 (1959; Zbl 0092.05101)] to which the authors do refer. This instability is partnered by that present in the computation of continued fraction coefficients, e.g.\ in the use of the \(q\)-\(d\)-algorithm [\textit{H. Rutishauser}, Der Quotienten-Differenzen-Algorithmus, Mitt\ Inst. Angew. Math. Techn. Hochschule Zürich. No.~7. Birkhäuser Verlag (1957; Zbl 0077.11103)] which is simply not mentioned. Analyses of stability may be referred to a number of representative cases just as has recently been done with regard to convergence ([\textit{K. A. Driver} and \textit{D. S. Lubinsky}, Aequationes Math. 42, No.~1, 85--106 (1991; Zbl 0746.41021), Approximation theory, Proc. Conf. Kecskemet/Hung. 1990, Colloq. Math. Soc. Janos Bolyai 58, 221--239 (1991; Zbl 0788.41010), Aequationes Math. 45, No.~1, 1--23 (1993; Zbl 0767.41019); \textit{M. Prevost}, Appl. Numer. Math. 17, No.~4, 461--469 (1995; Zbl 0840.65004)]; these papers are again simply not mentioned). Convergence acceleration of continued fractions is dealt with, but may be carried out by the use of stable methods which are once again not mentioned (for a gracious Zentralblatt review, see [\textit{P. Wynn}, Rev. Franc. Traitement Inform., Chiffres…
reviewed by: Peter Wynn
Mathematics Subject Classification ID: 40-02
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 30B70
Mathematics Subject Classification ID: 65B99
zbMATH DE Number: 5284831
zbMATH Keywords: continued fractions
zbMATH Keywords: three term recurrence
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: relational approximation
describes a project that uses: PlanetMath
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5284831