Sequence Transformations
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Editorial Board
R. L. Graham, Murray Hill
J. Stoer, WOrzburg
R. Varga, Kent (Ohio)
Jean-Pau I Delahaye
Sequence Transformations With an Introduction by Claude Brezinski
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Jean-Paul Oelahaye Laboratoire d'informatique Fondamentale de Lille Universite des Sciences et Techniques de Lille Flandres Artois U.F.R. d'I.E.EA, 8at. M3, F-59655 Villeneuve d'Ascq Cedex, France
With 164 Figures Mathematics Subject Classification (1980): 65805, 65899, 68C05,68C40 ISBN-13:978-3-642-64802-1 e-ISBN-13:978-3-642-61347-0 001: 10.1007/978-3-642-61347-0
Library of Congress Cataloging-in-Publication Data Delahaye, Jean-Paul. Sequence transformations. (Springer series in computational mathematics; 11) Includes bibliographies and index. 1. Sequences (Mathematics) 2. Transformations (Mathematics) 3. Numerical analysis-Acceleration of convergence. I. Title. II. Series. QA292.D451988 515'.24 88-4527 ISBN -13:978-3-642-64802-1 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988
Sofcover reprint of the hardcover 1st edition 1988
2145/3140-543210
CONTENTS Introduction and Historical Survey, by Claude Brezinski. Presentation General Notation . . . . . . . . . . . . . . . . . . . . .
IX . XVII XX
Chapter 1 The Various Kinds of Algorithmic Sequence Transformations Introduction 1 - Sequence Transformations . . . . . . . . . 2 - Algorithms for Sequences and Algorithmic Transformations . . . . . . . . . . . . . . . . . . . 3 - k-Normal Algorithms and k-Normal Transformations 4 - k-Memories Algorithms and k-Memories Transformations . . . . . . . . . . . . . . . . . . . . . 5 - k-Stationary Algorithms and k-Stationary Transformations . . . . . . . . . . . . . . . . . . . . 6 - Rational Transformations and Linear Transformations 7 - Diagram of Inclusions References . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 2 Decidability and Undecidability in the Limit Introduction 1 - Definitions and the Normalisation Theorem . . . . . 2 - Problems Concerning Convergence, Turbulence and Periodicity of Sequences . . . . . . . . . . . . . . . 3 - Algorithms for Counting the Number of Accumulation Points . . . . . . . . . . . . . . . . . . . . 4 - Algorithms for Determining the Period of an Asymptotically Periodic Sequence . . (a) Methods of Detector Coefficients (b) Methods Using Barycenters .. (c) Limitation Results . . . . . . . . 5 - Families of Sequences of Iterations . 6 - Two General Results Concerning the Decid
