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Extrapolation and Rational Approximation The Works of the Main Contributors

Extrapolation and Rational Approximation

Claude Brezinski • Michela Redivo-Zaglia

Extrapolation and Rational Approximation The Works of the Main Contributors

Claude Brezinski Laboratoire Paul Painlevé University of Lille Villeneuve d’Ascq, France

Michela Redivo-Zaglia Department of Mathematics “Tullio Levi-Civita” University of Padua Padua, Italy

ISBN 978-3-030-58417-7 ISBN 978-3-030-58418-4 (eBook) https://doi.org/10.1007/978-3-030-58418-4 Mathematics Subject Classification: 65B05, 40A15, 41A20, 65F10, 65H10 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1

General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mathematical Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Determinantal Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Schur’s Complement and Identity . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Designants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Extrapolation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Richardson’s Extrapolation Method . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Aitken Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The Shanks Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 The ε-Algorithm . . . . . . . . . . . . . . . . . . . . . . .