Computing the Zeros of Analytic Functions
Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros and analytic fuctions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computatio
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1727
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Peter Kravanja Marc Van Barel
Computing the Zeros of Analytic Functions
Springer
Authors Peter Kravanja Marc Van Barel Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200 A 3001 Heverlee, Belgium E-mail: [email protected] Marc.Van [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kravanja, Peter: Computing the zeros of analytic functions I Peter Kravanja ; Marc VanBarel. - Berlin ; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1727) ISBN 3-540-67162-5
Mathematics Subject Classification (2000): Primary: 65H05 Secondary: 65E05, 65HlO ISSN 0075-8434 ISBN 3-540-67162-5 Springer-Verlag Berlin Heidelberg New York 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a company in the BertelsmannSpringer publishing group. © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10724923 4113143/du
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Preface
In this book we consider the problem of computing zeros of analytic functions and several related problems in computational complex analysis. We start by studying the problem of computing all the zeros of an analytic function f that lie inside a positively oriented Jordan curve 'Y. Our principal means of obtaining information about the location of the zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along 'Y. This form involves the logarithmic derivative f' / f of f. Our approach could therefore be called a logarithmic residue based quadrature method. It can be seen as a continuation of the pioneering work done by Delves and Lyness. We shed new light on their approach by considering a different set of unknowns and by using the theory of formal orthogonal polynomials. Our algorithm computes not only approximations for the zeros but also their respective multiplicities. It does not require initial approximations for the zeros and we have found that it gives accurate results. The algor
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