Average-Case Analysis of Numerical Problems
The average-case analysis of numerical problems is the counterpart of the more traditional worst-case approach. The analysis of average error and cost leads to new insight on numerical problems as well as to new algorithms. The book provides a survey of r
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Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Klaus Ritter
Average-Case Analysis of Numerical Problems
Springer
Author Klaus Ritter Fachbereich Mathematik Technische Universitiit Darmstadt Schlossgartenstrasse 7 64289 Darmstadt, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ritter, Klaus: Average case analysis of numerical problems I Klaus Ritter. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris ; Singapore; Tokyo: Springer, ZOllO (Lecture notes in mathematics; 1733) ISBN 3-540-67449-7
Mathematics Subject Classification (2000): 60-02, 60Gxx, 62-02, 62Mxx, 65-02, 65Dxx, 65H05, 65KlO ISSN 0075-8434 ISBN 3-540-67449-7 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: 10725000 41/3143/du
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Acknowledgments I thank Jim Calvin, Fang Gensun, Stefan Heinrich, Norbert Hofmann, Dietrich Kolzow, Thomas Miiller-Gronbach, Erich Novak, Leszek Plaskota, Oleg Seleznjev, Joe Traub, Greg Wasilkowski, Art Werschulz, and Henryk Wosniakowski for valuable discussions and comments on my manuscript in its various stages. My special thanks are to my coauthors for the opportunity of joint research. This work was completed during my visit at the Computer Science Department, Columbia University, New York. I am grateful to Joe Traub for his invitation, and to the Alexander von Humboldt Foundation and the Lamont Doherty Earth Observatory for their support.
Contents Chapter I. Introduction 1. Overview 2. Historical and Bibliographical Remarks 3. Notation and Numbering
1 3 7 9
Chapter II. Linear Problems: Definitions and a Classical Example 1. Measures on Function Spaces 1.1. Borel Measures 1.2. Stochastic Processes, Random Functions 1.3. Covariance Kernel and Mean 1.4. Gaussian Measures 2. Linear Problems 2.1. Integration 2.2. Lp-Approximation 2.3. Linear Problems 2.4. Average Errors 2.5. Maximal Errors 2.
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