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Evaluating derivatives principles and techniques of algorithmic differentiation

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Evaluating

Derivatives

Evaluating

Derivatives

Principles and Techniques of

Algorithmic Differentiation

Second Edition

Andreas Griewank

Institut für Mathematik

Humboldt–Universität zu Berlin

Berlin, Germany

Andrea Walther

Institut für Wissenschaftliches Rechnen

Technische Universität Dresden

Dresden, Germany

Society for Industrial and Applied Mathematics • Philadelphia

10 9 8 7 6 5 4 3 2 1

reproduced, stored, or transmitted in any manner without the written permission of the

publisher. For information, write to the Society for Industrial and Applied Mathematics,

3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.

Trademarked names may be used in this book without the inclusion of a trademark symbol.

These names are used in an editorial context only; no infringement of trademark is intended.

Maple is a registered trademark of Waterloo Maple, Inc.

Mathematica is a registered trademark of Wolfram Research, Inc.

MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product informa-

tion, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA,

508-647-7000, Fax: 508-647-7101, info@mathworks.com, www.mathworks.com.

Library of Congress Cataloging-in-Publication Data

Griewank, Andreas.

Evaluating derivatives : principles and techniques of algorithmic differentiation. – 2nd ed. /

Andreas Griewank, Andrea Walther.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-898716-59-7 (alk. paper)

1. Differential calculus–Data processing. I. Walther, Andrea. II. Title.

QA304.G76 2008

515'.33--dc22

2008021064

is a registered trademark.

Contents

Rules ix

Preface xi

Prologue xv

Mathematical Symbols xix

1 Introduction 1

I Tangents and Gradients 13

2 A Framework for Evaluating Functions 15

2.1 TheLighthouseExample ...................... 16

2.2 Three-PartEvaluationProcedures ................. 18

2.3 Elemental Diﬀerentiability . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Generalizations and Connections . . . . . . . . . . . . . . . . . . 25

2.5 ExamplesandExercises ....................... 29

3 Fundamentals of Forward and Reverse 31

3.1 Forward Propagation of Tangents . . . . . . . . . . . . . . . . . . 32

3.2 Reverse Propagation of Gradients . . . . . . . . . . . . . . . . . . 37

3.3 Cheap Gradient Principle with Examples . . . . . . . . . . . . . 44

3.4 ApproximateErrorAnalysis..................... 50

3.5 SummaryandDiscussion ...................... 52

3.6 ExamplesandExercises ....................... 56

4 Memory Issues and Complexity Bounds 61

4.1 MemoryAllocationandOverwrites................. 61

4.2 RecordingPrevaluesontheTape.................. 64

4.3 SelectiveSavesandRestores..................... 70

4.4 ATemporalComplexityModel ................... 73

4.5 Complexity of Tangent Propagation . . . . . . . . . . . . . . . . 80

4.6 Complexity of Gradient Propagation . . . . . . . . . . . . . . . . 83

4.7 ExamplesandExercises ....................... 88

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