The Theory and Applications of Iteration Methods

448 Pages·2021·9.33 MB·english

by  Argyros

The Theory and
Applications of
Iteration Methods

The Theory and
Applications of
Iteration Methods
Second Edition
Ioannis K. Argyros
Second edition published 2022
by CRC Press
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and by CRC Press
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© 2022 Ioannis K. Argyros
First edition published by CRC Press 1993
CRC Press is an imprint of Taylor & Francis Group, LLC
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ISBN: 978-0-367-65101-5 (hbk)
ISBN: 978-0-367-65330-9 (pbk)
ISBN: 978-1-003-12891-5 (ebk)
DOI: 10.1201/9781003128915
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Dedication
Theauthordedicatesthisbooktohisbelovedparents,Anastasiaand
Konstantinos,wifeDiana,andchildrenChristopher,Gus,Michael,
andStacey

Contents
Preface.....................................................................................................................xiii
Acknowledgement...................................................................................................xv
Author....................................................................................................................xvii
Chapter1 TheConvergenceofAlgorithmicModels.......................................1
1.1 AlgorithmicModels................................................................1
1.2 ConvergenceCriteriaforAlgorithmicModels.......................3
1.3 Applications............................................................................9
1.4 Exercises...............................................................................34
References......................................................................................36
Chapter2 TheConvergenceofIterationSequences......................................39
2.1 TheGeneralConvergenceTheorem.....................................39
2.2 Convergenceofl-StepMethods...........................................41
2.3 ConvergenceofSingle-StepMethods..................................44
2.4 ConvergenceofSingle-StepMethodswith
DifferentiableIterationFunctions.........................................48
2.5 Applications..........................................................................52
2.6 Exercises.............................................................................100
References....................................................................................103
Chapter3 MonotoneConvergence...............................................................105
3.1 GeneralResults...................................................................105
3.2 AGeneralModelinLinearSpaces.....................................106
3.3 Applications........................................................................110
3.4 Exercises.............................................................................129
References....................................................................................132
Chapter4 Applicationsin:...........................................................................133
4.1 NeuralNetworks.................................................................133
4.2 ReliabilityEngineering.......................................................134
4.3 EconomicModels...............................................................142
4.4 Exercises.............................................................................146
References....................................................................................147
vii
viii Contents
Chapter5 ANovelSchemeFreefromDerivatives......................................149
5.1 Convergence.......................................................................150
5.2 Applications........................................................................154
5.3 Exercises.............................................................................157
References....................................................................................159
Chapter6 EfficientSixthConvergenceOrderMethod................................161
6.1 Localconvergence..............................................................162
6.2 Applications........................................................................168
6.3 Exercises.............................................................................172
References....................................................................................173
Chapter7 High-OrderIterativeMethods.....................................................175
7.1 LocalconvergenceAnalysisI.............................................175
7.2 LocalconvergenceanalysisII.............................................177
7.3 Applications........................................................................183
7.4 Applicationswithlargesystems.........................................187
7.5 Exercises.............................................................................190
References....................................................................................192
Chapter8 UnifiedLocalConvergenceofk−StepSolvers...........................195
8.1 Localconvergence..............................................................197
8.2 Applications........................................................................200
8.3 Exercises.............................................................................204
References....................................................................................206
Chapter9 BallComparisonBetweenThreeSixth-OrderMethods.............207
9.1 Localconvergence..............................................................208
9.2 Applications........................................................................216
9.3 Exercises.............................................................................220
References....................................................................................221
Chapter10 ConstrainedGeneralizedEquations ...........................................223
10.1 Notationandauxiliaryresults.............................................225
10.2 ThelocalNewtonmethod...................................................226
10.3 Applications........................................................................235
10.4 Exercises.............................................................................236
References....................................................................................238
Chapter11 Inexact Gauss–Newton Method for Solving Least Squares
Problems......................................................................................241
11.1 Majorizingsequences.........................................................243
Contents ix
11.2 ConvergenceAnalysisforAlgorithmTGNU(I)................246
11.3 ConvergenceAnalysisforAlgorithmTGNU(II)...............252
11.4 Applications........................................................................254
11.5 Exercises.............................................................................256
References....................................................................................257
Chapter12 The Kantorovich’s Theorem on Newton’s Method for Solving
GeneralizedEquation..................................................................259
12.1 Preliminaries.......................................................................261
12.2 Kantorovich’stheoremforNewton’smethod....................262
12.3 Applications........................................................................268
12.4 Exercises.............................................................................270
References....................................................................................272
Chapter13 AnInverseFreeBroyden’sMethod.............................................273
13.1 Preliminaries:regularlycontinuousdd...............................274
13.2 Semi-localconvergenceanalysisofBM.............................276
13.3 Applications........................................................................283
13.4 Exercises.............................................................................286
References....................................................................................288
Chapter14 ComplexityofaHomotopyMethodforLocatinganApproximate
Zero..............................................................................................289
14.1 Convergenceanalysis..........................................................290
14.2 SpecialCases......................................................................294
14.3 Applications........................................................................295
14.4 Exercises.............................................................................297
References....................................................................................299
Chapter15 InexactMethodsforFindingZeroswithMultiplicity.................301
15.1 Auxiliaryresults..................................................................303
15.2 Ballconvergence.................................................................305
15.3 Applications........................................................................309
15.4 Exercises.............................................................................310
References....................................................................................311
Chapter16 Multi-StepHighConvergenceOrderMethods............................313
16.1 Localconvergenceanalysis................................................314
16.2 Applications........................................................................317
16.3 Exercises.............................................................................320
References....................................................................................322