Contemporary Algorithms: Theory and Applications. Volume I

450 Pages·2022·32.31 MB·English

by  Christopher I. Argyros

Mathematics Research Developments

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Mathematics Research Developments

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Christopher I. Argyros, Samundra Regmi,
Ioannis K. Argyros and Santhosh George

Contemporary Algorithms

Theory and Applications

Volume I
Copyright © 2022 by Nova Science Publishers, Inc.
https://doi.org/10.52305/IHML8594

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Published by Nova Science Publishers, Inc. † New York
The first author dedicates this book to his beloved grandparents
Jolanda, Mihallaq, Anastasia and Konstantinos.
The second author dedicates this book to his mother Madhu Kumari
Regmi and Father Moti Ram Regmi.
The third author dedicates this book to his wonderful children
Christopher, Gus, Michael, and lovely wife Diana.
The fourth author dedicatesthis book to his lovely wife Rose.

Contents
GlossaryofSymbols xv
Preface xvii
1 BallConvergenceforHighOrderMethods 1
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. LocalConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 ContinuousAnalogsofNewton-TypeMethods 13
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Semi-localConvergenceI . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3. Semi-localConvergenceII . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 InitialPointsforNewton’sMethod 25
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2. Semi-localConvergenceResult . . . . . . . . . . . . . . . . . . . . . . . . 27
3. MainResult . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4. OntheConvergenceRegion . . . . . . . . . . . . . . . . . . . . . . . . . 30
5. APrioriErrorBoundsandQuadraticConvergenceofNewton’sMethod . . 31
6. LocalConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 SeventhOrder Methods 37
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2. LocalConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 38
3. NumericalExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 ThirdOrder Schemes 49
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2. BallConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
viii Contents
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 FifthandSixthOrderMethods 61
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2. BallConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7 SixthOrderMethods 73
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2. BallConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8 Extended Jarratt-TypeMethods 83
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2. ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9 MultipointPointSchemes 91
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2. LocalConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10 FourthOrderMethods 101
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11 InexactNewtonAlgorithm 113
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2. ConvergenceofNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3. NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12 Halley’sMethod 119
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2. ConvergenceofHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
13 Newton’sAlgorithmforSingularSystems 125
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2. ConvergenceofNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129