This book provides different avenues to study algorithms. It also brings new techniques and methodologies to problem solving in computational sciences, engineering, scientific computing and medicine (imaging, radiation therapy) to mention a few. A plethora of algorithms which are universally applicable are presented in a sound, analytical way. The chapters are written independently of each other, so they can be understood without reading earlier chapters. But some knowledge of analysis, linear algebra, and some computing experience is required. The organization and content of this book cater to senior undergraduate, graduate students, researchers, practitioners, professionals, and academicians in the aforementioned disciplines. It can also be used as a reference book and includes numerous references and open problems.
Author(s): Christopher I. Argyros
Series: Mathematics Research Developments
Publisher: Nova Science Publishers
Year: 2022
Language: English
Pages: 447
City: New York
Mathematics Research Developments
Contemporary AlgorithmsTheory and Applications
Contents
Glossary of Symbols
Preface
Chapter 1Ball Convergence for High OrderMethods
1. Introduction
2. Local Convergence Analysis
3. Numerical Examples
4. Conclusion
References
Chapter 2Continuous Analogs of Newton-TypeMethods
1. Introduction
2. Semi-local Convergence I
3. Semi-local Convergence II
4. Conclusion
References
Chapter 3Initial Points for Newton’s Method
1. Introduction
2. Semi-local Convergence Result
3. Main Result
4. On the Convergence Region
5. A Priori Error Bounds and Quadratic Convergence of Newton’sMethod
6. Local Convergence
7. Numerical Examples
8. Conclusion
References
Chapter 4Seventh Order Methods
1. Introduction
2. Local Convergence Analysis
3. Numerical Example
4. Conclusion
References
Chapter 5Third Order Schemes
1. Introduction
2. Ball Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 6Fifth and Sixth Order Methods
1. Introduction
2. Ball Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 7Sixth Order Methods
1. Introduction
2. Ball Convergence
3. Conclusion
References
Chapter 8Extended Jarratt-Type Methods
1. Introduction
2. Convergence Analysis
3. Conclusion
References
Chapter 9Multipoint Point Schemes
1. Introduction
2. Local Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 10Fourth Order Methods
1. Introduction
2. Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 11Inexact Newton Algorithm
1. Introduction
2. Convergence of NA
3. Numerical Examples
4. Conclusion
References
Chapter 12Halley’s Method
1. Introduction
2. Convergence of HA
3. Conclusion
References
Chapter 13Newton’s Algorithm for SingularSystems
1. Introduction
2. Convergence of NA
3. Conclusion
References
Chapter 14Gauss-Newton-Algorithm
1. Introduction
2. Semi-Local Convergence
3. Local Convergence
4. Conclusion
References
Chapter 15Newton’s Algorithm on RiemannianManifolds
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 16Gauss-Newton-Kurchatov Algorithmfor Least Squares Problems
1. Introduction
2. Convergence of GNKA
3. Conclusion
References
Chapter 17Uniqueness of the Solution ofEquations in Banach Space: I
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 18Uniqueness of the Solution ofEquations in Banach Space: II
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 19Convergence of Newton’s Algorithmfor Sections on RiemannianManifolds
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 20Newton Algorithm on Lie Groups: I
1. Introduction
2. Two versions of NA
2.1. The Differential of the Map F
3. Conclusion
References
Chapter 21Newton Algorithm on Lie Groups: II
1. Introduction
2. Convergence Criteria
3. Conclusion
References
Chapter 22Two-Step Newton Method under L−Average Conditions
1. Introduction
2. Semi-Local Convergence of TSNM
3. Conclusion
References
Chapter 23Unified Methods for SolvingEquations
1. Introduction
2. Ball Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 24Eighth Convergence OrderDerivative Free Method
1. Introduction
2. Ball Convergence
3. Conclusion
References
Chapter 25m−Step Methods
1. Introduction
2. Local Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 26Third Order Schemes for SolvingEquations
1. Introduction
2. Ball Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 27Deformed Newton Method forSolving Equations
1. Introduction
2. Local Convergence ofMethod (27.3)
3. Semi-local Convergence ofMethod (27.3)
4. Numerical Examples
5. Conclusion
References
Chapter 28On the Newton-KantorovichTheorem
1. Introduction
2. Convergence Analysis
3. Concluding Remarks and Applications
4. Conclusion
References
Chapter 29Kantorovich-Type Extensions forNewton Method
1. Introduction
2. Semi-Local Convergence for Newton-Like Methods
3. Numerical Examples
4. Conclusion
References
Chapter 30Improved Convergence for theKing-Werner Method
1. Introduction
2. Convergence Analysis of King-Werner-Type Methods (30.2)and (30.3)
3. Numerical Examples
4. Conclusion
References
Chapter 31Extending the Applicability ofKing-Werner-Type Methods
1. Introduction
2. Majorizing Sequences for King-Werner-TypeMethods (31.3)and (31.4)
3. Convergence Analysis of King-Werner-Type Methods (31.3)and (31.4)
4. Numerical Examples
5. Conclusion
References
Chapter 32Parametric Efficient Family ofIterative Methods
1. Introduction
2. Convergence Analysis of Method (32.2)
3. Numerical Examples
4. Conclusion
References
Chapter 33Fourth Order Derivative FreeScheme with Three Parameters
1. Introduction
2. Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 34Jarratt-Type Methods
1. Introduction
2. Convergence Analysis
3. Conclusion
References
Chapter 35Convergence Radius of an EfficientIterative Method with FrozenDerivatives
1. Introduction
2. Convergence for Method (35.2)
3. Numerical Examples
4. Conclusion
References
Chapter 36Efficient Sixth Convergence OrderMethods under GeneralizedContinuity
1. Introduction
2. Local Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 37Fifth Order Methods underGeneralized Conditions
1. Introduction
2. Local Analysis
3. Numerical Examples
4. Conclusion
References
Chapter 38Two Fourth Order Solvers forNonlinear Equations
1. Introduction
2. Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 39Kou’s Family of Schemes
1. Introduction
2. Local Analysis
3. Numerical Examples
4. Conclusion
References
Chapter 40Multi-Step Steffensen-Line Methods
1. Introduction
2. Semi-Local Convergence
3. Conclusion
References
Chapter 41Newton-Like Scheme for SolvingInclusion Problems
1. Introduction
2. Semi-Local Convergence
3. Conclusion
References
Chapter 42Extension of Newton-Secant-LikeMethod
1. Introduction
2. Majorizing Sequences
3. Convergence for Method (42.2)
4. Conclusion
References
Chapter 43Inexact Newton-Like Method forInclusion Problems
1. Introduction
2. Convergence of INLM
3. Conclusion
References
Chapter 44Semi-Smooth Newton-TypeAlgorithms for Solving VariationalInclusion Problems
1. Introduction
2. Preliminaries
3. Convergence
4. Conclusion
References
Chapter 45Extended Inexact Newton-LikeAlgorithm under KantorovichConvergence Criteria
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 46Kantorovich-Type Results UsingNewton’s Algorithms for GeneralizedEquations
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 47Developments of Newton’s Methodunder H¨older Conditions
1. Introduction
2. Convergence
3. Conclusion
References
Chapter 48Ham-Chun Fifth Convergence OrderSolver
1. Introduction
2. Ball Convergence
3. Numerical Examples
4. Conclusion
References
Chapter 49A Novel Method Free fromDerivatives of Convergence Order
1. Introduction
2. Convergence
3. Example
4. Conclusion
References
Chapter 50Newton-Kantorovich Scheme forSolving Generalized Equations
1. Introduction
2. Background
3. Convergence Analysis
4. Conclusion
References
About the Authors
Christopher I. Argyros
Samundra Regmi
Ioannis K. Argyros
Dr. Santhosh George
Index
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