## Table of Contents

Preface

Chapter 1. Ball Convergence for High Order Methods

1. Introduction

2. Local Convergence Analysis

3. Numerical Examples

4. Conclusion

Chapter 2. Continuous Analogs of Newton-Type Methods

1. Introduction

2. Semilocal convergence I

3. Semilocal Convergence II

4. Conclusion

Chapter 3. Initial Points for Newton’s Method

1. Introduction

2. Semilocal Convergence Result

3. Main Result

4. On the Convergence Region

5. A Priori Error Bounds and Quadratic Convergence of Newton’s Method

6. Local Convergence

7. Numerical Examples

8. Conclusion

Chapter 4. Seventh Order Methods

1. Introduction

2. Local Convergence Analysis

3. Numerical Example

4. Conclusion

Chapter 5. Third Order Schemes

1. Introduction

2. Ball Convergence

3. Numerical Examples

4. Conclusion

Chapter 6. Fifth and Sixth Order Methods

1. Introduction

2. Ball Convergence

3. Numerical Examples

4. Conclusion

Chapter 7. Sixth Order Methods

1. Introduction

2. Ball Convergence

3. Conclusion

Chapter 8. Extended Jarratt-Type Methods

1. Introduction

2. Convergence Analysis

3. Conclusion

Chapter 9. Multipoint Point Schemes

1. Introduction

2. Local Convergence

3. Numerical Examples

4. Conclusion

Chapter 10. Fourth Order Methods

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

Chapter 11. Inexact Newton Algorithm

1. Introduction

2. Convergence of NA

3. Numerical Examples

4. Conclusion

Chapter 12. Halley’s Method

1. Introduction

2. Convergence of HA

3. Conclusion

Chapter 13. Newton’s Algorithm for Singular Systems

1. Introduction

2. Convergence of NA

3. Conclusion

Chapter 14. Gauss-Newton-Algorithm

1. Introduction

2. Semi-Local Convergence

3. Local Convergence

4. Conclusion

Chapter 15. Newton’s Algorithm on Riemannian Manifolds

1. Introduction

2. Convergence

3. Conclusion

Chapter 16. Gauss-Newton-Kurchatov Algorithm for Least Squares Problems

1. Introduction

2. Convergence of GNKA

3. Conclusion

Chapter 17. Uniqueness of the Solution of Equations in Banach Space: I

1. Introduction

2. Convergence

3. Conclusion

Chapter 18. Uniqueness of the Solution of Equations in Banach Space: II

1. Introduction

2. Convergence

3. Conclusion

Chapter 19. Convergence of Newton’s Algorithm for Sections on Riemannian Manifolds

1. Introduction

2. Convergence

3. Conclusion

Chapter 20. Newton Algorithm on Lie Groups: I

1. Introduction

2. Two versions of NA

2.1. The Differential of the Map F

3. Conclusion

Chapter 21. Newton Algorithm on Lie Groups: II

1. Introduction

2. Convergence Criteria

3. Conclusion

Chapter 22. Two-Step Newton Method under L− Average Conditions

1. Introduction

2. Semi-Local Convergence of TSNM

3. Conclusion

Chapter 23. Unified Methods for Solving Equations

1. Introduction

2. Ball Convergence

3. Numerical Examples

4. Conclusion

Chapter 24. Eighth Convergence Order Derivative Free Method

1. Introduction

2. Ball Convergence

3. Conclusion

Chapter 25. m−Step Methods

1. Introduction

2. Local Convergence

3. Numerical Examples

4. Conclusion

Chapter 26. Third Order Schemes for Solving Equations

1. Introduction

2. Ball Convergence

3. Numerical Examples

4. Conclusion

Chapter 27. Deformed Newton Method for Solving Equations

1. Introduction

2. Local Convergence of Method (27.3)

3. Semilocal Convergence of Method (27.3)

4. Numerical Examples

5. Conclusion

Chapter 28. On the Newton–Kantorovich Theorem

1. Introduction

2. Convergence Analysis

3. Concluding Remarks and Applications

4. Conclusion

Chapter 29. Kantorovich-Type Extensions for Newton Method

1. Introduction

2. Semi-Local Convergence for Newton-Like Methods

3. Numerical Examples

4. Conclusion

Chapter 30. Improved Convergence for the King-Wermer Method

1. Introduction

2. Convergence Analysis of King-Werner-Type Methods (30.2) and (30.3)

3. Numerical Examples

4. Conclusion

Chapter 31. Extending the Applicability of King-Werner-Type Methods

1. Introduction

2. Majorizing Sequences for King-Werner-Type Methods (31.3) and (31.4)

3. Convergence Analysis of King-Werner-Type Methods (31.3) and (31.4)

4. Numerical Examples

5. Conclusion

Chapter 32. Parametric Efficient Family of Iterative Methods

1. Introduction

2. Convergence Analysis of Method (32.2)

3. Numerical Examples

4. Conclusion

Chapter 33. Fourth Order Derivative Free Scheme with Three Parameters

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

Chapter 34. Jarratt-Type Methods

1. Introduction

2. Convergence Analysis

3. Conclusion

Chapter 35. Convergence Radius of an Efficient Iterative Method with Frozen Derivatives

1. Introduction

2. Convergence for Method (35.2)

3. Numerical Examples

4. Conclusion

Chapter 36. Efficient Sixth Convergence Order Methods under Generalized Continuity

1. Introduction

2. Local Convergence

3. Numerical Examples

4. Conclusion

Chapter 37. Fifth Order Methods under Generalized Conditions

1. Introduction

2. Local Analysis

3. Numerical Examples

4. Conclusion

Chapter 38. Two Fourth Order Solvers for Nonlinear Equations

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

Chapter 39. Kou’s Family of Schemes

1. Introduction

2. Local Analysis

3. Numerical Examples

4. Conclusion

Chapter 40. Multi-Step Steffensen-Line Methods

1. Introduction

2. Semi-Local Convergence

3. Conclusion

Chapter 41. Newton-Like Scheme for Solving Inclusion Problems

1. Introduction

2. Semi-Local Convergence

3. Conclusion

Chapter 42. Extension of Newton-Secant-Like Method

1. Introduction

2. Majorizing Sequences

3. Convergence for Method (42.2)

4. Conclusion

Chapter 43. Inexact Newton-Like Method for Inclusion Problems

1. Introduction

2. Convergence of INLM

3. Conclusion

Chapter 44. Semi-Smooth Newton-Type Algorithms for Solving Variational Inclusion Problems

1. Introduction

2. Preliminaries

3. Convergence

4. Conclusion

Chapter 45. Extended Inexact Newton-Like Algorithm under Kantorovich Convergence

Criteria

1. Introduction

2. Convergence

3. Conclusion

Chapter 46. Kantorovich-Type Results Using Newton’s Algorithms for Generalized Equations

1. Introduction

2. Convergence

3. Conclusion

Chapter 47. Developments of Newton’s Method under Hӧlder Conditions

1. Introduction

2. Convergence

3. Conclusion

Chapter 48. Ham-Chun Fifth Convergence Order Solver

1. Introduction

2. Ball Convergence

3. Numerical Examples

4. Conclusion

Chapter 49. A Novel Method Free from Derivatives of Convergence Order

1. Introduction

2. Convergence

3. Example

4. Conclusion

Chapter 50. Newton-Kantorovich Scheme for Solving Generalized Equations

1. Introduction

2. Background

3. Convergence Analysis

4. Conclusion

Glossary of Symbols

Index