## Table of Contents

**Preface**

**Chapter 1. Correcting and Extending the Applicability of Two Fast Algorithms**

1. Introduction

2. Semi-Local Convergence

3. Conclusion

**Chapter 2. On the Solution of Generalized Equations in Hilbert Space**

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

**Chapter 3. Gauss-Newton Algorithm for Convex Composite Optimization**

1. Introduction

2. Convergence of GNA

3. Conclusion

**Chapter 4. Local Convergence of Newtonâ€™s Algorithm of Riemannian Manifolds**

1. Introduction

2. Convergence

3. Conclusion

**Chapter 5. Newtonâ€™s Algorithm on Riemannian Manifolds with Values in a Cone**

1. Introduction

2. Semi-Local Convergence

3. Conclusion

**Chapter 6. Gauss-Newton Algorithm on Riemannian Manifolds under L-Average Lipschitz Conditions**

1. Introduction

2. Semi-Local Convergence

3. Conclusion

**Chapter 7. Newtonâ€™s Method with Applications to Interior Point Algorithms of Mathematical Programming**

1. Introduction

2. An Improved Newtonâ€“Kantorovich Theorem

3. Applications to Interior-Point Algorithm

4. Conclusion

**Chapter 8. Newtonâ€™s Method for Solving Nonlinear Equations Using Generalized Inverses: ****Part I Outer Inverses**

1. Introduction

2. Convergence

3. Conclusion

**Chapter 9. Newtonâ€™s Method for Solving Nonlinear Equations Using Generalized Inverses: ****Part II Matrices**

1. Introduction

2. Local Convergence

3. Conclusion

**Chapter 10. Newtonâ€™s Method for Solving Nonlinear Equations Using Generalized Inverses: ****Part III Ball of Convergence for Nonisolated Solutions**

1. Introduction

2. Convergence of Method (10.2)

3. Conclusion

**Chapter 11. On an Efficient Steffensen-Like Method to Solve Equations**

1. Introduction

2. Analysis

3. Conclusion

**Chapter 12. Convergence Analysis for King-Werner-Like Methods**

1. Introduction

2. Semi-Local Convergence of Method (12.2)

3. Local Convergence of Method (12.2)

4. Numerical Examples

5. Conclusion

**Chapter 13. Multi-Point Family of High Order Methods**

1. Introduction

2. Local Convergence

3. Numerical Examples

4. Conclusion

**Chapter 14. Ball Convergence Theorems for Some Third-Order Iterative Methods**

1. Introduction

2. Local Convergence for Method (14.2)

3. Local Convergence of Method (14.3)

4. Numerical Examples

5. Conclusion

**Chapter 15. Convergence Analysis of Frozen Steffensen-Type Methods under Generalized Conditions**

1. Introduction

2. Semi-Local Convergence Analysis

3. Conclusion

**Chapter 16. Convergence of Two-Step Iterative Methods for Solving Equations with Applications**

1. Introduction

2. Semi-Local Convergence Analysis

3. Local Convergence Analysis

4. Numerical Examples

5. Conclusion

**Chapter 17. Three Step Jarratt-Type Methods under Generalized Conditions**

1. Introduction

2. Local Analysis

3. Numerical Examples

4. Conclusion

**Chapter 18. Extended Derivative Free Algorithms of Order Seven**

1. Introduction

2. Local Analysis

3. Numerical Examples

4. Conclusion

**Chapter 19. Convergence of Fifth OrderMethods for Equations under the Same Conditions**

1. Introduction

2. Local Convergence

3. Numerical Examples

4. Conclusion

**Chapter 20. A Novel Eighth Convergence Order Scheme with Derivatives and Divided Difference**

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

**Chapter 21. Homocentric Ball for Newtonâ€™s and the Secant Method**

1. Introduction

2. Local Convergence

3. Semi-Local Convergence

4. Numerical Examples

5. Conclusion

**Chapter 22. A Tenth Convergence Order Method under Generalized Conditions**

1. Introduction

2. Convergence

3. Numerical Examples

4. Conclusion

**Chapter 23. Convergence of Chebyshevâ€™s Method**

1. Introduction

2. Semi-Local Convergence Analysis

3. Local Convergence Analysis

4. Numerical Experiments

5. Conclusion

**Chapter 24. Gauss-Newton Algorithms for Optimization Problems**

1. Introduction

2. Convergence

3. Conclusion

**Chapter 25. Two-Step Methods under General Continuity Conditions**

1. Introduction

2. Majorizing Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 26. A Noor-Waseem Third Order Method to Solve Equations**

1. Introduction

2. Majorizing Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 27. Generalized Homeier Method**

1. Introduction

2. Local Convergence

3. Numerical Experiments

4. Conclusion

**Chapter 28. A Xiao-Yin Third Order Method for Solving Equations**

1. Introduction

2. Majorizing Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 29. Fifth Order Scheme**

1. Introduction

2. Scalar Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 30. Werner Method**

1. Introduction

2. Majorizing Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 31. Yadav-Singh Method of Order Five**

1. Introduction

2. Semi-Local Convergence

3. Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 32. Convergence of a P+1 Step Method of Order 2P+1 with Frozen Derivatives**

1. Introduction

2. Local Convergence

3. Numerical Experiments

4. Conclusion

**Chapter 33. Efficient Fifth Order Scheme**

1. Introduction

2. Ball Convergence

3. Numerical Experiments

4. Conclusion

**Chapter 34. Sharma-Gupta Fifth Order Method**

1. Introduction

2. Convergence

3. Numerical Experiments

4. Conclusion

**Chapter 35. Seventh Order Method for Equations**

1. Introduction

2. Convergence

3. Numerical Experiments

4. Conclusion

**Chapter 36. Newton-Like Method**

1. Introduction

2. Mathematical Background

3. Majorizing Sequences

4. Semi-Local Convergence

5. Numerical Experiments

6. Conclusion

**Chapter 37. King-Type Methods**

1. Introduction

2. Majorizing Sequences

3. Semi-Local Convergence

4. Numerical Experiments

5. Conclusion

**Chapter 38. Single Step Third Order Method**

1. Introduction

2. Semi-Local Analysis

3. Local Convergence

4. Numerical Example

5. Conclusion

**Chapter 39. Newton-Type Method for Non-Differentiable Inclusion Problems**

1. Introduction

2. Majorizing Sequences

3. Analysis

4. Conclusion

**Chapter 40. Extended Kantorovich-Type Theory for Solving Nonlinear Equations Iteratively: ****Part I Newtonâ€™s Method**

1. Introduction

2. Convergence of NM

3. Conclusion

**Chapter 41. Extended Kantorovich-Type Theory for Solving Nonlinear Equations Iteratively: ****Part II Newtonâ€™s Method**

1. Introduction

2. Convergence of NLM

3. Conclusion

**Chapter 42. Updated and Extended Convergence Analysis for Secant-Type Iterations**

1. Introduction

2. Convergence of STI

3. Conclusion

**Chapter 43. Updated Halleyâ€™s and Chebyshevâ€™s Iterations**

1. Introduction

2. Semi-Local Convergence Analysis for HI and CI

3. Conclusion

**Chapter 44. Updated Iteration Theory for Non Differentiable Equations**

1. Introduction

2. Convergence

3. Conclusion

**Chapter 45. On Generalized Halley-Like Methods for Solving Nonlinear Equations**

1. Introduction

2. Majorizing Convergence Analysis

3. Semi-Local Analysis

4. Special Cases

5. Conclusion

**Chapter 46. Extended Semi-Local Convergence of Steffensen-Like Methods for Solving ****Nonlinear Equations**

1. Introduction

2. Majorizing Real Sequences

3. Convergence

4. Conclusion

**Glossary of Symbols**

**Index**