Research in Mathematics at Cameron University


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Series: Mathematics Research Developments

BISAC: MAT003000

Numerous problems from diverse disciplines can be converted using mathematical modeling to an equation defined on suitable abstract spaces usually involving the n-dimensional Euclidean space or Hilbert space or Banach Space or even more general spaces. The solution of these equations is sought in closed form. But this is possible only in special cases. That is why researchers and practitioners use iterative algorithms, which seem to be the only alternative.

Due to the explosion of technology, faster and faster computers become available. This development simply means that new optimized algorithms should be developed to take advantage of these improvements. That is exactly where we come in with our book containing such algorithms with applications in problems from numerical analysis and economics but also from other areas such as biology, chemistry, physics, parallel computing, and engineering. The book is an outgrowth of scientific research conducted over two years.

This book can be used by senior undergraduate students, graduate students, researchers, and practitioners in the aforementioned areas in the classroom or as reference material. Readers should know the fundamentals of numerical-functional analysis, economic theory, and Newtonian physics. Some knowledge of computers and contemporary programming shall be very helpful to readers.

Table of Contents


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Chapter 1. The History of Newton’s Method and Extended Classical Results

Chapter 2. Extended Global Convergence of Iterative Methods

Chapter 3. Extended Gauss-Newton-Approximate Projection Methods of Constrained Nonlinear Least Squares Problems

Chapter 4. Convergence Analysis of Inexact Gauss-Newton Like for Solving Systems

Chapter 5. Local Convergence of the Gauss-Newton Scheme on Hilbert Spaces Under a Restricted Convergence Domain

Chapter 6. Ball Convergence for Inexact Newton-type Conditional Gradient Solver for Constrained Systems

Chapter 7. Newton-like Methods with Recursive Approximate Inverses

Chapter 8. Updated Mesh Independence Principle

Chapter 9. Ball Convergence for Ten Solvers Under the Same Set of Conditions

Chapter 10. Extended Newton’s Solver for Generalized Equations Using a Restricted Convergence Domain

Chapter 11. Extended Newton’s Method for Solving Generalized Equations: Kantorovich’s Approach

Chapter 12. Extended Robust Convergence Analysis of Newton’s Method for Cone Inclusion Problems in Banach Spaces

Chapter 13. Extended and Robust Kantorovich’s Theorem on the Inexact Newton’s Method with Relative Residual Error Tolerance

Chapter 14. Extended Local Convergence for Iterative Schemes Using the Gauge Function Theory

Chapter 15. Improved Local Convergence of Inexact Newton Methods under Average Lipschitz-type Conditions

Chapter 16. Semi-Local Convergence of Newton’s Method Using the Gauge Function Theory: An Extension

Chapter 17. Extending the Semi-Local Convergence of Newton’s Method Using the Gauge Theory

Chapter 18. Global Convergence for Chebyshev’s Method

Chapter 19. Extended Convergence of Efficient King-Werner-Type Methods of Order 1+√2

Chapter 20. Extended Convergence for Two Chebyshev-Like Methods

Chapter 21. Extended Convergence Theory for Newton-Like Methods of Bounded Deterioration

Chapter 22. Extending the Kantorovich Theorem for Solving Equations Using Telescopic Series

Chapter 23. Extended ω-Convergence Conditions for the Newton-Kantorovich Method

Chapter 24. Extended Semilocal Convergence Analysis for Directional Newton Method

Chapter 25. Extended Convergence of Damped Newton’s Method

Chapter 26. Extended Convergence Analysis of a One-Step Intermediate Newton Iterative Scheme for Nonlinear Equations

Chapter 27. Enlarging the Convergence Domain of Secant-Type Methods

Chapter 28. Two-Step Newton-Type Method for Solving Equations

Chapter 29. Two-Step Secant-Type Method for Solving Equations

Chapter 30. Unified Convergence for General Iterative Schemes

Chapter 31. Extending the Applicability of Gauss-Newton Method for Convex Composite Optimization

Chapter 32. Local Convergence Comparison Between Newton’s and the Secant Method: Part-I

Chapter 33. Convergence Comparison Between Newton’s and Secant Method: Part-II

Chapter 34. Extended Convergence Domains for a Certain Class of Fredholm Hammerstein Equations

Chapter 35. Extended Convergence of the Gauss-Newton-Kurchatov Method

Chapter 36. Extended Semi-Local Convergence of Newton’s Method under Conditions on the Second Derivative

Chapter 37. Extended Convergence for the Secant Method Under Mysovskii-like Conditions

Glossary of Symbols

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