1st Edition
Iterative Methods and Their Dynamics with Applications A Contemporary Study
Iterative processes are the tools used to generate sequences approximating solutions of equations describing real life problems. Intended for researchers in computational sciences and as a reference book for advanced computational method in nonlinear analysis, this book is a collection of the recent results on the convergence analysis of numerical algorithms in both finite-dimensional and infinite-dimensional spaces and presents several applications and connections with fixed point theory. It contains an abundant and updated bibliography and provides comparisons between various investigations made in recent years in the field of computational nonlinear analysis.
The book also provides recent advancements in the study of iterative procedures and can be used as a source to obtain the proper method to use in order to solve a problem. The book assumes a basic background in Mathematical Statistics, Linear Algebra and Numerical Analysis and may be used as a self-study reference or as a supplementary text for an advanced course in Biosciences or Applied Sciences. Moreover, the newest techniques used to study the dynamics of iterative methods are described and used in the book and they are compared with the classical ones.
Halley’s method
Introduction
Semilocal convergence of Halley’s method
Numerical examples
Basins of attraction
References
Newton’s method for k-Fréchet differentiable operators
Introduction
Semilocal convergence analysis for Newton’s method
Numerical examples
References
Nonlinear Ill-posed Equations
Introduction
Convergence Analysis
Error Bounds
Implementation of adaptive choice rule
Numerical Example
References
Sixth-order iterative methods
Introduction
Scheme for constructing sixth-order iterative methods
Sixth-order iterative methods contained in family USS
Numerical Work
Dynamics for method SG
References
Local convergence and basins of attraction of a two-step Newton-like method for equations with solutions of multiplicity greater than one
Introduction
Local convergence
Basins of attraction
Numerical examples
References
Extending the Kantorovich theory for solving equations
Introduction
First convergence improvement
Second convergence improvement
References
Robust convergence for inexact Newton method
Introduction
Standard results on convex functions
Semilocal converngence
Special cases and applications
References
Inexact Gauss-Newton-like method for least square problems
Introduction
Auxiliary Results
Local convergence analysis
Applications and Examples
References
Lavrentiev Regularization Methods for Ill-posed Equations
Introduction
Basic assumptions and some preliminary results
Error Estimates
Numerical Examples
References
King-Werner-type methods of order 1+sqrt(2)
Introduction
Majorizing sequences for King-Werner-type methods (1.3) and (1.4)
Convergence analysis of King-Werner-type methods
Numerical examples
References
Generalized equations and Newton’s method
Introduction
Preliminaries
Semilocal Convergence
References
Newton’s method for generalized equations using restricted domains
Introduction
Preliminaries
Local convergence
Special Cases
References
Secant-like methods
Introduction
Semilocal Convergence analysis of the secant method I
Semilocal Convergence analysis of the secant method II
Local Convergence analysis of the secant method I
Local Convergence analysis of the secant method II
Numerical examples
References
King-Werner-like methods free of derivatives
Introduction
Semilocal convergence
Local convergence
Numerical examples
References
Müller’s method
Convergence ball for method (1.2)
Numerical examples
References
Generalized Newton Method with applications
Introduction
Preliminaries
Semilocal Convergence
References
Newton-secant methods with values in a cone
Introduction
Convergence of the Newton-secant method
References
Gauss-Newton method with applications to convex optimization
Introduction
Gauss-Newton Algorithm and Quasi-Regularity condition
Semilocal convergence for GNA
Specializations and Numerical Examples
References
Directional Newton methods and restricted domains
Introduction
Semilocal convergence analysis
References
Gauss-Newton method for convex optimization
Introduction
Gauss-Newton Algorithm and Quasi-Regularity condition
Semi-local convergence
Numerical Examples
References
Ball Convergence for eighth order method
Introduction
Local convergence analysis
Numerical Examples
References
Expanding Kantorovich’s theorem for solving generalized equations
Introduction
Preliminaries
Semilocal Convergence
References
Biography
Ioannis Konstantinos Argyros, Angel Alberto Magreñán
