Iterative methods and their dynamics with applications : a contemporary study

Description

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. Content: Cover Half title Title Copyright Dedication Preface Contents List of Figures List of Tables Symbol Description Chapter 1 Halley's method 1.1 Introduction 1.2 Semilocal convergence of Halley's method 1.3 Numerical examples 1.4 Basins of attraction 1.4.1 2 roots 1.4.2 3 roots 1.4.3 4 roots References Chapter 2 Newton's method for k-Fréchet differentiable operators 2.1 Introduction 2.2 Semilocal convergence analysis for Newton's method 2.2.1 Uniqueness of solution 2.2.2 Special choices for function g 2.2.2.1 Choice 1 2.2.2.2 Choice 2 2.3 Numerical examples Chapter 5 Local convergence and basins of attraction of a two-step Newton-like method for equations with solutions of multiplicity greater than one5.1 Introduction 5.2 Local convergence 5.3 Basins of attraction 5.3.1 Basins of F(x) = (x -- 1)2(x + 1) 5.3.2 Basins of F(x) = (x -- 1)3(x + 1) 5.3.3 Basins of F(x) = (x -- 1)4(x + 1) 5.4 Numerical examples References Chapter 6 Extending the Kantorovich theory for solving equations 6.1 Introduction 6.2 First convergence improvement 6.3 Second convergence improvement References Chapter 7 Robust convergence for inexact Newton method 7.1 Introduction7.2 Standard results on convex functions 7.3 Semilocal convergence 7.4 Special cases and applications References Chapter 8 Inexact Gauss-Newton-like method for least square problems 8.1 Introduction 8.2 Auxiliary results 8.3 Local convergence analysis 8.4 Applications and examples References Chapter 9 Lavrentiev Regularization methods for Ill-posed equations 9.1 Introduction 9.2 Basic assumptions and some preliminary results 9.3 Error estimates 9.3.1 Apriori parameter choice 9.3.2 Aposteriori parameter choice 9.4 Numerical examples References Chapter 10 King-Werner-type methods of order 1 + √210.1 Introduction 10.2 Majorizing sequences for King-Werner-type methods 10.3 Convergence analysis of King-Werner-type methods 10.4 Numerical examples References Chapter 11 Generalized equations and Newton's method 11.1 Introduction 11.2 Preliminaries 11.3 Semilocal convergence References Chapter 12 Newton's method for generalized equations using restricted domains 12.1 Introduction 12.2 Preliminaries 12.3 Local convergence 12.4 Special cases References Chapter 13 Secant-like methods 13.1 Introduction