Iterative Algorithms I

It is a well-known fact that iterative methods have been studied concerning problems where mathematicians cannot find a solution in a closed form. There exist methods with different behaviors when they are applied to different functions and methods with higher order of convergence, methods with great zones of convergence, methods which do not require the evaluation of any derivative, and optimal methods among others. It should come as no surprise, therefore, that researchers are developing new iterative methods frequently.

Once these iterative methods appear, several researchers study them in different terms: convergence conditions, real dynamics, complex dynamics, optimal order of convergence, etc. These phenomena motivated the authors to study the most used and classical ones, for example Newton’s method, Halley¡¯s method and/or its derivative-free alternatives.

Related to the convergence of iterative methods, the most well-known conditions are the ones created by Kantorovich, who developed a theory which has allowed many researchers to continue and experiment with these conditions. Many authors in recent years have studied modifications of these conditions related, for example, to centered conditions, omega-conditions and even convergence in Hilbert spaces.

In this monograph, the authors present their complete work done in the past decade in analyzing convergence and dynamics of iterative methods. It is the natural outgrowth of their related publications in these areas. Chapters are self-contained and can be read independently. Moreover, an extensive list of references is given in each chapter in order to allow the reader to use the previous ideas. For these reasons, the authors think that several advanced courses can be taught using this book.

The book’s results are expected to help find applications in many areas of applied mathematics, engineering, computer science and real problems. As such, this monograph is suitable to researchers, graduate students and seminar instructors in the above subjects. The authors believe it would also make an excellent addition to all science and engineering libraries.

Table of Contents

Chapter 1 Secant-Type Methods
Chapter 2 Efficient Steffensen-Type Algorithms For Solving Nonlinear Equations
Chapter 3 On The Semilocal Convergence Of Halley’Smethod Under A Center-Lipschitz Condition On The Second Fr´Echet Derivative
Chapter 4 An Improved Convergence Analysis Of Newton’S Method For Twice Fr´Echet Differentiable Operators
Chapter 5 Expanding The Applicability Of Newton’S Method Using Smale’S A-Theory
Chapter 6 Newton-Type Methods On Riemannianmanifolds Under Kantorovich-Type Conditions
Chapter 7 Improved Local Convergence Analysis Of Inexact Gauss-Newton Like Methods
Chapter 8 Expending The Applicability Of Lavrentiev Regularizationmethods For Ill-Posed Problems
Chapter 9 A Semilocal Convergence For A Uniparametric Family Of Efficient Secant-Like Methods
Chapter 10 On The Semilocal Convergence Of A Two-Step Newton-Like Projection Method For Ill-Posed Equations
Chapter 15 On The Semilocal Convergence Of Modified Newton-Tikhonov Regularizationmethod For Nonlinear Ill-Posed Problems
Chapter 16 Local Convergence Analysis Of Proximal Gauss-Newtonmethod For Penalized Nonlinear Least Squares Problems
Chapter 17 On The Convergence Of A Damped Newtonmethod Withmodified Right-Hand Side Vector
Chapter 18 Local Convergence Of Inexact Newton-Like Method Under Weak Lipschitz Conditions
Chapter 19 Expanding The Applicability Of Secant Method With Applications
Chapter 20 Expanding The Convergence Domain For Chun-Stanica-Neta Family Of Third Order Methods In Banach Spaces
Chapter 21 Local Convergence Of Modified Halley-Like Methods With Less Computation Of Inversion
Chapter 22 Local Convergence For An Improved Jarratt-Type Method In Banach Space
Chapter 23 Enlarging The Convergence Domain Of Secant-Like Methods For Equations