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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-Frechet 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 Muller'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