The exponential growth of technology forces all disciplines to adjust accordingly, so they can meet the demands of a very dynamic world that heavily depends upon it. Therefore, mathematics cannot be an exception. In fact, mathematics should be the first to adjust and in fact it is. In this volume, which is a continuation of the previous three under the same title, we present state-of-the-art iterative methods for solving equations related to concrete problems from diverse areas such as applied mathematics, mathematical: biology, chemistry, economics, physics and also engineering to mention a few. Most of these methods are new and a few are old but still very popular. One major problem with iterative methods is that the convergence domain is small in general. We have introduced a technique that finds a smaller set than before containing the iterates leading to tighter Lipschitz functions than before. This way and under the same computational effort, we derive: weaker sufficient convergence criteria (leading to a wider choice of initial points); tighter error bounds on the distances involved (i.e., fewer iterates are needed to obtain a desired predetermined accuracy), and a more precise information on the location of the solution. These advantages are considered major achievements in computational disciplines. The volume requires knowledge of linear algebra, numerical functional analysis and familiarity with contemporary computing programing. It can be used by researchers, practitioners, senior undergraduate and graduate students as a source material or as a required textbook in the classroom.

Table of Contents:
Preface; Acknowledgements; A Family of Cubically Convergent Methods; Characterization of Some Newton-Like Methods; Newtons Method Defined on Not Necessarily Bounded Domains; Local Convergence of a Class of Multi-Point SuperHalley Methods; ULMs-Like Method under Weak Convergence Conditions; Newton like Method Free of Bilinear Operators; On the Local Convergence of a Secant like Method; Extending the Applicability of Newtons Method; A Derivative Free Solver in Banach Space; Convergence for Two Optimal Eighth-Order Methods; Local Convergence of an Eighth-Order Method; An Efficient Class of Fourth-Order Jarratt-Type Methods; Unified Local Convergence for a Certain Family of Methods; Ball Convergence for EI15th-Order Variants of Hansenpatricks Family; Extended Local Convergence of Newton-Type Methods; Local Analysis of an Ostrowski-Like Method; Local Convergence for Multipoint Methods; Choosing Good Starting Points for the Convergence of Newtons Method; Local Convergence of a Tri-Parametric Eighth 18DER Method; Ball Convergence for Ostrowski-Like Method with Accelerated Eighth Order Convergence; A Traub-Steffensen-Like Composition for Banach Space Valued Operators; A Novel Traub-Steffensen Three Step Iterative Method Free Of Derivatives; Extended Semilocal Convergence of the NHSS Method under Generalized Lipschitz Condition; A Novel Traub-Steffensen Three Step Iterative Method Free Of Derivatives; Extended Local Convergence of an Efficient Sixth Order Method; Traub-Steffensen-Type Solvers for Nonlinear Equations; Unified Local Convergence for Third Order Methods; Improved Error Bounds for Newton-Type Solvers-I; Improved Error Bounds for Newton-Type Solvers-II; Comparing the Extended Yamamotos Error Bounds for Newtons Solver; Extended Newtons Method for Nonsmooth Operators; Extended and Unified Convergence Theory for Iterative Processes; Comparison of Two Sixth Order Solvers Using the First Derivative; Efficient Third Convergence Order Method for Solving Nonlinear Systems; On the Local Convergence of an Efficient Third Convergence Order Method; Extended Convergence for Newton-Like Solvers; Ball Convergence of Schröder-Like Methods for Multiple Roots; Ball Convergence of Eight Order Methods for Multiple Roots; Fourth Order Newton-Type Methods for Roots of Multiplicity; Gauss-Newton Solvers with Projections for Solving Least Squares Problems; Extended Ball Convergence Results for Newtons Solver under Hölder-Like Conditions; Ball Convergence for Traub-Steffensen-Chebyshev Solver; Extended Ball Convergence of the Gauss-Newton Solver for Injective-Overdetermined Systems of Equations; Extended Ball Convergence for Newton Conditional Gradient Solver; Extended Ball Convergence of the Gauss- Newton-Like Solver for Injective-Verdetermined Systems of Equations; Extended Ball Convergence for Inexact Newton-Like Conditional Gradient Solver; Extended Ball Convergence of the Gauss-Newton Solver; Extended Semi-Local Convergence of the Gauss-Newton Solver for Convex Composite Optimization; Extended Local Convergence Analysis of a Proximal Gauss-Newton Procedure; Extended Local Convergence of the Gauss-Newton Method; Solvers for Problems with Small Divisors; A Two Step Iterative Scheme with a Free Parameter; A Family of Unified High Convergence Order Methods; Three Step Seventh Order Method for Solving Equations; A Unified Family of Jarratt-Like Iterative Methods in Banach Space; Extended Local Convergence for Newton Simpsons 3/8th Solver; On the Harmonic Mean and Midpoint Newtons Solver; Comparing the Local Convergence of Three Newtonmean-Type Solvers; Comparing Two High Convergent Order Methods; Generalized Solvers for Equations; On the Harmonic Mean and Midpoint Newtons Solver-II; Extended Jarratt-Type Solver; Extended Chebyshev-Type Solver without Second Derivatives; Newton-Type Solvers Using Fifth Order Quadrature Formulas; Extending the Local Convergence of an Efficient Sixth Order Method; Increased and Extended Local Convergence for Some Iterative Methods; Extended Newton-Traub-Type Methods; Extended Newton-Traub Methods; Solvers of Convergence Three and Four with a Free Parameter; Extended Fourth Order Weighted Newton Solver; Extended Sixth Order Schemes with Parameters; Extended Newton-Jarratt Scheme; Chebyshev-Halley-Type Methods with Parameters; Ostrowski-Chun Like Schemes with Parametrs; Extended Seventh Order Method with Divided Differences; Derivative-Free Methods 1: Order Six; Derivative-Free Methods II: Order Seven; High-Order and Efficient Solvers in Banach Spaces; Derivative Free Method III: Order Seven; Derivative Free Methods IV: Order Six; Extended Semilocal Convergence of a Sixth Order Jarratt-Type Method in Banach Space; Extended Inexact Gauss-Newton-Like Schemes for Injective Overdetermined Equations; Index.