Extended convergence analysis of Newton-Potra solver for equations

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator

Convergence analysis of a two-step method for the nonlinear least squares problem with decomposition of operator

In this article, we propose a two-step method for the nonlinear least squares problem with the decomposition of the operator. We investigate the convergence of this method by applying the classical Lipschitz condition for the first- and second-order derivatives of the differentiable part and for the first-order differences of the non-differentiable part of the decomposition. The convergence order as well as the convergence radius of the method are studied and the uniqueness ball of the solution of the nonlinear least squares problem is examined. Finally, we carry out numerical experiments on a set of test problems

Improving Convergence Analysis of the Newton–Kurchatov Method under Weak Conditions

The technique of using the restricted convergence region is applied to study a semilocal convergence of the Newton–Kurchatov method. The analysis is provided under weak conditions for the derivatives and the first order divided differences. Consequently, weaker sufficient convergence criteria and more accurate error estimates are retrieved. A special case of weak conditions is also considered

Two-Step Solver for Nonlinear Equations

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence 1 + 2 . We study the local and a semilocal convergence using weaker conditions in order to extend the applicability of the solver. Finally, we present the numerical example that confirms the theoretical results

On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space

A plethora of methods are used for solving equations in the finite-dimensional Euclidean space. Higher-order derivatives, on the other hand, are utilized in the calculation of the local convergence order. However, these derivatives are not on the methods. Moreover, no bounds on the error and uniqueness information for the solution are given either. Thus, the advantages of these methods are restricted in their application to equations with operators that are sufficiently many times differentiable. These limitations motivate us to write this paper. In particular, we present the more interesting semi-local convergence analysis not given previously for two sixth-order methods that are run under the same set of conditions. The technique is based on the first derivative that only appears in the methods. This way, these methods are more applicable for addressing equations and in the more general setting of Banach space-valued operators. Hence, the applicability is extended for these methods. This is the novelty of the paper. The same technique can be used in other methods. Finally, examples are used to test the convergence of the methods

Perturbed Newton Methods for Solving Nonlinear Equations with Applications

Symmetries play an important role in the study of a plethora of physical phenomena, including the study of microworlds. These phenomena reduce to solving nonlinear equations in abstract spaces. Therefore, it is important to design iterative methods for approximating the solutions, since closed forms of them can be found only in special cases. Several iterative methods were developed whose convergence was established under very general conditions. Numerous applications are also provided to solve systems of nonlinear equations and differential equations appearing in the aforementioned areas. The ball convergence analysis was developed for the King-like and Jarratt-like families of methods to solve equations under the same set of conditions. Earlier studies have used conditions up to the fifth derivative, but they failed to show the fourth convergence order. Moreover, no error distances or results on the uniqueness of the solution were given either. However, we provide such results involving the derivative only appearing on these methods. Hence, we have expanded the usage of these methods. In the case of the Jarratt-like family of methods, our results also hold for Banach space-valued equations. Moreover, we compare the convergence ball and the dynamical features both theoretically and in numerical experiments

On the Convergence of Two-Step Kurchatov-Type Methods under Generalized Continuity Conditions for Solving Nonlinear Equations

The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one

Newton-Type Methods for Solving Equations in Banach Spaces: A Unified Approach

A plethora of quantum physics problems are related to symmetry principles. Moreover, by using symmetry theory and mathematical modeling, these problems reduce to solving iteratively finite differences and systems of nonlinear equations. In particular, Newton-type methods are introduced to generate sequences approximating simple solutions of nonlinear equations in the setting of Banach spaces. Specializations of these methods include the modified Newton method, Newton’s method, and other single-step methods. The convergence of these methods is established with similar conditions. However, the convergence region is not large in general. That is why a unified semilocal convergence analysis is developed that can be used to handle these methods under even weaker conditions that are not previously considered. The approach leads to the extension of the applicability of these methods in cases not covered before but without new conditions. The idea is to replace the Lipschitz parameters or other parameters used by smaller ones to force convergence in cases not possible before. It turns out that the error analysis is also extended. Moreover, the new idea does not depend on the method. That is why it can also be applied to other methods to also extend their applicability. Numerical applications illustrate and test the convergence conditions

On the Convergence of Two-Step Kurchatov-Type Methods under Generalized Continuity Conditions for Solving Nonlinear Equations

The study of the microworld, quantum physics including the fundamental standard models are closely related to the basis of symmetry principles. These phenomena are reduced to solving nonlinear equations in suitable abstract spaces. Such equations are solved mostly iteratively. That is why two-step iterative methods of the Kurchatov type for solving nonlinear operator equations are investigated using approximation by the Fréchet derivative of an operator of a nonlinear equation by divided differences. Local and semi-local convergence of the methods is studied under conditions that the first-order divided differences satisfy the generalized Lipschitz conditions. The conditions and speed of convergence of these methods are determined. Moreover, the domain of uniqueness is found for the solution. The results of numerical experiments validate the theoretical results. The new idea can be used on other iterative methods utilizing inverses of divided differences of order one

Extended convergence analysis of Newton-Potra solver for equations

In the paper a local and a semi-local convergence of combined iterative process for solving nonlinear operator equations is investigated. This solver is built based on Newton solver and has R-convergence order 1.839.... The radius of the convergence ball and convergence order of the investigated solver are determined in an earlier paper. Modifications of previous conditions leads to extended convergence domain. These advantages are obtained under the same computational effort. Numerical experiments are carried out on the test examples with nondifferentiable operator