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In this study, we introduce an iterative approach exhibiting sixth-order convergence for the solution of nonlinear equations. The method attains sixth-order convergence by using three evaluations of the function and two evaluations of the first-order derivative per iteration. We examined the theoretical convergence of our method through the convergence theorem, which substantiates the convergence order. Furthermore, we analyzed the local convergence of our proposed technique by employing a hypothesis that involves the first-order derivative of the function Θ alongside the Lipschitz conditions. To evaluate the performance and efficacy of our iterative method, we provide a comparative analysis against existing methods based on various standard numerical problems. Finally, graphical comparisons employing basins of attraction are presented to illustrate the dynamic behavior of the iterative method in the complex plane. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new Steffensen-type method with the advantage that its behavior is very similar to Newton's method; therefore, it is a very remarkable way of avoiding the drawback that Newton's method presents for nondifferentiable operators. In our study, we perform an exhaustive comparative study between the semilocal convergence of Newton's method and the derivative-free point-to-point iterative process considered; in the case of differentiable operators, we use the majoring sequences and the majorant principle. In the nondifferentiable case, we impose conditions on the starting point and on the nonlinear operator to obtain a semilocal convergence result for the iterative process considered. In both cases, we complete the theoretical convergence proofs with a dynamical study and a numerical test. In the case of differentiable operators, this study confirms that the accessibility and numerical behavior of both iterative processes, Newton's method and the derivative-free point-to-point iterative process considered, are very similar. [ABSTRACT FROM AUTHOR]

It is well known that the Steffensen-type methods approximate the derivative appearing in Newton's scheme by means of the first-order divided difference operator. The generalized multistep Steffensen iterative method consists of composing the method with itself m times. Specifically, the divided difference is held constant for every m steps before it is updated. In this work, we introduce a modification to this method, in order to accelerate the convergence order. In the proposed, scheme we compute the divided differences in first and second step and use the divided difference from the second step in the following m−1 steps. We perform an exhaustive study of the computational efficiency of this scheme and also introduce memory to this method to speed up convergence without performing new functional evaluations. Finally, some numerical examples are studied to verify the usefulness of these algorithms. [ABSTRACT FROM AUTHOR]

In this study, we suggest a new iterative family of iterative methods for approximating the roots with multiplicity in nonlinear equations. We found a lack in the approximation of multiple roots in the case that the nonlinear operator be non-differentiable. So, we present, in this paper, iterative methods that do not use the derivative of the non-linear operator in their iterative expression. With our new iterative technique, we find better numerical results of Planck's radiation, Van Der Waals, Beam designing, and Isothermal continuous stirred tank reactor problems. Divided difference and weight function approaches are adopted for the construction of our schemes. The convergence order is studied thoroughly in the Theorems 1 and 2, for the case when multiplicity p ≥ 2 . The obtained numerical results illustrate the preferable outcomes as compared to the existing ones in terms of absolute residual errors, number of iterations, computational order of convergence (COC), and absolute error difference between two consecutive iterations. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a new family of iterative methods for approximating the solution of nonlinear systems for non-differentiable operators. The novelty of this family is that it is a m-step generalization of the Steffensen-type method by updating the divided difference operator in the first two steps but not in the following ones. This procedure allows us to increase both the order of convergence and the efficiency index with respect to that obtained in the family that updates divided differences only in the first step. We perform a semilocal convergence study that allows us to fix the convergence domain and uniqueness for real applied problems, where the existence of a solution is not known a priori. After this study, some numerical tests are developed to apply the semilocal convergence theoretical results obtained. Finally, mediating the dynamic planes generated by the different numerical methods that compose the family, we study the symmetry of the basins of attraction generated by each solution, the shape of these basins, and the convergence to each root of a polynomial function. [ABSTRACT FROM AUTHOR]

In this work, a uniparametric generalization of the iterative method due to Kurchatov is presented. The iterative model presented is derivative‐free and approximates the solution of nonlinear equations when the operator is non‐differenciable. As the accessibility of the Kurchatov method is usually a problem in the application of the method, since the set of initial guesses that guarantee the convergence of the method is small, the main objective of this work is to improve the Kurchatov iterative method in its accessibility while maintaining and even increasing its speed of convergence. For this purpose, we introduce a variable parameter in the iterative function of the Kurchatov method that allows us to get a better approximation of the derivative by using a symmetric uniparameteric first‐order divided difference operator. We perform a complex dynamic study that corroborate the improvements in the accessibility region. Moreover, a complete analysis of the local and semilocal convergence is established for the new uniparametric iterative method. Finally, we apply the theoretical results to solve a nonlinear integral equation showing the usefulness of the work. [ABSTRACT FROM AUTHOR]

The objective in this paper is the expansion of the utilization for a fifth convergence order scheme without derivatives for finding solutions of Banach space valued equations. Conditions of the first order divided difference of the operator involved are only imposed. In this way the use of the scheme is expanded, since in earlier articles the derivatives until order four that do appear in the iterative method are required for setting convergence. Our technique also provides bounds on error distances as well as information about the location of the solutions not given in earlier works. Experiments with concrete problems complete this study. [ABSTRACT FROM AUTHOR]