In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter $\alpha$ to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods.

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

The secant method is the most used iterative method to solve an operator equation where the operator involved is nondifferentiable. A known problem that arises when applying this method is its accessibility. Then, we try to improve it by using the technique of decomposition of the operator involved, so that the operator is in turn the sum of two operators, one differentiable and the other continuous but not differentiable. For this, we use a family of Newton‐secant‐type iterative methods that arises from Newton's method and from a well‐known family of secant‐type iterative methods. We study the accessibility of the new methods in two different ways: From the convergence balls of the methods, obtained from a local study of the convergence, and from a dynamic study of the methods. Some examples related to chemistry are also presented to prove the theoretical results.

In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use different convergence planes to show family behavior. Finally, the new results are used to solve some applications related to chemistry.