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The purpose of this paper is to present a new family of iterative methods with eighth‐order convergence for solving the nonlinear equation.

The paper uses a family of eighth‐order iterative methods for solving nonlinear equation based on Kou's seventh‐order method.

This family of methods is preferable to Ostrowski's, Grau's and Kou's methods in high‐precision computations.

This paper only deals with the nonlinear equations.

This paper is concerned with the iterative methods for finding a simple root of the nonlinear equation f(x)=0. One of the reasons for discussing the solution of nonlinear equation is that many methods for high‐dimensional optimization problems involve solving a sub‐problem which is a one‐dimensional search problem. And the nonlinear finite element problem, the boundary‐value problems appearing in Kinetic theory of gases, elasticity and other applied areas are also reduced to solving such an equation.

New methods of this family require three evaluations of the function and one evaluation of its first derivative and without using the second derivatives per iteration. This new family of methods as a new example agrees with Kung‐Traub's conjecture for n=4 and achieves its optimal convergence order 2ⁿ⁻¹.

The purpose of this article is to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically.

The scheme is generated by using King-type iterative methods. By employing the Traub-Steffensen technique, the proposed class is designed into the derivative-free family.

The proposed class requires three functional evaluations at each stage of computation to attain fourth-order convergency. Moreover, it can be observed that the theoretical convergency results of family are symmetrical for particular cases of multiplicity of zeros. This further motivates the authors to present the result in general, which confirms the convergency order of the methods. It is also worth mentioning that the authors can obtain already existing methods as particular cases of the family for some suitable choice of free disposable parameters. Finally, the authors include a wide variety of benchmark problems like van der Waals's equation, Planck's radiation law and clustered root problem. The numerical comparisons are included with several existing algorithms to confirm the applicability and effectiveness of the proposed methods.

The numerical results demonstrate that the proposed scheme performs better than the existing methods in terms of CPU timing and absolute residual errors.

The purpose of the article is to construct a new class of higher-order iterative techniques for solving scalar nonlinear problems.

The scheme is generalized by using the power-mean notion. By applying Neville's interpolating technique, the methods are formulated into the derivative-free approaches. Further, to enhance the computational efficiency, the developed iterative methods have been extended to the methods with memory, with the aid of the self-accelerating parameter.

It is found that the presented family is optimal in terms of Kung and Traub conjecture as it evaluates only five functions in each iteration and attains convergence order sixteen. The proposed family is examined on some practical problems by modeling into nonlinear equations, such as chemical equilibrium problems, beam positioning problems, eigenvalue problems and fractional conversion in a chemical reactor. The obtained results confirm that the developed scheme works more adequately as compared to the existing methods from the literature. Furthermore, the basins of attraction of the different methods have been included to check the convergence in the complex plane.

The presented experiments show that the developed schemes are of great benefit to implement on real-life problems.