The complex dynamical analysis of the parametric fourth-order Kim's iterative family is made on quadratic polynomials, showing the MATLAB codes generated to draw the fractal images necessary to complete the study. The parameter spaces associated with the free critical points have been analyzed, showing the stable (and unstable) regions where the selection of the parameter will provide us the excellent schemes (or dreadful ones).

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A new family of two-steps fourth-order iterative methods for solving nonlinear equations is introduced based on the weight functions procedure. This family is optimal in the sense of Kung-Traub conjecture and it is extended to design a class of iterative schemes with four step and seventh order of convergence. We are interested in analyzing the dynamical behavior of different elements of the fourth-order class. This analysis gives us important information about the stability of these members of the family. The methods are also tested with nonlinear functions and compared with other known schemes. The results show the good features of the introduced class. KEYWORDS complex dynamics, iterative methods, weight functions Comp and Math Methods. 2019;1:e1023.wileyonlinelibrary.com/journal/cmm4

Based on the third-order Traub's method, two iterative schemes with memory are introduced. The proper inclusion of accelerating parameters allows the introduction of memory. Therefore, the order of convergence of the iterative methods increases from 3 up to 3.73 without new functional evaluations. One of them includes derivatives and the other one is derivative-free. The stability of the methods with memory is analyzed and their basins of attraction are compared to check the differences between them. The methods are applied to solve two nonlinear problems in Chemistry, such as the fractional conversion of the nitrogen-hydrogen feed that gets converted to ammonia and the Colebrook-White equation. KeywordsNonlinear equation • Iterative method with memory • Derivative-free • Complex dynamics • Basin of attraction • Chemistry applications This research was partially supported by Ministerio de Economía y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.

In this paper, we analyze the stability of a parametric family of iterative methods with memory for solving nonlinear equations. This family is obtained from an optimal class of fourth-order schemes without memory designed by means of weight functions procedure. By studying the real fixed and critical points of the rational function resulting from the application of the family with memory on quadratic polynomials, the best elements of the family, in terms of absence of chaotic behavior, are selected. Finally, a numerical study is performed verifying the dynamical theoretical results.