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In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application oriented problems. Finally on the basis of numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order and efficiency and also that the stability results provide the most efficient member of each class of iterative schemes.Keywords Systems of nonlinear equations • Order of convergence • Multipoint iterative methods • Stability analysis.

In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a=1, the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

We propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of the required root or f′(x)=0. On the contrary, the previous modifications either diverge or fail to work. In addition, we also extended the same idea for an iterative method with memory. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed method as well as that of some of the existing methods. Numerical experiments are performed on engineering problems, such as fractional conversion in a chemical reactor, Planck’s radiation law problem, Van der Waal’s problem, trajectory of an electron in between two parallel plates. The numerical results reveal that the proposed schemes are of utmost importance to be applied on various real–life problems. Basins of attraction also support this aspect.