4th-order iterative methods for the numerical solution of nonlinear equations and their dynamics

In this study, some new iterative methods which can be used for the numerical solution of nonlinear equations of the form $f(x)=0$, where $f: \mathbb{R} \longrightarrow \mathbb{R}$ is a function of a real variable $x$, were developed. The methods developed are 4th-order convergent, multi-point methods without memory. The methods are optimal since they require three new function evaluations in getting each new approximation to a solution of the nonlinear equation $f(x)=0$. Convergence analysis of the newly developed multi-point methods was performed and error equations showing that they are 4th-order convergent were obtained. In order to investigate the effectiveness of the methods, comparisons were made with some widely known methods that are available in the relevant field, using some test problems. Comparisons based on numerical results have shown that newly developed methods are as effective as existing methods and gives better results in some test problems. As with similar methods in the literature, these methods can also be used in the numerical solution of nonlinear equations of the form $f(z)=0$, where $f:\mathbb{C} \rightarrow \mathbb{C}$ is a function of a complex variable $z$ and, in this respect, in order to compare them with the existing methods in the literature, the dynamics of the methods were also examined and their effectiveness was discussed by using attractive basins for some polynomial equations in the complex plane.