Dynamics of some one-point third-order methods for the solution of nonlinear equations

The article of record as published may be found at https://doi.org/10.5899/2018/cna-00362In this paper we have considered 32 one-point methods of cubic order to obtain simple zeros of a nonlinear function. These schemes are constructed by decomposition of previously known schemes. We have used the idea of basins of attractions to compare the performance of these methods with Halley's method on 4 polynomial functions and one non-polynomial function. Based on 3 quantitative criteria, namely average number of iterations per point, CPU time required and the number of points for which the method diverge, we have found 4 methods that performed close to best. We also show that decomposing good methods does not necessarily lead to a better one or even to a scheme as good as the original. We found only one example that gave reasonable results and it is the only one with repelling extraneous fixed points on the imaginary axis