New modifications of Hansen-Patrick's family with optimal fourth and eighth orders of convergence

In this paper, we present two new second-derivative free classes of iterative methods based on Hansen-Patrick's family (Hansen and Patrick, 1977 [4]) for solving nonlinear equations numerically. In terms of computational cost, both families require only three and four functional evaluations to achieve optimal fourth and eighth orders of convergence, respectively. Moreover, the local convergence analysis of the proposed methods is also given using hypotheses only on the first derivative and Lipschitz constants. Furthermore, the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical examples and comparisons with some existing methods are included to confirm the theoretical results and high computational efficiency.