Optimal Newton-Secant like methods without memory for solving nonlinear   equations with its dynamics
  We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane. 
Mehdi Salimi, Taher Lotfi, Somayeh Sharifi, Stefan Siegmund
math.NA