  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.
