A family of fifth-order iterations composed of Newton and third-order methods

In this paper, we present a family of fifth-order methods formed by the composition of Newton and third-order methods for solving non-linear equations. Per iteration the new methods require two evaluations of the function, one of its first derivative and one of its second derivative. Analysis of efficiency shows that the new methods can compete with Newton's method and the classical third-order methods. Numerical results show that the methods have definite practical utility.