An improvement of Chebyshev–Halley methods free from second derivative

In this paper, a family of modified Chebyshev–Halley’s methods free from second derivative is presented. Per iteration the new methods require three evaluations of the function and one of its first derivatives. A detailed convergence analysis of the new methods shows that the new methods are at least fifth-order convergent and especially, the modified super-Halley’s method is sixth-order convergent. Numerical examples are given to illustrate the efficiency and performance of the new methods.