Improved Chebyshev-Halley methods with sixth and eighth order convergence

We present a second derivative free family of modified Chebyshev-Halley methods with increasing order of convergence for solving nonlinear equations. The idea is based on the recent development by Li et al. (2014). Analysis of convergence shows that the family possesses at least sixth order of convergence. In a particular case even eighth order of convergence is achieved. Per iteration each method of the family requires four evaluations, namely three functions and one first derivative. That means, the eighth order method is optimal in the sense of Kung-Traub hypothesis. Numerical tests are performed, which confirm the theoretical results regarding order of convergence and efficiency of the new methods.