On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives

Abstract

The one-parameter Chebyshev-Halley family is an important family of one-point third order iterative methods which requires one function, one first and one second derivative evaluations. The famous Chebyshev, Halley and Super-Halley methods are its members. Nedzhibov-Hasanov-Petkov (Numer. Alg. 42:127–136, 2006) approximated the second derivative present in the Chebyshev-Halley family to obtain a two-parameter Chebyshev-Halley-like family of two-point iterative methods free from second derivative. Only one member of this family known as the famous Jarratt method is fourth order and satisfies the Kung-Traub Conjecture. Other members are third order. Recent advancement in this field of numerical analysis have made it possible to develop fourth order methods from third order ones with the same number of function evaluations using weight functions. In this work, we develop a two-parameter Chebyshev-Halley-like family of two-point fourth order methods using weight functions. We compare the special members of the new family with the old one through an numerical example to illustrate the efficiency of the new family.