Highly efficient classes of Chebyshev-Halley type methods free from second-order derivative

In this paper, we present new highly efficient fourth-order optimal families of Chebyshev-Halley type methods free from second-order derivative. In terms of computational cost, each member of the families requires one function and two first-order derivative evaluations per iteration, so that their efficiency indices are 1.587. On the account of the results obtained, it is found that in majority of the problems tested here, our proposed methods are efficient and show better performance than classical Jarratt's method and other existing Jarratt-type methods when the accuracy is tested in multi-precision digits.