The improvement of modified newton's method

A new variant of Newton's method based on mid-point method has been developed and their convergence properties have been discussed. The order of convergence of the proposed method is five. Starting with a suitably chosen X₀, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its fifth order of convergence. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods. Analysis of efficiency shows that the new method can compete with Newton's method and the classical third order methods. Numerical results show that the method has definite practical utility.