Some iterative methods free from second derivatives for nonlinear equation Muhammad Aslam Noor Dept. of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

To overcome these drawback, presenting Introduction In the implementation of methods, one has to evaluate the second derivative of the functions, which is itself a serious and difficult problem. To overcome these drawback, presenting some modifications of the method in his previous paper by an appropriate finite difference scheme.

Ⅰ Iterative Methods For , a simple zero Define the sequence such that for every , with He proposed an arbitrary such that with a parameter . Define Then It is similar to Newton’s method with

Ⅱ Algorithm 2.1 Taking Then It implies that

Ⅱ Algorithm 2.2 In algorithm 2.1, taking Then If is negligible, these algorithm collapse to Newton’s method. But if not, second derivative form interrupts our computation.

Ⅵ New Approaches Taking and using Taylor series (which is that by the form of Newton’s method)

Ⅶ Algorithm 2.3 By Algorithm 2.1, It is for free from the second derivative

Ⅸ Algorithm 2.4 A kind of second-order algebraic equation with the variable of

Ⅹ Algorithm 2.5 Taking and neglecting term, Two-step Newton’s method

Ⅰ Algorithm 2.6 Taking From the Algorithm 2.4, one can obtain several new flexible methods for solving the nonlinear equations.

Ⅱ Theorem 2.1 For a simple zero of , if is sufficiently closed to , then Algorithm 2.4 satisfies It means that, the order of convergence is two.