Some substantial modifications and improvements for derivative-free iterative methods and derivative-free transformation for multiple zeros

In this paper we present some substantial modifications and improvements for some derivative-free iterative formulae. We propose some new methods which are convergent iterative formulae of order 1+2≈2.414 with only two evaluations of f per step. Thus the new methods enjoy the efficiency index 1+2≈1.554. Especially, a new method of self-accelerating regula falsi type with global convergence for finding a simple root p of a nonlinear equation f(x) = 0 in the interval [a, b] is offered in this paper. The new method of self-accelerating regula falsi type is shown to be convergent with order 1+2≈2.414 for both the sequences of diameters {(bn − an)} and the iterative points {xn}. The new method has been tested on a series of published examples. The numerical results demonstrate that the new method is more effective. Furthermore, we introduce a derivative-free transformation for multiple zeros and consider its error analysis and then demonstrate its superiority by comparing it with the traditional ones.