Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation

Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation

In this work we show a general procedure to obtain optimal derivative free iterative methods [4] for nonlinear equations f(x) = 0, applying polynomial interpolation to a generic optimal derivative free iterative method of lower order. Let us consider an optimal method of order q = 2n−1, v = φn(x), that uses n functional evaluations. Performing a Newton step w = v − f(v) f ′(v) one obtains a met...

A Family of Optimal Derivative Free Iterative Methods with Eighth-Order Convergence for Solving Nonlinear Equations

In this paper, modification of Steffensen’s method with eight-order convergence is presented. We propose a family of optimal three-step methods with eight-order convergence for solving the simple roots of nonlinear equations by using the weight function and interpolation methods. Per iteration this method requires four evaluations of the function which implies that the efficiency index of the d...

THIRD-ORDER AND FOURTH-ORDER ITERATIVE METHODS FREE FROM SECOND DERIVATIVE FOR FINDING MULTIPLE ROOTS OF NONLINEAR EQUATIONS

In this paper, we present two new families of third-order and fourth-order methods for finding multiple roots of nonlinear equations. Each of them requires one evaluation of the function and two of its first derivative per iteration. Several numerical examples are given to illustrate the performance of the presented methods.    

Seventh-order iterative algorithm free from second derivative for solving algebraic nonlinear equations

Nonmonotone derivative-free methods for nonlinear equations

In this paper we study nonmonotone globalization techniques, in connection with iterative derivative-free methods for solving a system of nonlinear equations in several variables. First we define and analyze a class of nonmonotone derivative-free linesearch techniques for uncon-strained minimization of differentiable functions. Then we introduce a globalization scheme, which combines nonmonoton...