A fourth-order derivative-free algorithm for nonlinear equations

Abstract

a b s t r a c tA two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results show that the algorithm is effective and is preferable to some existing derivative-free methods in terms of computation cost.

Multipoint methods for solving nonlinear equations: A survey

“…Methods which are either very similar to (35) or arise from (35) were considered in the papers [35,47,55,57]. Previously considered two-point methods use quadratically convergent methods (either Newton's or Traub-Steffensen's iteration) in the first step.…”

A New SPH Iterative Method for Solving Nonlinear Equations

“…Most of those methods were based on the well-known Newton's method, which is easy to implement and has quadratical convergence under fair assumptions, however, it requires to compute F (x) with F (x) = 0 in each calculation step, and sometimes it is difficult to provide the derivatives of function when the function is complicated. To overcome this problem, some derivative-free iterative methods have been proposed [Peng et al (2011); Yun (2011); Cordero et al (2013)].…”

On developing a higher-order family of double-Newton methods with a bivariate weighting function

“…Petković et al [25] recently collected and updated the state of the art of multipoint methods. Other works on multipoint methods can be found in [3][4][5]11,13,14,16,20,24,27]. The principal aim of this paper is to design a family of high-order methods costing only two derivatives and two functions.…”