A new general eighth-order family of iterative methods for solving nonlinear equations

A New SPH Iterative Method for Solving Nonlinear Equations

“…A variety of complex problems in different fields of science and engineering require finding the solution of a nonlinear equation (or the system of nonlinear equations) of the form F (x) = 0, in order to solve this equation researchers proposed many iterative methods [Basto et al (2006); Kanwar et al (2008); Chen and Neta (2009); Liu and Wang (2012); Khan et al (2012); Rahman et al (2010)]. 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.…”

A note on the paper “A new general eighth-order family of iterative methods for solving nonlinear equations”

“…Khan et al assert that if the conditions of the following theorem hold, then the iterative method (1.1) has convergence order eight (see Theorem in Section 4 in [1]). However, we prove that this is not true and prove that its convergence order is five.…”

“…aðÀbÞ means a  10 b . In each table, COC stands for computational order of convergence (see [1]) which is given by…”

“…Lett. 25:2262-2266, 2012. Accordingly, we first show that their convergence theorem includes major errors, and it is attempted to provide the correct error equation of their method having fifth-order not eighth-order convergence.…”

Two weighted eight-order classes of iterative root-finding methods

“…The most of multistep iterative methods modify Newton's scheme to solve nonlinear equations with a highorder of convergence (see [3,4,5,6,7]). However, they frequently use derivatives, which is a serious disadvantage.…”