Accurate fourteenth-order methods for solving nonlinear equations

Abstract

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton's method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton's iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed fourstep methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

“…Note that nMax is the number of full steps we are going to apply our high order iterative method Eq. (6). Due to the fact that the order of Eq.…”

“…We assume that the function f in this vicinity which is also known as the real open interval D, has a simple root and the first derivative of the function in this interval does not vanish [1]. Due to the applications of such equations, in the past few years many iterative methods have been proposed for approximating their solutions; see for example [2][3][4][5][6][7].…”

A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem

“…More limited is the quantity of optimal eighth-order derivative-free iterative methods, mainly designed by using divided differences of high order (see, for example, [10,11,12]). Moreover, when only first-order divided differences are used, it is necessary to introduce several weight functions with one, two or more variables, all of them quotients of function f evaluated in different steps of the process (see, for instance [13,14,15] and the references therein).…”

A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

“…In 2011, Sargolzaei and Soleymani [7] presented new fourteenth order convergent iterative methods to approximate the simple roots of nonlinear equations. The three-step eighthorder construction can be viewed as a variant of Newton's method by using Hermite interpolation to reduce the number of function evaluations.…”