Changbum Chun

Some sixth-order variants of Ostrowski root-finding methods

Abstract In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixth-order convergent. Several numerical examples are given to illustrate the performance of some of the presented methods.

SOME FOURTH-ORDER ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS

Abstract In this paper we present some fourth-order iterative methods for solving nonlinear equations, which contains the well-known King’s fourth-order family as a particular case. Per iteration the methods require two function and one first derivative evaluations. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

Some improvements of Jarratt’s method with sixth-order convergence

Abstract In this paper, we present a one-parameter family of variants of Jarratt’s fourth-order method for solving nonlinear equations. It is shown that the order of convergence of each family member is improved from four to six even though it adds one evaluation of the function at the point iterated by Jarratt’s method per iteration. Several numerical examples are given to illustrate the performance of the presented methods.

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.

A new iterative method for solving nonlinear equations

Abstract In this paper, we present a new efficient iterative method for solving nonlinear equations improving Newton–Raphson method. This method is based on a modification of the proposal of Abbasbandy on improving the order of accuracy of Newton–Raphson method [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887–893]. The convergence properties of the method and some other methods are discussed. Some numerical illustrations are given to show that the proposed method behaves equal or better performance compared to the other methods of same kind.

A family of composite fourth-order iterative methods for solving nonlinear equations

In this paper, we present a family of new fourth-order iterative methods for solving nonlinear equations. Per iteration the methods consisting of the family require only two evaluations of the function and one evaluation of its derivative. Several numerical examples are given to illustrate the efficiency and performance of some of the presented methods.

Some fourth-order modifications of Newton’s method

Abstract In this paper, we construct some fourth-order modifications of Newton’s method for solving nonlinear equations. Any two existing fourth-order methods can be effectively used to give rise to new fourth-order methods. Per iteration the new methods require two evaluations of the function and one of its first-derivative. Numerical examples are given to show the performance of the presented methods.

A one-parameter fourth-order family of iterative methods for nonlinear equations

Abstract In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub–Ostrowski’s fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.

Some second-derivative-free variants of super-Halley method with fourth-order convergence

In this paper, we present some new variants of super-Halley method for solving nonlinear equations. These methods are free from second derivatives and require one function and two first derivative evaluations per iteration. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

Some variants of King’s fourth-order family of methods for nonlinear equations

Abstract In this paper, we present fourth-order families of new variants of King’s fourth-order family of methods for solving nonlinear equations. Per iteration the obtained methods require two evaluations of the given function and one of its first derivative. The classical Traub–Ostrowski method is obtained as a special variant of King’s method. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.