A New Modification of Newton Method with Cubic Convergence

Applied Mathematics, 2015

Journal of Computational and Applied Mathematics, 2008

In this paper we consider constructing some higher-order modifications of Newton's method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods.

BASRA JOURNAL OF SCIENCE

In this study, we suggest and analyze two new one-parameter families of an efficient iterative methods free from higher derivatives for solving nonlinear equations based on Newton theorem of calculus and Bernstein quadrature formula, Bernoulli polynomial basis, Taylor’s expansion and some numerical techniques. We prove that the new iterative methods reach orders of convergence ten with six and eight with four functional evaluations per iteration, which implies that the efficiency index of the new iterative methods is (10)1/6 1.4678 and (8)1/4 1.6818 respectively. Numerical examples are provided to show the efficiency and performance of our iterative methods, compare to Newton’s method and other relevant methods.

Mathematics and Computers in Simulation, 2017

A family of optimal quadratic-order multiple-zero finders with a weight function of the principal kth root of a derivative-to-derivative ratio and their basins of attraction Young, Hee Geum; Kim, Young Ik; Neta, Beny Elsevier Y.H. Geum, Y.I. Kim, B. Neta, "A family of optimal quartic-order multiple-zero finders with a weight function of the principle kth root of a derivative-to-derivative ratio and their basins of attraction," Mathematics and Computers in Simulation, v.136, (2017),

Communications in Numerical Methods in Engineering, 2005

In this paper, Newton-like iteration methods for solving non-linear equations or improving the existing iteration methods are proposed. The iteration formulae are obtained by applying the homotopy perturbation method which contains the well-known Newton iteration formula in logic, so those improving the Newton method. The orders of convergence of some of those iteration formulae are derived analytically and by applying symbolic computation of Maple. Some numerical illustrations are given. Copyright

International Journal of Computer Mathematics, 2009

Journal of Computational and Applied Mathematics, 2007

In this paper, we present some new modifications of Newton's method for solving non-linear equations. Analysis of convergence shows that these methods have order of convergence five. Numerical tests verifying the theory are given and based on these methods, a class of new multistep iterations is developed.

Applied Mathematics, 2013

In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.

Journal of Information and Optimization Sciences, 2018

In this paper, we suggest and analyze new higher order family of iterative methods for solving nonlinear equations by using variational iteration technique. We give several examples to illustrate the efficiency of the proposed methods. Comparison with other similar methods is also given. New methods can be considered as an alternative of the existing methods. This technique can be used to suggest a wide class of new iterative methods for solving nonlinear equations.

International Journal of Differential Equations and Applications, 2013

The aim of this paper is generalization fourth order Jarratt formula iterative methods for solving system of nonlinear equations (SNLE) of n-dimension with n-variables. We present three algorithms for solving (SNLE). We prove that these algorithms have convergence. Several numerical examples are tested of the new iterative methods. These new algorithms may be viewed as an extensions and generalizations of the existing methods for solving the system of nonlinear equations.