Abstract
In this paper, a published algorithm is investigated that proposes a three-step iterative method for solving nonlinear equations. This method is con- sidered to be ecient with third order of convergence and an improvement to previous methods. This paper proves that the order of convergence of the previous scheme is two, and the eciency index is less than the corresponding Newton's method. In addition, the three-step iterative method of the scheme is imple- mented, and the previously published numerical results are found to be incorrect. Furthermore, this paper presents a new three-step iterative method with third order of convergence for solving nonlinear equations. The same numerical examples previously presented in literature are used in this study to correct those results and to illustrate the eciency and performance of the new method.
Highlights
Nonlinear equations, Efficiency index, Order of convergence [4] derived optimal iterative methods with eighth and sixteenth orders of convergence for solving nonlinear equations
The following two-step iterative method for solving nonlinear equation (1) is suggested
Noor and Noor [8] derived a three-step iterative method for solving nonlinear equation (1), but the algorithm was quite different than algorithm 3 and is given as follows: Algorithm 4 For a given x0, compute the approximate solution xn+1 by the iterative scheme: Predictor Steps: yn
Summary
1 Introduction assume that f (x) has a simple root at α, and γ is an initial guess sufficiently close to α. Algorithm 3 For a given x0, compute the approximate solution xn+1 by the iterative scheme: Predictor Steps: yn. This suggests the following one-step iterative method for solving the the nonlinear equation (1). Algorithm 1 For a given x0, compute the approximate solution xn+1 by the iterative scheme: xn+1 This algorithm is well-known as Newton’s method which has a second order of convergence. Using this relation, the following two-step iterative method for solving nonlinear equation (1) is suggested. Noor and Noor [8] derived a three-step iterative method for solving nonlinear equation (1), but the algorithm was quite different than algorithm 3 and is given as follows: Algorithm 4 For a given x0, compute the approximate solution xn+1 by the iterative scheme: Predictor Steps:. Algorithm 3 is shown with a cubic order of convergence while the Noor and Noor algorithm, Algorithm 4, has a quadratic convergence
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