# A multi-point iterative method for solving nonlinear equations with optimal order of convergence

• Mehdi Salimi
• N. M. A. Nik Long
• Somayeh Sharifi
• Bruno Antonio Pansera
Original Paper Area 2

## Abstract

In this study, a three-point iterative method for solving nonlinear equations is presented. The purpose is to upgrade a fourth order iterative method by adding one Newton step and using a proportional approximation for last derivative. Per iteration this method needs three evaluations of the function and one evaluation of its first derivatives. In addition, the efficiency index of the developed method is $$\root 4 \of {8}\approx 1.682$$ which supports the Kung-Traub conjecture on the optimal order of convergence. Moreover, numerical and graphical comparison of the proposed method with other existing methods with the same order of convergence are given.

## Keywords

Multi-point iterative methods Simple root Order of convergence Kung and Traub’s conjecture Efficiency index

65H05 37F10

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© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

## Authors and Affiliations

• Mehdi Salimi
• 1
• N. M. A. Nik Long
• 2
• Somayeh Sharifi
• 3
• Bruno Antonio Pansera
• 4
1. 1.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany
2. 2.Department of Mathematics, Faculty of ScienceUniversiti Putra MalaysiaSerdangMalaysia
3. 3.MEDAlics, Research Center at Università per Stranieri Dante AlighieriReggio CalabriaItaly
4. 4.Department of Law and EconomicsUniversity Mediterranea of Reggio CalabriaReggio CalabriaItaly