A new fourth-order iterative method for finding multiple roots of nonlinear equations

2009 ◽  
Vol 215 (3) ◽  
pp. 1288-1292 ◽  
Author(s):  
Li Shengguo ◽  
Liao Xiangke ◽  
Cheng Lizhi
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Roots

scholarly journals A family of optimal fourth‐order methods for multiple roots of nonlinear equations

2018 ◽  
Vol 43 (14) ◽  
pp. 7869-7884 ◽  
Author(s):  
Fiza Zafar ◽  
Alicia Cordero ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Roots

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

scholarly journals A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Harmonic Mean ◽  
Optimal Order ◽  
Multiple Roots ◽  
Multiple Zeros ◽  
Asymptotic Error ◽  
Two Parameter

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

scholarly journals A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Multiple Root ◽  
Multiple Roots ◽  
Asymptotic Error ◽  
Error Constant ◽  
Asymptotic Error Constant

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

scholarly journals A Newton Type Iterative Method with Fourth-order Convergence

2017 ◽  
Vol 12 (1) ◽  
pp. 87-95
Author(s):  
Jivandhar Jnawali
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton's Method ◽  
Second Order ◽  
Secant Method ◽  
Order Derivative ◽  
Single Variable ◽  
Order Convergence ◽  
Numerical Examples

The aim of this paper is to propose a fourth-order Newton type iterative method for solving nonlinear equations in a single variable. We obtained this method by combining the iterations of contra harmonic Newton’s method with secant method. The proposed method is free from second order derivative. Some numerical examples are given to illustrate the performance and to show this method’s advantage over other compared methods.Journal of the Institute of Engineering, 2016, 12 (1): 87-95

Fifth-order iterative method for finding multiple roots of nonlinear equations

2010 ◽  
Vol 57 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Xiaowu Li ◽  
Chunlai Mu ◽  
Jinwen Ma ◽  
Linke Hou
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Multiple Roots ◽  
Fifth Order

scholarly journals Optimal Methods for Finding Simple and Multiple Roots of Nonlinear Equations and their Basins of Attraction

2020 ◽  
Vol 37 (1-2) ◽  
pp. 14-29
Author(s):  
Prem Bahadur Chand
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Multiple Root ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Multiple Roots ◽  
Order Method ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Simple And Multiple Roots

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

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