scholarly journals A fourth order method for finding a simple root of univariate function

2015 ◽  
Vol 34 (2) ◽  
pp. 197-211
Author(s):  
D. Sbibih ◽  
Abdelhafid Serghini ◽  
A. Tijini ◽  
A. Zidna
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Simple Root ◽  
Newton's Method ◽  
Order Method ◽  
Numerical Examples ◽  
Quadratic Interpolation ◽  
Univariate Function ◽  
Fourth Order Method

In this paper, we describe an iterative method for approximating asimple zero $z$ of a real defined function. This method is aessentially based on the idea to extend Newton's method to be theinverse quadratic interpolation. We prove that for a sufficientlysmooth function $f$ in a neighborhood of $z$ the order of theconvergence is quartic. Using Mathematica with its high precisioncompatibility, we present some numerical examples to confirm thetheoretical results and to compare our method with the others givenin the literature.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

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.

A New SPH Iterative Method for Solving Nonlinear Equations

2019 ◽  
Vol 17 (01) ◽  
pp. 1843005 ◽  
Author(s):  
Rahmatjan Imin ◽  
Ahmatjan Iminjan
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basic Principle ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Taylor Series Expansion ◽  
Numerical Examples ◽  
First Order ◽  
Kernel Approximation

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

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

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.

scholarly journals After notes on Chebyshev’s iterative method

2017 ◽  
Vol 2 (1) ◽  
pp. 1-12 ◽  
Author(s):  
S. Amat ◽  
S. Busquier
Keyword(s):  
Global Convergence ◽  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Geometric Interpretation ◽  
Higher Order ◽  
Order Method ◽  
Chebyshev’S Method ◽  
Higher Order Method ◽  
Dynamical Properties

AbstractThis paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.

scholarly journals Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 942 ◽  
Author(s):  
Prem B. Chand ◽  
Francisco I. Chicharro ◽  
Neus Garrido ◽  
Pankaj Jain
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Weight Functions ◽  
Order Method ◽  
Eighth Order ◽  
Numerical Examples ◽  
Fourth Order Method ◽  
Cubic Polynomials

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.

scholarly journals Some Class of Third- and Fourth-Order Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Basins Of Attraction ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
Multivariate Extension ◽  
New Class ◽  
Fourth Order Method

The object of the present work is to give the new class of third- and fourth-order iterative methods for solving nonlinear equations. Our proposed third-order method includes methods of Weerakoon and Fernando (2000), Homeier (2005), and Chun and Kim (2010) as particular cases. The multivariate extension of some of these methods has been also deliberated. Finally, some numerical examples are given to illustrate the performances of our proposed methods by comparing them with some well existing third- and fourth-order methods. The efficiency of our proposed fourth-order method over some fourth-order methods is also confirmed by basins of attraction.

scholarly journals Eighteenth Order Convergent Method for Solving Non-Linear Equations

2017 ◽  
Vol 10 (1) ◽  
pp. 144-150 ◽  
Author(s):  
V.B Vatti ◽  
Ramadevi Sri ◽  
M.S Mylapalli
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Subject Classification ◽  
Order Convergence ◽  
Numerical Examples ◽  
Convergent Method ◽  
Non Linear Equations

In this paper, we suggest and discuss an iterative method for solving nonlinear equations of the type f(x)=0 having eighteenth order convergence. This new technique based on Newton’s method and extrapolated Newton’s method. This method is compared with the existing ones through some numerical examples to exhibit its superiority. AMS Subject Classification: 41A25, 65K05, 65H05.

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

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