scholarly journals A New Sixth Order Method for Nonlinear Equations inR

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Sukhjit Singh ◽  
D. K. Gupta
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Performance Measure ◽  
Sixth Order ◽  
Order Method ◽  
Numerical Examples ◽  
Real Roots ◽  
New Iterative Method ◽  
Number Of Iterations

A new iterative method is described for finding the real roots of nonlinear equations inR. Starting with a suitably chosenx0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newton’s method and other sixth order methods considered.

scholarly journals Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

2018 ◽  
Vol 24 (1) ◽  
pp. 3
Author(s):  
Himani Arora ◽  
Juan Torregrosa ◽  
Alicia Cordero
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Sixth Order ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Boundary Problems ◽  
Newton Step ◽  
The Family ◽  
Efficiency And Reliability

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals A New Iterative Method for Finding Approximate Inverses of Complex Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
M. Kafaei Razavi ◽  
A. Kerayechian ◽  
M. Gachpazan ◽  
S. Shateyi
Keyword(s):  
Iterative Method ◽  
Order Of Convergence ◽  
Complex Matrices ◽  
Approximate Inverse ◽  
Numerical Examples ◽  
Convergence Behavior ◽  
Approximate Inverses ◽  
Nonsingular Matrices ◽  
New Iterative Method

This paper presents a new iterative method for computing the approximate inverse of nonsingular matrices. The analytical discussion of the method is included to demonstrate its convergence behavior. As a matter of fact, it is proven that the suggested scheme possesses tenth order of convergence. Finally, its performance is illustrated by numerical examples on different matrices.

scholarly journals On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
H. Montazeri ◽  
F. Soleymani ◽  
S. Shateyi ◽  
S. S. Motsa
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
The Other ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Numerical Tests ◽  
Programming Package ◽  
New Iterative Method

We consider a system of nonlinear equationsF(x)=0. A new iterative method for solving this problem numerically is suggested. The analytical discussions of the method are provided to reveal its sixth order of convergence. A discussion on the efficiency index of the contribution with comparison to the other iterative methods is also given. Finally, numerical tests illustrate the theoretical aspects using the programming package Mathematica.

scholarly journals Some novel Newton-type methods for solving nonlinear equations

2019 ◽  
Vol 38 (3) ◽  
pp. 111-123
Author(s):  
Morteza Bisheh-Niasar ◽  
Abbas Saadatmandi
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Sixth Order ◽  
New Method ◽  
Order Convergence ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Newton Iterative Method

The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

scholarly journals New Iterative Method for Solving Nonlinear Equations

2014 ◽  
Vol 11 (4) ◽  
pp. 1649-1654 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Equation ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
New Method ◽  
New Iterative Method

The aim of this paper is to propose an efficient three steps iterative method for finding the zeros of the nonlinear equation f(x)=0 . Starting with a suitably chosen , the method generates a sequence of iterates converging to the root. The convergence analysis is proved to establish its five order of convergence. Several examples are given to illustrate the efficiency of the proposed new method and its comparison with other methods.

scholarly journals Introduction to a family of Thukral k-order method for finding multiple zeros of nonlinear equations

2017 ◽  
Vol 13 (3) ◽  
pp. 7230-7237
Author(s):  
R Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Computational Cost ◽  
Order Method ◽  
Multiple Zeros ◽  
New Iterative Method

A new one-point k-order iterative method for finding zeros of nonlinear equations having unknown multiplicity is introduced.  In terms of computational cost the new iterative method requires k+1 evaluations of functions per iteration.  It is shown that the new iterative method has a convergence of order k.

scholarly journals A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

scholarly journals Analyzing a New Third Order Iterative Method for Solving Nonlinear Problems

2021 ◽  
pp. 72-90
Author(s):  
M. S. Ndayawo ◽  
B. Sani
Keyword(s):  
Iterative Method ◽  
Series Expansion ◽  
Nonlinear Problems ◽  
Third Order ◽  
Numerical Examples ◽  
Adomian Method ◽  
The Individual ◽  
New Iterative Method ◽  
Number Of Iterations ◽  
Taylor’S Series

In this paper, we propose and analyse a new iterative method for solving nonlinear equations. The method is constructed by applying Adomian method to Taylor’s series expansion. Using one-way analysis of variance (ANOVA), the method is being compared with other existing methods in terms of the number of iterations and solution to convergence between the individual methods used. Numerical examples are used in the comparison to justify the efficiency of the new iterative method.

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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