Some New Iterative Techniques for the Problems Involving Nonlinear Equations

2019 ◽  
Vol 17 (07) ◽  
pp. 1950037 ◽  
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Arif Rafiq
Keyword(s):  
Mathematical Models ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Graphical Analysis ◽  
Decomposition Technique ◽  
Convergence Criteria ◽  
Iterative Techniques ◽  
New Methods

We develop some new iterative methods, using decomposition technique, for solving the problems which involve nonlinear equations. Importantly, these methods include the generalization of some well-known existing methods. We prove the convergence criteria of our newly proposed methods. Various test examples are considered to validate the efficiency of our new methods. We also give the numerical as well as graphical analysis for two mathematical models to endorse the performance of these methods.

scholarly journals Some New Iterative Methods for Nonlinear Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said ◽  
Muhammad Waseem
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
System Of Equations ◽  
Decomposition Technique ◽  
Methods Comparison ◽  
Numerical Examples ◽  
New Methods ◽  
And Performance

We suggest and analyze some new iterative methods for solving the nonlinear equationsf(x)=0using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

scholarly journals Decomposition Technique and a Family of Efficient Schemes for Nonlinear Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Farooq Ahmed Shah ◽  
Maslina Darus ◽  
Imran Faisal ◽  
Muhammad Arsalan Shafiq
Keyword(s):  
Applied Sciences ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Decomposition Technique ◽  
Unified Framework ◽  
New Methods ◽  
New Family ◽  
And Performance

Various problems of pure and applied sciences can be studied in the unified framework of nonlinear equations. In this paper, a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique. The convergence of the new methods is proven. For the implementation and performance of the new methods, some examples are solved and the results are compared with some existing 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.

SOME NEWTON-TYPE ITERATIVE METHODS WITH AND WITHOUT MEMORY FOR SOLVING NONLINEAR EQUATIONS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350078 ◽  
Author(s):  
XIAOFENG WANG ◽  
TIE ZHANG
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Type Method ◽  
New Methods ◽  
Hermite Interpolating Polynomial ◽  
With Memory ◽  
Theoretical Results ◽  
Very High ◽  
High Computational Efficiency

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, [Formula: see text] and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

scholarly journals Extended Semilocal Convergence for the Newton- Kurchatov Method

2020 ◽  
Vol 53 (1) ◽  
pp. 85-91
Author(s):  
H.P. Yarmola ◽  
I. K. Argyros ◽  
S.M. Shakhno
Keyword(s):  
Iterative Methods ◽  
Error Estimates ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Semilocal Convergence ◽  
Convergence Criteria ◽  
Precise Location ◽  
Lipschitz Constants ◽  
Restricted Region

We provide a semilocal analysis of the Newton-Kurchatov method for solving nonlinear equations involving a splitting of an operator. Iterative methods have a limited restricted region in general. A convergence of this method is presented under classical Lipschitz conditions.The novelty of our paper lies in the fact that we obtain weaker sufficient semilocal convergence criteria and tighter error estimates than in earlier works. We find a more precise location than before where the iterates lie resulting to at least as small Lipschitz constants. Moreover, no additional computations are needed than before. Finally, we give results of numerical experiments.

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals Optimal High-Order Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Artidiello ◽  
A. Cordero ◽  
Juan R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Order ◽  
Function Technique ◽  
Numerical Tests ◽  
New Methods ◽  
Theoretical Results ◽  
Made In

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

scholarly journals New Family of Iterative Methods for Solving Nonlinear Models

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Faisal Ali ◽  
Waqas Aslam ◽  
Kashif Ali ◽  
Muhammad Adnan Anwar ◽  
Akbar Nadeem
Keyword(s):  
Mathematical Models ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Nonlinear Models ◽  
Graphical Analysis ◽  
Governing Equations ◽  
Iterative Schemes ◽  
New Family ◽  
Special Cases ◽  
Improved Performance

We introduce a new family of iterative methods for solving mathematical models whose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.

scholarly journals Modified Homotopy Perturbation Technique for the Approximate Solution of Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Farooq Ahmed Shah
Keyword(s):  
Approximate Solution ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Homotopy Perturbation Method ◽  
Perturbation Technique ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
New Methods ◽  
And Performance ◽  
Homotopy Perturbation Technique

We use a new modified homotopy perturbation method to suggest and analyze some new iterative methods for solving nonlinear equations. This new modification of the homotopy method is quite flexible. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations.

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