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 Local Comparison between Two Ninth Convergence Order Algorithms for Equations

Algorithms ◽  
2020 ◽  
Vol 13 (6) ◽  
pp. 147
Author(s):  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Error Estimates ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Radius Of Convergence ◽  
Convergence Order ◽  
Convergence Criteria ◽  
First Derivative ◽  
Local Comparison ◽  
Uniqueness Of The Solution

A local convergence comparison is presented between two ninth order algorithms for solving nonlinear equations. In earlier studies derivatives not appearing on the algorithms up to the 10th order were utilized to show convergence. Moreover, no error estimates, radius of convergence or results on the uniqueness of the solution that can be computed were given. The novelty of our study is that we address all these concerns by using only the first derivative which actually appears on these algorithms. That is how to extend the applicability of these algorithms. Our technique provides a direct comparison between these algorithms under the same set of convergence criteria. This technique can be used on other algorithms. Numerical experiments are utilized to test the convergence criteria.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals On a Family of High-Order Iterative Methods under Kantorovich Conditions and Some Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
S. Amat ◽  
C. Bermúdez ◽  
S. Busquier ◽  
M. J. Legaz ◽  
S. Plaza
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
High Order ◽  
High Order Iterative Methods

This paper is devoted to the study of a class of high-order iterative methods for nonlinear equations on Banach spaces. An analysis of the convergence under Kantorovich-type conditions is proposed. Some numerical experiments, where the analyzed methods present better behavior than some classical schemes, are presented. These applications include the approximation of some quadratic and integral equations.

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 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.

SOME HIGH ORDER ITERATIVE METHODS FOR NONLINEAR EQUATIONS BASED ON THE MODIFIED HOMOTOPY PERTURBATION METHODS

2010 ◽  
Vol 03 (03) ◽  
pp. 395-408
Author(s):  
Bilian Chen ◽  
Yajun Xie ◽  
Changfeng Ma
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Perturbation Methods ◽  
Systems Of Nonlinear Equations ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Nonlinear Equation Systems ◽  
High Order Iterative Methods

In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria of the present methods. Some numerical examples are given to illustrate the performance and efficiency of the proposed methods.

scholarly journals A Unified Convergence Analysis for Some Two-Point Type Methods for Nonsmooth Operators

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 701
Author(s):  
Sergio Amat ◽  
Ioannis Argyros ◽  
Sonia Busquier ◽  
Miguel Ángel Hernández-Verón ◽  
María Jesús Rubio
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Initial Guess ◽  
Local Analysis ◽  
Convergence Criteria ◽  
Point Type

The aim of this paper is the approximation of nonlinear equations using iterative methods. We present a unified convergence analysis for some two-point type methods. This way we compare specializations of our method using not necessarily the same convergence criteria. We consider both semilocal and local analysis. In the first one, the hypotheses are imposed on the initial guess and in the second on the solution. The results can be applied for smooth and nonsmooth operators.

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