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.

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.

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.

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 Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

2021 ◽  
Vol 1 (2) ◽  
pp. 68-85
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
General Setting ◽  
Convergence Criteria ◽  
Recurrent Functions ◽  
Solving Equations ◽  
Order Four ◽  
Methods Numerical

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

Ball convergence of some fourth and sixth-order iterative methods

2016 ◽  
Vol 09 (02) ◽  
pp. 1650034
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Sixth Order ◽  
Chebyshev Method ◽  
Local Convergence Analysis ◽  
Appl Math

We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355–367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665–1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1–8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131–143; M. A. Hernández, Chebyshev’s approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501–515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412–420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821 ; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunič, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and [Formula: see text]-order for modified Chebyshev–Halley methods, Numer. Algorithms 64(1) (2013) 105–126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study.

scholarly journals Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1942
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Boundary Value Problems ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Single Step ◽  
Convergence Criterion ◽  
Convergence Criteria ◽  
Operator Equations

A plethora of sufficient convergence criteria has been provided for single-step iterative methods to solve Banach space valued operator equations. However, an interesting question remains unanswered: is it possible to provide unified convergence criteria for single-step iterative methods, which are weaker than earlier ones without additional hypotheses? The answer is yes. In particular, we provide only one sufficient convergence criterion suitable for single-step methods. Moreover, we also give a finer convergence analysis. Numerical experiments involving boundary value problems and Hammerstein-like integral equations complete this paper.

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