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.

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.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

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 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.

On the Ostrowski Method for Solving Equations

2021 ◽  
Vol 2 ◽  
pp. 3
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Christopher I. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Convergence Domain ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Ostrowski’S Method

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique  to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

scholarly journals Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 233 ◽  
Author(s):  
Ioannis Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Small Region ◽  
Convergence Region ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton’s, or Stirling’s, or Steffensen’s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

scholarly journals Group Splitting with SOR/AOR Methods for Solving Boundary Value Problems: A Computational Comparison

2021 ◽  
Vol 14 (3) ◽  
pp. 905-914
Author(s):  
Abdulkafi Mohammed Saeed ◽  
Najah Mohammad AL-harbi
Keyword(s):  
Iterative Method ◽  
Boundary Value Problems ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Elliptic Partial Differential Equations ◽  
The Other ◽  
Alternative Methods ◽  
Group Methods ◽  
Computational Comparison

Many researchers are working on the explicit group methods as the alternative methods for solving several boundary value problems. These methods have been shown to be much faster than the other point iterative methods in solving the elliptic partial differential equations (EPDEs), which is due to the formers’ overall lower computational complexities. This paper is concerned with the application of a suitable Explicit Group (EG) iterative method for solving EPDEs. This study will compare several iterative methods such that S5-point-SOR,4 Point-EGSOR, 5S-point-AOR, and 4 Point-EGAOR. Numerical experiments were carried out to confirm our results by using MATLAB software. The results reveal that 4 Point-EGAOR is the most superior method among these 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.

scholarly journals Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 89
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Order Derivative ◽  
First Derivative

We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods.

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