scholarly journals Local Convergence of a Family of Weighted-Newton Methods

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 103
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
J.A. Machado ◽  
Ali Alshomrani
Keyword(s):  
Banach Space ◽  
Fourth Order ◽  
Local Convergence ◽  
Real Life ◽  
Newton Methods ◽  
Convergence Radius ◽  
Life Problems ◽  
Scope Of Application ◽  
Error Estimations ◽  
Derivatives Of

This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of order one. The convergence radius, error estimations, and results on uniqueness also depend on this derivative. The scope of application of the method is extended, since no derivatives of higher order are required as in previous works. Finally, we demonstrate the applicability of the proposed method in real-life problems and discuss a case where previous studies cannot be adopted.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

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 Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

2021 ◽  
Vol 5 (4) ◽  
pp. 165
Author(s):  
Mohamed Abdelhakem ◽  
Toqa Alaa-Eldeen ◽  
Dumitru Baleanu ◽  
Maryam G. Alshehri ◽  
Mamdouh El-Kady
Keyword(s):  
Error Analysis ◽  
Galerkin Method ◽  
Chebyshev Polynomials ◽  
Real Life ◽  
Operational Matrices ◽  
Integer Order ◽  
First Derivative ◽  
The Core ◽  
Life Problems ◽  
Derivatives Of

An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

An Optimal Reconstruction of Chebyshev–Halley-Type Methods with Local Convergence Analysis

2019 ◽  
Vol 17 (05) ◽  
pp. 1940017
Author(s):  
Ali Saleh Alshomrani ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl
Keyword(s):  
Local Convergence ◽  
Real Life ◽  
Eighth Order ◽  
Dynamical Study ◽  
First Order ◽  
Life Problems ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Scalar Equations ◽  
Basins Of Attractions

Our principle aim in this paper is to present a new reconstruction of classical Chebyshev–Halley schemes having optimal fourth and eighth-order of convergence for all parameters [Formula: see text] unlike in the earlier studies. In addition, we analyze the local convergence of them by using hypotheses requiring the first-order derivative of the involved function [Formula: see text] and the Lipschitz conditions. In addition, we also formulate their theoretical radius of convergence. Several numerical examples originated from real life problems demonstrate that they are applicable to a broad range of scalar equations, where previous studies cannot be used. Finally, a dynamical study of them also demonstrates that bigger and more promising basins of attractions are obtained.

scholarly journals A Novel Picture Fuzzy n-Banach Space with Some New Contractive Conditions and Their Fixed Point Results

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Awais Asif ◽  
Hassen Aydi ◽  
Muhammad Arshad ◽  
Zeeshan Ali
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Linear Space ◽  
Fuzzy Set ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Real Life ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Picture Fuzzy Set

A picture fuzzy n-normed linear space (NPF), a mixture of a picture fuzzy set and an n-normed linear space, is a proficient concept to cope with uncertain and unpredictable real-life problems. The purpose of this manuscript is to present some novel contractive conditions based on NPF. By using these contractive conditions, we explore some fixed point theorems in a picture fuzzy n-Banach space (BPF). The discussed modified results are more general than those in the existing literature which are based on an intuitionistic fuzzy n-Banach space (BIF) and a fuzzy n-Banach space. To express the reliability and effectiveness of the main results, we present several examples to support our main theorems.

scholarly journals Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1375
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
Fouad Othman Mallawi
Keyword(s):  
Local Convergence ◽  
Real Life ◽  
High Order ◽  
Systems Of Nonlinear Equations ◽  
Convergence Criteria ◽  
Restricted Area ◽  
Iterative Schemes ◽  
First Order ◽  
Life Problems ◽  
Iterative Procedures

In this study, we suggested the local convergence of three iterative schemes that works for systems of nonlinear equations. In earlier results, such as from Amiri et al. (see also the works by Behl et al., Argryos et al., Chicharro et al., Cordero et al., Geum et al., Guitiérrez, Sharma, Weerakoon and Fernando, Awadeh), authors have used hypotheses on high order derivatives not appearing on these iterative procedures. Therefore, these methods have a restricted area of applicability. The main difference of our study to earlier studies is that we adopt only the first order derivative in the convergence order (which only appears on the proposed iterative procedure). No work has been proposed on computable error distances and uniqueness in the aforementioned studies given on Rk. We also address these problems too. Moreover, by using Banach space, the applicability of iterative procedures is extended even further. We have examined the convergence criteria on several real life problems along with a counter problem that completes this study.

scholarly journals A CONVERGENCE ANALYSIS FOR NEWTON-LIKE METHODS IN BANACH SPACE UNDER WEAK HYPOTHESES AND APPLICATIONS

1999 ◽  
Vol 30 (4) ◽  
pp. 253-261
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Banach Space ◽  
Lipschitz Continuity ◽  
Real Life ◽  
Approximate Solutions ◽  
Continuity Condition ◽  
Point Boundary ◽  
Hölder Continuous ◽  
Life Problems ◽  
Convergence Results ◽  
Bending Of Beams

In this study we use Newton-like methods to approximate solutions of nonlinear equations in a Banach space setting. Most convergence results for Newton-like methods involve some type of a Lipschitz continuity condition on the Frechet-derivative of the operator involved However there are many interesting real life problems already in the literature where the operator can only satisfy a Holder continuity condition. That is why here we chose the Frcchet-derivativc of the operator involved to be only Holder continuous, which allows us to consider a wider range of problems than before. Special choices of our parameters reduce our results to earlier ones. An error analysis is also provided for our method. At the end of our study, we provide applications to show that our results apply where earlier results do not. In paricular we solve a two point boundary value problem appearing in physics in connection with the problem of bending of beams.

scholarly journals Local Convergence Balls for Nonlinear Problems with Multiplicity and Their Extension to Eighth-Order Convergence

2019 ◽  
Vol 2019 ◽  
pp. 1-17
Author(s):  
Ramandeep Behl ◽  
Eulalia Martínez ◽  
Fabricio Cevallos ◽  
Ali S. Alshomrani
Keyword(s):  
Local Convergence ◽  
Chemical Engineering ◽  
Real Life ◽  
Nonlinear Problems ◽  
Engineering Problem ◽  
Test Problems ◽  
Order Convergence ◽  
Eighth Order ◽  
Life Problems ◽  
Fourth Order Method

The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m≥1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion chemical engineering problem, and two standard academic test problems, are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they not only have faster convergence towards the desired zero of the involved function but also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.

scholarly journals Ball convergence theorem for inexact Newton methods in Banach space

2020 ◽  
Vol 29 (2) ◽  
pp. 113-120
Author(s):  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Newton Methods ◽  
Convergence Conditions ◽  
Inexact Newton Methods ◽  
Numerical Examples ◽  
Ball Convergence ◽  
Local Convergence Analysis

We present a local convergence analysis for inexact Newton methods in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Frèchet-derivative of the operator involved. Numerical examples are also provided in this study.

The Psychologist in the Medical School: An Example of Solving Real-Life Problems

1970 ◽  
Author(s):  
Matisyohu Weisenberg ◽  
Carl Eisdorfer ◽  
C. Richard Fletcher ◽  
Murray Wexler
Keyword(s):  
Medical School ◽  
Real Life ◽  
Life Problems
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