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 Efficient Three-Step Class of Eighth-Order Multiple Root Solvers and Their Dynamics

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 837
Author(s):  
R. A. Alharbey ◽  
Munish Kansal ◽  
Ramandeep Behl ◽  
J. A. Tenreiro Machado
Keyword(s):  
General Class ◽  
Real Life ◽  
Multiple Root ◽  
Order Convergence ◽  
Eighth Order ◽  
Dynamical Study ◽  
New Methods ◽  
Life Problems ◽  
Special Cases ◽  
New Strategy

This article proposes a wide general class of optimal eighth-order techniques for approximating multiple zeros of scalar nonlinear equations. The new strategy adopts a weight function with an approach involving the function-to-function ratio. An extensive convergence analysis is performed for the eighth-order convergence of the algorithm. It is verified that some of the existing techniques are special cases of the new scheme. The algorithms are tested in several real-life problems to check their accuracy and applicability. The results of the dynamical study confirm that the new methods are more stable and accurate than the existing schemes.

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 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 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 Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4)=f(x,u,u′,u′′)

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 246 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Real Life ◽  
Order Ordinary Differential Equation ◽  
New Techniques ◽  
First Order ◽  
New Methods ◽  
Life Problems ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Order Four ◽  
Primary Contribution

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.

scholarly journals On the Influence of Center-Lipschitz Conditions in the Convergence Analysis of Multi-Point Iterative Methods

2018 ◽  
Author(s):  
Ioannis K Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Convergence Region ◽  
Numerical Examples ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to extend the local as well as the semi-local convergence analysis of multi-point iterative methods using center Lipschitz conditions in combination with our idea, of the restricted convergence region. It turns out that this way a finer convergence analysis for these methods is obtained than in earlier works and without additional hypotheses. Numerical examples favoring our technique over earlier ones completes this article.

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 A stable class of modified Newton-like methods for multiple roots and their dynamics

2020 ◽  
Vol 21 (6) ◽  
pp. 603-621
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Real Life ◽  
Chemical Reactor ◽  
Function Technique ◽  
Continuous Stirred Tank Reactor ◽  
Parallel Plates ◽  
Nonlinear Functions ◽  
Eighth Order ◽  
Multiple Zeros ◽  
Life Problems ◽  
Global Carbon

AbstractThere have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

scholarly journals Design and analysis of a faster King-Werner-type derivative free method

2022 ◽  
Vol 40 ◽  
pp. 1-18
Author(s):  
J. R. Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Examples ◽  
Weak Center ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
Methods Numerical

We introduce a new faster  King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local  convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.

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