Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space

Problems from numerous disciplines such as applied sciences, scientific computing, applied mathematics, engineering to mention some can be converted to solving an equation. That is why, we suggest higher-order iterative method to solve equations with Banach space valued operators. Researchers used the suppositions involving seventh-order derivative by Chen, S.P. and Qian, Y.H. But, here, we only use suppositions on the first-order derivative and Lipschitz constrains. In addition, we do not only enlarge the applicability region of them but also suggest computable radii. Finally, we consider a good mixture of numerical examples in order to demonstrate the applicability of our results in cases not covered before.

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Ball Convergence for a Multi-Step Harmonic Mean Newton-Like Method in Banach Space

International Journal of Computational Methods ◽

10.1142/s0219876219400188 ◽

2019 ◽

Vol 17 (05) ◽

pp. 1940018 ◽

Cited By ~ 1

Author(s):

Ramandeep Behl ◽

Ali Saleh Alshormani ◽

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Nonlinear Equations ◽

Local Convergence ◽

Harmonic Mean ◽

Order Derivative ◽

Systems Of Nonlinear Equations ◽

Numerical Examples ◽

First Order ◽

Ball Convergence ◽

Local Convergence Analysis

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) “On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations,” Algorithms 8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply.

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Convergence of Higher Order Jarratt-Type Schemes for Nonlinear Equations from Applied Sciences

Symmetry ◽

10.3390/sym13071162 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1162

Author(s):

Ramandeep Behl ◽

Ioannis K. Argyros ◽

Fouad Othman Mallawi ◽

Christopher I. Argyros

Keyword(s):

Applied Sciences ◽

Banach Space ◽

Euclidean Space ◽

Numerical Experiments ◽

Higher Order ◽

Physical Systems ◽

First Order ◽

Uniqueness Of The Solution ◽

Jarratt’S Method ◽

Local Convergence Analysis

Symmetries are important in studying the dynamics of physical systems which in turn are converted to solve equations. Jarratt’s method and its variants have been used extensively for this purpose. That is why in the present study, a unified local convergence analysis is developed of higher order Jarratt-type schemes for equations given on Banach space. Such schemes have been studied on the multidimensional Euclidean space provided that high order derivatives (not appearing on the schemes) exist. In addition, no errors estimates or results on the uniqueness of the solution that can be computed are given. These problems restrict the applicability of the methods. We address all these problems by using the first order derivative (appearing only on the schemes). Hence, the region of applicability of existing schemes is enlarged. Our technique can be used on other methods due to its generality. Numerical experiments from chemistry and other disciplines of applied sciences complete this study.

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Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽

10.3390/math8020179 ◽

2020 ◽

Vol 8 (2) ◽

pp. 179

Author(s):

Ramandeep Behl ◽

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Nonlinear Equations ◽

Local Convergence ◽

Nonlinear Problems ◽

Fréchet Derivative ◽

High Order ◽

Order Derivative ◽

First Order ◽

Lipschitz Constants ◽

Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

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Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽

10.3390/math6110260 ◽

2018 ◽

Vol 6 (11) ◽

pp. 260 ◽

Cited By ~ 3

Author(s):

Janak Sharma ◽

Ioannis Argyros ◽

Sunil Kumar

Keyword(s):

Iterative Method ◽

Iterative Methods ◽

Higher Order ◽

High Order ◽

Convergence Order ◽

Eighth Order ◽

First Derivative ◽

Ball Convergence ◽

Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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A New SPH Iterative Method for Solving Nonlinear Equations

International Journal of Computational Methods ◽

10.1142/s0219876218430053 ◽

2019 ◽

Vol 17 (01) ◽

pp. 1843005 ◽

Cited By ~ 2

Author(s):

Rahmatjan Imin ◽

Ahmatjan Iminjan

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Basic Principle ◽

Newton’S Method ◽

Newton's Method ◽

Quadratic Convergence ◽

Taylor Series Expansion ◽

Numerical Examples ◽

First Order ◽

Kernel Approximation

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

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A Newton Type Iterative Method with Fourth-order Convergence

Journal of the Institute of Engineering ◽

10.3126/jie.v12i1.16729 ◽

2017 ◽

Vol 12 (1) ◽

pp. 87-95

Author(s):

Jivandhar Jnawali

Keyword(s):

Iterative Method ◽

Nonlinear Equations ◽

Fourth Order ◽

Newton's Method ◽

Second Order ◽

Secant Method ◽

Order Derivative ◽

Single Variable ◽

Order Convergence ◽

Numerical Examples

The aim of this paper is to propose a fourth-order Newton type iterative method for solving nonlinear equations in a single variable. We obtained this method by combining the iterations of contra harmonic Newton’s method with secant method. The proposed method is free from second order derivative. Some numerical examples are given to illustrate the performance and to show this method’s advantage over other compared methods.Journal of the Institute of Engineering, 2016, 12 (1): 87-95

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Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

Mathematics ◽

10.3390/math8010092 ◽

2020 ◽

Vol 8 (1) ◽

pp. 92

Author(s):

Ramandeep Behl ◽

Sonia Bhalla ◽

Ioannis K. Argyros ◽

Sanjeev Kumar

Keyword(s):

Nonlinear Systems ◽

Applied Mathematics ◽

Real Life ◽

Higher Order ◽

Local Analysis ◽

First Order ◽

The Family ◽

Lipschitz Constants ◽

Theoretical Results ◽

Real World Problems

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efficient and shows better performance as compared to the existing one.

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Ball convergence for a family of eight-order iterative schemes under hypotheses only of the first-order derivative

International Journal of Computer Mathematics ◽

10.1080/00207160.2019.1646904 ◽

2019 ◽

Vol 97 (1-2) ◽

pp. 444-454

Author(s):

Ali Saleh Alshomrani ◽

Ramandeep Behl ◽

Ioannis K. Argyros

Keyword(s):

Order Derivative ◽

Iterative Schemes ◽

First Order ◽

Ball Convergence

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Higher Order Two-Point Efficient Family of Halley Type Methods for Simple Roots

International Journal of Computational Methods ◽

10.1142/s0219876216410164 ◽

2016 ◽

Vol 13 (04) ◽

pp. 1641016 ◽

Cited By ~ 1

Author(s):

Ramandeep Behl ◽

S. S. Motsa

Keyword(s):

Nonlinear Equations ◽

Computational Cost ◽

Basins Of Attraction ◽

Higher Order ◽

The Other ◽

Sixth Order ◽

Order Derivative ◽

Initial Value ◽

First Order ◽

The Family

In this paper, we proposed a new highly efficient two-point sixth-order family of Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. In terms of computational cost, each member of the family requires two-function and two first-order derivative evaluations per iteration. On the account of the results obtained, it is found that our proposed methods are efficient and show better performance than existing sixth-order methods available in the literature. Further, it is also noted that larger basins of attraction belong to our methods as compared to the existing ones. On the other hand, the existing methods are slower and have darker basins while some of them are too sensitive upon the choice of the initial value.

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