Symmetry

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16 pages, 2221 KiB

Open AccessArticle

A Convex Combination Approach for Mean-Based Variants of Newton’s Method

by Alicia Cordero, Jonathan Franceschi, Juan R. Torregrosa and Anna C. Zagati

Symmetry 2019, 11(9), 1106; https://doi.org/10.3390/sym11091106 - 2 Sep 2019

Cited by 5 | Viewed by 2445

Several authors have designed variants of Newton’s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton’s method (MBN) are re-analyzed from the [...] Read more.

Several authors have designed variants of Newton’s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton’s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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11 pages, 423 KiB

Open AccessArticle

A Variant of Chebyshev’s Method with 3αth-Order of Convergence by Using Fractional Derivatives

by Alicia Cordero, Ivan Girona and Juan R. Torregrosa

Symmetry 2019, 11(8), 1017; https://doi.org/10.3390/sym11081017 - 6 Aug 2019

Cited by 15 | Viewed by 2988

Abstract

In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev’s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo [...] Read more.

In this manuscript, we propose several iterative methods for solving nonlinear equations whose common origin is the classical Chebyshev’s method, using fractional derivatives in their iterative expressions. Due to the symmetric duality of left and right derivatives, we work with right-hand side Caputo and Riemann–Liouville fractional derivatives. To increase as much as possible the order of convergence of the iterative scheme, some improvements are made, resulting in one of them being of

3 α

-th order. Some numerical examples are provided, along with an study of the dependence on initial estimations on several test problems. This results in a robust performance for values of

α

close to one and almost any initial estimation. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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11 pages, 259 KiB

Open AccessArticle

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

by R. A. Alharbey, Ioannis K. Argyros and Ramandeep Behl

Symmetry 2019, 11(8), 1002; https://doi.org/10.3390/sym11081002 - 3 Aug 2019

Viewed by 1963

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

15 pages, 337 KiB

Open AccessArticle

Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

by Ioannis K Argyros, Santhosh George, Chandhini Godavarma and Alberto A Magreñán

Symmetry 2019, 11(8), 981; https://doi.org/10.3390/sym11080981 - 2 Aug 2019

Viewed by 2277

Abstract

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations [...] Read more.

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton–Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299–315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection–diffusion equations further validate the theoretical results. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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30 pages, 1470 KiB

Open AccessArticle

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

by R. A. Alharbey, Munish Kansal, Ramandeep Behl and J. A. Tenreiro Machado

Symmetry 2019, 11(7), 837; https://doi.org/10.3390/sym11070837 - 26 Jun 2019

Cited by 5 | Viewed by 2239

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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17 pages, 1403 KiB

Open AccessArticle

Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

by Janak Raj Sharma, Sunil Kumar and Ioannis K. Argyros

Symmetry 2019, 11(6), 766; https://doi.org/10.3390/sym11060766 - 5 Jun 2019

Cited by 14 | Viewed by 2305

Abstract

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, [...] Read more.

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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21 pages, 362 KiB

Open AccessArticle

Strong Convergence of a System of Generalized Mixed Equilibrium Problem, Split Variational Inclusion Problem and Fixed Point Problem in Banach Spaces

by Mujahid Abbas, Yusuf Ibrahim, Abdul Rahim Khan and Manuel de la Sen

Symmetry 2019, 11(5), 722; https://doi.org/10.3390/sym11050722 - 27 May 2019

Cited by 11 | Viewed by 2740

Abstract

The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone operators, and fixed-point problems of a countable [...] Read more.

The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone operators, and fixed-point problems of a countable family of left Bregman, strongly asymptotically non-expansive mappings in uniformly convex and uniformly smooth Banach spaces. A strong convergence theorem for the above problems are established. As an application, we solve a generalized mixed equilibrium problem, split Hammerstein integral equations, and a fixed-point problem, and provide a numerical example to support better findings of our result. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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11 pages, 308 KiB

Open AccessArticle

Sixteenth-Order Optimal Iterative Scheme Based on Inverse Interpolatory Rational Function for Nonlinear Equations

by Mehdi Salimi and Ramandeep Behl

Symmetry 2019, 11(5), 691; https://doi.org/10.3390/sym11050691 - 19 May 2019

Cited by 3 | Viewed by 2366

Abstract

The principal motivation of this paper is to propose a general scheme that is applicable to every existing multi-point optimal eighth-order method/family of methods to produce a further sixteenth-order scheme. By adopting our technique, we can extend all the existing optimal eighth-order schemes [...] Read more.

The principal motivation of this paper is to propose a general scheme that is applicable to every existing multi-point optimal eighth-order method/family of methods to produce a further sixteenth-order scheme. By adopting our technique, we can extend all the existing optimal eighth-order schemes whose first sub-step employs Newton’s method for sixteenth-order convergence. The developed technique has an optimal convergence order regarding classical Kung-Traub conjecture. In addition, we fully investigated the computational and theoretical properties along with a main theorem that demonstrates the convergence order and asymptotic error constant term. By using Mathematica-11 with its high-precision computability, we checked the efficiency of our methods and compared them with existing robust methods with same convergence order. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

11 pages, 287 KiB

Open AccessArticle

Modified Optimal Class of Newton-Like Fourth-Order Methods for Multiple Roots

by Munish Kansal, Ramandeep Behl, Mohammed Ali A. Mahnashi and Fouad Othman Mallawi

Symmetry 2019, 11(4), 526; https://doi.org/10.3390/sym11040526 - 11 Apr 2019

Cited by 6 | Viewed by 2356

Abstract

Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergence analysis that demonstrated the establishment of fourth-order [...] Read more.

Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergence analysis that demonstrated the establishment of fourth-order convergence of the developed methods. It is interesting to note that some existing schemes are found to be the special cases of our proposed scheme. Numerical experiments have been performed on a good number of problems arising from different disciplines such as the fractional conversion problem of a chemical reactor, continuous stirred tank reactor problem, and Planck’s radiation law problem. Computational results demonstrates that suggested methods are better and efficient than their existing counterparts. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

17 pages, 829 KiB

Open AccessFeature PaperArticle

An Efficient Class of Traub-Steffensen-Like Seventh Order Multiple-Root Solvers with Applications

by Janak Raj Sharma, Deepak Kumar and Ioannis K. Argyros

Symmetry 2019, 11(4), 518; https://doi.org/10.3390/sym11040518 - 10 Apr 2019

Cited by 9 | Viewed by 2041

Abstract

Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing [...] Read more.

Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing a new family of higher order derivative-free solvers for computing multiple zeros by using a simple approach. The stability of the techniques is checked through complex geometry shown by drawing basins of attraction. Applicability is demonstrated on practical problems, which illustrates the efficient convergence behavior. Moreover, the comparison of numerical results shows that the proposed derivative-free techniques are good competitors of the existing techniques that require derivative evaluations in the iteration. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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14 pages, 4917 KiB

Open AccessArticle

Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

by Ramandeep Behl, M. Salimi, M. Ferrara, S. Sharifi and Samaher Khalaf Alharbi

Symmetry 2019, 11(2), 239; https://doi.org/10.3390/sym11020239 - 15 Feb 2019

Cited by 26 | Viewed by 4375

Abstract

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable [...] Read more.

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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9 pages, 240 KiB

Open AccessArticle

Two-Step Solver for Nonlinear Equations

by Ioannis K. Argyros, Stepan Shakhno and Halyna Yarmola

Symmetry 2019, 11(2), 128; https://doi.org/10.3390/sym11020128 - 23 Jan 2019

Cited by 8 | Viewed by 2461

Abstract

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence

1 + \sqrt{2}

. We study the local and a semilocal convergence using weaker conditions in order [...] Read more.

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence

1 + \sqrt{2}

. We study the local and a semilocal convergence using weaker conditions in order to extend the applicability of the solver. Finally, we present the numerical example that confirms the theoretical results. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

13 pages, 327 KiB

Open AccessArticle

Local Convergence of a Family of Weighted-Newton Methods

by Ramandeep Behl, Ioannis K. Argyros, J.A. Tenreiro Machado and Ali Saleh Alshomrani

Symmetry 2019, 11(1), 103; https://doi.org/10.3390/sym11010103 - 17 Jan 2019

Cited by 1 | Viewed by 2850

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Symmetry with Operator Theory and Equations)

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