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Mathematics
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Multipoint Methods for the Solution of Nonlinear Equations
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Multipoint Methods for the Solution of Nonlinear Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (15 January 2021) | Viewed by 11699

Special Issue Editors

Prof. Dr. Young I. Kim

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Guest Editor
Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea
Interests: mathematical computing, dynamical systems; numerical analysis; complex dynamics
Prof. Dr. Beny Neta

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Guest Editor
Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, USA
Interests: partial differential equations; numerical analysis; parallel computing; finite elements
Prof. Dr. Juan Ramón Torregrosa Sánchez

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Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ángel Alberto Magreñán

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Department of Mathematics and Computation, University of La Rioja, Madre de Dios 53, 26006 Logroño, Spain
Interests: applied mathematics; mathematical problems; new trends in mathematical education; E-learning
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

A number of real-life natural phenomena, such as the movement of a hurricane, the spinning of a tennis ball, and the turbulent flow of a waterfall splashing against the rocks, are governed by a set of nonlinear integro-differential equations or algebraic equations. As the exact solution of such a nonlinear governing equation is generally infeasible, we are seeking its approximate solution accurately under the prescribed error bound. Iterative multipoint methods are widely employed to efficiently find the relevant approximate solution. Such iterative methods can be regarded as discrete dynamical systems by treating the iteration index as the evolution time variable. This aspect of the dynamics underlying the iterative methods would induce the interesting limit behavior of the periodic and chaotic character.

The main objective of this Special Issue is for authors working in various scientific disciplines to publish their research works, as well as to share their strategic experiences and developments in designing efficient (either optimal in the sense of Kung-Traub or higher-order convergent with less number of function evaluations) iterative multipoint methods for solving the nonlinear problems under consideration. The importance of collecting the latest innovative articles and pursuing new technologies is increasingly recognized to enhance “Multipoint Methods for the Solution of Nonlinear Equations”.

Prof. Dr. Young I. Kim
Prof. Dr. Beny Neta
Prof. Dr. Juan R. Torregrosa
Prof. Dr. Ángel Alberto Magreñán
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Iterative root-finding methods
  • Stability analysis
  • Fixed-point theory in the Banach space
  • Discrete dynamics
  • Parameter space
  • Dynamical plane
  • Basin of attraction
  • Initial-boundary value problems
  • Kung-Traub conjecture
  • Efficiency index
  • Complex dynamics on the Riemann sphere
  • Limit behavior of a long-term orbit

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Published Papers (5 papers)

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Research

19 pages, 1034 KiB  
Article
An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros
by Ramandeep Behl, Sonia Bhalla, Ángel Alberto Magreñán and Alejandro Moysi
Mathematics 2021, 9(5), 546; https://doi.org/10.3390/math9050546 - 5 Mar 2021
Cited by 13 | Viewed by 2354
Abstract
In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, [...] Read more.
In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed for particular values of multiple roots. Afterward, it concludes in general. Moreover, the effectiveness of the presented methods are certified on some applications of nonlinear equations and compared with the earlier derivative and derivative-free schemes. The obtained results depict better performance than the existing methods. Full article
(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)
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17 pages, 1784 KiB  
Article
Computational Bifurcations Occurring on Red Fixed Components in the λ-Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map
by Young Hee Geum and Young Ik Kim
Mathematics 2020, 8(5), 763; https://doi.org/10.3390/math8050763 - 11 May 2020
Cited by 2 | Viewed by 1841
Abstract
Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. [...] Read more.
Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The λ -parameters for periodic orbits in the parameter plane are painted in different colors depending on their periods, and the bounded or chaotic ones are colored black to illustrate λ -dependent connected components. When a red fixed component in the parameter plane branches into a q-periodic component, we encounter geometric bifurcation phenomena whose characteristics determine the desired boundary equation and bifurcation point. Computational results along with illustrated components support the bifurcation phenomena underlying this paper. Full article
(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)
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13 pages, 831 KiB  
Article
Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders
by Min-Young Lee and Young Ik Kim
Mathematics 2020, 8(1), 51; https://doi.org/10.3390/math8010051 - 1 Jan 2020
Cited by 1 | Viewed by 1781
Abstract
Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of [...] Read more.
Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper. Full article
(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)
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10 pages, 263 KiB  
Article
Extending the Applicability of Stirling’s Method
by Cristina Amorós, Ioannis K. Argyros, Á. Alberto Magreñán, Samundra Regmi, Rubén González and Juan Antonio Sicilia
Mathematics 2020, 8(1), 35; https://doi.org/10.3390/math8010035 - 31 Dec 2019
Cited by 3 | Viewed by 2052
Abstract
Stirling’s method is considered as an alternative to Newton’s method when the latter fails to converge to a solution of a nonlinear equation. Both methods converge quadratically under similar convergence criteria and require the same computational effort. However, Stirling’s method has shortcomings too. [...] Read more.
Stirling’s method is considered as an alternative to Newton’s method when the latter fails to converge to a solution of a nonlinear equation. Both methods converge quadratically under similar convergence criteria and require the same computational effort. However, Stirling’s method has shortcomings too. In particular, contractive conditions are assumed to show convergence. However, these conditions limit its applicability. The novelty of our paper lies in the fact that our convergence criteria do not require contractive conditions. Hence, we extend its applicability of Stirling’s method. Numerical examples illustrate our new findings. Full article
(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)
14 pages, 314 KiB  
Article
Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications
by Francisco I. Chicharro, Alicia Cordero, Neus Garrido and Juan R. Torregrosa
Mathematics 2019, 7(12), 1194; https://doi.org/10.3390/math7121194 - 5 Dec 2019
Cited by 4 | Viewed by 2496
Abstract
A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the [...] Read more.
A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods. Full article
(This article belongs to the Special Issue Multipoint Methods for the Solution of Nonlinear Equations)
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