Special Issue "Iterative Methods for Solving Nonlinear Equations and Systems"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 30 June 2019

Special Issue Editors

Guest Editor
Prof. Dr. Juan R. Torregrosa

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022-València, Spain
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Guest Editor
Prof. Dr. Alicia Cordero

Inst. Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022-València, Spain
Website | E-Mail
Guest Editor
Dr. Fazlollah Soleymani

Institute for Advanced Studies in Basic Sciences, Zanjan, Zanjan, Iran
Website | E-Mail
Interests: Numerical linear algebra; option pricing PDEs; computational methods for SDEs; iterative methods

Special Issue Information

Dear Colleagues,

Solving nonlinear equations and systems is a non-trivial task that involves many areas of Science and Technology. Usually, it is not affordable in a direct way, and iterative algorithms play a fundamental role in their approach. This is an area of research that has experienced exponential growth in the last years.

The main theme of this Special Issue, which is not the unique, is the design, analysis of convergence, and stability and application of new iterative schemes for solving nonlinear problems to practical problems. This includes methods with and without memory, with derivatives or derivative-free, with real or complex dynamics associated with them and an analysis of their convergence that can be local, semi-local, or global.

Prof. Dr. Juan R. Torregrosa
Prof. Dr. Alicia Cordero
Dr. Fazlollah Soleymani
Guest Editors

Manuscript Submission Information

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Keywords

  • nonlinear problems
  • iterative methods
  • convergence
  • efficiency
  • chaotic behavior
  • complex or real dynamics

Published Papers (11 papers)

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Research

Open AccessArticle
A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots
Mathematics 2019, 7(4), 339; https://doi.org/10.3390/math7040339
Received: 15 January 2019 / Revised: 21 March 2019 / Accepted: 22 March 2019 / Published: 9 April 2019
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Abstract
The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function [...] Read more.
The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application
Mathematics 2019, 7(4), 322; https://doi.org/10.3390/math7040322
Received: 26 February 2019 / Revised: 24 March 2019 / Accepted: 26 March 2019 / Published: 30 March 2019
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Abstract
The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the [...] Read more.
The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Open AccessArticle
Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations
Mathematics 2019, 7(3), 306; https://doi.org/10.3390/math7030306
Received: 21 January 2019 / Revised: 10 March 2019 / Accepted: 18 March 2019 / Published: 26 March 2019
Cited by 1 | PDF Full-text (1297 KB) | HTML Full-text | XML Full-text
Abstract
Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique [...] Read more.
Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Open AccessArticle
Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications
Mathematics 2019, 7(3), 299; https://doi.org/10.3390/math7030299
Received: 3 March 2019 / Revised: 20 March 2019 / Accepted: 21 March 2019 / Published: 24 March 2019
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Abstract
The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of [...] Read more.
The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
Study of a High Order Family: Local Convergence and Dynamics
Mathematics 2019, 7(3), 225; https://doi.org/10.3390/math7030225
Received: 10 December 2018 / Revised: 22 February 2019 / Accepted: 25 February 2019 / Published: 28 February 2019
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Abstract
The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater [...] Read more.
The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We investigate the dynamics of the method. To validate the theoretical results obtained, a real-world application related to chemistry is provided. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Open AccessArticle
Extended Local Convergence for the Combined Newton-Kurchatov Method Under the Generalized Lipschitz Conditions
Mathematics 2019, 7(2), 207; https://doi.org/10.3390/math7020207
Received: 4 February 2019 / Revised: 19 February 2019 / Accepted: 20 February 2019 / Published: 23 February 2019
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Abstract
We present a local convergence of the combined Newton-Kurchatov method for solving Banach space valued equations. The convergence criteria involve derivatives until the second and Lipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notion of the restricted convergence [...] Read more.
We present a local convergence of the combined Newton-Kurchatov method for solving Banach space valued equations. The convergence criteria involve derivatives until the second and Lipschitz-type conditions are satisfied, as well as a new center-Lipschitz-type condition and the notion of the restricted convergence region. These modifications of earlier conditions result in a tighter convergence analysis and more precise information on the location of the solution. These advantages are obtained under the same computational effort. Using illuminating examples, we further justify the superiority of our new results over earlier ones. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions
Mathematics 2019, 7(1), 89; https://doi.org/10.3390/math7010089
Received: 17 December 2018 / Revised: 5 January 2019 / Accepted: 9 January 2019 / Published: 16 January 2019
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Abstract
We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these [...] Read more.
We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
A Few Iterative Methods by Using [1,n]-Order Padé Approximation of Function and the Improvements
Mathematics 2019, 7(1), 55; https://doi.org/10.3390/math7010055
Received: 15 November 2018 / Revised: 28 December 2018 / Accepted: 30 December 2018 / Published: 7 January 2019
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Abstract
In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [1,n]-order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the [...] Read more.
In this paper, a few single-step iterative methods, including classical Newton’s method and Halley’s method, are suggested by applying [ 1 , n ] -order Padé approximation of function for finding the roots of nonlinear equations at first. In order to avoid the operation of high-order derivatives of function, we modify the presented methods with fourth-order convergence by using the approximants of the second derivative and third derivative, respectively. Thus, several modified two-step iterative methods are obtained for solving nonlinear equations, and the convergence of the variants is then analyzed that they are of the fourth-order convergence. Finally, numerical experiments are given to illustrate the practicability of the suggested variants. Henceforth, the variants with fourth-order convergence have been considered as the imperative improvements to find the roots of nonlinear equations. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
A Third Order Newton-Like Method and Its Applications
Mathematics 2019, 7(1), 31; https://doi.org/10.3390/math7010031
Received: 8 October 2018 / Revised: 21 November 2018 / Accepted: 7 December 2018 / Published: 30 December 2018
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Abstract
In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method [...] Read more.
In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
Open AccessArticle
An Efficient Family of Optimal Eighth-Order Multiple Root Finders
Mathematics 2018, 6(12), 310; https://doi.org/10.3390/math6120310
Received: 13 November 2018 / Revised: 3 December 2018 / Accepted: 5 December 2018 / Published: 7 December 2018
Cited by 3 | PDF Full-text (1028 KB) | HTML Full-text | XML Full-text
Abstract
Finding a repeated zero for a nonlinear equation f(x)=0, f:IRR has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified [...] Read more.
Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I R R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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Open AccessArticle
Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space
Mathematics 2018, 6(12), 306; https://doi.org/10.3390/math6120306
Received: 16 October 2018 / Revised: 25 November 2018 / Accepted: 28 November 2018 / Published: 5 December 2018
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Abstract
Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order [...] Read more.
Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order convergence and its non-dependence on the initial value. For some special cases of divergence for the H-H-H method, we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybrid second order method for orthogonal projection onto parametric curve in an n-dimensional Euclidean space (hereafter, our method). Our method essentially utilizes hybrid iteration, so it converges faster than current methods with a second order convergence and remains independent from the initial value. We provide some numerical examples to confirm robustness and high efficiency of the method. Full article
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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