Research

14 pages, 787 KiB

Open AccessArticle

A New Optimal Family of Schröder’s Method for Multiple Zeros

by Ramandeep Behl, Arwa Jeza Alsolami, Bruno Antonio Pansera, Waleed M. Al-Hamdan, Mehdi Salimi and Massimiliano Ferrara

Mathematics 2019, 7(11), 1076; https://doi.org/10.3390/math7111076 - 08 Nov 2019

Cited by 8 | Viewed by 2078

Abstract

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the [...] Read more.

Here, we suggest a high-order optimal variant/modification of Schröder’s method for obtaining the multiple zeros of nonlinear uni-variate functions. Based on quadratically convergent Schröder’s method, we derive the new family of fourth -order multi-point methods having optimal convergence order. Additionally, we discuss the theoretical convergence order and the properties of the new scheme. The main finding of the present work is that one can develop several new and some classical existing methods by adjusting one of the parameters. Numerical results are given to illustrate the execution of our multi-point methods. We observed that our schemes are equally competent to other existing methods. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

28 pages, 1613 KiB

Open AccessArticle

Approximating Fixed Points of Bregman Generalized α-Nonexpansive Mappings

by Kanikar Muangchoo, Poom Kumam, Yeol Je Cho, Sompong Dhompongsa and Sakulbuth Ekvittayaniphon

Mathematics 2019, 7(8), 709; https://doi.org/10.3390/math7080709 - 06 Aug 2019

Cited by 2 | Viewed by 3038

Abstract

In this paper, we introduce a new class of Bregman generalized

α

-nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized

α

-nonexpansive mappings in Banach [...] Read more.

In this paper, we introduce a new class of Bregman generalized

α

-nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized

α

-nonexpansive mappings in Banach spaces. A numerical example is given to illustrate the main results of fixed point approximation using Halpern’s algorithm. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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24 pages, 423 KiB

Open AccessArticle

A Modified Self-Adaptive Conjugate Gradient Method for Solving Convex Constrained Monotone Nonlinear Equations for Signal Recovery Problems

by Auwal Bala Abubakar, Poom Kumam, Aliyu Muhammed Awwal and Phatiphat Thounthong

Mathematics 2019, 7(8), 693; https://doi.org/10.3390/math7080693 - 01 Aug 2019

Cited by 17 | Viewed by 3370

Abstract

In this article, we propose a modified self-adaptive conjugate gradient algorithm for handling nonlinear monotone equations with the constraints being convex. Under some nice conditions, the global convergence of the method was established. Numerical examples reported show that the method is promising and [...] Read more.

In this article, we propose a modified self-adaptive conjugate gradient algorithm for handling nonlinear monotone equations with the constraints being convex. Under some nice conditions, the global convergence of the method was established. Numerical examples reported show that the method is promising and efficient for solving monotone nonlinear equations. In addition, we applied the proposed algorithm to solve sparse signal reconstruction problems. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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27 pages, 474 KiB

Open AccessArticle

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

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

Mathematics 2019, 7(6), 492; https://doi.org/10.3390/math7060492 - 29 May 2019

Cited by 2 | Viewed by 1907

Abstract

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of [...] Read more.

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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13 pages, 297 KiB

Open AccessArticle

Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

by Ioannis K. Argyros, Ángel Alberto Magreñán, Lara Orcos and Íñigo Sarría

Mathematics 2019, 7(5), 463; https://doi.org/10.3390/math7050463 - 23 May 2019

Cited by 3 | Viewed by 2122

Abstract

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued [...] Read more.

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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11 pages, 586 KiB

Open AccessArticle

On a Variational Method for Stiff Differential Equations Arising from Chemistry Kinetics

by Sergio Amat, María José Legaz and Juan Ruiz-Álvarez

Mathematics 2019, 7(5), 459; https://doi.org/10.3390/math7050459 - 21 May 2019

Cited by 11 | Viewed by 2665

Abstract

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test [...] Read more.

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test its numerical performance through some experiments. The main advantage with respect to other implicit methods is that our approach has a global convergence. The other approaches need to ensure convergence of the iterative scheme used to approximate the associated nonlinear equations that appear for the implicitness. Notice that these iterative methods, for these nonlinear equations, have bounded basins of attraction. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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22 pages, 749 KiB

Open AccessArticle

Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks

by Chang-Cheng Wei, Haidar Ali, Muhammad Ahsan Binyamin, Muhammad Nawaz Naeem and Jia-Bao Liu

Mathematics 2019, 7(4), 368; https://doi.org/10.3390/math7040368 - 23 Apr 2019

Cited by 25 | Viewed by 2889

Abstract

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical [...] Read more.

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks

H D N 3 (r)

,

T H D N 3 (r)

,

R H D N 3 (r)

,

C H D N 3 (r)

, and compute exact results for topological indices which are based on degrees of end vertices. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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14 pages, 293 KiB

Open AccessArticle

Convergence Analysis of Weighted-Newton Methods of Optimal Eighth Order in Banach Spaces

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

Mathematics 2019, 7(2), 198; https://doi.org/10.3390/math7020198 - 19 Feb 2019

Cited by 2 | Viewed by 1981

Abstract

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, [...] Read more.

We generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study their local convergence. In a previous study, the Taylor expansion of higher order derivatives is employed which may not exist or may be very expensive to compute. However, the hypotheses of the present study are based on the first Fréchet-derivative only, thereby the application of methods is expanded. New analysis also provides the radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of derivatives of higher order. Moreover, the order of convergence for the methods is verified by using computational order of convergence or approximate computational order of convergence without using higher order derivatives. Numerical examples are provided to verify the theoretical results and to show the good convergence behavior. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

13 pages, 268 KiB

Open AccessArticle

Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

by Ioannis Argyros, Stepan Shakhno and Yurii Shunkin

Mathematics 2019, 7(1), 99; https://doi.org/10.3390/math7010099 - 18 Jan 2019

Cited by 6 | Viewed by 2870

Abstract

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of [...] Read more.

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

13 pages, 246 KiB

Open AccessArticle

Resistance Distance in the Double Corona Based on R-Graph

by Li Zhang, Jing Zhao, Jia-Bao Liu and Salama Nagy Daoud

Mathematics 2019, 7(1), 92; https://doi.org/10.3390/math7010092 - 17 Jan 2019

Cited by 4 | Viewed by 3184

Abstract

Let

G_{0}

be a connected graph on n vertices and m edges. The R-graph

R (G_{0})

of

G_{0}

is a graph obtained from

G_{0}

by adding a new vertex corresponding to each edge of [...] Read more.

Let

G_{0}

be a connected graph on n vertices and m edges. The R-graph

R (G_{0})

of

G_{0}

is a graph obtained from

G_{0}

by adding a new vertex corresponding to each edge of

G_{0}

and by joining each new vertex to the end points of the edge corresponding to it. Let

G_{1}

and

G_{2}

be graphs on

n_{1}

and

n_{2}

vertices, respectively. The R-graph double corona

G_{0}^{(R)} \circ {G_{1}, G_{2}}

of

G_{0}

,

G_{1}

and

G_{2}

, is the graph obtained by taking one copy of

R (G_{0})

, n copies of

G_{1}

and m copies of

G_{2}

and then by joining the i-th old-vertex of

R (G_{0})

to every vertex of the i-th copy of

G_{1}

and the j-th new vertex of

R (G_{0})

to every vertex of the j-th copy of

G_{2}

. In this paper, we consider resistance distance in

G_{0}^{(R)} \circ {G_{1}, G_{2}}

. Moreover, we give an example to illustrate the correction and efficiency of the proposed method. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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13 pages, 317 KiB

Open AccessArticle

Extending the Applicability of Two-Step Solvers for Solving Equations

by Ioannis K. Argyros and Stepan Shakhno

Mathematics 2019, 7(1), 62; https://doi.org/10.3390/math7010062 - 08 Jan 2019

Cited by 4 | Viewed by 1988

Abstract

We present a local convergence of two-step solvers for solving nonlinear operator equations under the generalized Lipschitz conditions for the first- and second-order derivatives and for the first order divided differences. In contrast to earlier works, we use our new idea of center [...] Read more.

We present a local convergence of two-step solvers for solving nonlinear operator equations under the generalized Lipschitz conditions for the first- and second-order derivatives and for the first order divided differences. In contrast to earlier works, we use our new idea of center average Lipschitz conditions, through which, we define a subset of the original domain that also contains the iterates. Then, the remaining average Lipschitz conditions are at least as tight as the corresponding ones in earlier works. This way, we obtain weaker sufficient convergence criteria, larger radius of convergence, tighter error estimates, and better information on the solution. These extensions require the same effort, since the new Lipschitz functions are special cases of the ones in earlier works. Finally, we give a numerical example that confirms the theoretical results, and compares favorably to the results from previous works. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

13 pages, 308 KiB

Open AccessArticle

Convergence Ball and Complex Geometry of an Iteration Function of Higher Order

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

Mathematics 2019, 7(1), 28; https://doi.org/10.3390/math7010028 - 29 Dec 2018

Cited by 3 | Viewed by 1973

Abstract

Higher-order derivatives are used to determine the convergence order of iterative methods. However, such derivatives are not present in the formulas. Therefore, the assumptions on the higher-order derivatives of the function restrict the applicability of methods. Our convergence analysis of an eighth-order method [...] Read more.

Higher-order derivatives are used to determine the convergence order of iterative methods. However, such derivatives are not present in the formulas. Therefore, the assumptions on the higher-order derivatives of the function restrict the applicability of methods. Our convergence analysis of an eighth-order method uses only the derivative of order one. The convergence results so obtained are applied to some real problems, which arise in science and engineering. Finally, stability of the method is checked through complex geometry shown by drawing basins of attraction of the solutions. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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17 pages, 357 KiB

Open AccessArticle

New Iterative Methods for Solving Nonlinear Problems with One and Several Unknowns

by Ramandeep Behl, Alicia Cordero, Juan R. Torregrosa and Ali Saleh Alshomrani

Mathematics 2018, 6(12), 296; https://doi.org/10.3390/math6120296 - 01 Dec 2018

Cited by 5 | Viewed by 2963

Abstract

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free [...] Read more.

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free parameter/s at both substeps, as well as small residual errors and asymptotic error constants. In addition, we generalize these schemes to nonlinear systems preserving the order of convergence. Regarding the applicability of the proposed techniques, we choose some real-world problems, namely chemical fractional conversion and the trajectory of an electron in the air gap between two parallel plates, in order to study the multi-factor effect, fractional conversion of species in a chemical reactor, Hammerstein integral equation, and a boundary value problem. Moreover, we find that our proposed schemes run better than or equal to the existing ones in the literature. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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8 pages, 249 KiB

Open AccessArticle

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

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

Mathematics 2018, 6(11), 260; https://doi.org/10.3390/math6110260 - 17 Nov 2018

Cited by 6 | Viewed by 2173

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

10 pages, 248 KiB

Open AccessArticle

Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

by Ioannis K. Argyros and Santhosh George

Mathematics 2018, 6(11), 233; https://doi.org/10.3390/math6110233 - 30 Oct 2018

Cited by 1 | Viewed by 1775

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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