Local convergence of seventh-order iterative method under weak conditions and its applications

Authors

Dongdong Ruan, Xiaofeng Wang, Yan Wang

Type: Article

Publication Date: 2025-04-23

Citations: 0

DOI: https://doi.org/10.1108/ec-08-2024-0775

View

Abstract

Purpose The purpose of this paper is to study local convergence and applications of a new seventh-order iterative method. Design/methodology/approach The order of convergence for the method is proved by using Taylor expansions. In addition, local convergence is studied under Lipschitz conditions with the first derivative. Findings By using Taylor expansions, we can show the convergence order of the method is seven. The specific domains of convergence and the solutions of nonlinear equations can be obtained by applying the method to practical physics problems and nonlinear systems. In this way, the uniqueness of the solution and error estimates also are analyzed. Originality/value In the proof of convergence order, Taylor expansions require third or higher derivatives. The applicability of the method is restricted. In order to extend the applicability of the method, local convergence is studied under Lipschitz conditions with the first derivative. Finally, in order to prove the applicability of the method, the method is applied to some physical problems and nonlinear systems.

Topics

Locations

Engineering Computations
View

Local Convergence of a Family of Iterative Methods with Sixth and Seventh Order Convergence Under Weak Conditions

2018-04-10

Tianbao Liu , Xiwen Qin , Peng Wang

In this paper, we study a local convergence analysis of a family of iterative methods with sixth and seventh order convergence for nonlinear equations, which was established by [Cordero et … In this paper, we study a local convergence analysis of a family of iterative methods with sixth and seventh order convergence for nonlinear equations, which was established by [Cordero et al. [2010] in “A family of iterative methods with sixth and seventh order convergence for nonlinear equations,” Math. Comput. Model. 52, 1190–1496]. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative. In our work, we make an attempt to study and establish a local convergence theorem by using only hypotheses the first derivative of the function and Lipschitz constants. We can also obtain error bounds and radii of convergence based on our results. Hence, the applicability of the methods is expanded. Moreover, we consider some different numerical examples and obtain the radii of convergence centered at the solution for different parameter values [Formula: see text] of the family. Furthermore, the basins of attraction of the family with different parameter values are also studied, which allow us to distinguish between the good and bad members of the family in terms of convergence and stable properties, and help us find the members with better or the best stable behavior.

Extended Local Convergence for Seventh order method with $\psi$-continuity condition in Banach Spaces

2021-12-12

Akanksha Saxena , J. P. Jaiswal , Kamal Raj Pardasani

In this article, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations. The point worth noting in our paper is that our analysis … In this article, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations. The point worth noting in our paper is that our analysis requires a weak hypothesis where the Fr\'echet derivative of the nonlinear operator satisfies the $\psi$-continuity condition and extends the applicability of the computation when both Lipschitz and H\"{o}lder conditions fail. The convergence in this study is shown under the hypotheses on the first order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some numerical examples are provided to show the performance of the method presented in this contribution over some existing schemes.

View PDF Chat

Extended Local Convergence for Seventh order method with $ψ$-continuity condition in Banach Spaces

2021-01-01

Akanksha Saxena , J. P. Jaiswal , Kamal Raj Pardasani

View PDF Chat

Local Convergence of an Optimal Eighth Order Method under Weak Conditions

2015-08-19

Ioannis K. Argyros , Ramandeep Behl , S. S. Motsa

We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence … We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

View PDF Chat

Broadening the convergence domain of Seventh-order method satisfying Lipschitz and Hölder conditions

2021-01-01

Akanksha Saxena , J. P. Jaiswal , Kamal Raj Pardasani

In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fréchet derivative belongs to the Lipschitz class. … In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fréchet derivative belongs to the Lipschitz class. The significance of our work is that it avoids the standard practice of Taylor expansion thereby, extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, this study provides radii of balls of convergence, the error bounds in terms of distances in addition to the uniqueness of the solution. Furthermore, generalization of this analysis satisfying Hölder continuity condition is provided since it is more relaxed than Lipschitz continuity condition. We have considered some numerical examples and computed the radii of the convergence balls.

View PDF Chat

Iterative methods of order four and five for solving non-linear system : A study on local convergence

2022-08-27

Jinny Ann John , Jayakumar Jayaraman

In this paper, we develop the local convergence analysis of Newton-like fourth and fifth order iterative methods for solving a system of non-linear equations. Earlier studies as in Petkovic (Multipoint … In this paper, we develop the local convergence analysis of Newton-like fourth and fifth order iterative methods for solving a system of non-linear equations. Earlier studies as in Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam,2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Kalyanasundaram et al (International Journal of Applied and Computational Mathematics 3 3:2213-2230, 2017) shows that the local convergence was proved using Taylor series expansion which involved the computation of derivatives of order higher than one. For the fourth and fifth order iterative methods under consideration in this paper, it is required that the functions should be at least five and six times differentiable respectively so that the method is applicable to find the solution. This restricts the applicability of the method and also the cost in finding the solution increases as it involves the computation of higher order derivatives. The local convergence analysis derived in this paper uses Lipschitz and ω-continuity conditions which involves only first derivative (present in the method) to prove the convergence. Moreover, the present study also provides details about the radii of domain of convergence and also estimates on error bounds. Therefore, it is evident that the present study enhances the applicability of the methods under consideration. The obtained results have been verified with suitable numerical illustrations.

View PDF Chat

An Extension on the Local Convergence for the Multi-Step Seventh Order Method with ψ-Continuity Condition in the Banach Spaces

2022-11-30

M.T. Darvishi , R. H. Al-Obaidi , Akanksha Saxena +2 more

The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires … The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article.

View PDF Chat

The Local Convergence of a Three-Step Sixth-Order Iterative Approach with the Basin of Attraction

2024-06-14

K. L. Saraswathi Devi , M. Prashanth , Eulalia Martı́nez +1 more

In this study, we introduce an iterative approach exhibiting sixth-order convergence for the solution of nonlinear equations. The method attains sixth-order convergence by using three evaluations of the function and … In this study, we introduce an iterative approach exhibiting sixth-order convergence for the solution of nonlinear equations. The method attains sixth-order convergence by using three evaluations of the function and two evaluations of the first-order derivative per iteration. We examined the theoretical convergence of our method through the convergence theorem, which substantiates the convergence order. Furthermore, we analyzed the local convergence of our proposed technique by employing a hypothesis that involves the first-order derivative of the function Θ alongside the Lipschitz conditions. To evaluate the performance and efficacy of our iterative method, we provide a comparative analysis against existing methods based on various standard numerical problems. Finally, graphical comparisons employing basins of attraction are presented to illustrate the dynamic behavior of the iterative method in the complex plane.

View PDF Chat

Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

2016-09-29

Ioannis K. Argyros , Ramandeep Behl , S. S. Motsa

In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. … In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.

View PDF Chat

An Efficient Iterative Method with Order of Convergence Seven for Nonlinear Equations

2012-11-01

Yanbo Hu , Liang Fang , Li Fang Guo +1 more

In this paper, we present a modified seventh-order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two … In this paper, we present a modified seventh-order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives at each step. Therefore the efficiency index of the presented method is 1.47577 which is better than that of classical Newton’s method 1.41421. Some numerical results demonstrate the efficiency and performance of the presented method.

A Modified Newton-type Method with Order of Convergence Seven for Solving Nonlinear Equations

2014-04-21

Liang Fang , Lin Pang

In this paper, we mainly study the iterative method for nonlinear equations. We present and analyze a modified seventh-order convergent Newton-type method for solving nonlinear equations. The method is free … In this paper, we mainly study the iterative method for nonlinear equations. We present and analyze a modified seventh-order convergent Newton-type method for solving nonlinear equations. The method is free from second derivatives. Some numerical results illustrate that the proposed method is more efficient and performs better than the classicalNewton's method.

Local Convergence of an Efﬁcient High Convergence Order Method Using Hypothesis Only on the First Derivative

2015-11-20

Ioannis K. Argyros , Ramandeep Behl , S. S. Motsa

We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier … We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only ﬁrst order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the ﬁrst order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction.

View PDF Chat

Order of Convergence and Dynamics of Newton–Gauss-Type Methods

2023-02-13

Ramya Sadananda , Santhosh George , Ioannis K. Argyros +1 more

On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in … On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of extreme importance, because higher order means that fewer iterations are carried out to achieve a predetermined error tolerance. In order to enhance the practicality of these methods by Zhongli Liu, the convergence analysis is carried out without the application of Taylor expansion and requires the operator to be only two times differentiable, unlike the earlier studies. A semilocal convergence analysis is provided. Furthermore, numerical experiments verifying the convergence criteria, comparative studies and the dynamics are discussed for better interpretation.

View PDF Chat

Broadening the convergence domain of Seventh-order method satisfying Lipschitz and H\"{o}lder conditions

2021-12-07

Akanksha Saxena , J. P. Jaiswal , Kamal Raj Pardasani

In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fr\'echet derivative belongs to the Lipschitz class. … In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fr\'echet derivative belongs to the Lipschitz class. The significance of our work is that it avoids the standard practice of Taylor expansion thereby, extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, this study provides radii of balls of convergence, the error bounds in terms of distances in addition to the uniqueness of the solution. Furthermore, generalization of this analysis satisfying H\"{o}lder continuity condition is provided since it is more relaxed than Lipschitz continuity condition. We have considered some numerical examples and computed the radii of the convergence balls.

View PDF Chat

An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics

2014-01-01

Anuradha Singh , J. P. Jaiswal

The prime objective of this paper is to design a new family of optimal eighth-order iterative methods by accelerating the order of convergence of the existing seventh-order method without using … The prime objective of this paper is to design a new family of optimal eighth-order iterative methods by accelerating the order of convergence of the existing seventh-order method without using more evaluations for finding simple root of nonlinear equations. Numerical comparisons have been carried out to demonstrate the efficiency and performance of the proposed method. Finally, we have compared new method with some existing eighth-order methods by basins of attraction and observed that the proposed scheme is more efficient.

View PDF Chat

A NOVEL AND PRECISE SEVENTH-ORDER NEWTON'S ITERATIVE METHOD FOR SOLVING NONLINEAR EQUATIONS

2015-09-15

Xiaofeng Wang , Dongyang Shi

In this paper, we study the problem of solving nonlinear equations. By using Taylor formulas and cupling method, we get a novel and robust three-step seventh-order iterative scheme. The contributed … In this paper, we study the problem of solving nonlinear equations. By using Taylor formulas and cupling method, we get a novel and robust three-step seventh-order iterative scheme. The contributed without memory method does not need to calculate higher order derivatives and has a large radius of convergence and higher efficiency of calculation.

Efficient sixth order iterative method free from higher derivatives for nonlinear equations

2022-01-01

In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new … In this paper, we proposed new iterative sixth order convergence method for solving nonlinear equations. The combination of the Taylor series and composition approach is used to derive the new method. Numerous methods have been developed by many researchers whenever the function’s second and higher order derivatives exist in the neighbourhood of the root. Computing the second and higher derivative of a function is a very cumbersome and time consuming task. In terms of low computation cost, the newly proposed method finds the best approximation to the root of non-linear equations by evaluating the function and its first derivative. The proposed method has been theoretically demonstrated to have sixth-order convergence. The proposed method has an efficiency index of 1.56. Several comparisons of the proposed method with the various existing iterative method of the same order have been performed on the number of problems. Finally, the computational results suggest that the newly proposed method is efficient compared to the well-known existing methods.

A new family of twentieth order convergent methods with applications to nonlinear systems in engineering

2022-12-31

Hameer Akhtar Abro , Muhammad Mujtaba Shaikh

A new family of iterative methods with a strong converging order of twenty to solve nonlinear equations and systems is presented in this study. A simple strategy of blending some … A new family of iterative methods with a strong converging order of twenty to solve nonlinear equations and systems is presented in this study. A simple strategy of blending some existing methods is used to develop the proposed family. The theoretical order of convergence is derived by employing Taylor’s series. The performance of the iterative methods in the proposed family is examined by applying the methods on real-world engineering problems. A nonlinear equation modeled by NASA for launching “Wind” satellite and some other complex applied systems, such as combustion problem, tank-reactor problem, kinematic synthesis mechanism, neurophysiology application and one boundary-value problem, have been solved to check the performance of the proposed family against other methods under similar test conditions. All the numerical results show that the proposed family converges very fast in complex and difficult problems as compared to other well-known methods. The methods in the proposed family have an efficiency improvement of 11.99% over the classical Newton method for scalar nonlinear equations.

View PDF Chat

Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

2024-01-08

Gagan Deep , Ioannis K. Argyros , Gaurav Verma +3 more

In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence … In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.

View PDF Chat

Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations

2022-09-21

Christopher I. Argyros , Ioannis K. Argyros , Samundra Regmi +2 more

The semi-local convergence is presented for a one parameter seventh order method to obtain solutions of Banach space valued nonlinear models. Existing works utilized hypotheses up to the eighth derivative … The semi-local convergence is presented for a one parameter seventh order method to obtain solutions of Banach space valued nonlinear models. Existing works utilized hypotheses up to the eighth derivative to prove the local convergence. But these high order derivatives are not on the method and they may not exist. Hence, the earlier results can only apply to solve equations containing operators that are at least eight times differentiable although this method may converge. That is why, we only apply the first derivative in our convergence result. Therefore, the results on calculable error estimates, convergence radius and uniqueness region for the solution are derived in contrast to the earlier proposals dealing with the less challenging local convergence case. Hence, we enlarge the applicability of these methods. The methodology used does not depend on the method and it is very general. Therefore, it can be used to extend other methods in an analogous way. Finally, some numerical tests are performed at the end of the text, where the convergence conditions are fulfilled.

View PDF Chat

The alternating direction implicit difference scheme and extrapolation method for a class of three dimensional hyperbolic equations with constant coefficients

2025-01-01

Zhaoxiang Zhang , Xuehua Yang , Song Wang

Seventh-order derivative-free iterative method for solving nonlinear systems

2015-01-19

Xiaofeng Wang , Tie Zhang , Weiyi Qian +1 more

On Newton-type methods with cubic convergence

2004-09-17

Herbert H. H. Homeier

A new continuation Newton-like method and its deformation

2000-06-01

Xinyuan Wu

A family of Steffensen type methods with seventh-order convergence

2012-06-01

Xiaofeng Wang , Tie Zhang

On the semilocal convergence of Newton–Kantorovich method under center-Lipschitz conditions

2013-07-13

J.M. Gutiérrez , Á. Alberto Magreñán , Natalia Romero

Improved local analysis for a certain class of iterative methods with cubic convergence

2011-10-28

Hongmin Ren , Ioannis K. Argyros

Families of Newton-like methods with fourth-order convergence

2012-11-12

Divya Jain

Families of fourth-order methods are presented which are obtained by existing third-order methods applied in succession with the secant method. Families of fourth-order methods are presented which are obtained by existing third-order methods applied in succession with the secant method.

A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

2004-09-14

Ioannis K. Argyros

A semilocal convergence analysis for directional Newton methods

2010-07-08

Ioannis K. Argyros

A semilocal convergence analysis for directional Newton methods in $n$-variables is provided in this study. Using weaker hypotheses than in the elegant related work by Y. Levin and A. Ben-Israel … A semilocal convergence analysis for directional Newton methods in $n$-variables is provided in this study. Using weaker hypotheses than in the elegant related work by Y. Levin and A. Ben-Israel and introducing the center-Lipschitz condition we provide under the same computational cost as in Levin and Ben-Israel a semilocal convergence analysis with the following advantages: weaker convergence conditions; larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the zero of the function. A numerical example where our results apply to solve an equation but not the ones in Levin and Ben-Israel is also provided in this study.

View PDF Chat

Extending the Applicability of a Seventh Order Method Without Inverses of Derivatives Under Weak Conditions

2019-12-20

Ioannis K. Argyros , Santhosh George

Numerical Methods for Grid Equations

1989-01-01

A. A. Samarskiĭ , Evgenii S. Nikolaev

b] three parameters /1, /2, /3, where /1 and /2 (for D = B = E) are bounds on the symmetric part of the operator A:where Al = 0.5(A -A*) … b] three parameters /1, /2, /3, where /1 and /2 (for D = B = E) are bounds on the symmetric part of the operator A:where Al = 0.5(A -A*) is the skew-symmetric part of A. Choosing T from the minimum norm condition for the transition or resolving operator, in all cases we obtain an increase in the number of iterations in comparison with the case A = A * .

Extended ball convergence of a seventh order derivative free method for solving system of equations with applications

2022-06-21

Ioannis K. Argyros , Debasis Sharma , Christopher I. Argyros +2 more

Convergence ball of a new fourth-order method for finding a zero of the derivative

2024-01-01

Xiaofeng Wang , Dongdong Ruan

<abstract>There are numerous applications for finding zero of derivatives in function optimization. In this paper, a two-step fourth-order method was presented for finding a zero of the derivative. In the … <abstract>There are numerous applications for finding zero of derivatives in function optimization. In this paper, a two-step fourth-order method was presented for finding a zero of the derivative. In the research process of iterative methods, determining the ball of convergence was one of the important issues. This paper discussed the radii of the convergence ball, uniqueness of the solution, and the measurable error distances. In particular, in contrast to Wang's method under hypotheses up to the fourth derivative, the local convergence of the new method was only analyzed under hypotheses up to the second derivative, and the convergence order of the new method was increased to four. Furthermore, different radii of the convergence ball was determined according to different weaker hypotheses. Finally, the convergence criteria was verified by three numerical examples and the new method was compared with Wang's method and the same order method by numerical experiments. The experimental results showed that the convergence order of the new method is four and the new method has higher accuracy at the same cost, so the new method is finer.</abstract>

On the convergence of a new fourth-order method for finding a zero of a derivative

2024-01-01

Dongdong Ruan , Xiaofeng Wang

<abstract>On the basis of Wang's method, a new fourth-order method for finding a zero of a derivative was presented. Under the hypotheses that the third and fourth order derivatives of … <abstract>On the basis of Wang's method, a new fourth-order method for finding a zero of a derivative was presented. Under the hypotheses that the third and fourth order derivatives of nonlinear function were bounded, the local convergence of a new fourth-order method was studied. The error estimate, the order of convergence, and uniqueness of the solution were also discussed. In particular, Herzberger's matrix method was used to obtain the convergence order of the new method to four. By comparing the new method with Wang's method and the same order method, numerical illustrations showed that the new method has a higher order of convergence and accuracy.</abstract>

Local convergence analysis of a novel derivative-free method with and without memory for solving nonlinear systems

2025-02-18

Xiaofeng Wang , Ning Shang