Simple Roots Of Nonlinear Equations Research Articles

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.

Iterative method improving Newton's method with higher order

In the present article, we have constructed and looked at improving two- and three-step iterative methods to locate simple roots of non-linear equations. It appears that the three-step iterative method converges to the eighth order. The whole aim of this research is to derive and present a new modified iterative method of higher order and to obtain less iteration than the classical Newton method for solving nonlinear equations of simple roots. The present technique has to evaluate three functions and two first derivatives in each iteration. The advantage of the proposed method has been observed to have at least better performance effective and more stability by comparing with the other methods for the same or less than order. Also, noted that our method gives better results in terms of the number of iterations. To demonstrate the efficacy and popularity, different numerical illustrations are provided for the recommended techniques.

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.

Stability Analysis of Simple Root Seeker for Nonlinear Equation

In this paper, the stability of a class of Liu–Wang’s optimal eighth-order single-parameter iterative methods for solving simple roots of nonlinear equations was studied by applying them to arbitrary quadratic polynomials. Under the Riemann sphere and scaling theorem, the complex dynamic behavior of the iterative method was analyzed by fractals. We discuss the stability of all fixed points and the parameter spaces starting from the critical points with the Mathematica software. The dynamical planes of the elements with good and bad dynamical behavior are given, and the optimal parameter element with stable behavior was obtained. Finally, a numerical experiment and practical application were carried out to prove the conclusion.

A New Fifth Order Derivative Free Newton-Type Method for Solving Nonlinear Equations

The present paper deals with fifth order convergent Newton-t ype with and without derivative iterative methods for estimating a simple root of nonlinear equations. The error equations are used to establish the fifth order of convergence of the propo sed iterative methods. Finally, various numerical comparisons are made using MATLAB to demonstrate the performence of the developed methods.

Five-point thirty-two optimal order iterative method for solving non-linear equations

A five-point thirty-two convergence order derivative-free iterative method to find simple roots of non-linear equations is constructed. Six function evaluations are performed to achieve optimal convergence order 26-1 = 32 conjectured by Kung and Traub [1]. Secant approximation to the derivative is computed around the initial guess. High order convergence is attained by constructing polynomials of quotients for functional values.

Basin attractors for derivative-free methods to find simple roots of nonlinear equations

Many methods exist for solving nonlinear equations. Several of these methods are derivative-free. One of the oldest is the secant method where the derivative is replaced by a divided difference. Clearly such method will need an additional starting value. Here we consider several derivative-free methods and compare them using the idea of basin of attraction.

A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameterpwherepis a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameterp. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

Computers in mathematical research: the study of three-point root-finding methods

This paper is motivated by results of extensive comparative study of three-point methods for solving nonlinear equations presented in the paper (C. Chun, B. Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algor. 74, 1169–1201, 2017). In the first part, we show that the best ranking method, constructed by Sharma and Arora in 2016, is actually a special case of the family of three-point methods presented in the paper (J. Džunic, M. S. Petkovic, L. D. Petkovic, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217, 7612–7619, 2011). Since the mentioned Chun-Neta’s comparative study was carried out by testing algebraic polynomials of low degree (not higher than 6), in the second part we have extended comparative study to higher-order polynomials, polynomials with random coefficients, and polynomials with clusters of zeros. The performed experiments confirmed results of Chun and Neta. As in the cited paper of Chun and Neta, we have also applied the quality study based on computer visualization plotting the basins of attraction.

Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Solving nonlinear equations is one of the most important problems in numerical analysis, and has a wide range of application in various aspects, as well as many branches of science, engineering, physics, computing, astronomy, finance, .. . Generally, it is difficult to find the exact root of the nonlinear equations, and so iterative methods become the efficient way to obtain approximate solutions. Recently Kou et. al. presented a class of new variants of Ostrowski's method with order of convergence equals seven (OSM7) for solving simple roots of nonlinear equations proposed in [7]. Ostrowski's method (OSM7) has efficiency index equals to ∜7≈1.626, it has four functions per iteration but its order of convergence is seven that means it's not optimal method. In this paper, we have proposed new two improvements of Ostrowski's method (OSM7) to make it an optimal eight family and to increase its efficiency index. The first improvement has obtained by multiplied the third step of (OSM7) by product of two weight functions with some special conditions and the second improvement has obtained by multiplied the third step of (OSM7) by summation of two others weigh functions with special conditions, too. Using weight functions in every improvement helped us to improve the order of convergence of (OSM7) from seven to eight without changing the number of function evaluations to be an optimal family. New two optimal families have efficiency index equals to ∜8≈1.682. Some numerical examples are provided to show the good performance of the new methods.

A variant of McDougall-Wotherspoon method for finding simple roots of nonlinear equations

Recently, McDougall and Wotherspoon have developed a simple modification to the standard Newton method for finding simple roots of nonlinear equations. Their method is based on the arithmetic mean and achieves approximately 2.4 convergence order. This research presents a new iteration scheme based on the harmonic mean with the same convergence order and the comparison with the original McDougall-Wotherspoon method. Several test examples have been used and the results show good numerical behavior of the new method.

Improved Newton-Raphson Methods for Solving Nonlinear Equations

In this paper, we mainly study the numerical algorithms for simple root of nonlinear equations based on Newton-Raphson method. Two modified Newton-Raphson methods for solving nonlinear equations are suggested. Both of the methods are free from second derivatives. Numerical examples are made to show the performance of the presented methods, and to compare with other ones. The numerical results illustrate that the proposed methods are more efficient and performs better than Newton-Raphson method.

Two modified algorithms with order of convergence five for finding a simple root of nonlinear equations

Two modified algorithms with order of convergence five for finding a simple root of nonlinear equations

Optimal eighth order convergent iteration scheme based on Lagrange interpolation

In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. The scheme requires three evaluations of the function and one evaluation of the first derivative per iteration. Numerical examples are included to confirm the theoretical results and to show the competitive performance of the proposed iteration scheme.

A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics

A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics

A family of Newton-Halley type methods to find simple roots of nonlinear equations and their dynamics

In this work, a new family of Newton-Chebyshev type methods for solving nonlinear equations is presented. The dynamics of the Newton-Chebyshev family for the class of quadratic polynomials is analyzed and the convergence is established. We find the fixed and critical points. The stable and unstable behaviors are studied. The parameter space associated with the family is studied and finally, some dynamical planes that show different aspects of the dynamics of this family are presented.

Comparative study of eighth-order methods for finding simple roots of nonlinear equations

Recently, there were many papers discussing the basins of attraction of various methods and ideas how to choose the parameters appearing in families of methods and weight functions used. Here, we collected many of the eighth-order schemes scattered in the literature and presented a quantitative comparison. We have used the average number of function evaluations per point, the CPU time, and the number of black points to compare the methods. Based on seven examples, we found that the best method based on the three criteria is SA8 due to Sharma and Arora.

Higher Order Two-Point Efficient Family of Halley Type Methods for Simple Roots

In this paper, we proposed a new highly efficient two-point sixth-order family of Halley type methods that do not require any second-order derivative evaluation for obtaining simple roots of nonlinear equations, numerically. In terms of computational cost, each member of the family requires two-function and two first-order derivative evaluations per iteration. On the account of the results obtained, it is found that our proposed methods are efficient and show better performance than existing sixth-order methods available in the literature. Further, it is also noted that larger basins of attraction belong to our methods as compared to the existing ones. On the other hand, the existing methods are slower and have darker basins while some of them are too sensitive upon the choice of the initial value.

A New Approach for Solving Nonlinear Equations by Using of Integer Nonlinear Programming

One of the most important issues in numerical calculations is finding simple roots of nonlinear equations. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering, to achieve the result need to solve a nonlinear equation. Thus, the formulation of a recursive relationship with high order of convergence and low time complexity is very important. This paper provides a modification to the Weerakoon-Fernando and Parhi-Gupta methods. It is shown that, in each iterate, the improved method requires three evaluations of the function and two evaluations of the first derivatives of function. The proposed with the Kou et al., Neta, Parhi-Gupta, Thukral and Mir et al. methods have been applied to a collection of 12 test problem. The results show that proposed approach significantly reduces the number of function calls when compared to the above methods. The numerical examples show that the proposed method is more efficiency than other methods in this class, such as sixth-order method of Parhi-Gupta or eighth-order method of Mir et al. and Thukral. We show that the order of convergence the proposed method is 9 and also, the modified method has the efficiency of .