Optimal Eighth-order Methods Research Articles

A stable class of modified Newton-like methods for multiple roots and their dynamics

Abstract There have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions

It is well known that the optimal iterative methods are of more significance than the non-optimal ones in view of their efficiency and convergence speed. There are only a few number of optimal iterative methods for finding multiple zeros with eighth order of convergence. In this paper, we propose a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity. We present an extensive convergence analysis which confirms theoretically eighth-order convergence of the presented scheme. Moreover, we consider several real life problems that contain simple as well as multiple zeros in order to compare our proposed methods with the existing eighth-order iterative schemes. Some dynamical aspects of the presented methods are also discussed. Finally, we conclude on the basis of obtained numerical results that the proposed family of iterative methods perform better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

Generating function method for constructing new iterations

In this paper we propose a generating function method for constructing new two- and three-point iterations with p (3 ≤ p ≤ 8) order of convergence. This approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases. The necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters τn and αn. We also propose some generating functions for τn and αn. We give the extension of a class of optimal fourth-order Jarratt’s type iterations with a≠23. We develop a unified representation of all optimal eighth-order methods. Several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods.

An eighth-order family of optimal multiple root finders and its dynamics

There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.

A novel family of weighted-Newton optimal eighth order methods with dynamics

In this paper, we present a family of eighth order methods for solving nonlinear equations. Formula is composed of three steps, namely; Newton iteration in the first step and weighted-Newton iterations in second and third steps. Hence the name weighted-Newton methods. In terms of computational cost, the family requires three evaluations of function and one of first derivative. Therefore, it is optimal in the sense of Kung–Traub conjecture and has efficiency index 1.682 which is better than that of Newton method of efficiency index 1.414 and many other higher order methods. Numerical examples are considered to support that the method thus obtained is competitive with other similar robust methods. Moreover, basins of attraction are presented to demonstrate the performance in complex plane.

Ball Convergence for Two Optimal Eighth-Order Methods Using Only the First Derivative

We present a local convergence analysis of two optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation defined on the real line or the complex plane. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness of the solution with results based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

Comparison of several families of optimal eighth order methods

Several families of optimal eighth order methods to find simple roots are compared to the best known eighth order method due to Wang and Liu (2010). We have tried to improve their performance by choosing the free parameters of each family using two different criteria.

A new family of optimal eighth order methods with dynamics for nonlinear equations

We propose a simple yet efficient family of three-point iterative methods for solving nonlinear equations. Each method of the family requires four evaluations, namely three functions and one derivative, per full iteration and possesses eighth order of convergence. Thus, the family is optimal in the sense of Kung–Traub conjecture and has the efficiency index 1.682 which is better than that of Newton’s and many other higher order methods. Various numerical examples are considered to check the performance and to verify the theoretical results. Computational results including the elapsed CPU-time, confirm the efficient and robust character of proposed technique. Moreover, the presented basins of attraction also confirm better performance of the methods as compared to other established methods in literature.

On the new family of optimal eighth order methods developed by Lotfi et al.

Recently Lotfi et al. (Numer. Algor. 68, 261---288, 5) have developed a new family of optimal order eight for the solution of nonlinear equations. They have experimented with 3 members of the family and compared them to other eighth order methods. One of the best known eight order method was not included. They also did not mention the best choice of parameters in the methods used and why. The basins of attraction were given for several examples without a quantitative comparison. It will be shown how to choose the best parameters in all these methods, and to quantitatively compare the methods.

Ball comparison between two optimal eight-order methods under weak conditions

We present a local convergence analysis of two families of optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as Chun and Lee (Appl Math Comput 223:506–519, 2013), and Chun and Neta (Appl Math Comput 245:86–107, 2014) the convergence order of these methods was given under hypotheses reaching up to the eighth derivative of the function although only the first derivative appears in these methods. In this paper, we expand the applicability of these methods by showing convergence using only the first derivative. Moreover, we compare the convergence radii and provide computable error estimates for these methods using Lipschitz constants.

An Efficient Class of Multipoint Root-Solvers With andWithout Memory for Nonlinear Equations

A new class of optimal eighth-order iterative methods without memory for solving nonlinear equations is presented. Based on this new class of methods, we present an efficient class of multipoint methods with memory by suitable variation of a free parameter in each iterative step. Consequently, the order of convergence is increased without any additional function evaluations from 8 to 12. Numerical comparisons are made to show the performance of the presented methods.

An analysis of a family of Maheshwari-based optimal eighth order methods

In this paper we analyze an optimal eighth-order family of methods based on Maheshwari’s fourth order method. This family of methods uses a weight function. We analyze the family using the information on the extraneous fixed points. Two measures of closeness of an extraneous points set to the imaginary axis are considered and applied to the members of the family to find its best performer. The results are compared to a modified version of Wang–Liu method.

An Analysis of a King-based Family of Optimal Eighth-order Methods

In this paper we analyze an optimal eighth-order family of methods based on King's fourth order method to solve a nonlinear equation. This family of methods was developed by Thukral

Geometric construction of eighth-order optimal families of Ostrowski's method.

Based on well-known fourth-order Ostrowski's method, we proposed many new interesting optimal families of eighth-order multipoint methods without memory for obtaining simple roots. Its geometric construction consists in approximating f n′ at zn in such a way that its average with the known tangent slopes f n′ at xn and yn is the same as the known weighted average of secant slopes and then we apply weight function approach. The adaptation of this strategy increases the convergence order of Ostrowski's method from four to eight and its efficiency index from 1.587 to 1.682. Finally, a number of numerical examples are also proposed to illustrate their accuracy by comparing them with the new existing optimal eighth-order methods available in the literature. It is found that they are very useful in high precision computations. Further, it is also noted that larger basins of attraction belong to our methods although the other methods are slow and have darker basins while some of the methods are too sensitive upon the choice of the initial value.

On a New Iterative Scheme without Memory with Optimal Eighth Order

The purpose of this paper is to derive and discuss a three-step iterative expression for solving nonlinear equations. In fact, we derive a derivative-free form for one of the existing optimal eighth-order methods and preserve its convergence order. Theoretical results will be upheld by numerical experiments.

Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations

Several optimal eighth order methods to obtain simple roots are analyzed. The methods are based on two step, fourth order optimal methods and a third step of modified Newton. The modification is performed by taking an interpolating polynomial to replace either f(zn) or f′(zn). In six of the eight methods we have used a Hermite interpolating polynomial. The other two schemes use inverse interpolation. We discovered that the eighth order methods based on Jarratt’s optimal fourth order methods perform well and those based on King’s or Kung–Traub’s methods do not. In all cases tested, the replacement of f(z) by Hermite interpolation is better than the replacement of the derivative, f′(z).

Efficient optimal eighth-order derivative-free methods for nonlinear equations

This paper opens the issue of solving a nonlinear equation by iterative methods in which no derivative is required per iteration. Hence, an optimal class of three-step methods without memory is demonstrated, where each of its elements (methods) requires only four function evaluations per iteration to achieve the eighth order of convergence. In order to verify the effectiveness of the proposed methods, we employ numerical examples. The attained results are consistent with the developed theory. We finally present an algorithm to capture all the real simple solutions of nonlinear functions in an interval.

A triparametric family of three-step optimal eighth-order methods for solving nonlinear equations

A new triparametric family of three-step optimal eighth-order iterative methods free from second derivatives are proposed in this paper, to find a simple root of nonlinear equations. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants.