Generating Optimal Eighth Order Methods for Computing Multiple Roots

An excellent numerical technique for multiple roots

In this manuscript, we present a new general family of optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity using weight functions. An extensive convergence analysis is presented to verify the optimal eighth order convergence of the new family. Some special cases of the family are also presented which require only three functions and one derivative evaluation at each iteration to reach optimal eighth order convergence. A variety of numerical test functions along with some real-world problems such as beam designing model and Van der Waals’ equation of state are presented to ensure that the newly developed family efficiently competes with the other existing methods. The dynamical analysis of the proposed methods is also presented to validate the theoretical results by using graphical tools, termed as the basins of attraction.

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.

This article concerned with the issue of solving a nonlinear equation with the help of iterative method where no any derivative evaluation is required per iteration.Therefore, this work contributes to a new class of optimal eighth-order Steffensen-type methods.Theoretical proof has been given to reveal the eighth-order convergence.Numerical comparisons have been carried out to show the effectiveness of contributed scheme.

In this paper, using the variant of Frontini-Sormani method, some higher order methods for finding the roots (simple and multiple) of nonlinear equations are proposed. In particular, we have constructed an optimal fourth order method and a family of sixth order method for finding a simple root. Further, an optimal fourth order method for finding a multiple root of a nonlinear equation is also proposed. We have used different weight functions to a cubically convergent For ntini-Sormani method for the construction of these methods. The proposed methods are tested on numerical examples and compare the results with some existing methods. Further, we have presented the basins of attraction of these methods to understand their dynamics visually.

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

In this paper, some sixth-order modifications of Jarratt method for solving single variable nonlinear equations are proposed. Per iteration, they consist of two function and two first derivative evaluations. The convergence analyses of the presented iterative methods are provided theoretically and a comparison with other existing famous iterative methods of different orders is given. Numerical examples include some of the newest and the most efficient optimal eighth-order schemes, such as Petkovic (SIAM J Numer Anal 47:4402---4414, 2010), to put on show the accuracy of the novel methods. Finally, it is also observed that the convergence radii of our schemes are better than the convergence radii of the optimal eighth-order methods.

We have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method of eighth-order for multiple zeros with known or unknown multiplicity. Some scholars from the worldwide have tried to present optimal or non-optimal multipoint eighth-order iteration functions. But, unfortunately, none of them get success in this direction and attained maximum sixth-order convergence in the case of multiple zeros with known multiplicity m. Motivated and inspired by this fact, we propose an optimal scheme with eighth-order convergence based on weight function approach. In addition, an extensive convergence study is discussed in order to demonstrate the eighth-order convergence of the present scheme. Moreover, we also show the applicability of our scheme on some real life and academic problems. These problems illustrate that our methods are more efficient among the available multiple root finding techniques.

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

In this paper, a family of Newton-type iterative methods with memory is obtained for solving nonlinear equations, which uses some special self-accelerating parameters. To this end, we first present two optimal fourth-order iterative methods with memory for solving nonlinear equations. Then we give a novel way to construct the self-accelerating parameter and obtain a family of Newton-type iterative methods with memory. The self-accelerating parameters have the properties of simple structure and easy calculation, which do not increase the computational cost of the iterative methods. The convergence order of the new iterative method is increased from 4 to . Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the new methods. Experiment results show that, compared with the existing methods, the new iterative methods with memory have the advantage of costing less computing time.

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basins of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basins of attraction.

In this article, we first construct a family of optimal 2-step iterative methods for finding a single root of the nonlinear equation using the procedure of weight function. We then extend these methods for determining all roots simultaneously. Convergence analysis is presented for both cases to show that the order of convergence is 4 in case of the single-root finding method and is 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The dynamical behavior is presented to analyze the stability of fixed and critical points of the rational operator of one-point iterative methods. The computational cost, basins of attraction, efficiency, log of the residual, and numerical test examples show that the newly constructed methods are more efficient as compared with the existing methods in the literature.

We develop an eighth order family of methods, consisting of three steps and three parameters, for solving nonlinear equations. Per iteration the methods require four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that the family is eighth-order convergent which is also substantiated through the numerical work. Computational results ascertain that family of methods are efficient and demonstrate equal or better performance as compared with other well known methods.

In this paper, we propose a family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity with the introduction of two free parameters and three univariate weight functions. Also numerical experiments have applied to a number of academical test functions and chemical problems for different special schemes from this family that satisfies the conditions given in convergence result.

A Hybrid Intelligent Method for Estimating Distribution Network Reconfigurations