Methods For Finding Multiple Roots Research Articles

Mean-based iterative methods for finding multiple roots in nonlinear chemistry problems

Mean-based iterative methods for finding multiple roots in nonlinear chemistry problems

A Family of Optimal Eighth Order Iteration Functions for Multiple Roots and Its Dynamics

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.

Convergence Analysis of the Modified Chebyshev’s Method for Finding Multiple Roots

In this paper, an estimate of the radius of convergence ball of the modified Chebyshev’s method for finding multiple roots of nonlinear equations is provided under the hypotheses that the (m + 1)st derivative f(m+ 1) of function f is Holder continuous and bounded. The unique ball of a solution is also established. Finally, some examples are provided to show the effectiveness of our results.

A family of fifth-order iterative methods for finding multiple roots of nonlinear equations

A family of fifth-order iterative methods for finding multiple roots of nonlinear equations

An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence

In this work, we construct a family of seventh order iterative methods for finding multiple roots of a nonlinear function. The scheme consists of three steps, of which the first is Newton’s step and last two are the weighted-Newton steps. Hence, the name of the scheme is ‘weighted-Newton methods’. Theoretical results are studied exhaustively along with the main theorem describing convergence analysis. Stability and convergence domain of the proposed class are also demonstrated by means of using a graphical technique, namely, basins of attraction. Boundaries of these basins are fractal like shapes through which basins are symmetric. Efficacy is demonstrated through numerical experimentation on variety of different functions that illustrates good convergence behavior. Moreover, the theoretical result concerning computational efficiency is verified by computing the elapsed CPU time. The overall comparison of numerical results including accuracy and CPU-time shows that the new methods are strong competitors for the existing methods.

An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

Stability analysis of fourth-order iterative method for finding multiple roots of non-linear equations

Abstract The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.

Optimal Fourth-Order Weerakoon–Fernando-Type Methods for Multiple Roots and Their Dynamics

In this paper, we present optimal fourth-order methods for finding multiple roots of non-linear equations, where the multiplicity is known in advance. These methods are based on the third-order method given by Weerakoon and Fernando for simple roots. The dynamical behavior of these methods around multiple roots is studied using basin of attraction in complex plane. We also present numerical examples to confirm our theoretical results.

Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics

In this paper, we propose a family of iterative methods for finding multiple roots, with known multiplicity, by means of the introduction of four univariate weight functions. With the help of these weight functions, that play an important role in the development of higher order convergent iterative techniques, we are able to construct three-point eight-order optimal multiple-root finders. Also, numerical experiments have been applied to a number of test equations for different special schemes from this family satisfying the conditions given in the convergence analysis. We have also compared the basins of attraction of some proposed and known methods in order to check the wideness of the sets of converging initial points for each problem.

On the optimality of some multi-point methods for finding multiple roots of nonlinear equation

This paper deals with the problem of determining the multiple roots of nonlinear equations, where the multiplicity of the roots is known. The paper contains some remarks on the optimality of the recently published methods [B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations, Nonlinear Anal. Model. Control, 18(2):143–152, 2013] and [X. Zhou, X. Chen, Y. Song, Families of third- and fourth-order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219(11):6030–6038, 2013]. Separate analysis of odd and even multiplicity, has shown the cases where those methods lose their optimal convergence properties. Numerical experiments are made and they support theoretical analysis.

A sixth order transformation method for finding multiple roots of nonlinear equations and basin attractors for various methods

In this contribution, a sixth-order transformation method is proposed and analyzed for finding multiple roots of nonlinear equations, when the multiplicity of the root is not known explicitly. The proposed method does not require the evaluation of second derivative. The basins of attraction of existing transformation methods and the proposed method are presented to demonstrate their performance.

A new family of multi-point iterative methods for finding multiple roots of nonlinear equations

In this paper, new three-point eighth-order iterative methods for solving nonlinear equations are constructed. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on \(n\) evaluations, could achieve optimal convergence order \(2^{n-1} .\) Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\). Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.

General Family of Third Order Methods for Multiple Roots of Nonlinear Equations and Basin Attractors for Various Methods

A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of f, f′, and f′′ each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.

New Families of Third-Order Iterative Methods for Finding Multiple Roots

Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.

On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition

It is very important to enlarge the convergence ball of an iterative method. Recently, the convergence radius of the modified Newton method for finding multiple roots of nonlinear equations has been presented by Ren and Argyros when the involved function is Holder and center---Holder continuous. Different from the technique and the hypothesis used by them, in this paper, we also investigate the convergence radius of the modified Newton method under the condition that the derivative $f^{(m)}$ of function f satisfies the center---Holder continuous condition. The radius given here is larger than that given by Ren and Argyros. The uniqueness ball of solution is also discussed. Some examples are given to show applications of our theorem.

Modified Chebyshev-Halley type method and its variants for computing multiple roots

We present two families of third order methods for finding multiple roots of nonlinear equations. One family is based on the Chebyshev-Halley scheme (for simple roots) and includes Halley, Chebyshev and Chun-Neta methods as particular cases for multiple roots. The second family is based on the variant of Chebyshev-Halley scheme and includes the methods of Dong, Homeier, Neta and Li et al. as particular cases. The efficacy is tested on a number of relevant numerical problems. It is observed that the new methods of the families are equally competitive with the well known special cases of the families.

A cubic-order variant of Newton’s method for finding multiple roots of nonlinear equations

A second-derivative-free iteration method is proposed below for finding a root of a nonlinear equation f ( x ) = 0 with integer multiplicity m ≥ 1 : x n + 1 = x n − f ( x n − μ f ( x n ) / f ′ ( x n ) ) + γ f ( x n ) f ′ ( x n ) , n = 0 , 1 , 2 , … . We obtain the cubic order of convergence and the corresponding asymptotic error constant in terms of multiplicity m , and parameters μ and γ . Various numerical examples are presented to confirm the validity of the proposed scheme.

New third and fourth order nonlinear solvers for computing multiple roots

We present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by Kou et al. [J. Kou, Y. Li, X. Wang, A family of fourth-order methods for solving non-linear equations, Appl. Math. Comput. 188 (2007) 1031–1036]. Further investigation gives rise to new third and fourth order families of methods which do not require second derivative. The fourth order family has optimal order, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. Computational results ascertain that the present methods are competitive with other similar robust methods.

Variational iteration technique for finding multiple roots of nonlinear equations

In this paper, we suggest and analyze some new iterative methods for finding multiple roots of nonlinear equations by using the variational iteration technique. The technique generates the methods of higher order. Here, the new methods are of second and third order convergence. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar methods is also given. Key words: Variational iteration, multiple roots, Newton method, Iterative methods.

Traub’s accelerating generator of iterative root-finding methods

An accelerating generator of iterative methods for finding multiple roots, based on Traub’s differential–difference recurrence relation, is presented. It is proved that this generator yields an iteration function of order r + 1 starting from arbitrary iteration function of order r . In this way, it is possible to construct various iterative formulas of higher order for finding single roots of nonlinear equations and all simple or multiple roots of algebraic polynomials, simultaneously. For demonstration, two iterative methods of the fourth order in ordinary (real or complex) arithmetic and an iterative method in interval arithmetic are presented.