Fourth-order Family Research Articles

Design and dynamical behavior of a fourth order family of iterative methods for solving nonlinear equations

<abstract><p>In this paper, a new fourth-order family of iterative schemes for solving nonlinear equations has been proposed. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. For studying the stability of this class, the rational function resulting from applying the iterative expression to a low degree polynomial was analyzed. The dynamics of this rational function allowed us to better understand the performance of the iterative methods of the class. In addition, the critical points have been calculated and the parameter spaces and dynamical planes have been presented, in order to determine the regions with stable and unstable behavior. Finally, some parameter values within and outside the stability region were chosen. The performance of these methods in the numerical section have confirmed not only the theoretical order of convergence, but also their stability. Therefore, the robustness and wideness of the attraction basins have been deduced from these numerical tests, as well as comparisons with other existing methods of the same order of convergence.</p></abstract>

DYNAMICS OF A FAMILY OF RATIONAL OPERATORS OF ARBITRARY DEGREE

In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family Oa,n,k of degree n+k operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of Oa,n,k and discuss for which parameters n and k these operators would be suitable from the numerical point of view.

An optimal class of fourth-order multiple-root finders of Chebyshev-Halley type and their basins of attraction

In this paper, we propose a family of optimal fourth-order of Chebyshev-Halley type methods free from second-order derivative for finding the multiple roots. The new methods are tested and compared...

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.

Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications

A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.

Local Convergence of a Family of Weighted-Newton Methods

This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of order one. The convergence radius, error estimations, and results on uniqueness also depend on this derivative. The scope of application of the method is extended, since no derivatives of higher order are required as in previous works. Finally, we demonstrate the applicability of the proposed method in real-life problems and discuss a case where previous studies cannot be adopted.

On generalized Halley-like methods for solving nonlinear equations

Generalized Halley-like one-parameter families of order three and four for finding multiple root of a nonlinear equation are constructed and studied. This presentation is, actually, a mixture of theoretical results, algorithmic aspects, numerical experiments, and computer graphics. Starting from the proposed class of third order methods and using an accelerating procedure, we construct a new fourth order family of Halley's type. To analyze convergence behavior of two presented families, we have used two methodologies: (i) testing by numerical examples and (ii) dynamic study using basins of attraction.

Ball convergence of a stable fourth-order family for solving nonlinear systems under weak conditions

We present a local convergence analysis of fourth-order methods in order to approximate a locally unique solution of a nonlinear equation in Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the fifth 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.

A biparametric extension of King’s fourth-order methods and their dynamics

A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending King’s fourth-order family of methods. With the introduction of an error corrector having a weight function dependent on a function-to-function ratio, higher-order convergence is obtained. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes.

An analysis of a Khattri’s 4th order family of methods

In this paper we analyze an optimal fourth-order family of methods suggested by Khattri and Babajee, (2013). We analyze the family using the information on the extraneous fixed points. Two measures of closeness to the imaginary axis of the set of extraneous points are considered and applied to the members of the family to find its best performer. The results are compared to three best members of King’s family of methods.

New two-parameter Chebyshev–Halley-like family of fourth and sixth-order methods for systems of nonlinear equations

The two-parameter Chebyshev–Halley-like family of optimal two-point fourth-order methods proposed by Babajee (2015), is further extended to solve systems of nonlinear equations. This two-step fourth-order family is further extended to obtain a two-parameter family of sixth-order methods which requires only one extra function evaluation. The performance of some special members of the proposed families using only single inverse per iteration have been tested through numerical examples and the results show that these are effective and comparable to existing methods both in order and efficiency.

An optimal fourth-order family of methods for multiple roots and its dynamics

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

On the convergence of an optimal fourth-order family of methods and its dynamics

In this paper, we present the study of the semilocal and local convergence of an optimal fourth-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior. Parameter spaces are shown and the study of the stability of all the fixed points is presented.

A new fourth-order family for solving nonlinear problems and its dynamics

In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.

The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor

The thin film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in [21] to be analogous to the (second-order) porous medium family. They are 2-Wasserstein (=d 2) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d 2 gradient flow of the generalized entropy E(v) by Otto [41]. The identity aI(v) = bE(v) + |∇ d 2 E(v)|2/2 implies a Hess d 2 I(v *) = Hess d 2 E(v *)(b + Hess d 2 E(v *)) formally, when the equation is rescaled and linearized around the resulting self-similar critical profile v *. We couple this relation with the diagonalization of Hess d 2 E(v *) for the porous medium flow computed in [46]. This yields information about the leading- and higher-order asymptotics of the equation on R N which—outside of special cases—was inaccessible previously.

Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane

The real dynamics of a family of fourth-order iterative methods is studied when it is applied on quadratic polynomials. A Scaling Theorem is obtained and the conjugacy classes are analyzed. The convergence plane is used to obtain the same kind of information as from the parameter space, and even more, in complex dynamics.

Different anomalies in a Jarratt family of iterative root-finding methods

In this paper, the behavior of a Jarratt family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior of this fourth-order family of methods. Parameter spaces are shown and the study of the stability of all the fixed points is presented. Dynamical planes for members with good and bad dynamical behavior are also provided.

New modified optimal families of King’s and Traub-Ostrowski’s methods

Based on quadratically convergent Schroder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.

Optimal equi-scaled families of Jarratt's method

In this paper, we present many new fourth-order optimal families of Jarratt's method and Ostrowski's method for computing simple roots of nonlinear equations numerically. The proposed families of Jarratt's method having the same scaling factor of functions as that of Jarratt's method (i.e. quadratic scaling factor of functions in the numerator and denominator of the correction factor) are the main finding of this paper. It is observed that the body structures of our proposed families of Jarratt's method are simpler than those of the original families of Jarratt's method. The efficiency of these methods is tested on a number of relevant numerical problems. Furthermore, numerical examples suggest that each member of the proposed families can be competitive to other similar robust methods available in the literature.

Families of third and fourth order methods for multiple roots of nonlinear equations

This paper presents two families of higher-order iterative methods for solving multiple roots of nonlinear equations. One is of order three and the other is of order four. The presented iterative families all require two evaluations of the function and one evaluation of its first derivative, thus the latter is of optimal order. The third-order family contains several iterative methods known already. And, different from the optimal fourth-order methods for multiple roots known already, the presented fourth-order family use the modified Newton’s method as its first step. Local convergence analyses and some special cases of the presented families are given. We also carry out some numerical examples to show their performance.