Family Of Iterative Methods Research Articles

A new family of twentieth order convergent methods with applications to nonlinear systems in engineering

A new family of iterative methods with a strong converging order of twenty to solve nonlinear equations and systems is presented in this study. A simple strategy of blending some existing methods is used to develop the proposed family. The theoretical order of convergence is derived by employing Taylor’s series. The performance of the iterative methods in the proposed family is examined by applying the methods on real-world engineering problems. A nonlinear equation modeled by NASA for launching “Wind” satellite and some other complex applied systems, such as combustion problem, tank-reactor problem, kinematic synthesis mechanism, neurophysiology application and one boundary-value problem, have been solved to check the performance of the proposed family against other methods under similar test conditions. All the numerical results show that the proposed family converges very fast in complex and difficult problems as compared to other well-known methods. The methods in the proposed family have an efficiency improvement of 11.99% over the classical Newton method for scalar nonlinear equations.

Family of fourth-order optimal classes for solving multiple-root nonlinear equations

We present a new iterative procedure for solving nonlinear equations with multiple roots with high efficiency. Starting from the arithmetic mean of Newton’s and Chebysev’s methods, we generate a two-step scheme using weight functions, resulting in a family of iterative methods that satisfies the Kung and Traub conjecture, yielding an optimal family for different choices of weight function. We have performed an in-depth analysis of the stability of the family members, in order to select those members with the highest stability for application in solving mathematical chemistry problems. We show the good characteristics of the selected methods by applying them on four relevant chemical problems.

Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations

In this paper, the weight function technique is utilized to develop new family of two-step fourth order convergence iterative methods for approximating the solution of nonlinear equations. The methods require the evaluation of three distinct functions evaluation per iteration circle and as such, are optimal in agreement with the Kung-Traub’s conjecture. This developed family of methods is further extended to design another new family of four-step ninth order convergence with efficiency index EI = 1.5518. We carried out the convergence analysis of the two families of iterative methods. This analysis provided us with information about the flexibility of the weight function in the method used in constructing other new families of iterative methods. The methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations. The results obtained from computation experience are compared with some of its existing contemporary methods.

A study on the local convergence and complex dynamics of Kou’s family of iterative methods

A study on the local convergence and complex dynamics of Kou’s family of iterative methods

Ball Convergence of a Parametric Efficient Family of Iterative Methods for Solving Nonlinear Equations

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

On High-Order Iterative Schemes for the Matrix pth Root Avoiding the Use of Inverses

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained.

A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

Iteration Methods with an Auxiliary Function for Nonlinear Equations

Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function g x to solve the nonlinear equation f x = 0 . In this paper, we introduce the expansion of g x with the inclusion of weights w i such that ∑ i = 1 p w i = 1 and knots τ i ∈ 0,1 in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function g x , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.

New technique for the approximation of the zeros of nonlinear scientific models

Abstract Most of the problems in mathematical and engineering sciences can be studied in the context of nonlinear equations. In this paper, we develop a new family of iterative methods for the approximation of the zeros of mathematical models whose governing equations are nonlinear in nature. The proposed methods are based on decomposition technique due to Daftardar-Gejji and Jaffri [1]. The new family gives several iterative schemes as special cases. The convergence analysis of proposed methods is also presented. In order to determine the performance of newly developed methods, numerical as well as graphical analysis of four complex mathematical models from diverse fields of science and engineering are considered. We also consider some complex polynomials to visualize the roots through polynomiography in the context of proposed methods.

Local and Semilocal Convergence of a Family of Multi-point Weierstrass-Type Root-Finding Methods

Weierstrass (Sitzungsber Konigl Preuss Akad Wiss Berlin II:1085–1101, 1891) introduced his famous iterative method for numerical finding all zeros of a polynomial simultaneously. Kyurkchiev and Ivanov (Ann Univ Sofia Fac Math Mech 78:132–136, 1984) constructed a family of multi-point root-finding methods which are based on the Weierstrass method. The purpose of this research is threefold: (1) to develop a new simple approach for the study of the local convergence of the multi-point simultaneous iterative methods; (2) to present a new local convergence result for this family which improves in several directions the result of Kyurkchiev and Ivanov; (3) to provide semilocal convergence results for Kyurkchiev–Ivanov’s family of iterative methods.

Complex dynamics of a sixth and seventh order family of root finding methods

This paper intends to investigate the dynamical behavior of a parameter based sixth and seventh order convergent family of iterative methods in the complex plane. The fixed points of the rational function obtained from this family are described with their stability. Further to this, the stability regions for the fixed points are displayed using graphical tools. Parameter planes for independent free critical points are presented to determine stable and poor members of the family. Dynamical planes are also provided for various iterative methods of the family having important numerical properties.

A family of high order iterations for calculating the sign of a matrix

For the computation of a matrix sign function, a family of iterative methods is proposed. for some cases of the accelerator parameters, this method is proven to converge globally with higher rates of convergence. Its stability is discussed analytically as well. to illustrate the efficiency of the proposed methods from the family, several tests of various matrix sizes are considered to worked out.

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.

A new efficient parametric family of iterative methods for solving nonlinear systems

ABSTRACT A bi-parametric family of iterative schemes for solving nonlinear systems is presented. We prove for any value of parameters the sixth-order of convergence of any members of the class. The efficiency and computational efficiency indices are studied for this family and compared with that of the other known schemes with similar structure. In the numerical section, we solve, after discretizating, the nonlinear boundary problem described by the Fisher's equation. This numerical example confirms the theoretical results and show the performance of the proposed schemes.

Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros

There are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King's family of iterative methods for nonlinear equations. The use of King's correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros.

HIGHER-ORDER FAMILIES OF MULTIPLE ROOT FINDING METHODS SUITABLE FOR NON-CONVERGENT CASES AND THEIR DYNAMICS

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

A family of fifth-order convergent methods for solving nonlinear equations using variational iteration technique

In this paper, we suggest and analyze new higher order family of iterative methods for solving nonlinear equations by using variational iteration technique. We give several examples to illustrate the efficiency of the proposed methods. Comparison with other similar methods is also given. New methods can be considered as an alternative of the existing methods. This technique can be used to suggest a wide class of new iterative methods for solving nonlinear equations.

On optimal parameter of Laguerre's family of zero-finding methods

ABSTRACTA one parameter Laguerre's family of iterative methods for solving nonlinear equations is considered. This family includes the Halley, Ostrowski and Euler methods, most frequently used one-point third-order methods for finding zeros. Investigation of convergence quality of these methods and their ranking is reduced to searching optimal parameter of Laguerre's family, which is the main goal of this paper. Although methods from Laguerre's family have been extensively studied in the literature for more decades, their proper ranking was primarily discussed according to numerical experiments. Regarding that such ranking is not trustworthy even for algebraic polynomials, more reliable comparison study is presented by combining the comparison by numerical examples and the comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. This combined approach has shown that Ostrowski's method possesses the best convergence behaviour for most polynomial equations.