Simple Roots Of Nonlinear Equations Research Articles

A New Approach for Solving Nonlinear Equations by Using of Integer Nonlinear Programming

One of the most important issues in numerical calculations is finding simple roots of nonlinear equations. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering, to achieve the result need to solve a nonlinear equation. Thus, the formulation of a recursive relationship with high order of convergence and low time complexity is very important. This paper provides a modification to the Weerakoon-Fernando and Parhi-Gupta methods. It is shown that, in each iterate, the improved method requires three evaluations of the function and two evaluations of the first derivatives of function. The proposed with the Kou et al., Neta, Parhi-Gupta, Thukral and Mir et al. methods have been applied to a collection of 12 test problem. The results show that proposed approach significantly reduces the number of function calls when compared to the above methods. The numerical examples show that the proposed method is more efficiency than other methods in this class, such as sixth-order method of Parhi-Gupta or eighth-order method of Mir et al. and Thukral. We show that the order of convergence the proposed method is 9 and also, the modified method has the efficiency of .

New Predictor-Corrector Iterative Methods with Twelfth-Order Convergence for Solving Nonlinear Equations

In this paper, we propose and analyze new two efficient iterative methods for finding the simple roots of nonlinear equations. These methods based on a Jarratt's method, Householder's method and Chun&Kim's method by using a predictor-corrector technique. The error equations are given theoretically to show that the proposed methods have twelfth-order convergence. Several numerical examples are given to illustrate the efficiency and robustness of the proposed methods. Comparison with other well-known iterative methods is made.

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

Abstract In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to 8 1 4 ≈ 1.682 ${8^{{\textstyle{1 \over 4}}}} \approx 1.682$ . We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

An improved Regula falsi method for finding simple roots of nonlinear equations

In this note, an improved Regula falsi (IRF) method based on the classical Regula falsi (RF) method is proposed. The method is verified on a number of test examples and numerical results. The proposed method has faster convergence rate with respect to the classical Regula falsi method.

A Derivative-Free Method of Eighth-Order For Finding Simple Root of Nonlinear Equations

In this paper we have constructed an optimal eighth-order method with four function evaluations to solve the non-linear equations. The proposed method is a three-step method in which no derivative is required. Our scheme is optimal in the sense of Kung and Traub. Moreover, some test functions have been also included to confirm the superiority of the proposed method. At the end, we have presented the basins of attraction of some existing methods along with our proposed method to illustrate their performances.

Two Optimal General Classes of Iterative Methods with Eighth-Order

Two new three-step classes of optimal iterative methods to approximate simple roots of nonlinear equations, satisfying the Kung-Traub's conjecture, are designed. The development of the methods and their convergence analysis are provided joint with a generalization of both processes. In order to check the goodness of the theoretical results, some concrete methods are extracted and numerical and dynamically compared with some known methods.

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.

An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics

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.

New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permittingf′(x)=0in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to 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.

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).

MODIFIED OPTIMAL FAMILIES OF FOURTH-ORDER JARRATT'S METHOD

In this paper, we present a two parameter modified families of Jarratt’s method having quartic convergence for computing simple roots of nonlinear equations, permitting f (x) = 0 in the vicinity of the required root. The present approach of deriving this optimal class of Jarratt’s method is based on applying weight function approach. Jarratt’s method and Zhou et al. family of methods [1] for simple roots can be seen as special cases of our proposed scheme. Each member of the class requires three functional evaluations per iteration. The performance of proposed multipoint methods is compared with its closest competitors namely, King’s method, Ostrowski’s method and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be more effective and comparable to the similar robust methods available in the literature. AMS Subject Classification: 65xx, 65Hxx

On a family of Halley-like methods to find simple roots of nonlinear equations

There are many methods for solving a nonlinear algebraic equation. Here we introduce a family of Halley-like methods and show that Euler–Chebyshev and BSC are just members of the family. We discuss the conjugacy maps and the effect of the extraneous roots on the basins of attraction.

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.

A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence

This paper presents a new family of twelfth-order methods for solving simple roots of nonlinear equations which greatly improves the order of convergence and the computational efficiency of the Newton’s method and some other known methods.

Three-step iterative methods with eighth-order convergence for solving nonlinear equations

A family of eighth-order iterative methods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency index of the developed methods is 1.682. Some numerical examples illustrate that the algorithms are more efficient and performs better than the other methods.

Basins of attraction for several methods to find simple roots of nonlinear equations

There are many methods for solving a nonlinear algebraic equation. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss several third and fourth order methods to find simple zeros. The relationship between the basins of attraction and the corresponding conjugacy maps will be discussed in numerical experiments. The effect of the extraneous roots on the basins is also discussed.

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.

A General Family of Fifth-Order Methods for Finding Simple Roots of Nonlinear Equations

In this paper, a new fifth-order family of methods free from second derivative is obtained. The new iterative family contains the King’s, which is one of the most well-known family of methods for solving nonlinear equations, and some other known methods as its particular case. To illustrate the efficiency and performance of proposed family, several numerical examples are presented. Numerical results illustrate better efficiency and performance of the presented methods in comparison with the other compared fifth-order methods. Therefore, the proposed family can be effectively used for solving nonlinear equations.

Some New Variants of Cauchy′s Methods for Solving Nonlinear Equations

We present and analyze some variants of Cauchy′s methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well‐known existing methods and give better numerical results in many aspects.

A family of optimal iterative methods with fifth and tenth order convergence for solving nonlinear equations

In this paper, we propose a two-step method with fifth-order convergence and a family of optimal three-step method with tenth-order convergence for finding the simple roots of nonlinear equations. The optimal efficiency indices are all found to be $5^{\frac{1}{3}}\approx1.71$ and $10^{\frac{1}{4}}\approx1.78.$ Some numerical examples illustrate that the algorithms are more efficient and performs better than other methods.