Simple Roots Of Nonlinear Equations Research Articles

A Family of Optimal Derivative Free Iterative Methods with Eighth-Order Convergence for Solving Nonlinear Equations

In this paper, modification of Steensen's method with eight-order convergence is presented. We propose a family of optimal three-step methods with eight-order convergence for solving the simple roots of nonlinear equations by using the weight function and interpola- tion methods. Per iteration this method requires four evaluations of the function which implies that the eciency index of the developed meth- ods is 1.682. Some numerical examples illustrate that the algorithm is more ecient and performs better than other methods.

Accurate fourteenth-order methods for solving nonlinear equations

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton's method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton's iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

A family of modified Ostrowski’s methods with optimal eighth order of convergence

In this paper, we derive a new family of eighth-order methods for obtaining simple roots of nonlinear equations by using the weight function method. Each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which are optimal according to the Kung and Traub’s conjecture (1974) [2]. Numerical comparisons are made to show the performance of the derived method, as is shown in the numerical section.

A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots

A uniparametric family of three-step eighth-order multipoint iterative methods requiring only a first derivative are proposed in this paper to find simple roots of nonlinear equations. Development and convergence analysis on the proposed methods is described along with numerical experiments including comparison with existing methods.

A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function

This paper proposes a biparametric family of three-step eighth-order multipoint iterative methods with optimal efficiency index in the sense of Kung-Traub for simple roots of nonlinear equations. We employ their third-step weighting function decomposed into King’s linear fractional function and a two-variable function to construct such a family of optimal methods. Development and convergence analysis on the proposed methods is fully described in addition to numerical experiments including comparison with existing methods.

Construction of optimal order nonlinear solvers using inverse interpolation

There is a vast literature on finding simple roots of nonlinear equations by iterative methods. These methods can be classified by order, by the information used or by efficiency. There are very few optimal methods, that is methods of order 2 m requiring m + 1 function evaluations per iteration. Here we give a general way to construct such methods by using inverse interpolation and any optimal two-point method. The presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence.

A new general fourth-order family of methods for finding simple roots of nonlinear equations

In this paper, a new fourth-order family of methods free from second derivative is obtained. This new iterative family contains the King's family, 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 fourth-order methods. Due to that, they can be effectively used for solving nonlinear equations.

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

Modified Ostrowski’s method with eighth-order convergence and high efficiency index

In this paper, based on Newton’s method, we derive a modified Ostrowski’s method with an eighth-order convergence for solving the simple roots of nonlinear equations by Hermite interpolation methods. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency index of the developed method is 1.682, which is optimal according to Kung and Traub’s conjecture Kung and Traub (1974) [2]. Numerical comparisons are made to show the performance of the derived method, as shown in the illustrative examples.

The Neural-Network Approaches to Solve Nonlinear Equation

In this paper, we proposed two neural - network a pproaches for solving nonlinear equations or polynomials . The first method is suitable for finding simple roots of nonlinear equation or polynomial, and the second approach is fit to finding both the multiple and simple roots of nonlinear equations or polynomial, which were not well solved by the other methods . The convergence of algorithm proposed was researched. The convergence theorem provides the theory criterion selecting learning rate of neural network. The specific examples showed that the proposed method can find the simple or multiple roots of nonlinear equations or polynomials at a very rapid convergence and very high accuracy with less computation.

Eighth-order methods with high efficiency index for solving nonlinear equations

In this paper, we derive a new family of eighth-order methods for solving simple roots of nonlinear equations by using weight function methods. Per iteration these methods require three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency indexes are 1.682. Numerical comparisons are made to show the performance of the derived methods, as shown in the illustration examples.

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816–1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14–19] are the special cases of the new improvements.

A one-parameter family of second-order iteration methods

A new one-parameter family of iteration methods for finding simple roots of nonlinear equations is developed. Error analysis providing the second-order convergence is given. Numerical experience shows that the method is comparable to the well-known Newton’s method.

A family of third-order methods to solve nonlinear equations by quadratic curves approximation

A one-parameter family of iteration functions for finding the simple roots of nonlinear equations is presented. The iteration process is based on one-point approximation by the quadratic equation x 2 + ay 2 + bx + cy + d = 0, where the unknowns b, c and d are determined in terms of a. Different choices of a correspond to different approximating quadratic curves, viz. parabola, circle, ellipse and hyperbola. Euler, Chebyshev, Halley, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family are cubically convergent except Newton’s which is quadratically convergent.