A family of Newton-Chebyshev type methods to find simple roots of nonlinear equations and their dynamics (original) (raw)

Some Higher-Order Families of Methods for Finding Simple Roots of Nonlinear Equations

2011

In this paper, a new fifth-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 fifth-order methods. Due to that, they can be effectively used for solving nonlinear equations.

Comparative Study of Methods of Various Orders for Finding Simple Roots of Nonlinear Equations

Journal of Applied Analysis and Computation, 2019

Recently there were many papers discussing the basins of attraction of various methods and ideas how to choose the parameters appearing in families of methods and weight functions used. Here we collected many of the results scattered and put a quantitative comparison of methods of orders from 2 to 7. We have used the average number of function-evaluations per point, the CPU time and the number of black points to compare the methods. We also include the best eighth order method. Based on 7 examples, we show that there is no method that is best based on the 3 criteria. We found that the best eighth order method, SA8, and CLND are at the top.

A Simple Hybrid Method for Finding the Root of Nonlinear Equations

In this paper, we proposed a simple modification of McDougall and Wotherspoon [11] method for approximating the root of univariate function. Our modification is based on the approximating the derivative in the corrector step of the proposed McDougall and Wotherspoon Newton like method using secant method. Numerical examples demonstrate the efficiency of the proposed method.

New Variants of Newton's Method for Solving Nonlinear Equations

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 2023

Two Newton-type iterative techniques have been created in this work to locate the true root of univariate nonlinear equations. One of these can be acquired by modifying the double Newton's method in a straightforward manner, while the other can be gotten by modifying the midpoint Newton's method. The iterative approach developed by McDougall and Wortherspoon is employed for the change. The study demonstrates that the modified double Newton's approach outperforms the current one in terms of both convergence order and efficiency index, even though both methods assess the same amount of functions and derivatives every iteration. In comparison to the midpoint Newton's technique, which has a convergence order of 3, the modified midpoint Newton's method has a convergence order of 5.25 and requires two extra functions to be evaluated per iteration. In order to evaluate the effectiveness of recently introduced approaches with current methods, some numerical examples are shown in the final section.

Modified families of Newton, Halley and Chebyshev methods

Applied Mathematics and Computation, 2007

This paper presents new families of Newton-type iterative methods (Newton, Halley and Chebyshev methods) for finding simple zero of univariate non-linear equation, permitting f 0 ðxÞ ¼ 0 in the vicinity of the root. Newton-type iterative methods have well-known geometric interpretation and admit their geometric derivation from a parabola. These algorithms are comparable to the well-known powerful classical methods of Newton, Halley and Chebyshev respectively, and these can be seen as special cases of these families. The efficiency of the presented methods is demonstrated by numerical examples.

Chebyshev-like root-finding methods with accelerated convergence

Numerical Linear Algebra with Applications, 2009

Iterative methods for the simultaneous determination of simple or multiple complex zeros of a polynomial, based on a cubically convergent Chebyshev method, are considered. Using Newton's and Halley's corrections the convergence of the basic method of the fourth order is increased to five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is also studied for the proposed methods. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. Finally, modified methods of Chebyshev's type for finding multiple zeros and single-step methods based on the Gauss-Seidel approach are constructed.

A variant of McDougall-Wotherspoon method for finding simple roots of nonlinear equations

Scientific Publications of the State University of Novi Pazar, 2018

Recently, McDougall and Wotherspoon have developed a simple modification to the standard Newton method for finding simple roots of nonlinear equations. Their method is based on the arithmetic mean and achieves approximately 2.4 convergence order. This research presents a new iteration scheme based on the harmonic mean with the same convergence order and the comparison with the original McDougall-Wotherspoon method. Several test examples have been used and the results show good numerical behavior of the new method.

A Modification On Newton'S Method For Solving Systems Of Nonlinear Equations

2009

In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.

Improved Newton’s Method for Solving Nonlinear Equations (1998)

An iterative scheme is introduced improving Newton's method which is widely used for solving nonlinear equations. The method is developed for both functions of one variable and two variables. Proposed scheme replaces the rectangular approximation of the indefinite integral involved in Newton's Method by a trapezium. It is shown that 'the order of convergence of the new method is at least three for functions of one variable. Computational results overwhelmingly support this theory and the computational order of convergence is even more than three for certain functions. Algorithms constructed were implemented by using the high level computer language Turbo Pascal (Ver. 7) Key words: Convergence, Newton's method, Improved Newton's method, Nonlinear equations, Root finding, Order of convergence, Iterative methods