Steffensen type methods for solving nonlinear equations

[EN] In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski's method and in a sixth-order improved Ostrowski's method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton's method. (C) 2010 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnología MTM2010-18539Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2012). Steffensen type methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 236(12):3058-3064. https://doi.org/10.1016/j.cam.2010.08.043S305830642361

A Modified Newton-type Method with Order of Convergence Seven for Solving Nonlinear Equations

In this paper, we mainly study the iterative method for nonlinear equations. We present and analyze a modified seventh-order convergent Newton-type method for solving nonlinear equations. The method is free from second derivatives. Some numerical results illustrate that the proposed method is more efficient and performs better than the classicalNewton's method

High order nonlinear solver

J. Computational Methods in Science and Engineering, 8 No. 4-6, (2008), 245–250.An eighth order method for finding simple zeros of nonlinear functions is developed. The method requires two function- and three derivative-evaluation per step. If we define informational efficiency of a method as the order per function evaluation, we find that our method has informational efficiency of 1.6

Newton Homotopy Continuation Method for Solving Nonlinear Equations using Mathematica

In this paper, we solve the nonlinear equations by using a classical method and a powerful method.  A powerful method known as homotopy continuation method (HCM) is used to solve the problem of classical method.  We use Newton-HCM to solve the divergence problem that the classical Newton’s method always faces. The divergence problem occurs when a bad initial guess is used. The problem with Newton’s method happens when the derivative of given function at initial point equal to zero. The division by zero makes the scheme become nonsense. Thus, an approach used to solve this mathematical problem by using Newton-HCM. The results are implemented by mathematical software known as Mathematica 7.0. The results obtained indicate the ability of Newton-HCM to solve this mathematical problem

Improved Newton-Raphson Methods for Solving Nonlinear Equations

In this paper, we mainly study the numerical algorithms for simple root of nonlinear equations based on Newton-Raphson method. Two modified Newton-Raphson methods for solving nonlinear equations are suggested. Both of the methods are free from second derivatives. Numerical examples are made to show the performance of the presented methods, and to compare with other ones. The numerical results illustrate that the proposed methods are more efficient and performs better than Newton-Raphson method

Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowski's method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, 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 is optimal according to Kung and Traub's conjecture. Numerical comparisons are made to show the performance of the new family. © 2011 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics. 235(10):3189-3194. https://doi.org/10.1016/j.cam.2011.01.004S318931942351

A New Sixth Order Method for Nonlinear Equations in R

A new iterative method is described for finding the real roots of nonlinear equations in R. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newton’s method and other sixth order methods considered

Solving Polynomial Equations using Modified Super Ostrowski Homotopy Continuation Method

Homotopy continuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations.  HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions.   The starting value problem has been solved using Super Ostrowski homotopy continuation method for the initial guesses between . Nevertheless, Super Ostrowski homotopy continuation method was only used to find out real roots of nonlinear equations.  In this paper, we employ the Modified Super Ostrowski-HCM to solve several real life applications which involves polynomial equations by expanding the range of starting values. The results indicate that the Modified Super Ostrowski-HCM performs better than the standard Super Ostrowski-HCM. In other words, the complex roots of polynomial equations can be found even the starting value is real with this proposed scheme