Family Of Sixth-order Methods Research Articles

Local convergence for a family of sixth order methods with parameters

Local convergence of a family of sixth order methods for solving Banach space valued equations is considered in this article. The local convergence analysis is provided using only the first derivative in contrast to earlier works on the real line using the seventh derivative. This way the applicability is expanded for these methods. Numerical examples complete the article.

A family of sixth-order methods with six stages

A family of sixth-order methods with six stages

A family of sixth-order methods with six stages

A family of sixth-order methods with six stages

An Efficient Family of Traub-Steffensen-Type Methods for Solving Systems of Nonlinear Equations

Based on Traub-Steffensen method, we present a derivative free three-step family of sixth-order methods for solving systems of nonlinear equations. The local convergence order of the family is determined using first-order divided difference operator for functions of several variables and direct computation by Taylor's expansion. Computational efficiency is discussed, and a comparison between the efficiencies of the proposed techniques with the existing ones is made. Numerical tests are performed to compare the methods of the proposed family with the existing methods and to confirm the theoretical results. It is shown that the new family is especially efficient in solving large systems.

A sixth order method for nonlinear equations

In this paper, a sixth order method is developed by extending a third order method of Weerakoon and Fernando [S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93] for finding the real roots of nonlinear equations in R . Starting with a suitably chosen x 0 , the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first derivatives per iteration. This implies that efficiency index of our method is 1.565. Our method is comparable with the methods of Neta [B. Neta, A sixth order family of methods for nonlinear equations, Intern. J. Computer Math. 7 (1979) 157–161] and that of Kou and Li [Jisheng Kou, Yitian Li, An improvement of Jarratt method, Appl. Math. Comput. 189 (2007) 1816–1821]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev–Halley type methods [Jisheng Kou, Xiuhua Wang, Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations, Appl. Math. Comput. 190 (2007) 1839–1843; Jisheng Kou, On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations, Appl. Math. Comput. 190 (2007) 126–131]. The efficacy of the method is tested on a number of numerical examples. It is observed that our method takes less number of iterations than Newton’s method and the method of Weerakoon and Fernando. On comparison with the other sixth order methods, it behaves either similarly or better for the examples considered.

A family of improved Jarratt multipoint methods

We derive a one-parameter family of sixth-order methods for solving equations which is based on the Jarratt fourth-order multipoint method. Further, we show that a particular choice of the parameter results in fifth-order convergence. Each member of the family requires two evaluations of the given function and two of its derivative per iteration. Numerical examples are presented and the performance is compared with the Jarratt method.

A family of modified Ostrowski methods with accelerated sixth order convergence

Based on Ostrowski fourth order multipoint method, we derive a one-parameter family of sixth order methods for solving equations. Each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration. Numerical examples are presented and performance is compared with Ostrowski method.

A new family of higher order methods for solving equations

Neta's three step sixth order family of methods for solving nonlinear equations require 3 function and 1 derivative evaluation per iteration. Using exactly the same information another three step method can be obtained with convergence rate 10.81525 which is much better than the sixth order.

A note on neta's family of sixth-order methods for solving equations

It has been shown that some three-step methods exist which, as well as Neta's methods, require one derivative and three function evaluations per iteration, but have an asymptotic convergence rate 7 which is better than the 6 of Neta.