jarratt method
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Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 540 ◽  
Author(s):  
Zhang Yong ◽  
Neha Gupta ◽  
J. P. Jaiswal ◽  
Kalyanasundaram Madhu

In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local ω -continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section.


2015 ◽  
Vol 08 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George

We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441–456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study.


2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Rajni Sharma ◽  
Ashu Bahl

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.


2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yueqing Zhao ◽  
Rongfei Lin ◽  
Zdenek Šmarda ◽  
Yasir Khan ◽  
Jinbiao Chen ◽  
...  

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.


2013 ◽  
Vol 2013 (1) ◽  
pp. 194 ◽  
Author(s):  
Ioannis K Argyros ◽  
Yeol Cho ◽  
Sanjay Khattri

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.


2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
R. Thukral

We present two new families of Jarratt-type methods for solving nonlinear equations. It is proved that the order of convergence of each family member is improved from four to six by the addition of one function evaluation. Per iteration, these new methods require two evaluations of the function and two evaluations of the first-order derivatives. In fact, the efficiency index of these methods is 1.565. Numerical comparisons are made with other existing methods to show the performance of the presented methods.


2010 ◽  
Vol 57 (4) ◽  
pp. 441-456 ◽  
Author(s):  
Xiuhua Wang ◽  
Jisheng Kou ◽  
Chuanqing Gu

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