Local convergence of Super Halley’s method under weaker conditions on Fréchet derivative in Banach spaces

2017 ◽  
Vol 28 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Abhimanyu Kumar ◽  
D. K. Gupta
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
Halley’S Method ◽  
Halley's Method ◽  
Weaker Conditions

scholarly journals Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

2018 ◽  
Vol 15 (06) ◽  
pp. 1850048
Author(s):  
Sukhjit Singh ◽  
Dharmendra Kumar Gupta ◽  
Randhir Singh ◽  
Mehakpreet Singh ◽  
Eulalia Martinez
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
First Order ◽  
Weaker Conditions ◽  
Fifth Order ◽  
Derivatives Of ◽  
Text Condition

The convergence analysis both local under weaker Argyros-type conditions and semilocal under [Formula: see text]-condition is established using first order Fréchet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Hölder conditions are particular cases of the [Formula: see text]-condition. Examples can be constructed for which the Lipchitz and Hölder conditions fail but the [Formula: see text]-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.

On the convergence of Halley’s method for simultaneous computation of polynomial zeros

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Petko D. Proinov ◽  
Stoil I. Ivanov
Keyword(s):  
Convergence Theorem ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Halley’S Method ◽  
Halley's Method ◽  
Semilocal Convergence Theorem

AbstractIn this paper we study the convergence of Halley’s method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley’s method for simultaneous computation of polynomial zeros.

Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

2017 ◽  
Vol 10 (02) ◽  
pp. 1750086
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Frechet Derivative ◽  
Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

On a sixth-order Jarratt-type method in Banach spaces

2015 ◽  
Vol 08 (04) ◽  
pp. 1550065 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Sixth Order ◽  
Convergence Conditions ◽  
Type Method ◽  
Frechet Derivative ◽  
The Third ◽  
Local Convergence Analysis ◽  
Jarratt Method

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.

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