scholarly journals Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces

2015 ◽  
Vol 273 ◽  
pp. 205-213 ◽  
Author(s):  
A. Cordero ◽  
M.A. Hernández-Verón ◽  
N. Romero ◽  
J.R. Torregrosa
Keyword(s):  
Banach Spaces ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Order Method ◽  
Fifth Order

scholarly journals Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Liang Chen ◽  
Chuanqing Gu ◽  
Yanfang Ma
Keyword(s):  
Banach Spaces ◽  
Newton’S Method ◽  
Newton's Method ◽  
Recurrence Relations ◽  
A Priori ◽  
Semilocal Convergence ◽  
Numerical Application ◽  
A Priori Error Bounds ◽  
Modified Newton’S Method ◽  
Fifth Order

We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.

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 R-ORDER CONVERGENCE OF A THIRD ORDER METHOD IN BANACH SPACES UNDER MILD DIFFERENTIABILITY CONDITIONS

2009 ◽  
Vol 06 (02) ◽  
pp. 291-306 ◽  
Author(s):  
P. K. PARIDA ◽  
D. K. GUPTA
Keyword(s):  
Banach Spaces ◽  
Recurrence Relations ◽  
Hölder Continuity ◽  
A Priori ◽  
Continuity Condition ◽  
Order Method ◽  
Holder Continuity ◽  
Third Order ◽  
A Priori Error Bounds ◽  
Two Parameters

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F(x)=0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity condition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails.

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