Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition

2016 ◽  
Vol 74 (3) ◽  
pp. 643-657 ◽  
Author(s):  
Xiuhua Wang ◽  
Jisheng Kou
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Root Finding

Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under Hölder Continuity Condition

2020 ◽  
Vol 87 (1-2) ◽  
pp. 56
Author(s):  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Banach Spaces ◽  
Error Bound ◽  
Order Of Convergence ◽  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Theoretical Development ◽  
Numerical Examples ◽  
Seventh Order

The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x<sup>*</sup> and its uniqueness along with prior error bound are established. Also, the <em>R</em>-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.

SEMILOCAL CONVERGENCE OF A STIRLING-LIKE METHOD IN BANACH SPACES

2010 ◽  
Vol 07 (02) ◽  
pp. 215-228 ◽  
Author(s):  
S. K. PARHI ◽  
D. K. GUPTA
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Recurrence Relations ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Numerical Examples

The aim of this paper is to establish the semilocal convergence of a third order Stirling–like method employed for solving nonlinear equations in Banach spaces by using the first Fréchet derivative, which satisfies the Lipschitz continuity condition. This makes it possible to avoid the evaluation of higher order Fréchet derivatives which are computationally difficult at times or may not even exist. The recurrence relations are used for convergence analysis. A convergence theorem is given for deriving error bounds and the domains of existence and uniqueness of solutions. The R order of the method is also established to be equal to 3. Finally, two numerical examples are worked out, and the results obtained are compared with the existing results. It is observed that our convergence analysis is more effective.

Convergence of Traub's iteration under ω continuity condition in Banach spaces

2021 ◽  
pp. 61-79
Author(s):  
Debasis Sharma ◽  
◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda ◽  
◽  
...  
Keyword(s):  
Banach Spaces ◽  
Continuity Condition

Deformed super-Halley’s iteration in Banach spaces and its local and semilocal convergence

2019 ◽  
Vol 30 (3-4) ◽  
pp. 413-431
Author(s):  
M. Prashanth ◽  
Abhimanyu Kumar ◽  
D. K. Gupta ◽  
S. S. Mosta
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence

scholarly journals On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 540 ◽  
Author(s):  
Zhang Yong ◽  
Neha Gupta ◽  
J. P. Jaiswal ◽  
Kalyanasundaram Madhu
Keyword(s):  
Nonlinear Operator ◽  
A Priori ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Fréchet Derivative ◽  
Third Order ◽  
Frechet Derivative ◽  
Initial Iterate ◽  
Jarratt Method ◽  
Third Derivative

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.

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