Semilocal convergence of a family of third-order Chebyshev-type methods under a mild differentiability condition

2010 ◽  
Vol 87 (15) ◽  
pp. 3405-3419 ◽  
Author(s):  
P. K. Parida ◽  
D. K. Gupta
Keyword(s):  
Semilocal Convergence ◽  
Third Order ◽  
Differentiability Condition ◽  
Mild Differentiability Condition

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.

scholarly journals A Third Order Newton-Like Method and Its Applications

2018 ◽  
Author(s):  
D. R. Sahu ◽  
R. P. Agarwal ◽  
Y. J. Cho ◽  
V. K. Singh
Keyword(s):  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
First Derivative ◽  
The Third

In this paper, we study the third order semilocal convergence of the Newton-like method for finding the approximate solution of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis under ω-continuity condition, which is weaker than the Lipschitz and Hölder continuity conditions. Second, we apply our approach to solve Fredholm integral equations, where the first derivative of involved operator not necessarily satisfy the Hölder and Lipschitz continuity conditions. Finally, we also prove that the R-order of the method is 2p + 1 for any p $\in$ (0,1].

Semilocal Convergence Theorem for a Newton-like Method

2017 ◽  
Vol 7 (3) ◽  
pp. 482-494
Author(s):  
Rong-Fei Lin ◽  
Qing-Biao Wu ◽  
Min-Hong Chen ◽  
Lu Liu ◽  
Ping-Fei Dai
Keyword(s):  
Error Estimate ◽  
Nonlinear Equations ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Third Order ◽  
Convergence Theorems ◽  
Numerical Examples ◽  
Semilocal Convergence Theorem

AbstractThe semilocal convergence of a third-order Newton-like method for solving nonlinear equations is considered. Under a weak condition (the so-called γ-condition) on the derivative of the nonlinear operator, we establish a new semilocal convergence theorem for the Newton-like method and also provide an error estimate. Some numerical examples show the applicability and efficiency of our result, in comparison to other semilocal convergence theorems.

scholarly journals Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Abhimanyu Kumar ◽  
Dharmendra K. Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Local Convergence ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Convergence Domain ◽  
Differentiability Condition ◽  
Type Integral ◽  
Weaker Conditions ◽  
Theoretical Results

The semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.

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.

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