scholarly journals Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

2016 ◽  
Vol 19 (1) ◽  
pp. 293-302
Author(s):  
Xiuhua Wang ◽  
Jisheng Kou
Keyword(s):  
Uniqueness Theorem ◽  
Banach Spaces ◽  
Semilocal Convergence ◽  
Second Derivative

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

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 Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yonghui Ling ◽  
Xiubin Xu ◽  
Shaohua Yu
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Nonlinear Operator ◽  
Convergence Criterion ◽  
Second Derivative ◽  
Operator Equations ◽  
Convergence Ball ◽  
Nonlinear Operator Equations ◽  
Steffensen’S Method ◽  
Convergence Problems

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

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.

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