Semilocal Convergence of a Computationally Efficient Eighth-Order Method in Banach Spaces Under w-Continuity Condition

2016 ◽  
Vol 42 (2) ◽  
pp. 819-826 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Order Method ◽  
Computationally Efficient ◽  
Eighth Order

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.

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.

Semilocal convergence of a sixth-order method in Banach spaces

2012 ◽  
Vol 61 (3) ◽  
pp. 413-427 ◽  
Author(s):  
Lin Zheng ◽  
Chuanqing Gu
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Sixth Order ◽  
Order Method
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