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.

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 &omega;-continuity condition, which is weaker than the Lipschitz and H&ouml;lder continuity conditions. Second, we apply our approach to solve Fredholm integral equations, where the first derivative of involved operator not necessarily satisfy the H&ouml;lder and Lipschitz continuity conditions. Finally, we also prove that the R-order of the method is 2p + 1 for any p $\in$ (0,1].

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 Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Nonsmooth Equations ◽  
Order Convergence ◽  
Linear Combinations ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Seventh Order ◽  
The Given

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

Semilocal Convergence Analysis for MMN-HSS Methods under Hölder Conditions

2017 ◽  
Vol 7 (2) ◽  
pp. 396-416
Author(s):  
Yang Li ◽  
Xue-Ping Guo
Keyword(s):  
Theoretical Analysis ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Positive Definite ◽  
Systems Of Nonlinear Equations ◽  
Newton Methods ◽  
Numerical Examples ◽  
Sparse Systems ◽  
Lipschitz Conditions ◽  
Semilocal Convergence Theorem

AbstractMulti-step modified Newton-HSS (MMN-HSS) methods, which are variants of inexact Newton methods, have been shown to be competitive for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices. Previously, we established these MMN-HSS methods under Lipschitz conditions, and we now present a semilocal convergence theorem assuming the nonlinear operator satisfies milder Hölder continuity conditions. Some numerical examples demonstrate our theoretical analysis.

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

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 31 ◽  
Author(s):  
D. Sahu ◽  
Ravi Agarwal ◽  
Vipin Singh
Keyword(s):  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Approximate Solutions ◽  
Continuity Condition ◽  
Convergence Theory ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
First Derivative

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature.

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.

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