A new semilocal convergence theorem for a fast iterative method with nondifferentiable operators

2009 ◽  
Vol 34 (1-2) ◽  
pp. 39-46 ◽  
Author(s):  
Hongmin Ren ◽  
Ioannis K. Argyros
Keyword(s):  
Iterative Method ◽  
Convergence Theorem ◽  
Semilocal Convergence ◽  
Semilocal Convergence Theorem

scholarly journals On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1640
Author(s):  
Petko D. Proinov ◽  
Milena D. Petkova
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
Polynomial Zeros ◽  
Numerical Examples ◽  
New Family ◽  
Local Convergence Theorem ◽  
Semilocal Convergence Theorem

In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.

On the convergence of Halley’s method for simultaneous computation of polynomial zeros

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Petko D. Proinov ◽  
Stoil I. Ivanov
Keyword(s):  
Convergence Theorem ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Halley’S Method ◽  
Halley's Method ◽  
Semilocal Convergence Theorem

AbstractIn this paper we study the convergence of Halley’s method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley’s method for simultaneous computation of polynomial zeros.

scholarly journals Semilocal Convergence Theorem for the Inverse-Free Jarratt Method under New Hölder Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yueqing Zhao ◽  
Rongfei Lin ◽  
Zdenek Šmarda ◽  
Yasir Khan ◽  
Jinbiao Chen ◽  
...  
Keyword(s):  
Banach Space ◽  
Error Estimate ◽  
Convergence Analysis ◽  
Operator Equation ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Nonlinear Operator Equation ◽  
Jarratt Method ◽  
Semilocal Convergence Theorem

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

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 Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Xiubin Xu ◽  
Yuan Xiao ◽  
Tao Liu
Keyword(s):  
Newton Method ◽  
Convergence Theorem ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Inexact Newton Method ◽  
Convergence Criteria ◽  
First Derivative ◽  
Special Cases ◽  
Lipschitz Conditions ◽  
Semilocal Convergence Theorem

Under the hypothesis that the first derivative satisfies some kind of weak Lipschitz conditions, a new semilocal convergence theorem for inexact Newton method is presented. Unified convergence criteria ensuring the convergence of inexact Newton method are also established. Applications to some special cases such as the Kantorovich type conditions andγ-Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.

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