Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations

1990 ◽  
Vol 7 (1) ◽  
pp. 131-143 ◽  
Author(s):  
Tetsuro Yamamoto ◽  
Xiaojun Chen
Keyword(s):  
Iterative Methods ◽  
Error Estimates ◽  
Nonlinear Equations ◽  
Convergence Theorems ◽  
Ball Convergence

scholarly journals Extended Semilocal Convergence for the Newton- Kurchatov Method

2020 ◽  
Vol 53 (1) ◽  
pp. 85-91
Author(s):  
H.P. Yarmola ◽  
I. K. Argyros ◽  
S.M. Shakhno
Keyword(s):  
Iterative Methods ◽  
Error Estimates ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Semilocal Convergence ◽  
Convergence Criteria ◽  
Precise Location ◽  
Lipschitz Constants ◽  
Restricted Region

We provide a semilocal analysis of the Newton-Kurchatov method for solving nonlinear equations involving a splitting of an operator. Iterative methods have a limited restricted region in general. A convergence of this method is presented under classical Lipschitz conditions.The novelty of our paper lies in the fact that we obtain weaker sufficient semilocal convergence criteria and tighter error estimates than in earlier works. We find a more precise location than before where the iterates lie resulting to at least as small Lipschitz constants. Moreover, no additional computations are needed than before. Finally, we give results of numerical experiments.

scholarly journals A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1855 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Methods ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
High Order ◽  
Convergence Theorems ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Unified Convergence Analysis of Chebyshev–Halley Methods for Multiple Polynomial Zeros

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 135
Author(s):  
Stoil I. Ivanov
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Error Estimates ◽  
Local Convergence ◽  
Initial Conditions ◽  
Extended Version ◽  
Unified Approach ◽  
Polynomial Zeros ◽  
Convergence Results ◽  
Appl Math

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods.

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