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 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.

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 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.

Convergence of derivative free iterative methods

2019 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Local Convergence ◽  
Practical Interest ◽  
Systems Of Equations ◽  
Hölder Continuous ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Derivative Free ◽  
Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

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