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.

Ball convergence of some fourth and sixth-order iterative methods

2016 ◽  
Vol 09 (02) ◽  
pp. 1650034
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Sixth Order ◽  
Chebyshev Method ◽  
Local Convergence Analysis ◽  
Appl Math

We present a local convergence analysis for some families of fourth and sixth-order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies [V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990) 355–367; C. Chun, P. Stanica and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011) 1665–1675; J. M. Gutiérrez and M. A. Hernández, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998) 1–8; M. A. Hernández and M. A. Salanova, Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method, J. Comput. Appl. Math. 126 (2000) 131–143; M. A. Hernández, Chebyshev’s approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–455; M. A. Hernández, Second-derivative-free variant of the Chebyshev method for nonlinear equations, J. Optim. Theory Appl. 104(3) (2000) 501–515; J. L. Hueso, E. Martinez and C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth-order families of iterative methods for nonlinear systems, J. Comput. Appl. Math. 275 (2015) 412–420; Á. A. Magre nán, Estudio de la dinámica del método de Newton amortiguado, Ph.D. thesis, Servicio de Publicaciones, Universidad de La Rioja (2013), http://dialnet.unirioja.es/servlet/tesis?codigo=38821 ; J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970); M. S. Petkovic, B. Neta, L. Petkovic and J. Džunič, Multi-Point Methods for Solving Nonlinear Equations (Elsevier, 2013); J. F. Traub, Iterative Methods for the Solution of Equations, Automatic Computation (Prentice-Hall, Englewood Cliffs, NJ, 1964); X. Wang and J. Kou, Semilocal convergence and [Formula: see text]-order for modified Chebyshev–Halley methods, Numer. Algorithms 64(1) (2013) 105–126] have used hypotheses on the fourth Fréchet derivative of the operator involved. We use hypotheses only on the first Fréchet derivative in our local convergence analysis. This way, the applicability of these methods is extended. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples illustrating the theoretical results are also presented in this study.

scholarly journals General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1599
Author(s):  
Stoil I. Ivanov
Keyword(s):  
Error Estimates ◽  
Local Convergence ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Sufficient Conditions ◽  
Posteriori Error ◽  
Picard Iteration ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Simple And Multiple Roots

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

scholarly journals On the Influence of Center-Lipschitz Conditions in the Convergence Analysis of Multi-Point Iterative Methods

2018 ◽  
Author(s):  
Ioannis K Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Convergence Region ◽  
Numerical Examples ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to extend the local as well as the semi-local convergence analysis of multi-point iterative methods using center Lipschitz conditions in combination with our idea, of the restricted convergence region. It turns out that this way a finer convergence analysis for these methods is obtained than in earlier works and without additional hypotheses. Numerical examples favoring our technique over earlier ones completes this article.

scholarly journals Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 233 ◽  
Author(s):  
Ioannis Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Small Region ◽  
Convergence Region ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton’s, or Stirling’s, or Steffensen’s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations

2020 ◽  
pp. 102-109
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Error Bounds ◽  
Local Convergence ◽  
Not Given ◽  
Systems Of Equations ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

The local convergence analysis of iterative methods is important since it demonstrates the degree of diffculty for choosing initial points. In the present study, we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher order derivatives. This way we extend the applicability of these methods. The analysis includes computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study.

scholarly journals Local and Semilocal Convergence of Nourein’s Iterative Method for Finding All Zeros of a Polynomial Simultaneously

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1801 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Method ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
Posteriori Error ◽  
A Posteriori ◽  
Convergence Theorems ◽  
A Posteriori Error ◽  
Convergence Results ◽  
Local Convergence Analysis ◽  
Appl Math

In 1977, Nourein (Intern. J. Comput. Math. 6:3, 1977) constructed a fourth-order iterative method for finding all zeros of a polynomial simultaneously. This method is also known as Ehrlich’s method with Newton’s correction because it is obtained by combining Ehrlich’s method (Commun. ACM 10:2, 1967) and the classical Newton’s method. The paper provides a detailed local convergence analysis of a well-known but not well-studied generalization of Nourein’s method for simultaneous finding of multiple polynomial zeros. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with verifiable initial condition and a posteriori error bound) for the classical Nourein’s method. Each of the new semilocal convergence results improves the result of Petković, Petković and Rančić (J. Comput. Appl. Math. 205:1, 2007) in several directions. The paper ends with several examples that show the applicability of our semilocal convergence theorems.

scholarly journals A convergence analysis of SOR iterative methods for linear systems with weak H-matrices

2016 ◽  
Vol 14 (1) ◽  
pp. 747-760
Author(s):  
Cheng-yi Zhang ◽  
Zichen Xue ◽  
Shuanghua Luo
Keyword(s):  
Convergence Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Convergence Results ◽  
Diagonally Dominant Matrices ◽  
Diagonally Dominant ◽  
Necessary And Sufficient

AbstractIt is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

2017 ◽  
Vol 10 (02) ◽  
pp. 1750086
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Frechet Derivative ◽  
Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

scholarly journals LOCAL CONVERGENCE OF JARRATT-TYPE METHODS WITH LESS COMPUTATION OF INVERSION UNDER WEAK CONDITIONS

2017 ◽  
Vol 22 (2) ◽  
pp. 228-236
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Space Setting ◽  
Local Convergence Analysis ◽  
Methods Numerical

We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study.

scholarly journals Unified Convergence Analysis of Two-Step Iterative Methods for Solving Equations

2021 ◽  
Vol 1 (2) ◽  
pp. 68-85
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
General Setting ◽  
Convergence Criteria ◽  
Recurrent Functions ◽  
Solving Equations ◽  
Order Four ◽  
Methods Numerical

In this paper we consider unified convergence analysis of two-step iterative methods for solving equations in the Banach space setting. The convergence order four was shown using Taylor expansions requiring the existence of the fifth derivative not on this method. But these hypotheses limit the utilization of it to functions which are at least five times differentiable although the method may converge. As far as we know no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided differences which appear on the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study.

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