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 Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 371
Author(s):  
Petko D. Proinov
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Local Convergence ◽  
Initial Approximation ◽  
Vector Spaces ◽  
Polynomial Zeros ◽  
High Q ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.

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.

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 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.

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 On Local Convergence Analysis of Inexact Newton Method for Singular Systems of Equations under Majorant Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fangqin Zhou
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Local Convergence ◽  
Singular Systems ◽  
Inexact Newton Method ◽  
Systems Of Equations ◽  
Convergence Ball ◽  
Special Cases ◽  
Majorant Condition ◽  
Local Convergence Analysis

We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

On a new family of high-order iterative methods for the matrix p th root

2015 ◽  
Vol 22 (4) ◽  
pp. 585-595 ◽  
Author(s):  
S. Amat ◽  
J. A. Ezquerro ◽  
M. A. Hernández-Verón
Keyword(s):  
Iterative Methods ◽  
High Order ◽  
New Family ◽  
The Matrix ◽  
High Order Iterative Methods

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 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 Advances in the Semilocal Convergence of Newton’s Method with Real-World Applications

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 299 ◽  
Author(s):  
Ioannis Argyros ◽  
Á. Magreñán ◽  
Lara Orcos ◽  
Íñigo Sarría
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Lipschitz Constant ◽  
Semilocal Convergence ◽  
Convergence Domain ◽  
Special Cases ◽  
Real World Applications ◽  
Lipschitz Constants ◽  
Local Convergence Analysis ◽  
Theoretical Results

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence domain is extended; the error estimates are tighter and the information on the location of the solution is at least as precise as before. These advantages are obtained using the same information as before, since new Lipschitz constant are tighter and special cases of the ones used before. Numerical examples and applications are used to test favorable the theoretical results to earlier ones.

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