scholarly journals Local Convergence for an Efficient Eighth Order Iterative Method with a Parameter for Solving Equations Under Weak Conditions

2015 ◽  
Vol 2 (4) ◽  
pp. 565-574 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Iterative Method ◽  
Local Convergence ◽  
Eighth Order ◽  
Solving Equations

scholarly journals Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 25
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Order Method ◽  
Eighth Order ◽  
First Derivative ◽  
Taylor Expansions ◽  
Derivative Free ◽  
Class Of Functions

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

scholarly journals Local convergence for an eighth order method for solving equations and systems of equations

2019 ◽  
Vol 8 (1) ◽  
pp. 74-79
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Divided Difference ◽  
Dimensional Euclidean Space ◽  
Order Method ◽  
Systems Of Equations ◽  
Eighth Order ◽  
First Derivative ◽  
Solving Equations ◽  
Computable Error Bounds

AbstractThe aim of this study is to extend the applicability of an eighth convergence order method from thek−dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions.

A fifth-order iterative method for solving equations

1987 ◽  
Vol 49 (1-2) ◽  
pp. 129-137 ◽  
Author(s):  
Duong Thuy Vy
Keyword(s):  
Iterative Method ◽  
Solving Equations ◽  
Fifth Order

scholarly journals Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 158
Author(s):  
Ioannis K. Argyros ◽  
Stepan Shakhno ◽  
Roman Iakymchuk ◽  
Halyna Yarmola ◽  
Michael I. Argyros
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Local Convergence ◽  
Nonlinear Least Squares ◽  
Computational Effort ◽  
Least Squares Problems ◽  
Restricted Region ◽  
Convergence Orders ◽  
Operator Decomposition ◽  
Lipschitz Conditions

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

scholarly journals Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

Algorithms ◽  
2016 ◽  
Vol 9 (4) ◽  
pp. 65 ◽  
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl ◽  
Sandile Motsa
Keyword(s):  
Convergence Analysis ◽  
Local Convergence ◽  
Eighth Order ◽  
First Derivative ◽  
Order Scheme ◽  
Local Convergence Analysis

scholarly journals Local Convergence of an Optimal Eighth Order Method under Weak Conditions

Algorithms ◽  
2015 ◽  
Vol 8 (3) ◽  
pp. 645-655 ◽  
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl ◽  
S.S. Motsa
Keyword(s):  
Local Convergence ◽  
Order Method ◽  
Eighth Order

scholarly journals On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 114-127
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Type Method ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Original Domain ◽  
Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

Local convergence of parameter based method with six and eighth order of convergence

2020 ◽  
Vol 58 (4) ◽  
pp. 841-853
Author(s):  
Ali Saleh Alshomrani ◽  
Ramandeep Behl ◽  
P. Maroju
Keyword(s):  
Local Convergence ◽  
Order Of Convergence ◽  
Eighth Order

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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