scholarly journals On an Aitken-Steffensen-Newton type method

2018 ◽  
Vol 34 (1) ◽  
pp. 85-92
Author(s):  
ION PAVALOIU ◽  
Keyword(s):  
Local Convergence ◽  
Convergence Result ◽  
Convergence Order ◽  
Monotone Convergence ◽  
Type Method ◽  
Numerical Examples ◽  
Newton Iterations ◽  
Convergence Orders

We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the q-convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems. Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.

scholarly journals Convergence Analysis of a Three Step Newton-like Method for Nonlinear Equations in Banach Space under Weak Conditions

2016 ◽  
Vol 54 (2) ◽  
pp. 37-46
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Convergence Order ◽  
Order Method ◽  
Numerical Examples ◽  
Local Convergence Analysis

Abstract In the present paper, we study the local convergence analysis of a fifth convergence order method considered by Sharma and Guha in [15] to solve equations in Banach space. Using our idea of restricted convergence domains we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

scholarly journals Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

scholarly journals New Simpson type method for solving nonlinear equations

2021 ◽  
Vol 5 (1) ◽  
pp. 94-100
Author(s):  
U. K. Qureshi ◽  
◽  
A. A. Shaikhi ◽  
F. K. Shaikh ◽  
S. K. Hazarewal ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Numerical Methods ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Real World ◽  
Convergence Order ◽  
Type Method ◽  
Numerical Examples ◽  
The Real ◽  
Better Than

Finding root of a nonlinear equation is one of the most important problems in the real world, which arises in the applied sciences and engineering. The researchers developed many numerical methods for estimating roots of nonlinear equations. The this paper, we proposed a new Simpson type method with the help of Simpson 1/3rd rule. It has been proved that the convergence order of the proposed method is two. Some numerical examples are solved to validate the proposed method by using C++/MATLAB and EXCEL. The performance of proposed method is better than the existing ones.

scholarly journals Two-step secant type method with approximation of the inverse operator

2013 ◽  
Vol 5 (1) ◽  
pp. 150-155 ◽  
Author(s):  
S.M. Shakhno ◽  
H.P. Yarmola
Keyword(s):  
Numerical Experiment ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Quadratic Convergence ◽  
Inverse Operator ◽  
Convergence Order ◽  
Test Problems ◽  
Type Method ◽  
Consistent Approximation

The two-step secant type method with a consistent approximation of the inverse operator for solving nonlinear equations is proposed. The local convergence of the proposed method is studied and the quadratic convergence order is established. A numerical experiment is made on the test problems.

scholarly journals Local convergence of the Gauss-Newton-Kurchatov method under generalized Lipschitz conditions

2021 ◽  
Vol 13 (2) ◽  
pp. 305-314
Author(s):  
S.M. Shakhno ◽  
H.P. Yarmola
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Lipschitz Conditions ◽  
Theoretical Results

We investigate the local convergence of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems. This method is a combination of Gauss-Newton and Kurchatov methods and it is used for problems with the decomposition of the operator. The convergence analysis of the method is performed under the generalized Lipshitz conditions. The conditions of convergence, radius and the convergence order of the considered method are established. Given numerical examples confirm the theoretical results.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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.

scholarly journals Transmetric Density Estimation

2016 ◽  
Vol 5 (2) ◽  
pp. 35
Author(s):  
Sigve Hovda
Keyword(s):  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Density ◽  
Convergence Order ◽  
Graphical Display ◽  
Mean Integrated Squared Error ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean ◽  
Convergence Orders

<div>A transmetric is a generalization of a metric that is tailored to properties needed in kernel density estimation.  Using transmetrics in kernel density estimation is an intuitive way to make assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display.  This framework is required for discussing the estimators that are suggested by Hovda (2014).</div><div> </div><div>Asymptotic arguments for the bias and the mean integrated squared error is difficult in the general case, but some results are given when the transmetric is of the type defined in Hovda (2014).  An important contribution of this paper is that the convergence order can be as high as $4/5$, regardless of the number of dimensions.</div>

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.

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