scholarly journals Influence of the Center Condition on the Two-Step Secant Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Abhimanyu Kumar ◽  
D. K. Gupta ◽  
Shwetabh Srivastava
Keyword(s):  
Nonlinear Equation ◽  
Secant Method ◽  
Numerical Examples ◽  
Nonlinear Elliptic ◽  
Center Condition ◽  
Elliptic Differential Equations ◽  
Different Types ◽  
Center Conditions ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this paper is to present a new improved semilocal and local convergence analysis for two-step secant method to approximate a locally unique solution of a nonlinear equation in Banach spaces. This study is important because starting points play an important role in the convergence of an iterative method. We have used a combination of Lipschitz and center-Lipschitz conditions on the Fréchet derivative instead of only Lipschitz condition. A comparison is established on different types of center conditions and the influence of our approach is shown through the numerical examples. In comparison to some earlier study, it gives an improved domain of convergence along with the precise error bounds. Finally, some numerical examples including nonlinear elliptic differential equations and integral equations validate the efficacy of our approach.

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 Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals The Convergence Ball and Error Analysis of the Relaxed Secant Method

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Rongfei Lin ◽  
Qingbiao Wu ◽  
Minhong Chen ◽  
Lu Liu
Keyword(s):  
Nonlinear Equation ◽  
Error Analysis ◽  
Error Estimate ◽  
Secant Method ◽  
Speed Of Convergence ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonlinear Equation Systems

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.

scholarly journals Design and analysis of a faster King-Werner-type derivative free method

2022 ◽  
Vol 40 ◽  
pp. 1-18
Author(s):  
J. R. Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Convergence Analysis ◽  
Local Convergence ◽  
Numerical Examples ◽  
Weak Center ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results ◽  
Methods Numerical

We introduce a new faster  King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local  convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.

scholarly journals A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order1+2and Its Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Gustavo Fernández-Torres
Keyword(s):  
Nonlinear Equation ◽  
Newton’S Method ◽  
Newton's Method ◽  
Secant Method ◽  
New Method ◽  
Convergence Order ◽  
Dynamical Analysis ◽  
Numerical Examples ◽  
Modified Method

A geometric modification to the Newton-Secant method to obtain the root of a nonlinear equation is described and analyzed. With the same number of evaluations, the modified method converges faster than Newton’s method and the convergence order of the new method is1+2≈2.4142. The numerical examples and the dynamical analysis show that the new method is robust and converges to the root in many cases where Newton’s method and other recently published methods fail.

Ball convergence for a three-point method with optimal convergence order eight under weak conditions

2018 ◽  
Vol 13 (02) ◽  
pp. 2050048
Author(s):  
Ioannis K. Argyros ◽  
Munish Kansal ◽  
V. Kanwar
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Eighth Order ◽  
Numerical Examples ◽  
Optimal Convergence ◽  
New Class ◽  
First Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis of an optimal eighth-order family of Ostrowski like methods for approximating a locally unique solution of a nonlinear equation. Earlier studies [T. Lotfi, S. Sharifi, M. Salimi and S. Siegmund, A new class of three-point methods with optimal convergence and its dynamics, Numer. Algorithms 68 (2015) 261–288.] have shown convergence of these methods under hypotheses up to the eighth derivative of the function although only the first derivative appears in the method. In this study, we expand the applicability of these methods using only hypotheses up to the first derivative of the function. By this way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

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 Extended Convergence of a Two-Step-Secant-Type Method Under a Restricted Convergence Domain

2020 ◽  
Vol 45 (01) ◽  
pp. 155-164
Author(s):  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Computational Effort ◽  
Convergence Domain ◽  
Type Method ◽  
Precise Information ◽  
Numerical Examples ◽  
Special Cases ◽  
Lipschitz Constants ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions ◽  
Theoretical Results

We present a local as well as a semi-local convergence analysis of a two-step secant-type method for solving nonlinear equations involving Banach space valued operators. By using weakened Lipschitz and center Lipschitz conditions in combination with a more precise domain containing the iterates, we obtain tighter Lipschitz constants than in earlier studies. This technique lead to an extended convergence domain, more precise information on the location of the solution and tighter error bounds on the distances involved. These advantages are obtained under the same computational effort, since the new constants are special cases of the old ones used in earlier studies. The new technique can be used on other iterative methods. The numerical examples further illustrate the theoretical results.

scholarly journals Ball convergence theorem for inexact Newton methods in Banach space

2020 ◽  
Vol 29 (2) ◽  
pp. 113-120
Author(s):  
IOANNIS K. ARGYROS ◽  
GEORGE SANTHOSH
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Newton Methods ◽  
Convergence Conditions ◽  
Inexact Newton Methods ◽  
Numerical Examples ◽  
Ball Convergence ◽  
Local Convergence Analysis

We present a local convergence analysis for inexact Newton methods in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Frèchet-derivative of the operator involved. Numerical examples are also provided in this study.

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