A convergence of a Steffensen-like method for solving nonlinear equations in a Banach space

2017 ◽  
Vol 26 (2) ◽  
pp. 125-136
Author(s):  
IOANNIS K. ARGYROS ◽  
SANTHOSH GEORGE
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Unique Solution ◽  
Nonlinear Equations ◽  
Semilocal Convergence ◽  
Convergence Conditions ◽  
Numerical Example ◽  
Space Setting

We present a local as well as a semilocal convergence analysis of a Steffensen-like method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. This method generalizes and improves the sufficient convergence conditions of earlier methods. In particular, a numerical example is presented to show the advantages of our approach.

scholarly journals Extending the applicability of Newton's and Secant methods under regular smoothness

2021 ◽  
Vol 39 (6) ◽  
pp. 195-210
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
Shobha M. Erappa
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Unique Solution ◽  
Operator Equation ◽  
Local Convergence ◽  
Numerical Examples ◽  
Space Setting ◽  
Secant Methods ◽  
Iterative Procedures ◽  
Local Convergence Analysis

The concept of regular smoothness has been shown to be an appropriate and powerfull tool for the convergence of iterative procedures converging to a locally unique solution of an operator equation in a Banach space setting.  Motivated by earlier works, and optimization considerations, we present a tighter semi-local convergence analysis using our new idea of restricted convergence domains. Numerical examples complete 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.

Improved semilocal convergence analysis in Banach space with applications to chemistry

2017 ◽  
Vol 56 (7) ◽  
pp. 1958-1975 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Elena Giménez ◽  
Á. A. Magreñán ◽  
Í. Sarría ◽  
Juan Antonio Sicilia
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Semilocal Convergence

scholarly journals Two-Step Solver for Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 128 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Halyna Yarmola
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Semilocal Convergence ◽  
Numerical Example ◽  
Weaker Conditions ◽  
Theoretical Results ◽  
Nondifferentiable Operator

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence 1 + 2 . We study the local and a semilocal convergence using weaker conditions in order to extend the applicability of the solver. Finally, we present the numerical example that confirms the theoretical results.

scholarly journals Semilocal Convergence Theorem for the Inverse-Free Jarratt Method under New Hölder Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yueqing Zhao ◽  
Rongfei Lin ◽  
Zdenek Šmarda ◽  
Yasir Khan ◽  
Jinbiao Chen ◽  
...  
Keyword(s):  
Banach Space ◽  
Error Estimate ◽  
Convergence Analysis ◽  
Operator Equation ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Semilocal Convergence ◽  
Nonlinear Operator Equation ◽  
Jarratt Method ◽  
Semilocal Convergence Theorem

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

Local Convergence and the Dynamics of a Two-Step Newton-Like Method

2016 ◽  
Vol 26 (05) ◽  
pp. 1630012 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Á. Alberto Magreñán
Keyword(s):  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Unique Solution ◽  
Local Convergence ◽  
Multiplicity One ◽  
Local Convergence Analysis

We present the local convergence analysis and the study of the dynamics of a two-step Newton-like method in order to approximate a locally unique solution of multiplicity one of a nonlinear equation.

EXTENDING THE APPLICABILITY OF NEWTON'S METHOD ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

2013 ◽  
Vol 06 (03) ◽  
pp. 1350041
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Error Analysis ◽  
Convergence Analysis ◽  
Lipschitz Condition ◽  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Semilocal Convergence ◽  
Inclusion Problem ◽  
Convergence Conditions ◽  
Precise Information

We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math.13(2B) (2009) 633–656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton–Kantorovich theorem.

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