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Improved local convergence analysis of inexact Newton-like methods under the majorant condition

Ioannis K. ArgyrosSanthosh George — 2015

Applicationes Mathematicae

We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations. Using more precise majorant conditions than in earlier studies, we provide: a larger radius of convergence; tighter error estimates on the distances involved; and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.

On the semilocal convergence of a two-step Newton-like projection method for ill-posed equations

Ioannis K. ArgyrosSanthosh George — 2013

Applicationes Mathematicae

We present new semilocal convergence conditions for a two-step Newton-like projection method of Lavrentiev regularization for solving ill-posed equations in a Hilbert space setting. The new convergence conditions are weaker than in earlier studies. Examples are presented to show that older convergence conditions are not satisfied but the new conditions are satisfied.

Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions

Ioannis K. ArgyrosSanthosh George — 2015

Applicationes Mathematicae

We present a local multi-point convergence analysis for a family of super-Halley methods of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second Fréchet derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet derivative. Numerical examples are also provided.

Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Ioannis K. ArgyrosSanthosh George — 2017

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved....

Local convergence analysis of a modified Newton-Jarratt's composition under weak conditions

Ioannis K. ArgyrosSanthosh George — 2019

Commentationes Mathematicae Universitatis Carolinae

A. Cordero et. al (2010) considered a modified Newton-Jarratt's composition to solve nonlinear equations. In this study, using decomposition technique under weaker assumptions 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.

Local convergence for a family of iterative methods based on decomposition techniques

Ioannis K. ArgyrosSanthosh GeorgeShobha Monnanda Erappa — 2016

Applicationes Mathematicae

We present a local convergence analysis for a family of iterative methods obtained by using decomposition techniques. The convergence of these methods was shown before using hypotheses on up to the seventh derivative although only the first derivative appears in these methods. In the present study we expand the applicability of these methods by showing convergence using only the first derivative. Moreover we present a radius of convergence and computable error bounds based only on Lipschitz constants....

Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa ShobhaIoannis K. ArgyrosSanthosh George — 2014

Applicationes Mathematicae

We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is y δ with | | y - y δ | | δ , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where F ' ( x ) - 1 exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...

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