Displaying similar documents to “Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions”

Local convergence of two competing third order methods in Banach space

Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

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We present a local convergence analysis for two popular third order methods of approximating a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for both methods under the same conditions. A comparison is given between the two methods, as well as numerical examples.

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

Ioannis K. Argyros, Santhosh George (2017)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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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...

On the convergence and application of Stirling's method

Ioannis K. Argyros (2003)

Applicationes Mathematicae

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We provide new sufficient convergence conditions for the local and semilocal convergence of Stirling's method to a locally unique solution of a nonlinear operator equation in a Banach space setting. In contrast to earlier results we do not make use of the basic restrictive assumption in [8] that the norm of the Fréchet derivative of the operator involved is strictly bounded above by 1. The study concludes with a numerical example where our results compare favorably with earlier ones. ...

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

Santhosh George, Ioannis K. Argyros (2019)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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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...

An improved convergence analysis of Newton's method for twice Fréchet differentiable operators

Ioannis K. Argyros, Sanjay K. Khattri (2013)

Applicationes Mathematicae

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We develop local and semilocal convergence results for Newton's method in order to solve nonlinear equations in a Banach space setting. The results compare favorably to earlier ones utilizing Lipschitz conditions on the second Fréchet derivative of the operators involved. Numerical examples where our new convergence conditions are satisfied but earlier convergence conditions are not satisfied are also reported.

A convergence analysis of Newton's method under the gamma-condition in Banach spaces

Ioannis K. Argyros (2009)

Applicationes Mathematicae

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We provide a local as well as a semilocal convergence analysis for Newton's method to approximate a locally unique solution of an equation in a Banach space setting. Using a combination of center-gamma with a gamma-condition, we obtain an upper bound on the inverses of the operators involved which can be more precise than those given in the elegant works by Smale, Wang, and Zhao and Wang. This observation leads (under the same or less computational cost) to a convergence analysis with...

Expanding the applicability of two-point Newton-like methods under generalized conditions

Ioannis K. Argyros, Saïd Hilout (2013)

Applicationes Mathematicae

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We use a two-point Newton-like method to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. Using more precise majorizing sequences than in earlier studies, we present a tighter semi-local and local convergence analysis and weaker convergence criteria. This way we expand the applicability of these methods. Numerical examples are provided where the old convergence criteria do not hold but the new convergence criteria...

Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces

I. K. Argyros, D. González, S. K. Khattri (2016)

Commentationes Mathematicae Universitatis Carolinae

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We present a local convergence analysis of a one parameter Jarratt-type method. We use this method to approximate a solution of an equation in a Banach space setting. The semilocal convergence of this method was recently carried out in earlier studies under stronger hypotheses. Numerical examples are given where earlier results such as in [Ezquerro J.A., Hernández M.A., New iterations of R -order four with reduced computational cost, BIT Numer. Math. 49 (2009), 325–342] cannot be used...

Convergence domains under Zabrejko-Zinčenko conditions using recurrent functions

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

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We provide a semilocal convergence analysis for Newton-type methods using our idea of recurrent functions in a Banach space setting. We use Zabrejko-Zinčenko conditions. In particular, we show that the convergence domains given before can be extended under the same computational cost. Numerical examples are also provided to show that we can solve equations in cases not covered before.

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

Ioannis K. Argyros, Santhosh George, Shobha Monnanda Erappa (2016)

Applicationes Mathematicae

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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...

On the Halley method in Banach spaces

Ioannis K. Argyros, Hongmin Ren (2012)

Applicationes Mathematicae

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We provide a semilocal convergence analysis for Halley's method using convex majorants in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our results reduce and improve earlier ones in special cases.