Displaying similar documents to “On the convergence of Newton's method for analytic operators.”

Local convergence theorems for Newton's method from data at one point

Ioannis K. Argyros (2002)

Applicationes Mathematicae

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We provide local convergence theorems for the convergence of Newton's method to a solution of an equation in a Banach space utilizing only information at one point. It turns out that for analytic operators the convergence radius for Newton's method is enlarged compared with earlier results. A numerical example is also provided that compares our results favorably with earlier ones.

A weaker affine covariant Newton-Mysovskikh theorem for solving equations

Ioannis K. Argyros (2006)

Applicationes Mathematicae

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The Newton-Mysovskikh theorem provides sufficient conditions for the semilocal convergence of Newton's method to a locally unique solution of an equation in a Banach space setting. It turns out that under weaker hypotheses and a more precise error analysis than before, weaker sufficient conditions can be obtained for the local as well as semilocal convergence of Newton's method. Error bounds on the distances involved as well as a larger radius of convergence are obtained. Some numerical...

On a new method for enlarging the radius of convergence for Newton's method

Ioannis K. Argyros (2001)

Applicationes Mathematicae

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We provide new local and semilocal convergence results for Newton's method. We introduce Lipschitz-type hypotheses on the mth-Frechet derivative. This way we manage to enlarge the radius of convergence of Newton's method. Numerical examples are also provided to show that our results guarantee convergence where others do not.

New unifying convergence criteria for Newton-like methods

Ioannis K. Argyros (2002)

Applicationes Mathematicae

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We present a local and a semilocal analysis for Newton-like methods in a Banach space. Our hypotheses on the operators involved are very general. It turns out that by choosing special cases for the "majorizing" functions we obtain all previous results in the literature, but not vice versa. Since our results give a deeper insight into the structure of the functions involved, we can obtain semilocal convergence under weaker conditions and in the case of local convergence a larger convergence...

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

Ioannis K. Argyros, Santhosh George (2013)

Applicationes Mathematicae

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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.

On the convergence of two-step Newton-type methods of high efficiency index

Ioannis K. Argyros, Saïd Hilout (2009)

Applicationes Mathematicae

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We introduce a new idea of recurrent functions to provide a new semilocal convergence analysis for two-step Newton-type methods of high efficiency index. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar type, and a differential equation containing a Green's kernel are also provided. ...

A convergence analysis of Newton-like methods for singular equations using outer or generalized inverses

Ioannis K. Argyros (2005)

Applicationes Mathematicae

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The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...

Inexact Newton methods and recurrent functions

Ioannis K. Argyros, Saïd Hilout (2010)

Applicationes Mathematicae

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We provide a semilocal convergence analysis for approximating a solution of an equation in a Banach space setting using an inexact Newton method. By using recurrent functions, we provide under the same or weaker hypotheses: finer error bounds on the distances involved, and an at least as precise information on the location of the solution as in earlier papers. Moreover, if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained. Furthermore,...

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.

On the convergence of the secant method under the gamma condition

Ioannis Argyros (2007)

Open Mathematics

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We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.

A general semilocal convergence result for Newton’s method under centered conditions for the second derivative

José Antonio Ezquerro, Daniel González, Miguel Ángel Hernández (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein...

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...