Displaying similar documents to “On the convergence of two-step Newton-type methods of high efficiency index”

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

Ioannis K. Argyros (2001)

Applicationes Mathematicae

Similarity:

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.

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

Ioannis K. Argyros, Saïd Hilout (2013)

Applicationes Mathematicae

Similarity:

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...

A weaker affine covariant Newton-Mysovskikh theorem for solving equations

Ioannis K. Argyros (2006)

Applicationes Mathematicae

Similarity:

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...

Inexact Newton methods and recurrent functions

Ioannis K. Argyros, Saïd Hilout (2010)

Applicationes Mathematicae

Similarity:

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,...

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

Ioannis K. Argyros (2002)

Applicationes Mathematicae

Similarity:

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 new approach for finding weaker conditions for the convergence of Newton's method

Ioannis K. Argyros (2005)

Applicationes Mathematicae

Similarity:

The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the...

Convergence domains under Zabrejko-Zinčenko conditions using recurrent functions

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

Similarity:

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.

On the convergence of Newton's method under ω*-conditioned second derivative

Ioannis K. Argyros, Saïd Hilout (2011)

Applicationes Mathematicae

Similarity:

We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's...

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

Ioannis K. Argyros, Santhosh George (2013)

Applicationes Mathematicae

Similarity:

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.

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

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

Applicationes Mathematicae

Similarity:

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.

Improved ball convergence of Newton's method under general conditions

Ioannis K. Argyros, Hongmin Ren (2012)

Applicationes Mathematicae

Similarity:

We present ball convergence results for Newton's method in order to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. Our hypotheses involve very general majorants on the Fréchet derivatives of the operators involved. In the special case of convex majorants our results, compared with earlier ones, have at least as large radius of convergence, no less tight error bounds on the distances involved, and no less precise information on the uniqueness...

On the gap between the semilocal convergence domains of two Newton methods

Ioannis K. Argyros (2007)

Applicationes Mathematicae

Similarity:

We answer a question posed by Cianciaruso and De Pascale: What is the exact size of the gap between the semilocal convergence domains of the Newton and the modified Newton method? In particular, is it possible to close it? Our answer is yes in some cases. Using some ideas of ours and more precise error estimates we provide a semilocal convergence analysis for both methods with the following advantages over earlier approaches: weaker hypotheses; finer error bounds on the distances involved,...