Displaying similar documents to “An iterative method for computing zeros of operators satisfying autonomous differential equations.”

New unifying convergence criteria for Newton-like methods

Ioannis K. Argyros (2002)

Applicationes Mathematicae

Similarity:

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...

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

Ioannis K. Argyros, Santhosh George (2015)

Applicationes Mathematicae

Similarity:

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.

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.

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 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...

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.

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

Ioannis K. Argyros (2009)

Applicationes Mathematicae

Similarity:

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...

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.

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...