Convergence Analysis For Newton-like Methods Research Articles

A Unified Kantorovich-type Convergence Analysis of Newton-like Methods for Solving Generalized Equations under the Aubin Property

Numerous applications from diverse disciplines reduce to solving generalized equations in a Banach space setting. These equations are solved mostly iteratively, when a sequence is generated approximating a solution provided that certain conditions are valid on the starting point and the operators appearing on the method. In particular, Newton-like methods are developed whose specializations reduce to well known methods such as Newton, modified Newton, Secant, Kurchatov and Steffensen to mention a few. A unified semi-local analysis of these methods is presented using the contraction mapping principle under the Aubin property of a set valued operator, and generalized continuity assumption on the operators on these methods.

New improved convergence analysis for Newton-like methods with applications

We present a new semilocal convergence analysis for Newton-like methods using restricted convergence domains in a Banach space setting. The main goal of this study is to expand the applicability of these methods in cases not covered in earlier studies. The advantages of our approach include, under the same computational cost as previous studies, a more precise convergence analysis under the same computational cost on the Lipschitz constants involved. Numerical studies including a chemical application are also provided in this study.

On the convergence of Newton-like methods using restricted domains

We present a new semi-local convergence analysis for Newton-like methods in order to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. The new idea uses more precise convergence domains. This way the new sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, are also provided in this study.

Improved convergence analysis for Newton-like methods

We present a new semilocal convergence analysis for Newton-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. This way, we expand the applicability of these methods in cases not covered in earlier studies. The advantages of our approach include a more precise convergence analysis under the same computational cost on the Lipschitz constants involved. Applications are also given in this study to show that our estimates on the distances involved are tighter than the older ones.

On Newton-like methods of “bounded deterioration” using recurrent functions

We provide a semilocal convergence analysis for Newton-like methods of “bounded deterioration” in a Banach space setting. We establish tighter error bounds on the distances involved, and a more precise information on the location of the solution, under the same or weaker hypotheses than before (Argyros, Acta Math. Sin. (Engl. Ser.), 23:2087–2096, 2007; Deuflhard, Newton methods for nonlinear problems. Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics, vol. 35. Springer, Berlin, 2004; Ezquerro and Hernandez, IMA J. Numer. Anal., 22:187–205, 2002) using recurrent functions. Numerical examples are also provided involving polynomial, integral, and differential equations.

Improved generalized differentiability conditions for Newton-like methods

We provide a semilocal convergence analysis for Newton-like methods using the ω -versions of the famous Newton–Kantorovich theorem (Argyros (2004) [1], Argyros (2007) [3], Kantorovich and Akilov (1982) [13]). In the special case of Newton’s method, our results have the following advantages over the corresponding ones (Ezquerro and Hernaández (2002) [10], Proinov (2010) [17]) under the same information and computational cost: finer error estimates on the distances involved; at least as precise information on the location of the solution, and weaker sufficient convergence conditions. Numerical examples, involving a Chandrasekhar-type nonlinear integral equation as well as a differential equation with Green’s kernel are provided in this study.

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

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

Concerning the convergence of Newton-like methods under weak Hölder continuity conditions

The concept of majorizing sequences is employed to provide a convergence analysis for Newton-like methods in a Banach space. We use Hölder and center-Hölder continuity assumptions on the Fréchet-derivative of the operators involved. This way we show that our convergence conditions are weaker; error bounds on the distances involved finer and the location of the solution more precise than in earlier results.