Displaying similar documents to “On Newton's polygons, Gröbner bases and series expansions of perturbed polynomial programs”

Nearly irreducibility of polynomials and the Newton diagrams

Mateusz Masternak (2020)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C2. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

Uniform Convergence of the Newton Method for Aubin Continuous Maps

Dontchev, Asen (1996)

Serdica Mathematical Journal

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* This work was supported by National Science Foundation grant DMS 9404431. In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F(x) with a C^1 function f and a set-valued map F acting in Banach spaces, is locally convergent uniformly in the parameter y if and only if the map (f +F)^(−1) is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform Q-quadratic convergence provided that...

Newton methods for solving two classes of nonsmooth equations

Yan Gao (2001)

Applications of Mathematics

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The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since...

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...

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...

"On the Shoulders of Giants" A brief excursion into the history of mathematical programming

Rainer Tichatschke (2012)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Similar to many mathematical fields also the topic of mathematical programming has its origin in applied problems. But, in contrast to other branches of mathematics, we don't have to dig too deeply into the past centuries to find their roots. The historical tree of mathematical programming, starting from its conceptual roots to its present shape, is remarkably short, and to quote Isaak Newton, we can say: "We are standing on the shoulders of giants". ...

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

José Antonio Ezquerro, Daniel González, Miguel Ángel Hernández (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

Similarity:

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 ...