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

Ioannis K. Argyros (2005)

Applicationes Mathematicae

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 convergent nonlinear splitting via orthogonal projection

Jan Mandel (1984)

Aplikace matematiky

We study the convergence of the iterations in a Hilbert space V , x k + 1 = W ( P ) x k , W ( P ) z = w = T ( P w + ( I - P ) z ) , where T maps V into itself and P is a linear projection operator. The iterations converge to the unique fixed point of T , if the operator W ( P ) is continuous and the Lipschitz constant ( I - P ) W ( P ) < 1 . If an operator W ( P 1 ) satisfies these assumptions and P 2 is an orthogonal projection such that P 1 P 2 = P 2 P 1 = P 1 , then the operator W ( P 2 ) is defined and continuous in V and satisfies ( I - P 2 ) W ( P 2 ) ( I - P 1 ) W ( P 1 ) .

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

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 type.

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

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...

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