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

Ioannis K. Argyros (2009)

Applicationes Mathematicae

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 the following...

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 discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair, Eberhard Schock (1998)

Annales Polonici Mathematici

Many discrepancy principles are known for choosing the parameter α in the regularized operator equation ( T * T + α I ) x α δ = T * y δ , | y - y δ | δ , in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and T * y δ are approximated by Aₙ and z δ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable...

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

Christine Bernardi, Frédéric Hecht, Rüdiger Verfürth (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.

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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