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Local convergence for a family of iterative methods based on decomposition techniques

Ioannis K. Argyros, Santhosh George, Shobha Monnanda Erappa (2016)

Applicationes Mathematicae

We present a local convergence analysis for a family of iterative methods obtained by using decomposition techniques. The convergence of these methods was shown before using hypotheses on up to the seventh derivative although only the first derivative appears in these methods. In the present study we expand the applicability of these methods by showing convergence using only the first derivative. Moreover we present a radius of convergence and computable error bounds based only on Lipschitz constants....

Local convergence for a multi-point family of super-Halley methods in a Banach space under weak conditions

Ioannis K. Argyros, Santhosh George (2015)

Applicationes Mathematicae

We present a local multi-point convergence analysis for a family of super-Halley methods of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second Fréchet derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet derivative. Numerical examples are also provided.

Local convergence of a one parameter fourth-order Jarratt-type method in Banach spaces

I. K. Argyros, D. González, S. K. Khattri (2016)

Commentationes Mathematicae Universitatis Carolinae

We present a local convergence analysis of a one parameter Jarratt-type method. We use this method to approximate a solution of an equation in a Banach space setting. The semilocal convergence of this method was recently carried out in earlier studies under stronger hypotheses. Numerical examples are given where earlier results such as in [Ezquerro J.A., Hernández M.A., New iterations of R -order four with reduced computational cost, BIT Numer. Math. 49 (2009), 325–342] cannot be used to solve equations...

Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

Ioannis Argyros (1999)

Applicationes Mathematicae

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator...

Local convergence of two competing third order methods in Banach space

Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

We present a local convergence analysis for two popular third order methods of approximating a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for both methods under the same conditions. A comparison is given between the two methods, as well as numerical examples.

Local convergence theorems for Newton's method from data at one point

Ioannis K. Argyros (2002)

Applicationes Mathematicae

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.

Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses

Ioannis K. Argyros (2000)

Czechoslovak Mathematical Journal

We provide local convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear...

Local degeneracy of Markov chain Monte Carlo methods

Kengo Kamatani (2014)

ESAIM: Probability and Statistics

We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider the cumulative probit model. It is well known that the natural data augmentation (DA) procedure does not work well for this model and the so-called...

Local Discontinuous Galerkin methods for fractional diffusion equations

W. H. Deng, J. S. Hesthaven (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by β ∈[1, 2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion...

Local interpolation by a quadratic Lagrange finite element in 1D

Josef Dalík (2006)

Archivum Mathematicum

We analyse the error of interpolation of functions from the space H 3 ( a , c ) in the nodes a < b < c of a regular quadratic Lagrange finite element in 1D by interpolants from the local function space of this finite element. We show that the order of the error depends on the way in which the mutual positions of nodes a , b , c change as the length of interval [ a , c ] approaches zero.

Local minimizers of functionals with multiple volume constraints

Édouard Oudet, Marc Oliver Rieger (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a Γ-limit argument and illustrate this with numerical computations. Finally...

Local preconditioners for steady and unsteady flow applications

Eli Turkel, Veer N. Vatsa (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners:...

Local preconditioners for steady and unsteady flow applications

Eli Turkel, Veer N. Vatsa (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state. In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We consider two types of local preconditioners: Jacobi...

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