iterative methods

the solution sought is a limit of a sequence of elements, and each element can be generated based on the previous ones. Initial elements are supposed to be given. Usual iterative methods are the Newton method, the successive approximations, etc.

Estimating the radius of an attraction ball

Abstract Given a nonlinear mapping $$G:D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}^n$$ differentiable at a fixed point $$x^\ast$$, the Ostrowski theorem offers the sharp…

On the convergence of some quasi-Newton iterates studied by I. Păvăloiu

Abstract In 1986, I. Păvăloiu [6] has considered a Banach space and the fixed point problem \[x=\lambda D\left( x\right) +y,…

The inexact, inexact perturbed and quasi-Newton methods are equivalent models

Abstract A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method,…

Sufficient convergence conditions for certain accelerated successive approximations

Abstract We have recently characterized the q-quadratic convergence of the perturbed successive approximations. For a particular choice of the parameters, these…

Affine invariant conditions for the inexact perturbed Newton method

Abstract The high q-convergence orders of the inexact Newton iterates were characterized by Ypma in terms of some affine invariant…

Inexact perturbed Newton methods and applications to a class of Krylov solvers

Abstract Inexact Newton methods are variant of the Newton method in which each step satisfies only approximately the linear system…

On accelerating the convergence of the successive approximations method

Abstract No q-superlinear convergence to a fixed point $$x^\ast$$ of a nonlinear mapping $$G$$ may be attained by the successive approximations when…

A note on the quadratic convergence of the inexact Newton methods

Abstract We show that a new sufficient condition for the convergence with q-order two of the inexact Newton iterates may be…

On the high convergence orders of the Newton-GMBACK methods

Abstract GMBACK is a Krylov solver for linear systems in $$\mathbb{R}^n$$. We analyze here the high convergence orders (superlinear convergence)…

On an Aitken-Newton type method

Abstract We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes…