How many steps still left to x*?

Abstract The high speed of $$x_{k}\rightarrow x^\ast\in{\mathbb R}$$ is usually measured using the C-, Q- or R-orders: \tag{$C$} \lim \frac…

Methods of Newton and Newton-Krylov type

Book summaryLocal convergence results on Newton-type methods for nonlinear systems of equations are studied. Solving of large linear systems by…

On an Aitken-Steffensen-Newton type method

Abstract We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We…

The numerical approximation to positive solution for some reaction-diffusion problems

Abstract A one-dimensional reaction-diffusion problem, with the reaction term of the form $$u^{p}$$, $$p>1$$ is considered. For $$p=3$$, we deduce…

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,…

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…

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…