How many steps still left to x*?
We survey the convergence orders of sequences and show the orders of the basic iterative sequences for solving nonlinear equations…
Newton method
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…