𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Local convergence of Newton’s method under majorant condition

✍ Scribed by O.P. Ferreira

Publisher
Elsevier Science
Year
2011
Tongue
English
Weight
240 KB
Volume
235
Category
Article
ISSN
0377-0427

No coin nor oath required. For personal study only.

✦ Synopsis

A local convergence analysis of Newton's method for solving nonlinear equations, under a majorant condition, is presented in this paper. Without assuming convexity of the derivative of the majorant function, which relaxes the Lipschitz condition on the operator under consideration, convergence, the biggest range for uniqueness of the solution, the optimal convergence radius and results on the convergence rate are established. Besides, two special cases of the general theory are presented as applications.

📜 SIMILAR VOLUMES

On the local convergence of inexact Newt
On the local convergence of inexact Newton-type methods under residual control-type conditions
✍ Hongmin Ren; Ioannis K. Argyros 📂 Article 📅 2010 🏛 Elsevier Science 🌐 English ⚖ 322 KB

A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergenc

Relaxing Convergence Conditions for Newt
Relaxing Convergence Conditions for Newton's Method
✍ M.A. Hernández 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 100 KB

The classical Kantorovich theorem on Newton's method assumes that the first 5 w Ž . derivative of the operator involved satisfies a Lipschitz condition ⌫ FЈ x y 0 Ž .x5 5 5 FЈ y F L x y y . In this paper, we weaken this condition, assuming that 5 w Ž . Ž .x5 Ž5 5 . ⌫ FЈ x y FЈ x F x y x for a given

A cubically convergent Newton-type metho
A cubically convergent Newton-type method under weak conditions
✍ Liang Fang; Guoping He; Zhongyong Hu 📂 Article 📅 2008 🏛 Elsevier Science 🌐 English ⚖ 119 KB

Under weak conditions, we present an iteration formula to improve Newton's method for solving nonlinear equations. The method is free from second derivatives, permitting f (x) = 0 in some points and per iteration it requires two evaluations of the given function and one evaluation of its derivative.

Newton's method under mild differentiabi
Newton's method under mild differentiability conditions
✍ Herbert B. Keller 📂 Article 📅 1970 🏛 Elsevier Science 🌐 English ⚖ 437 KB

We study Newton's method for determining the solution off(x) = 0 whenf(x) is required only to be continuous and piecewise continuously differentiable in some sphere about the initial iterate, x ) is nonsingular, (lc) II J-l(x~m)[J(x) -J(y)][I ~ o~(I x -y f), for all x andy ~ D.

Convergence of Stirling's method under w
Convergence of Stirling's method under weak differentiability condition
✍ S. K. Parhi; D. K. Gupta 📂 Article 📅 2010 🏛 John Wiley and Sons 🌐 English ⚖ 138 KB

The aim of this paper is to establish the semilocal convergence analysis of Stirling's method used to find fixed points of nonlinear operator equations in Banach spaces. This is done by using recurrence relations under weak Hölder continuity condition on the first Fréchet derivative of the involved

Local convergence of inexact methods und
Local convergence of inexact methods under the Hölder condition
✍ Chong Li; Weiping Shen 📂 Article 📅 2008 🏛 Elsevier Science 🌐 English ⚖ 816 KB

We study the convergence properties for some inexact Newton-like methods including the inexact Newton methods for solving nonlinear operator equations on Banach spaces. A new type of residual control is presented. Under the assumption that the derivative of the operator satisfies the Hölder conditio