The convergence of the perturbed Newton method and its application for ill-conditioned problems
type: Item
instance of: scholarly article
title: The convergence of the perturbed Newton method and its application for ill-conditioned problems
zbMATH Open document ID: 1255.65095
published in: Applied Mathematics and Computation
publication date: 2012-06-11T00:00:00Z
review text: The paper deals with the numerical solution to nonlinear equations, in particular the case of multiple or almost multiple roots. For this kind of ill-posed problem, the authors propose a modification of the classical Newton method called ``perturbed Newton method''. The idea of this algorithm is calculating the inverse derivative on every Newton iteration at a point given by a combination of the current approximation and the result of an action of some given linear operator. This operator can be selected depending on the equation features. The authors provide a theoretical investigation of convergence and practical recommendations for the use of the proposed method. The perturbed Newton behaves in a way similar to the classic Newton: when the iterates are far from the root it behaves like a bisection and if they are close to the root the sequence becomes second order. Presented numerical examples show the flexibility of perturbed Newton for ill-posed problems and demonstrate its better numerical performance when compared to the classical Newton for ill-posed equations.
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 6045127
zbMATH Keywords: Newton method
zbMATH Keywords: nonlinear equations
zbMATH Keywords: ill-posed problems
zbMATH Keywords: multiple or almost multiple roots
zbMATH Keywords: perturbed Newton method
zbMATH Keywords: algorithm
zbMATH Keywords: convergence
zbMATH Keywords: numerical example
author: R. Peris
author: Antonio Marquina
author: V. Candela
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
full work available at URL: j.amc.2011.08.019
OpenAlex ID: W2026191987
cites work: Inverse Iteration, Ill-Conditioned Equations and Newton’s Method
cites work: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting
cites work: Solving Ill-Conditioned and Singular Linear Systems: A Tutorial on Regularization
cites work: A one-parameter family of quadratically convergent iteration formulae
cites work: A one-parameter family of second-order iteration methods
cites work: Generalized differentiability conditions for Newton's method
cites work: Newton's method under mild differentiability conditions with error analysis
cites work: The Newton method for operators with Hölder continuous first derivative
cites work: Q4034594
cites work: Quasi-Newton Methods, Motivation and Theory
cites work: Recurrence relations for a Newton-like method in Banach spaces
cites work: Historical developments in convergence analysis for Newton's and Newton-like methods
cites work: Q4818712
cites work: A variant of Newton's method with accelerated third-order convergence
cites work: The Newton-Kantorovich Theorem
cites work: Recurrence relations for rational cubic methods. I: The Halley method
cites work: Recurrence relations for rational cubic methods. II: The Chebyshev method
cites work: Q3343091
cites work: Newton-Kantorovich method and its global convergence
DOI: 10.1016/J.AMC.2011.08.019
description: scientific article; zbMATH DE number 6045127