Smale's point estimate theory for Newton's method on Lie groups
type: Item
instance of: scholarly article
title: Smale's point estimate theory for Newton's method on Lie groups
zbMATH Open document ID: 1170.65040
author: C. Li
author: Jean-Pierre Dedieu
author: J. H. Wang
published in: Journal of Complexity
publication date: 2009-06-11T00:00:00Z
review text: Many problems from applied mathematics can be formulated as computing zeroes of mappings or vector fields on a Riemannian manifold. The best known of them is the Newton's method, another one is Smale's \(\alpha\)- and \(\gamma\)-theory. On the other hand, some numerical problems such as symmetric eigenvalue problems, ordinary differential equations on manifolds can be considered as problems on Lie groups. In the present paper the authors study the convergence for a variant of the Newton's method defined as follows: \[ x_{n+1}=x_{n}\cdot \exp(-df^{-1}_{x_{n}}f(x_{n})) \] where \(df\) is the derivative of \(f\) and is defined in terms of the exponential map and independent of the Riemannian metric. Unlike the notion of the \(\gamma\)-condition on the Riemann manifolds, this condition is defined via the one parameter subgroup and so is independent of the Riemannian connection. They establish the generalized \(\alpha\)- and \(\gamma\)-theory. As an application two initial value problems are presented on the special orthogonal group SO\((N,\mathbb R)\).
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 5564312
zbMATH Keywords: Newton's method
zbMATH Keywords: Lie Group
zbMATH Keywords: \(\gamma\)-condition
zbMATH Keywords: Smale's point estimate theory
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: convergence
zbMATH Keywords: initial value problems
MaRDI profile type: Publication
full work available at URL: j.jco.2008.11.001
OpenAlex ID: W2013010587
Wikidata QID: Q115350163
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DOI: 10.1016/J.JCO.2008.11.001
description: scientific article; zbMATH DE number 5564312