Solving nonlinear equations from higher order derivations in linear stages
type: Item
instance of: scholarly article
title: Solving nonlinear equations from higher order derivations in linear stages
zbMATH Open document ID: 0716.17014
DOI: 10.1016/0001-8708(90)90088-5
published in: Advances in Mathematics
publication date: 1990-01-01T00:00:00Z
review text: This paper is concerned with effective algorithms to compute approximations to higher order derivations. Let \(R=C^{\infty}({\mathbb{R}}^ N)\) and let \(D_{\mu}=\partial /\partial x_{\mu}\), for \(1\leq \mu \leq N\). The data consists of M derivations \(F_ 1,...,F_ M\) of the form \(F_ j=\sum^{N}_{\mu =1}a_ j^{\mu}D_{\mu}\), where \(a_ j^{\mu}\in R\), \(1\leq j\leq M\). The problem is to find M derivations \(E_ 1,...,E_ M\), with polynomial coefficients with the property that the \(E_ j\) generate a nilpotent Lie algebra and such that trajectories of the nonlinear system \(\dot y(t)=\sum^{M}_{j=1}u_ j(t)F_ j(y(t))\), \(y(0)=0\) are close to trajectories of the system \(\dot x(t)=\sum^{M}_{j=1}u_ j(t)E_ j(x(t))\), \(x(0)=0\) in the sense that \(| y(t)-x(t)| \leq Ct^ r\), for small t. In general this estimate does not hold for \(r\geq 2.\) This paper gives an algorithm which computes the derivations \(E_ j\) for larger r, when possible. The basic idea is to introduce a data structure involving finite trees labeled by the derivations, and to impose a multiplication on trees so that the space of trees becomes an algebra homomorphic to the algebra of derivations generated by the \(F_ j\). A careful analysis of the algebra of trees then yields the algorithm.
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17-08
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 93B40
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 4179605
zbMATH Keywords: algorithms
zbMATH Keywords: approximations to higher order derivations
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: finite trees
author: Robert L. Grossman
author: Richard Gustavus Larson
MaRDI profile type: Publication
full work available at URL: 0001-8708(90)90088-5
OpenAlex ID: W2129417531
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description: scientific article; zbMATH DE number 4179605