A direct Newton method for calculus of variations
type: Item
instance of: scholarly article
title: A direct Newton method for calculus of variations
zbMATH Open document ID: 0997.65089
DOI: 10.1016/S0377-0427(01)00427-7
author: Yuri Levin
author: Mikhail Nediak
author: Adi Ben-Israel
published in: Journal of Computational and Applied Mathematics
publication date: 2002-05-21T00:00:00Z
review text: The paper deals with a direct Newton method for calculus of variations. Here the system of equations: \(f_i(x_1,x_2,\dots, x_n)= 0\), \(i= 1,2,\dots, m\) and Newton iterations for this system that use the Moore-Penrose inverse of the Jacobian matrix, is considered. Under standard assumptions, the Newton interactions converge quadratically to a stationary point of the sum-of-squares \(\sum_i f^2_i\). Approximating the derivatives \(\dot x\) as differences \((\Delta x/\Delta t)\) with \(\Delta t= h\), the Newton method to the system obtained by discretizing the integral \(\int^{t_1}_{t_0} (t,x,\dot x) dt\), is applied. Main result: The approximate solutions \(y_{h_n}\) of the discretized problem are shown to converge to a solution of the Euler-Lagrange boundary value problem \((d/dt)\partial L/\partial\dot x=\partial L/\partial x\) with the degree of approximation linear in \(h\), if the Lagrangian \(L(t,x,\dot x)\) is twice continuously differentiable. Higher continuous derivatives of \(L\) guarantee higher-order of approximation.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49M15
Mathematics Subject Classification ID: 49J15
zbMATH DE Number: 1743069
zbMATH Keywords: calculus of variations
zbMATH Keywords: Euler-Lagrange equation
zbMATH Keywords: Newton method
zbMATH Keywords: stationary point
zbMATH Keywords: discretized problem
zbMATH Keywords: approximate solutions
zbMATH Keywords: convergence
zbMATH Keywords: Moore-Penrose inverse
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
cites work: A Newton-Raphson method for the solution of systems of equations
cites work: Q4209222
cites work: Numerical derivatives and nonlinear analysis
cites work: A Newton method for systems of \(m\) equations in \(n\) variables.
cites work: Convergence of Newton-like methods for singular operator equations using outer inverses
cites work: Case studies in trajectory optimization: trains, planes, and other pastimes
cites work: Q3995594
Wikidata QID: Q126388857
description: scientific article; zbMATH DE number 1743069