Numerical integration of a class of ordinary differential equations on the general linear group of matrices
type: Item
instance of: scholarly article
title: Numerical integration of a class of ordinary differential equations on the general linear group of matrices
zbMATH Open document ID: 1037.65068
published in: Numerical Algorithms
publication date: 2004-01-14T00:00:00Z
review text: This paper is concerned with the numerical solution of initial value problems for matrix differential equations on \( { GL}(n)\) with the form \[ Y'(t) = Y^{-T} (t) F( Y(t) ), \quad t \geq t_0,\;Y(t_0)= Y_0, \] where \(f\) is assumed to be a continuous and globally Lipschitz matrix function. By using the theory of ordinary differential equations it can be seen that the solution \( Y(t)\) may have a escape for some finite \( \tau \) and then \( Y(t)\) approaches to a singular matrix when \( t \to \tau. \) The authors propose two alternative approaches to deal with this kind of problems when finite escapes are encountered. In the first one they use a low order explicit Runge-Kutta method on some modified scheme to avoid the unstable behaviour in the neighbourhood of the singularity. In the second one, they use the singular value decomposition of \( Y(t) = U(t) \Sigma (t) V(t)\) to integrate the components of \( U, \Sigma \) and \( V\). The paper ends with the results of a numerical experiment to test the behaviour of both approaches for some low order Runge-Kutta methods.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 2026778
zbMATH Keywords: differential Equations on Manifolds
zbMATH Keywords: Initial Value Problems
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: Singularities detection
zbMATH Keywords: singular value decomposition
zbMATH Keywords: numerical experiment
author: Nicoletta Del Buono
author: Luciano Lopez
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: b:numa.0000005404.80948.ae
OpenAlex ID: W1965169410
DOI: 10.1023/B:NUMA.0000005404.80948.AE
description: scientific article; zbMATH DE number 2026778