This paper concerns with the numerical solution of matrix differential systems evolving on the general linear group GL(n,R) and having the form Y = Y-T F(t,Y,Y-T,(YY)-Y-T) for t > t(0). Equations of this form arise, for instance, when one solves minimization problems by means of the gradient flow technique or in the context of smooth decomposition of time-depending matrices. The presence of the inverse of the solution in the right hand side and the topological structure of GL(n,R) make the numerical integration of these ODEs sometimes worrisome. In this paper we will consider some non standard numerical techniques for solving this kind of ODEs. In case F does not depend explicitly from the inverse Y-1, we will show how the numerical solution of this ODE may be obtained via the solution of a Riccati-type algebraic equation.
Geometric integration of ODEs on the general linear group of matrices
DEL BUONO, Nicoletta;LOPEZ, Luciano
2005-01-01
Abstract
This paper concerns with the numerical solution of matrix differential systems evolving on the general linear group GL(n,R) and having the form Y = Y-T F(t,Y,Y-T,(YY)-Y-T) for t > t(0). Equations of this form arise, for instance, when one solves minimization problems by means of the gradient flow technique or in the context of smooth decomposition of time-depending matrices. The presence of the inverse of the solution in the right hand side and the topological structure of GL(n,R) make the numerical integration of these ODEs sometimes worrisome. In this paper we will consider some non standard numerical techniques for solving this kind of ODEs. In case F does not depend explicitly from the inverse Y-1, we will show how the numerical solution of this ODE may be obtained via the solution of a Riccati-type algebraic equation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
