<html><head><title>MATRIX DIFFERENTIAL EQUATIONS AND INVERSE PRECONDITIONERS.</title> <META NAME="DC.creator.origin" CONTENT="Laboratoire de Mathmatiques, Orsay"> <META NAME="DC.type" CONTENT="preprint"> <META NAME="DC.Creator.PersonalName" CONTENT="CHEHAB, Jean-Paul"> <META NAME="DC.Creator.Email" CONTENT="Jean-Paul.CHEHAB@math.u-psud.fr"> <META NAME="DC.title" CONTENT="Matrix Differential Equations and Inverse Preconditioners."> <META NAME="DC.subject.keywords" CONTENT="Matrix Differential Equation; Numerical schemes; Numerical linear algebra; Preconditioning"> <META NAME="DC.subject.topic" CONTENT="mathematics"> <META NAME="DC.subject" SCHEME="msc1991" CONTENT=" 65F10; 65F35; 65L05; 65L12; 65L20; 65N06"> <META NAME="DC.language" CONTENT="GB"> <META NAME="DC.date" CONTENT="2002-10-16"> </head><body bgcolor="#FFFFFF"><p><CENTER><H3><FONT COLOR=#0000ff>2002 Pr&eacute;publication d'Orsay num&eacute;ro 2002-29 (16/10/2002)</FONT></CENTER></H3><br><br> <B><FONT COLOR=#ff00ff>MATRIX DIFFERENTIAL EQUATIONS AND INVERSE PRECONDITIONERS.</FONT></B><P> <B>CHEHAB, Jean-Paul</B> - Analyse Numrique et E.D.P., Universit Paris-Sud, Bt. 425, 91405 Orsay cedex et Universite Lille 1,M2, 59655 Villeneuve d'Ascq <br> <hr><br> <B>Mots Cl&eacute;s :</B> Matrix Differential Equation; Numerical schemes; Numerical linear algebra; Preconditioning<BR><BR> <B>Classification MSC :</B> 65F10; 65F35; 65L05; 65L12; 65L20; 65N06<BR><BR> <hr><br> <B>Resum&eacute; :</B> <TABLE CELLPADDING="5" BGCOLOR="#22CCFF"> <TR><TD>  Pour une matrice regulire ${\bf P}$ donne, on construit des quations diffrentielles matricielles du premier ordre dont la solution coincide avec ${\bf P}^{-1}$, pour une valeur finie ou non de la variable indpendante, selon les cas. On propose alors de construire itrativement des prconditioneurs  inverses de ${\bf P}$ par intgration numrique des ces quations. On donne des rsultats de convergence et on applique cette approche pour rsoudre des problmes elliptiques.</TD></TR></TABLE><P> <B>Abstract :</B><TABLE CELLPADDING="5" BGCOLOR="#AFAFAF"> <TR><TD>  In this article, we propose to compute the inverse of a regular matrix ${\bf P}$ as the solution of a first order matrix differential equation, for a given (finite as well as infinite) value of the independent variable. Preconditioners can then be built by approaching the solution of the equation. These approximations are obtained by numerical integration leading to the generation of a sequence of preconditioners. We give convergence results with a special focus on the symmetric positive definite case and on Euler's method; stability is also studied for small residual data in the general case. Numerical simulations on solution of elliptic PDE's illustrate the efficiency of the method. </TD></TR></TABLE><P> <B>Article : </B>  <A HREF="http://www.math.u-psud.fr/~biblio/ppo/2002/fic/ppo_2002_29.ps"> Fichier Postscript</A><BR> <B>Contact : </B><A HREF="mailto:Jean-Paul.CHEHAB@math.u-psud.fr">Jean-Paul.CHEHAB@math.u-psud.fr</A> <BR><HR> </body> </html> 
