From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Generalized numerical range and Linear Systems
Date: 20 Feb 2001 17:08:23 GMT
Newsgroups: sci.math.num-analysis
Summary: Linear systems of differential algebraic equations

In article <3A92365C.C328065E@unina.it>,
 Ciro Caramiello <caramiel@unina.it> writes:
|> I'd like to ask you something about the generalized dynamic problem:
|> 
|> B*d/dt(u(t))=A*u(t)      u(0)=u0            [1]
|> 
|> , in particular when B is singular.
|> 

snip (my newsreader requires that) question on solution theory analogous to ODE.

you are dealing with a so called DAE (differential algebraic equation) and
are happy enough to deal with the constant coefficient case. In principle.
if det(\lambda B+A) is not identically zero in \lambda, then it is solvable 
and can be reduced to a system of  ordinary differential equations using 
iterated diffentiations. then you can apply the solution theory you mentioned.
the number of differentiations needed is also  known, it is the so called index
of the system. look up a textbook on DAE's e.g. 
Brenan&Campbell&Petzold : Numerical Solution of Initial
value problems in differential algebraic equations, SIAM classics in applied
mathematics 14, 1996, ISBN 0-89871-353-6
have it and give further references
hope that helps
peter
============================================================================== 

From: Daniel Kressner <kreda@mathematik.tu-chemnitz.de>
Subject: Re: Generalized numerical range and Linear Systems
Date: Tue, 20 Feb 2001 18:44:30 +0100
Newsgroups: sci.math.num-analysis

Ciro Caramiello schrieb:
> 
> I'd like to ask you something about the generalized dynamic problem:
> 
> B*d/dt(u(t))=A*u(t)      u(0)=u0            (1)
> 
> , in particular when B is singular.
> 

For theoretical purposes, you could use the Kronecker canonical form
[1] to decouple your system (1). You get 4 different types of
systems:

1. ODE
2. algebraic constraints
3. underdetermined systems
4. overdetermined systems

In the case of regular pencils (mentioned by Peter) you won't
have systems of types 3 and 4, which are quite good news.

best regards,
Daniel

[1] Gantmacher: "Matrix Theory"