Krylov Subspace-Based Dimension Reduction
        of Large-Scale Linear Dynamical Systems

                   Roland W. Freund
               Department of Mathematics
            University of California, Davis
                  One Shields Avenue
                    Davis, CA 95616

       e-mail:  freund@math.ucdavis.edu
          web:  http://www.math.ucdavis.edu/~freund/

Krylov subspace methods can be used to generate Pade approximants
of the transfer functions of linear dynamical systems.  Although
this Krylov-Pade connection has been known for a long time, somewhat
surprisingly, it was not until the 1990s that this connection
was exploited to devise practical numerical procedures for dimension
reduction of large-scale linear dynamical systems.

In this talk, we first describe a problem arising in the simulation of
electronic circuits that had triggered the interest in Krylov 
subspace-based dimension reduction in the 1990s.  We then present
an overview of the current state-of-the art of this class of
dimension reduction algorithms.  In particular, we focus on recent
advances in structure-preserving reduction techniques.