Parallel Model Reduction of Large-Scale Unstable Systems Peter Benner * Institut f¨ ur Mathematik, MA 4-5 Technische Universit¨at Berlin D-10623 Berlin, Germany benner@math.tu-berlin.de Maribel Castillo † Enrique S. Quintana-Ort´ ı † Gregorio Quintana-Ort´ ı † Depto. de Ingenier´ ıa y Ciencia de Computadores Universidad Jaume I 12.071-Castell´on,Spain {castillo,quintana,gquintan}@icc.uji.es Abstract We discuss an efficient algorithm for model reduction of large-scale unstable systems and an implementation which allows to reduce models of order up to O(10 4 ) using parallel computing techniques. The major computational step involves the additive decomposition of a transfer function via a block diagonalization based on the solution of a Sylvester equation. The actual model reduction is then achieved by reducing the stable part using techniques based on balanced truncation, singular perturbation approximation, or optimal Hankel norm approximation. We will see that all core computational steps can be based on the sign function method. Numerical experiments on a cluster of Intel Pentium II processors show the efficiency of our methods. Topic area: Section 2. Algorithms Classic presentation 1 Introduction Consider the transfer function matrix (TFM) G(s)= C (sI − A) -1 B + D, and the associated, not necessarily minimal, realization of a linear time-invariant (LTI) system, ˙ x(t) = Ax(t)+ Bu(t), t> 0, y(t) = Cx(t)+ Du(t), t ≥ 0, (1) with A ∈ IR n×n , B ∈ IR n×m , C ∈ IR p×n , and D ∈ IR p×m . For simplicity we assume that the spectrum of A is dichotomic with respect to the imaginary axis, i.e., Re(λ) = 0 for all eigenvalues λ of A. This implies that the system (1) has no poles on the imaginary axis. The case with poles on the imaginary axis could be treated as well with the method described in * Supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin. † Supported by the project CTDIA/2002/122 of the Generalitat Valenciana and the project PI-1B2001-14 of the Fundaci´oCaixa-Castell´o/Bancaixa. 1