The Journal of Supercomputing, 20, 55–66, 2001 © 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Efficient Algorithms for the Block Hessenberg Form ENRIQUE S. QUINTANA-ORT ´ I, GREGORIO QUINTANA-ORT ´ I AND MARIBEL CASTILLO quintana, gquintan, castillo@icc.uji.es Univ. Jaime I, Dept. de Inform´ atica 12080–Castell´ on, Spain VICENTE HERN ´ ANDEZ vhernand@dsic.upv.es Univ. Polit´ ecnica de Valencia Dept. de Sistemas Inform´ aticos y Computaci´ on 46071–Valencia, Spain Abstract. We investigate in this paper the performance of parallel algorithms for computing the con- trollable part of a control linear system, with application to the computation of minimal realizations. Our approach is based on a method that transforms the matrices of the system to block Hessenberg form by using rank-revealing orthogonal factorizations. The experimental analysis on a high performance architecture includes two rank-revealing numerical tools: the SVD and the rank-revealing QR factorizations. Results are also reported, using the rank- revealing QR factorizations, on a parallel distributed architecture. Keywords: analysis of control linear systems, structural properties, rank-revealing orthogonal factorizations, parallel algorithms and architectures, mathematical software 1. Introduction A continuous time-invariant control linear system (CLS) of order n is defined in the state-space model by an ordinary differential equation and an algebraic equation of the form ˙ xt = Axt + But y t = Cxt  (1) In these equations, xt ∈  n is the vector with the internal states of the system, with x0= x 0 the initial state, ut ∈  m is the vector of inputs (or controls), used to determine the behavior of the system, and y t ∈  p is the vector of outputs; here, A ∈  n×n , B ∈  n×m , and C ∈  p×n . Systems of large dimension (n of order O1000) are not uncommon in chemical and space engineering applications, are standard for second order systems, and represent rather coarse grids when derived from the discretization of a partial differential equation [3, 12, 21]. Many applications arising in control theory related with the design of CLS require at an early (analysis) stage the identification of a minimal realization of the system; e.g., the linear-quadratic optimal control problem, pole-assignment, model reduc- tion, stabilization, etc. [1, 15]. Intuitively, a minimal realization of (1) is given by a CLS, of order r ≤ n, where the states that remain in the realization are those which are controllable via the inputs and have an influence on the outputs. A formal def- inition and algorithms can be found, e.g., in [15].