Efficient Algorithm for Simultaneous Reduction to the m-Hessenberg–Triangular–Triangular Form

This paper proposes an efficient algorithm for simultaneous reduction of three matrices. The algorithm is a blocked version of the algorithm described by Miminis and Page (1982) which reduces A to the m-Hessenberg form, and B and E to the triangular form. The m-Hessenberg– triangular–triangular form of matrices A, B and E is specially suitable for solving multiple shifted systems. Such shifted systems naturally occur in control theory when evaluating the transfer function of a descriptor system, or in interpolatory model reduction methods. They also arise as a result of discretization of the time-harmonic wave equation in heterogeneous media, or originate from structural dynamics engineering problems. The proposed blocked algorithm for the m-Hessenberg– triangular–triangular reduction is based on the aggregated Givens rotations, which are a generalization of the blocked algorithm for the Hessenberg–triangular reduction proposed by Kågström et al. (2008). Numerical tests confirmed that the blocked algorithm is up to 3.4 times faster than its non-blocked version based on regular Givens rotations only. As an illustration of its efficiency, two applications of the m-Hessenberg–triangular–triangular reduction coming from control theory are described: evaluation of the transfer function of a descriptor system at many complex values, and computation of the staircase form used to identify the controllable part of the system.