caa a22 4500 605496927 CHVBK 20210128100538.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s10543-014-0516-y doi (NATIONALLICENCE)springer-10.1007/s10543-014-0516-y Bosner Nela Department of Mathematics, University of Zagreb, Zagreb, Croatia aut Efficient algorithm for simultaneous reduction to the $$m$$ m -Hessenberg-triangular-triangular form [Elektronische Daten] [Nela Bosner] This paper proposes an efficient algorithm for simultaneous reduction of three matrices by using orthogonal transformations, where $$A$$ A is reduced to $$m$$ m -Hessenberg form, and $$B$$ B and $$E$$ E to triangular form. The algorithm is a blocked version of the algorithm described by Miminis and Paige (Int J Control 35:341-354, 1982). The $$m$$ m -Hessenberg-triangular-triangular form of matrices $$A$$ A , $$B$$ B and $$E$$ E is specially suitable for solving multiple shifted systems $$(\sigma E-A)X=B$$ ( σ E - A ) X = B . 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 discretizing the time-harmonic wave equation in heterogeneous media, or originate from structural dynamics engineering problems. The proposed blocked algorithm for the $$m$$ m -Hessenberg-triangular-triangular reduction is based on aggregated Givens rotations, and is a generalization of the blocked algorithm for the Hessenberg-triangular reduction proposed by Kågström et al. (BIT 48:563-584, 2008). Numerical tests confirm that the blocked algorithm is much faster than its non-blocked version based on regular Givens rotations only. As an illustration of its efficiency, two applications of the $$m$$ m -Hessenberg-triangular-triangular reduction 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. Springer Science+Business Media Dordrecht, 2014 $$m$$ m -Hessenberg-triangular-triangular form nationallicence Orthogonal transformations nationallicence Level 3 BLAS nationallicence Blocked algorithm nationallicence Solving shifted system nationallicence Transfer function evaluation nationallicence Staircase form nationallicence BIT Numerical Mathematics Springer Netherlands 55/3(2015-09-01), 677-703 0006-3835 55:3<677 2015 55 10543 https://doi.org/10.1007/s10543-014-0516-y text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10543-014-0516-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Bosner Nela Department of Mathematics, University of Zagreb, Zagreb, Croatia aut NATIONALLICENCE 773 0- BIT Numerical Mathematics Springer Netherlands 55/3(2015-09-01), 677-703 0006-3835 55:3<677 2015 55 10543