BIT Numerical Mathematics

, Volume 55, Issue 3, pp 677–703 | Cite as

Efficient algorithm for simultaneous reduction to the \(m\)-Hessenberg-triangular-triangular form

  • Nela BosnerEmail author


This paper proposes an efficient algorithm for simultaneous reduction of three matrices by using orthogonal transformations, where \(A\) is reduced to \(m\)-Hessenberg form, and \(B\) and \(E\) 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 \(m\)-Hessenberg-triangular–triangular form of matrices \(A\), \(B\) and \(E\) is specially suitable for solving multiple shifted systems \((\sigma E-A)X=B\). 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 \(m\)-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 \(m\)-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.


\(m\)-Hessenberg-triangular-triangular form Orthogonal transformations Level 3 BLAS Blocked algorithm  Solving shifted system Transfer function evaluation Staircase form 

Mathematics Subject Classification

15A21 15A06 65F05 65Y20 93B05 93B10 93B17 93B40 



The author wishes to thank the referees for giving many helpful suggestions, which inspired the development of the double-blocked algorithm for the \(m\)-Hessenberg-triangular-triangular reduction, and helped to improve the quality of the paper.


  1. 1.
    Ahuja, K., De Sturler, E., Gugercin, S., Chang, E.R.: Recycling BiCG with an application to model reduction. SIAM J. Sci. Comput. 34, A1925–A1949 (2012)Google Scholar
  2. 2.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J.J., Du Croz, J., Greenbaum, A., Hammarling, S.J., McKenney, A., Sorensen, D.C.: LAPACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  3. 3.
    Beattie, C.A., Drmač, Z., Gugercin, S.: A note on shifted Hessenberg systems and frequency response computation. ACM Trans. Math. Softw. 38, 12:1–12:16 (2011)Google Scholar
  4. 4.
    Bosner, N., Bujanović, Z., Drmač, Z.: Efficient generalized Hessenberg form and applications. ACM Trans. Math. Softw. 39, 19:1–19:19 (2013)Google Scholar
  5. 5.
    Chu, K.-W.E.: A controllability condensed form and a state feedback pole assignment algorithm for descriptor systems. IEEE Trans. Autom. Control 33, 366–370 (1988)zbMATHCrossRefGoogle Scholar
  6. 6.
    Dackland, K., Kågström, B.: Blocked algorithms and software for reduction of a regular matrix pair to generalized Schur form. ACM Trans. Math. Softw. 25, 425–454 (1999)zbMATHCrossRefGoogle Scholar
  7. 7.
    Dongarra, J.J., Du Croz, J., Duff, I., Hammarling, S.: A set of Level 3 basic linear algebra subprograms. ACM Trans. Math. Soft. 16, 1–17 (1990)Google Scholar
  8. 8.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. M. D. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  9. 9.
    Gugercin, S., Antoulas, A.C., Beattie, C.A.: H2 model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30, 609–638 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)zbMATHCrossRefGoogle Scholar
  11. 11.
    Kågström, B., Kressner, D., Quintana-Ortí, E.S., Quintana-Ortí, G.: Blocked algorithms for the reduction to Hessenberg-triangular form revisited. BIT 48, 563–584 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Karlsson, L., Kågström, B.: Parallel two-stage reduction to Hessenberg form using dynamic scheduling on shared-memory architectures. Parallel Comput. 37, 771–782 (2011)zbMATHCrossRefGoogle Scholar
  13. 13.
    Lang, B.: Using level 3 BLAS in rotation-based algorithms. SIAM J. Sci. Comput. 19, 626–634 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Laub, A.J., Linnemann, A.: Hessenberg and Hessenberg/triangular forms in linear system theory. Int. J. Control 44, 1523–1547 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Miminis, G.S.: Deflation in eigenvalue assignment of descriptor systems using state feedback. IEEE Trans. Autom. Control 38, 1322–1336 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Miminis, G.S., Paige, C.C.: An algorithm for pole assignment of time invariant linear systems. Int. J. Control 35, 341–354 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Miminis, G.S., Paige, C.C.: An algorithm for pole assignment of time invariant multi-input linear systems. In: Proceedings 21st IEEE Conference on Decision and Control, pp. 62–67 (1982)Google Scholar
  18. 18.
    Paige, C.C.: Properties of numerical algorithms related to computing controllability. IEEE Trans. Autom. Control 26, 130–138 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Parallelism in the Intel\(\textregistered {}\) Math Kernel Library.
  20. 20.
    Simoncini, V.: Restarted full orthogonalization method for shifted linear systems. BIT 43, 459–466 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Simoncini, V., Perotti, F.: On the numerical solution of \((\lambda ^{2}A+\lambda B+C)x=b\) and application to structural dynamics. SIAM J. Sci. Comput. 23, 1875–1897 (2002)Google Scholar
  22. 22.
  23. 23.
    van Dooren, P.M., Verhaegen, M.: On the use of unitary state-space transformations, Linear algebra and its role in systems theory, Proc. AMS-IMS-SIAM Conf., Brunswick/Maine 1984. Contemp. Math. 47, 447–463 (1985)CrossRefGoogle Scholar
  24. 24.
    van Gijzen, M.B., Erlangga, Y.A., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29, 1942–1958 (2007)Google Scholar
  25. 25.
    Van Zee, F.G., van de Geijn, R.A., Quintana-Orti, G.: Restructuring the tridiagonal and bidiagonal QR algorithms for performance. ACM Trans. Math. Softw. 40, 18:1–18:34 (2014)Google Scholar
  26. 26.
    Varga, A.: Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, pp. 392–395 (1992)Google Scholar
  27. 27.
    Varga, A.: Computation of irreducible generalized state-space realizations. Kybernetika 26, 89–106 (1990)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Varga, A., Van Dooren, P.M.: Task I.A—basic software tools for standard and generalized state-space systems and transfer matrix factorizations, SLICOT Working Note 17 (1999)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations