Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils

SIAM Journal on Matrix Analysis and Applications

Volumn 24, Issue 1, 2003, Pages 165-190

  Benner, Peter ^a   Byers, Ralph ^b   Mehrmann, Volker ^c   Xu, Hongguo ^b  

^a UNIVERSITY OF BREMEN   (Germany)

^b UNIVERSITY OF KANSAS   (United States)

^c TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Deflating subspace; Eigenvalue problem; Hamiltonian matrix; Skew Hamiltonian matrix; Skew Hamiltonian Hamiltonian matrix pencil

Indexed keywords

EID: 0037289010     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479800367439     Document Type: Article

References (39)

1. G. AMMAR, P. BENNER, AND V. MEHRMANN, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electron. Trans. Numer. Anal., 1 (1993), pp. 33-48.
2. G. AMMAR AND V. MEHRMANN, On Hamiltonian and symplectic Hessenberg forms, Linear Algebra Appl., 149 (1991), pp. 55-72.
3. P. BENNER, R. BYERS, H. FASSBENDER, V. MEHRMANN, AND D. WATKINS, Cholesky-like factorizations of skew-symmetric matrices, Electron. Trans. Numer. Anal., 11 (2000), pp. 85-93.
4. P. BENNER, R. BYERS, V. MEHRMANN AND H. XU, Numerical Computation of Deflating Subspaces of Embedded Hamiltonian pencils, Tech. Report SFB393/99-15, Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany, 1999; available online from http://www.tu-chemnitz.de/sfb393/sfb99pr.html.
5. ∞ control problems, in Dynamical Systems, Control, Coding, Computer Vision: New Trends, Interfaces, and Interplay, G. Picci and D. Gilliam, eds., Progr. Systems Control Theory 25, Birkhäuser, Basel, 1999, pp. 203-222.
6. P. BENNER, V. MEHRMANN, AND H. XU, A new method for computing the stable invariant subspace of a real Hamiltonian matrix, J. Comput. Appl. Math., 86 (1997), pp. 17-43.
7. P. BENNER, V. MEHRMANN, AND H. XU, A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils, Numer. Math., 78 (1998), pp. 329-358.
8. P. BENNER, V. MEHRMANN, AND H. XU, A note on the numerical solution of complex Hamiltonian and skew-Hamiltonian eigenvalue problems, Electron. Trans. Numer. Anal., 8 (1999), pp. 115-126.
9. P. BENNER, V. MEHRMANN, AND H. XU, Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices, Berichte aus der Technomathematik, Report 00-01, FB3 - Mathematik und Informatik, Universität Bremen, Bremen, Germany, 2000; available online from http://www.math.uni-bremen.de/zetem/berichte.html.
10. A. BOJANCZYK, G. GOLUB, AND P. VAN DOOREN, The periodic Schur decomposition; algorithms and applications, Proc. SPIE, 1770 (1992), pp. 31-42.
11. A. BUNSE-GERSTNER, Matrix factorization for symplectic QR-like methods, Linear Algebra Appl., 83 (1986), pp. 49-77.
12. A. BUNSE-GERSTNER, R. BYERS, AND V. MEHRMANN, Numerical methods for algebraic Riccati equations, in Proceedings of the Workshop on the Riccati Equation in Control, Systems, and Signals, S. Bittanti, ed., Como, Italy, 1989, pp. 107-116.
13. R. BYERS, Hamiltonian and Symplectic Algorithms for the Algebraic Riccati Equation, Ph.D. thesis, Cornell University, Ithaca, NY, 1983.
14. R. BYERS, A Hamiltonian QR algorithm, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 212-229.
15. H. FASSBENDER, D. MACKEY, N. MACKEY, AND H. Xu, Hamiltonian square roots of skew-Hamiltonian matrices, Linear Algebra Appl., 287 (1998), pp. 125-159.
16. F. GANTMACHER, Theory of Matrices, Vol. 1, Chelsea, New York, 1959.
17. G. GOLUB AND C. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
18. M. GREEN AND D. LIMEBEER, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995.
19. J. HENCH AND A. LAUB, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Automat. Control, 39 (1994), pp. 1197-1210.
20. P. LANCASTER, Strongly stable gyroscopic systems, Electron. J. Linear Algebra, 5 (1999), pp. 53-66.
21. P. LANCASTER AND M. TISMENETSKY, The Theory of Matrices, 2nd ed., Academic Press, Orlando, FL, 1985.
22. A. LAUB, Invariant subspace methods for the numerical solution of Riccati equations, in The Riccati Equation, S. Bittanti, A. Laub, and J. Willems, eds., Springer-Verlag, Berlin, 1991, pp. 163-196.
23. W.-W. LIN AND T.-C. HO, On Schur Type Decompositions for Hamiltonian and Symplectic Pencils, Tech. Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990.
24. W.-W. LIN, V. MEHRMANN, AND H. XU, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra Appl., 301-303 (1999), pp. 469-533.
25. C. MEHL, Compatible Lie and Jordan Algebras and Applications to Structured Matrices and Pencils, dissertation, Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany, 1998.
26. C. MEHL, Condensed forms for skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 454-476.
27. V. MEHRMANN, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, Lecture Notes in Control and Inform. Sci. 163, Springer-Verlag, Heidelberg, 1991.
28. V. MEHRMANN AND D. WATKINS, Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils, SIAM J. Sci. Comput., 22 (2001), pp. 1905-1925; also available online from http://www.tu-chemnitz.de/sfb393/sfb00pr.html.
29. V. MEHRMANN AND H. XU, Lagrangian Invariant Subspaces of Hamiltonian Matrices, Tech. Report SFB393/98-25, Fakultät für Mathematik, TU Chemnitz, Chemnitz, Germany, 1998; available online from http://www.tu-chemnitz.de/sfb393/sfb98pr.html.
30. J. OLSON, H. JENSEN, AND P. JØRGENSEN, Solution of large matrix equations which occur in response theory, J. Comput. Phys., 74 (1988), pp. 265-282.
31. C. PAIGE AND C. VAN LOAN, A Schur decomposition for Hamiltonian matrices, Linear Algebra Appl., 14 (1981), pp. 11-32.
32. R. PATEL, On computing the eigenvalues of a symplectic pencil, Linear Algebra Appl., 188/189 (1993), pp. 591-611.
33. P. PETKOV, N. CHRISTOV, AND M. KONSTANTINOV, Computational Methods for Linear Control Systems, Prentice-Hall, Hertfordshire, UK, 1991.
34. A.-M. SÄNDIG AND W. WENDLAND, Asymptotic expansions of elastic fields in domains with boundary and structural singularities, in Boundary Element Topics (Stuttgart, 1995), Springer, Berlin, 1997, pp. 419-444.
35. V. SIMA, Algorithms for Linear-Quadratic Optimization, Pure Appl. Math. 200, Marcel Dekker, New York, 1996.
36. G. STEWART AND J.-G. SUN, Matrix Perturbation Theory, Academic Press, New York, 1990.
37. P. VAN DOOREN, A generalized eigenvalue approach for solving Riccati equations, SEAM J. Sci. Statist. Comput., 2 (1981), pp. 121-135.
38. C. VAN LOAN, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl., 61 (1984), pp. 233-251.
39. K. ZHOU, J. DOYLE, AND K. GLOVER, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1995.