A structured staircase algorithm for skew-symmetric/symmetric pencils

Electronic Transactions on Numerical Analysis

Volumn 26, Issue , 2007, Pages 1-33

  Byers, Ralph ^a   Mehrmann, Volker ^b   Xu, Hongguo ^a  

^a UNIVERSITY OF KANSAS   (United States)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

H control; Linear quadratic control; Skew Hamiltonian Hamiltonian pencil; Skew symmetric symmetric pencil; Structured Kronecker canonical form; Structured staircase form

Indexed keywords

BOUNDARY CONDITIONS; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA; NUMERICAL METHODS; PROBLEM SOLVING;

HAMILTONIAN PENCIL; LINEAR QUADRATIC CONTROL; STRUCTURED KRONECKER CANONICAL FORMS; STRUCTURED STAIRCASE FORM;

ALGORITHMS;

EID: 34447513552     PISSN: None     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (47)

1. U. ASCHER, R. MATTHEIJ, AND R. RUSSELL, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, PA, second ed., 1995.
2. U. M. ASCHER AND L. R. PETZOLD, Computer Methods for Ordinary Differential and Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1998.
3. _, Numerical computation of deflating subspaces of skew Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 165-190.
4. ∞-control, to appear in: Linear Algebra Appl., (2007).
5. 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.
6. C. BISCHOF AND G. QUINTANA- OSTÍ, Algorithm 782: codes for rank-revealing QR factorizations of dense matrices, ACM Trans. Math. Software, 24 (1998), pp. 254-257.
7. A. BOJANCZYK, G. GOLUB, AND P. VAN DOOREN, The periodic Schur decomposition; algorithms and applications, in Proc. SPIE Conference, vol. 1770, 1992, pp. 31-42.
8. K. BRENAN, S. CAMPBELL, AND L. PETZOLD, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, second ed., 1996.
9. R. BYERS, T. GEERTS, AND V. MEHRMANN, Descriptor systems without controllability at infinity, SIAM J. Cont. Optim., 35 (1997), pp. 462-479.
10. R. BYERS, V. MEHRMANN, AND H. XU, A structured staircase algorithm for skew-symmetric/symmetric pencils, technical report, DFG Research Center MATHEON, Mathematics for key technologies in Berlin, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, Germany, 2005. url: http://www.matheon.de/.
11. D. CHU, X. LIU, AND V. MEHRMANN, A numerical method for computing the Hamiltonian Schur form, Numer. Math., 105 (2007), pp. 375-412.
12. D. CLEMENTS, Private communication, 2002.
13. J. DEMMEL AND B. KÅGSTRÖM, Computing stable eigendecompositions of matrix pencils, Linear Algebra Appl., 88 (1987), pp. 139-186.
14. _, The generalized Schur decomposition of an arbitrary pencil A - λB: robust software with error bounds and applications. Parti: theory and algorithms, ACM Trans. Math. Software, 19 (1993), pp. 160-174.
15. _, The generalized Schur decomposition of an arbitrary pencil A - λB: robust software with error bounds and applications. Part II: software and applications, ACM Trans. Math. Software, 19 (1993), pp. 175-201.
16. A. EDELMANN, E. ELMROTH, AND B. KÅGSTRÖ M, A geometric approach to perturbation theory of matrices and matrix pencils. Parti: Versal deformations, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 653-692.
17. _, A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratification-enhanced staircase algorithm., SIAM J. Matrix Anal. Appl., 20 (1999), pp. 667-699.
18. ∞ control, SIAM J. Cont. Optim., 35 (1997), pp. 1690-1710.
19. F. GANTMACHER, Theory of Matrices, vol. 2, Chelsea, New York, 1959.
20. F. GANTMACHER, Theory of Matrices, vol. 1, Chelsea, New York, 1959.
21. G. GOLUB AND C. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore, third ed., 1996.
22. J. HENCH AND A. LAUB, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Automat. Control, 39 (1994), pp. 1197-1210.
23. D. KRESSNER, Numerical Methods and Software for General and Structured Eigenvalue Problems, PhD thesis, TU Berlin, Institut für Mathematik, Berlin, Germany, 2004.
24. L. KRONECKER, Algebraische Reduction der Scharen bilinearer Formen, S.B. Akad. Berlin, (1890), pp. 1225-1237.
25. P. KUNKEL AND V. MEHRMANN, Canonical forms for linear differential-algebraic equations with variable coefficients, J. Comput. Appl. Math., 56 (1994), pp. 225-251.
26. _, A new look at pencils of matrix valued functions, Linear Algebra Appl., 212/213 (1994), pp. 215-248.
27. P. KUNKEL AND R. STÖVER, Symmetric collocation methods for linear differential-algebraic boundary value problems, Numer. Math., 91 (2002), pp. 475-501.
28. P. LANCASTER AND L. RODMAN, The Algebraic Riccati Equation, Oxford University Press, Oxford, 1995.
29. 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.
30. C. LUCAS, Algorithms for Cholesky and QR Factorizations, and the Semidefinite Generalized Eigenvalue Problem, PhD thesis, University of Manchester, Manchester, England, 2004.
31. D. MACKEY, N. MACKEY, C. MEHL, AND V. MEHRMANN, Palindromic polynomial eigenvalue problems: Good vibrations from good linearizations, technical report, DFG Research Center MATHEON, Mathematics for key technologies in Berlin, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, Germany, 2005. url: http://www.matheon. de/
32. to appear in SIAM J. Matrix Anal. Appl.
33. THE MATH WORKS, INC., COCHITUATE PLACE, 24 PRIME PARK WAY, NATICK, MASS, 01760, MATLAB Version 6.0.0.88, 2001.
34. C. MEHL, Condensed forms for skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 454-476.
35. V. MEHRMANN, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, no. 163 in Lecture Notes in Control and Information Sciences, Springer-Verlag, Heidelberg, July 1991.
36. V. MEHRMANN AND D. WATKINS, Polynomial eigenvalue problems with hamiltonian structure, Electr. Trans. Numer. Anal., 13 (2002), pp. 106-118. http://etna.math.kent.edu/vol.13.2002/pp106-118.dir/pp106-118.html
37. V. MEHRMANN AND H. XU, Structured Jordan canonical forms for structured matrices that are Hermitian, skew Hermitian or unitary with respect to indefinite inner products, Electr. J. Linear Algebra, 5 (1999), pp. 67-103.
38. _, Numerical methods in control, J. Comput. Appl. Math., 123 (2000), pp. 371-394.
39. P. PETKOV, N. CHRISTOV, AND M. KONSTANTINOV, Computational Methods for Linear Control Systems, Prentice-Hall, Hertfordshire, UK, 1991.
40. V. SIMA, Algorithms for Linear-Quadratic Optimization, vol. 200 of Pure and Applied Mathematics, Marcel Dekker, Inc., New York, NY, 1996.
41. G. STEWART, The efficient generation of random orthogonal matrices with an application to condition estimators, SIAM J. Numer. Anal., 17 (1980), pp. 403-409.
42. G. STEWART AND J.-G. SUN, Matrix Perturbation Theory, Academic Press, New York, 1990.
43. R. C. THOMPSON, The characteristic polynomial of a principal submatrix of a Hermitian pencil, Linear Algebra Appl., 14 (1976), pp. 135-177.
44. _, Pencils of complex and real symmetric and skew matrices, Linear Algebra Appl., 147 (1991), pp. 323-371.
45. P. VAN DOOREN, The computation of Kronecker's canonical form of a singular pencil, Linear Algebra Appl., 27 (1979), pp. 103-121.
46. _, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 121-135.
47. K. ZHOU, J. DOYLE, AND K. GLOVER, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1995.