Computing eigenspaces with specified eigenvalues of a regular matrix pair (A, B) and condition estimation: Theory, algorithms and software

Numerical Algorithms

Volumn 12, Issue 2, 1996, Pages 369-407

  Kågström, Bo ^a   Poromaa, Peter ^a  

^a UMEÅ UNIVERSITY   (Sweden)

Author keywords

Blocked algorithms; Condition estimation; Deflating subspaces; Direct reordering of eigenvalues; Eigenvectors; Error bounds; Generalized Schur form; Generalized Sylvester equation; LAPACK; Regular matrix pair (pencil)

Indexed keywords

EID: 0041103179     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02142813     Document Type: Article

References (32)

1. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. Sorensen, LAPACK Users' Guide, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 1995.
2. Z. Bai and J. Demmel, On swapping diagonal blocks in real Schur form, Lin. Alg. Appl. 186 (1993) 73-95.
3. Z. Bai, J. Demmel and A. McKenney, On computing condition numbers for the nonsymmetric eigenproblem, ACM Trans. Math. Software 19 (1993) 202-223.
4. H. Bart, I. Gohberg, M. Kaashoek and P. Van Dooren, Factorization of transfer functions, SIAM J. Control Optim. 18 (1980) 675-696.
5. A. Bojanczyk and P. Van Dooren, Reordering diagonal blocks in the real Schur form, in: Linear Algebra for Large Scale and Real-Time Applications, M.S. Moonen, G.H. Golub, and B.L.R. De Moor, editors, Kluwer Academic, Amsterdam, 1993, pp. 351-352.
6. J. Demmel, The condition number of equivalence transformations that block diagonalize matrix pencils, SIAM J. Num. Anal. 20 (1983) 599-610.
7. J. Demmel, On condition numbers and the distance to the nearest ill-posed problem Num. Math. 51 (1987) 251-289.
8. J. Demmel and B. Kågström, Stable eigendecompositions of matrix pencils A - λB, Report UMINF-118.84, Institute of Information Processing, University of Umeå, S-901 87 Umeå, Sweden, 1984.
9. J. Demmel and B. Kågström, Computing stable eigendecompositions of matrix pencils, Lin. Alg. Appl. 88/89 (1987) 139-186.
10. J. Demmel and B. Kågström, The generalized Schur decomposition of an arbitrary pencil A - λB: Robust software with error bounds and applications. Part I: Theory and algorithms, ACM Trans. Math. Software 19 (1993) 160-174.
11. J. Demmel and B. Kågström, 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) 175-201.
12. F. Gantmacher, The Theory of Matrices, Vol. 1 and 2 (transl.), Chelsea, New York, 1959.
13. G. Golub and C. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, MD, 1989.
14. W.W. Hager, Condition estimators, SIAM J. Sci. Stat. Comput. 5 (1984) 311-316.
15. N.J. Higham, ALGORITHM 674: Fortran codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software 15 (1989) 168.
16. N.J. Higham, Perturbation theory and backward error analysis for AX - XB = C, BIT 33 (1993) 124-136.
17. B. Kågström, A direct method for reordering eigenvalues in the generalized real Schur form of a regular matrix pair (A, B), in: Linear Algebra for Large Scale and Real-Time Applications, M.S. Moonen, G.H. Golub, and B.L.R. De Moor, editors, Kluwer Academic, Amsterdam, 1993, pp. 195-218.
18. B. Kågström, A perturbation analysis of the generalized Sylvester equation, SIAM J. Matrix Anal. Appl. 15 (1994) 1045-1060.
19. B. Kågström and P. Poromaa, Computing eigenspaces with specified eigenvalues of a regular matrix pair (A, B) and condition estimation: Theory, algorithms and software, Report UMINF-94.04, Department of Computing Science, Umeå University, S-901 87 Umeå, Sweden, 1994. Also as LAPACK Working Note LAWN 87.
20. B. Kågström and P. Poromaa, LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs, ACM Trans. Math. Software 22 (1996) 78-103.
21. B. Kågström and P. Van Dooren, A generalized state-space approach for the additive decomposition of a transfer matrix, Int. J. Numerical Linear Algebra with Applications 1 (1992) 165-181.
22. B. Kågström and L. Westin, Generalized Schur methods with condition estimators for solving the generalized Sylvester equation, IEEE Trans. Autom. Contr. 34 (1989) 745-751.
23. A. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Autom. Contr. AC-24 (1979) 913-921.
24. C. Moler and G. Stewart, An algorithm for the generalized matrix eigenvalue problem, SIAM J. Numer. Anal. 10 (1973) 241-256.
25. J.L. Rigal and J. Gaches, On the compatability of a given solution with the data of a linear system, J. Assoc. Comput. Mach. 14 (1967) 543-548.
26. G.W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Review 15 (1973) 727-764.
27. G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990.
28. J.-G. Sun, Perturbation expansions for invariant subspaces, Lin. Alg. Appl. 153 (1991) 85-97.
29. J.-G. Sun, Backward perturbation analysis of certain characteristic subspaces, Num. Math. 65 (1993) 357-382.
30. P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Stat. Comp. 2 (1981) 121-135.
31. P. Van Dooren, ALGORITHM 590: DSUBSP and EXCHQZ: Fortran routines for computing deflating subspaces with specified spectrum, ACM Trans. Math. Software 8 (1982) 376-382, and (Corrections) 4 (1983) 787.
32. J.H. Wilkinson, Kronecker's canonical form and the QZ algorithm, Lin. Alg. Appl. 28 (1979) 285-303.