The conditioning of linearizations of matrix polynomials

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 4, 2006, Pages 1005-1028

  Higham, Nicholas J ^a   Mackey, D Steven ^a   Tisseur, Françoise ^a  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Companion form; Condition number; Homogeneous form; Linearization; Matrix pencil; Matrix polynomial; Quadratic eigenvalue problem; Scaling; Vector space

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; HOMOGENIZATION METHOD; LINEARIZATION; MATRIX ALGEBRA; PROBLEM SOLVING; SCALING LAWS;

COMPANION FORM; CONDITION NUMBER; HOMOGENEOUS FORM; MATRIX PENCIL; MATRIX POLYNOMIAL; QUADRATIC EIGENVALUE PROBLEM; VECTOR SPACE;

POLYNOMIALS;

EID: 34547394717     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050628283     Document Type: Conference Paper

References (18)

1. A. L. ANDREW, K.-W. E. CHU, AND P. LANCASTER, Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 903-926.
2. J.-P. DEDIEU, Condition operators, condition numbers, and condition number theorem for the generalized eigenvalue problem, Linear Algebra Appl., 263 (1997), pp. 1-24.
3. J.-P. DEDIEU AND F. TISSEUR, Perturbation theory for homogeneous polynomial eigenvalue problems, Linear Algebra Appl., 358 (2003), pp. 71-94.
4. H.-Y. FAN, W.-W. LIN, AND P. VAN DOOREN, Normwise scaling of second order polynomial matrices, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 252-256.
5. P. FREITAS, M. GRINFIELD, AND P. KNIGHT, Stability of finite-dimensional systems with indefinite damping, Adv. Math. Sci. Appl., 17 (1997), pp. 435-446.
6. I. GOHBERG, P. LANCASTER, AND L. RODMAN, Matrix Polynomials, Academic Press, New York, 1982.
7. N. J. HIGHAM, D. S. MACKEY, N. MACKEY, AND F. TISSEUR, Symmetrie Linearizations for Matrix Polynomials, SIAM J. Matrix Anal. Appl., to appear.
8. N. J. HIGHAM, F. TISSEUR, AND P. M. VAN DOOREN, Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems, Linear Algebra Appl., 351-352 (2002), pp. 455-474.
9. T. ITOH, Damped vibration mode superposition method for dynamic response analysis, Earthquake Engrg. Struct. Dyn., 2 (1973), pp. 47-57.
10. T. R. KOWALSKI, Extracting a Few Eigenpairs of Symmetric Indefinite Matrix Pencils, Ph.D. thesis, Department of Mathematics, University of Kentucky, Lexington, KY, 2000.
11. P. LANCASTER, Quadratic eigenvalue problems, Linear Algebra Appl., 150 (1991), pp. 499-506.
12. P. LANCASTER AND M. TISMENETSKY, The Theory of Matrices, 2nd ed., Academic Press, London, 1985.
13. D. S. MACKEY, N. MACKEY, C. MEHL, AND V. MEHRMANN, Vector Spaces of Linearizations for Matrix Polynomials, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971-1004.
14. R. REEMTSEN AND S. GÖRNER, Numerical methods for semi-infinite programming: A survey, in Semi-Infinite Programming, R. Reemtsen and J.-J. Rückmann, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 195-275.
15. G. W. STEWART AND J. SUN, Matrix Perturbation Theory, Academic Press, London, 1990.
16. F. TISSEUR, Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl., 309 (2000), pp. 339-361.
17. F. TISSEUR AND K. MEERBERGEN, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235-286.
18. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, UK, 1965.